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* Beginning to convert Python backend to Typescript * Change all fetch() calls to fetch_from_backend switcher * wip converting query.py to query.ts * wip started utils.js conversion. Tested that OpenAI API call works * more progress on converting utils.py to Typescript * jest tests for query, utils, template.ts. Confirmed PromptPipeline works. * wip converting queryLLM in flask_app to TS * Tested queryLLM and StorageCache compressed saving/loading * wip execute() in backend.ts * Added execute() and tested w concrete func. Need to test eval() * Added craco for optional webpack config. Config'd for TypeScript with Node.js packages browserify'd * Execute JS code on iframe sandbox * Tested and working JS Evaluator execution. * wip swapping backends * Tested TypeScript backendgit status! :) woot * Added fetchEnvironAPIKeys to Flask server to fetch os.environ keys when running locally * Route Anthropic calls through Flask when running locally * Added info button to Eval nodes. Rebuilt react * Edits to info modal on Eval node * Remove/error out on Python eval nodes when not running locally. * Check browser compat and display error if not supported * Changed all example flows to use JS. Bug fix in query.ts * Refactored to LLMProvider to streamline model additions * Added HuggingFace models API * Added back Dalai call support, routing through Flask * Remove flask app calls and socketio server that are no longer used * Added Comment Nodes. Rebuilt react. * Fix PaLM temp=0 build, update package vers and rebuild react
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{"flow": {"nodes": [{"width": 312, "height": 311, "id": "prompt-jee-math", "type": "prompt", "data": {"prompt": "{prompt}", "n": 1, "llms": [{"key": "aa3c0f03-22bd-416e-af4d-4bf5c4278c99", "settings": {"system_msg": "You are MathGPT, a skilled mathematician who is capable of providing accurate answers to complex questions.\nYou are about to take a test with multiple choice answers and your goal is to provide as many correct answers as possible.\nComplex formulae will begin with the character sequence '\\(' and end with '\\)'.\nEach question will be followed by exactly 4 potential answers, each prefixed by the string (A), (B), (C), or (D). \nOnly one of these answers is correct. Work through your reasoning to solve the question extremely briefly, in less than 100 words.\nYou must end your response with the phrase \"Therefore, the correct answer is (Letter)\", where Letter is the uppercase letter of your choice.\n", "temperature": 1, "functions": [], "function_call": "", "top_p": 1, "stop": [], "presence_penalty": 0, "frequency_penalty": 0}, "name": "GPT3.5", "emoji": "\ud83d\ude42", "model": "gpt-3.5-turbo", "base_model": "gpt-3.5-turbo", "temp": 1, "formData": {"shortname": "GPT3.5", "model": "gpt-3.5-turbo", "system_msg": "You are MathGPT, a skilled mathematician who is capable of providing accurate answers to complex questions.\nYou are about to take a test with multiple choice answers and your goal is to provide as many correct answers as possible.\nComplex formulae will begin with the character sequence '\\(' and end with '\\)'.\nEach question will be followed by exactly 4 potential answers, each prefixed by the string (A), (B), (C), or (D). \nOnly one of these answers is correct. Work through your reasoning to solve the question extremely briefly, in less than 100 words.\nYou must end your response with the phrase \"Therefore, the correct answer is (Letter)\", where Letter is the uppercase letter of your choice.\n", "temperature": 1, "functions": "", "function_call": "", "top_p": 1, "stop": "", "presence_penalty": 0, "frequency_penalty": 0}}]}, "position": {"x": 448, "y": 224}, "selected": false, "positionAbsolute": {"x": 448, "y": 224}, "dragging": false}, {"width": 333, "height": 182, "id": "eval-jee-math", "type": "evaluator", "data": {"code": "function evaluate(response) {\n\tlet ideals = JSON.parse(response.meta['Ideal']);\n\treturn ideals.some(i => response.text.includes(i));\n}", "language": "javascript"}, "position": {"x": 820, "y": 150}, "positionAbsolute": {"x": 820, "y": 150}}, {"width": 228, "height": 196, "id": "vis-jee-math", "type": "vis", "data": {"input": "eval-jee-math"}, "position": {"x": 1200, "y": 250}, "positionAbsolute": {"x": 1200, "y": 250}}, {"width": 302, "height": 260, "id": "inspect-jee-math", "type": "inspect", "data": {"input": "prompt-jee-math"}, "position": {"x": 820, "y": 400}, "positionAbsolute": {"x": 820, "y": 400}}, {"width": 423, "height": 417, "id": "table-jee-math", "type": "table", "data": {"rows": [{"prompt": "The total number of functions, f : {1, 2, 3, 4} \u2192 {1, 2, 3, 4, 5, 6} such that f(1) + f(2) = f(3), is equal to\n(A) 60\n(B) 90\n(C) 108\n(D) 126\n", "ideal": "[\"the correct answer is (B)\", \"the correct answer is \\textbf{(B)}\", \"the correct answer is $\\boxed{\\textbf{(B)}\"]"}, {"prompt": "If \u03b1, \u03b2, \u03b3, \u03b4 are the roots of the equation x^4 + x^3 + x^2 + x + 1 = 0, then \u03b1^2021 + \u03b2^2021 + \u03b3^2021 + \u03b4^2021 is equal to\n(A) \u20134\n(B) \u20131\n(C) 1\n(D) 4\n", "ideal": "[\"the correct answer is (B)\", \"the correct answer is \\textbf{(B)}\", \"the correct answer is $\\boxed{\\textbf{(B)}\"]"}, {"prompt": "The number of q\u2208 (0, 4\u03c0) for which the system of linear equations\n3(sin 3\u03b8) x \u2013 y + z = 2\n3(cos 2\u03b8) x + 4y + 3z = 3\n6x + 7y + 7z = 9\nhas no solution, is\n(A) 6\n(B) 7\n(C) 8\n(D) 9\n", "ideal": "[\"the correct answer is (B)\", \"the correct answer is \\textbf{(B)}\", \"the correct answer is $\\boxed{\\textbf{(B)}\"]"}, {"prompt": "If \\(\\displaystyle \\lim_{n\\rightarrow\\infty}\\left(\\sqrt{n^2-n-1}+n\\alpha+\\beta\\right)=0 \\) then 8(\u03b1 + \u03b2) is equal to\n(A) 4\n(B) \u20138\n(C) \u20134\n(D) 8\n", "ideal": "[\"the correct answer is (C)\", \"the correct answer is \\textbf{(C)}\", \"the correct answer is $\\boxed{\\textbf{(C)}\"]"}, {"prompt": "If the absolute maximum value of the function \\(f(x) = (x^2 \u2013 2x + 7) e^{(4x^3-12x^2-180x + 31)} \\) in the interval [\u20133, 0] is f(\u03b1), then\n(A) \u03b1 = 0\n(B) \u03b1 = \u20133\n(C) \u03b1\u2208 (\u20131, 0)\n(D) \u03b1\u2208 (\u20133, \u20131]\n", "ideal": "[\"the correct answer is (B)\", \"the correct answer is \\textbf{(B)}\", \"the correct answer is $\\boxed{\\textbf{(B)}\"]"}, {"prompt": "The curve y(x) = ax^3 + bx^2 + cx + 5 touches the x-axis at the point P(\u20132, 0) and cuts the y-axis at the point Q, where y\u2032 is equal to 3. Then the local maximum value of y(x) is \n(A) \\(\\frac{27}{4} \\)\n(B) \\(\\frac{29}{4} \\)\n(C) \\(\\frac{37}{4} \\)\n(D) \\(\\frac{9}{2} \\)\n", "ideal": "[\"the correct answer is (A)\", \"the correct answer is \\textbf{(A)}\", \"the correct answer is $\\boxed{\\textbf{(A)}\"]"}, {"prompt": "The area of the region given by\n\\(A=\\left\\{\\left(x,y\\right);x^2\\leq y \\leq \\text{min}\\left\\{x+2, 4-3x \\right\\} \\right\\}\\) is\n(A) \\(\\frac{31}{8}\\)\n(B) \\(\\frac{17}{6}\\)\n(C) \\(\\frac{19}{6}\\)\n(D) \\(\\frac{27}{8}\\)\n", "ideal": "[\"the correct answer is (B)\", \"the correct answer is \\textbf{(B)}\", \"the correct answer is $\\boxed{\\textbf{(B)}\"]"}, {"prompt": "For any real number x, let [x] denote the largest integer less than equal to x. Let f be a real valued function defined on the interval [\u201310, 10] by \\(f\\left(x\\right)=\\left\\{\\begin{matrix}x-[x], \\text{ if } [x] \\text{ is odd} \\\\1+[x]-x, \\text{if }[x]\\text{ is even}\\end{matrix}\\right.\\)\nThen the value of \\(\\frac{\\pi^2}{10}\\displaystyle\\int\\limits_{-10}^{10}f\\left(x\\right)\\cos\\pi x\\ dx\\) is\n(A) 4\n(B) 2\n(C) 1\n(D) 0\n", "ideal": "[\"the correct answer is (A)\", \"the correct answer is \\textbf{(A)}\", \"the correct answer is $\\boxed{\\textbf{(A)}\"]"}, {"prompt": "The slope of the tangent to a curve C : y = y(x) at any point (x, y) on it is \\(\\frac{2e^{2x}-6e^{-x}+9}{2+9e^{-2x}}\\). If C passes through the points \\(\\left(0, \\frac{1}{2}+\\frac{\\pi}{2\\sqrt{2}}\\right)\\text{ and }\\left(\\alpha,\\frac{1}{2}e^{2\\alpha}\\right),\\) then e^\u03b1 is equal to\n(A) \\(\\frac{3+\\sqrt{2}}{3-\\sqrt{2}} \\)\n(B) \\(\\frac{3}{\\sqrt{2}}\\left(\\frac{3+\\sqrt{2}}{3-\\sqrt{2}} \\right)\\)\n(C) \\(\\frac{1}{\\sqrt{2}}\\left(\\frac{\\sqrt{2}+1}{\\sqrt{2}-1} \\right)\\)\n(D) \\(\\frac{\\sqrt{2}+1}{\\sqrt{2}-1}\\)\n", "ideal": "[\"the correct answer is (B)\", \"the correct answer is \\textbf{(B)}\", \"the correct answer is $\\boxed{\\textbf{(B)}\"]"}, {"prompt": "The general solution of the differential equation (x \u2013 y^2)dx + y(5x + y^2)dy = 0 is :\n(A) \\(\\left(y^2+x\\right)^4=C\\left|\\left(y^2+2x\\right)^3\\right|\\)\n(B) \\(\\left(y^2+2x\\right)^4=C\\left|\\left(y^2+x\\right)^3\\right|\\)\n(C) \\(\\left|\\left(y^2+x\\right)^3\\right|=C\\left(2y^2+x\\right)^4\\)\n(D) \\(\\left|\\left(y^2+2x\\right)^3\\right|=C\\left(2y^2+x\\right)^4\\)\n", "ideal": "[\"the correct answer is (A)\", \"the correct answer is \\textbf{(A)}\", \"the correct answer is $\\boxed{\\textbf{(A)}\"]"}, {"prompt": "A line, with the slope greater than one, passes through the point A(4, 3) and intersects the line\nx \u2013 y \u2013 2 = 0 at the point B. If the length of the line segment AB is \\(\\frac{\\sqrt{29}}{3}, \\ \\text{then B also lies on the line :}\\)\u00a0\n(A) 2x + y = 9\n(B) 3x \u2013 2y = 7\n(C) x + 2y = 6\n(D) 2x \u2013 3y = 3\n", "ideal": "[\"the correct answer is (C)\", \"the correct answer is \\textbf{(C)}\", \"the correct answer is $\\boxed{\\textbf{(C)}\"]"}, {"prompt": "Let the locus of the centre (\u03b1, \u03b2), \u03b2> 0, of the circle which touches the circle x^2 +(y \u2013 1)^2 = 1 externally and also touches the x-axis be L. Then the area bounded by L and the line y = 4 is :\n(A) \\(\\frac{32\\sqrt{2}}{3}\\)\n(B) \\(\\frac{40\\sqrt{2}}{3}\\)\n(C) \\(\\frac{64}{3}\\)\n(D) \\(\\frac{32}{3}\\)\n", "ideal": "[\"the correct answer is (C)\", \"the correct answer is \\textbf{(C)}\", \"the correct answer is $\\boxed{\\textbf{(C)}\"]"}, {"prompt": "Let P be the plane containing the straight line \\(\\frac{x-3}{9}=\\frac{y+4}{-1}=\\frac{z-7}{-5}\\) and perpendicular to the plane containing the straight lines \\(\\frac{x}{2}=\\frac{y}{3}=\\frac{z}{5} \\) and \\(\\frac{x}{3}=\\frac{y}{7}=\\frac{z}{8}.\\) If d is the distance P from the point (2, \u20135, 11), then d^2 is equal to :\n(A) \\(\\frac{147}{2}\\)\n(B) 96\n(C) \\(\\frac{32}{3}\\)\n(D) 54\n", "ideal": "[\"the correct answer is (C)\", \"the correct answer is \\textbf{(C)}\", \"the correct answer is $\\boxed{\\textbf{(C)}\"]"}, {"prompt": "Let ABC be a triangle such that \\(\\overrightarrow{BC}=\\overrightarrow{a},\\overrightarrow{CA}=\\overrightarrow{b},\\overrightarrow{AB}=\\overrightarrow{c},\\left|\\overrightarrow{a}\\right|=6\\sqrt{2},\\left|\\overrightarrow{b}\\right|=2\\sqrt{3}\\) and \\(\\overrightarrow{b}\\cdot\\overrightarrow{c}=12.\\) Consider the statements :\n\\(\\left(S1\\right):\\left|\\left(\\overrightarrow{a}\\times\\overrightarrow{b}\\right)+\\left(\\overrightarrow{c}\\times\\overrightarrow{b}\\right)\\right|-\\left|\\overrightarrow{c}\\right|=6\\left(2\\sqrt{2}-1\\right)\\)\n\\(\\left(S2\\right):\\angle ACB=\\cos^{-1}\\left(\\sqrt{\\frac{2}{3}}\\right) \\)\nThen\n(A) Both (S1) and (S2) are true\n(B) Only (S1) is true\n(C) Only (S2) is true\n(D) Both (S1) and (S2) are false\n", "ideal": "[\"the correct answer is (C)\", \"the correct answer is \\textbf{(C)}\", \"the correct answer is $\\boxed{\\textbf{(C)}\"]"}, {"prompt": "If the sum and the product of mean and variance of a binomial distribution are 24 and 128 respectively, then the probability of one or two successes is :\n(A) \\(\\frac{33}{2^{32}} \\)\n(B) \\(\\frac{33}{2^{29}} \\)\n(C) \\(\\frac{33}{2^{28}} \\)\n(D) \\(\\frac{33}{2^{27}} \\)\n", "ideal": "[\"the correct answer is (C)\", \"the correct answer is \\textbf{(C)}\", \"the correct answer is $\\boxed{\\textbf{(C)}\"]"}, {"prompt": "If the numbers appeared on the two throws of a fair six faced die are \u03b1 and \u03b2, then the probability that x^2 + \u03b1x + \u03b2> 0, for all x \u2208 R, is :\n(A) \\(\\frac{17}{36}\\)\n(B) \\(\\frac{4}{9}\\)\n(C) \\(\\frac{1}{2}\\)\n(D) \\(\\frac{19}{36}\\)\n", "ideal": "[\"the correct answer is (A)\", \"the correct answer is \\textbf{(A)}\", \"the correct answer is $\\boxed{\\textbf{(A)}\"]"}, {"prompt": "The number of solutions of |cos x| = sinx, such that \u20134\u03c0 \u2264 x \u2264 4\u03c0 is :\n(A) 4\n(B) 6\n(C) 8\n(D) 12\n", "ideal": "[\"the correct answer is (C)\", \"the correct answer is \\textbf{(C)}\", \"the correct answer is $\\boxed{\\textbf{(C)}\"]"}, {"prompt": "A tower PQ stands on a horizontal ground with base Q on the ground. The point R divides the tower in two parts such that QR = 15 m. If from a point A on the ground the angle of elevation of R is 60\u00b0 and the part PR of the tower subtends an angle of 15\u00b0 at A, then the height of the tower is :\n(A) \\(5\\left(2\\sqrt{3}+3\\right)\\text{ m}\\)\n(B) \\(5\\left(\\sqrt{3}+3\\right)\\text{ m}\\)\n(C) \\(10\\left(\\sqrt{3}+1\\right)\\text{ m}\\)\n(D) \\(10\\left(2\\sqrt{3}+1\\right)\\text{ m}\\)\n", "ideal": "[\"the correct answer is (A)\", \"the correct answer is \\textbf{(A)}\", \"the correct answer is $\\boxed{\\textbf{(A)}\"]"}, {"prompt": "Which of the following statements is a tautology?\n(A) \\(\\left(\\left(\\sim p\\right)\\vee q \\right)\\Rightarrow p\\)\n(B) \\(p\\Rightarrow \\left(\\left(\\sim p\\right)\\vee q\\right)\\)\n(C) \\(\\left(\\left(\\sim p\\right)\\vee q\\right)\\Rightarrow q\\)\n(D) \\(q\\Rightarrow \\left(\\left(\\sim p\\right)\\vee q\\right)\\)\n", "ideal": "[\"the correct answer is (D)\", \"the correct answer is \\textbf{(D)}\", \"the correct answer is $\\boxed{\\textbf{(D)}\"]"}, {"prompt": "For \\(z \\in \\mathbb{C}\\ \\text{if the minimum value of}\\ \\left(|z-3\\sqrt{2}| + |z-p\\sqrt{2}i|\\right)\\)\u00a0 is 5\u221a2, then a value of p is _________.\n(A) 3\n(B) \\(\\frac{7}{2}\\)\n(C) 4\n(D) \\(\\frac{9}{2}\\)\n", "ideal": "[\"the correct answer is (C)\", \"the correct answer is \\textbf{(C)}\", \"the correct answer is $\\boxed{\\textbf{(C)}\"]"}, {"prompt": "The number of real values of \u03bb, such that the system of linear equations\n2x \u2013 3y + 5z = 9\nx + 3y \u2013 z = \u201318\n3x \u2013 y + (\u03bb^2 \u2013 | \u03bb |)z = 16\nhas no solutions, is\n(A) 0\n(B) 1\n(C) 2\n(D) 4\n", "ideal": "[\"the correct answer is (C)\", \"the correct answer is \\textbf{(C)}\", \"the correct answer is $\\boxed{\\textbf{(C)}\"]"}, {"prompt": "The number of bijective functions f : {1, 3, 5, 7, \u2026, 99} \u2192 {2, 4, 6, 8, \u2026.., 100} such that \\(f(3)\\ge f(9)\\ge f(15)\\ge f(21)\\ge \u2026 \\ge f(99)\\) is_____.\n(A) \\(^{50}P_{17}\\)\n(B) \\(^{50}P_{33}\\)\n(C) \\(33!\\times 17!\\)\n(D) \\(\\frac{50!}{2}\\)\n", "ideal": "[\"the correct answer is (B)\", \"the correct answer is \\textbf{(B)}\", \"the correct answer is $\\boxed{\\textbf{(B)}\"]"}, {"prompt": "The remainder when (11)^1011 + (1011)^11 is divided by 9 is\n(A) 1\n(B) 4\n(C) 6\n(D) 8\n", "ideal": "[\"the correct answer is (D)\", \"the correct answer is \\textbf{(D)}\", \"the correct answer is $\\boxed{\\textbf{(D)}\"]"}, {"prompt": "The sum \\(\\sum_{n=1}^{21}\\frac{3}{(4n-1)(4n+3)}\\) is equal to\n(A) \\(\\frac{7}{87}\\)\n(B) \\(\\frac{7}{29}\\)\n(C) \\(\\frac{14}{87}\\)\n(D) \\(\\frac{21}{29}\\)\n", "ideal": "[\"the correct answer is (B)\", \"the correct answer is \\textbf{(B)}\", \"the correct answer is $\\boxed{\\textbf{(B)}\"]"}, {"prompt": "\\(\\displaystyle \\lim_{ x\\to \\frac{x}{4}}\\frac{8\\sqrt{2}-(\\cos x + \\sin x)^7}{\\sqrt{2}-\\sqrt{2}\\sin 2x}\\) is equal to\n(A) 14\n(B) 7\n(C) \\(14\\sqrt{2}\\)\n(D) \\(7\\sqrt{2}\\)\n", "ideal": "[\"the correct answer is (A)\", \"the correct answer is \\textbf{(A)}\", \"the correct answer is $\\boxed{\\textbf{(A)}\"]"}, {"prompt": "\\(\\displaystyle \\lim_{n \\to \\infty}\\frac{1}{2^n}\\left(\\frac{1}{\\sqrt{1-\\frac{1}{2^n}}} + \\frac{1}{\\sqrt{1-\\frac{2}{2^n}}} + \\frac{1}{\\sqrt{1-\\frac{3}{2^n}}} + \u2026 + \\frac{1}{\\sqrt{1-\\frac{2^n -1}{2^n}}}\\right)\\) is equal to\n(A)\n(B) 1\n(C) 2\n(D) \u20132\n", "ideal": "[\"the correct answer is (C)\", \"the correct answer is \\textbf{(C)}\", \"the correct answer is $\\boxed{\\textbf{(C)}\"]"}, {"prompt": "If A and B are two events such that \\(P(A)=\\frac{1}{3}, P(B)=\\frac{1}{5}\\ \\text{and}\\ P(A\\cup B)=\\frac{1}{2},\\ \\text{then}\\) \\(P(A|B\u2019) + P(B|A\u2019|)\\) is equal to\n(A) \\(\\frac{3}{4}\\)\n(B) \\(\\frac{5}{8}\\)\n(C) \\(\\frac{5}{4}\\)\n(D) \\(\\frac{7}{8}\\)\n", "ideal": "[\"the correct answer is (B)\", \"the correct answer is \\textbf{(B)}\", \"the correct answer is $\\boxed{\\textbf{(B)}\"]"}, {"prompt": "Let [t] denote the greatest integer less than or equal to t. Then the value of the integral \\(\\int_{-3}^{101}\\left([\\sin (\\pi x)]+e^{[\\cos(2\\pi x)]}\\right)dx\\) is equal to \n(A) \\(\\frac{52(1-e)}{e}\\)\n(B) \\(\\frac{52}{e}\\)\n(C) \\(\\frac{52(2+e))}{e}\\)\n(D) \\(\\frac{104}{e}\\)\n", "ideal": "[\"the correct answer is (B)\", \"the correct answer is \\textbf{(B)}\", \"the correct answer is $\\boxed{\\textbf{(B)}\"]"}, {"prompt": "Let the point P(\u03b1, \u03b2) be at a unit distance from each of the two lines L1 : 3x \u2013 4y + 12 = 0 and L2 : 8x + 6y + 11 = 0. If P lies below L1 and above L2, then 100(\u03b1 + \u03b2) is equal to \n(A) \u201314\n(B) 42\n(C) \u201322\n(D) 14\n", "ideal": "[\"the correct answer is (D)\", \"the correct answer is \\textbf{(D)}\", \"the correct answer is $\\boxed{\\textbf{(D)}\"]"}, {"prompt": "Let a smooth curve y = f(x) be such that the slope of the tangent at any point (x, y) on it is directly proportional to (-y/x). If the curve passes through the points (1, 2) and (8, 1), then \\(\\left|y\\left(\\frac{1}{8}\\right)\\right|\\ \\text{ is equal to}\\)\n(A) 2 loge2\n(B) 4\n(C) 1\n(D) 4 loge2\n", "ideal": "[\"the correct answer is (B)\", \"the correct answer is \\textbf{(B)}\", \"the correct answer is $\\boxed{\\textbf{(B)}\"]"}, {"prompt": "If the ellipse \\(\\frac{x^2}{a^2}+\\frac{y^2}{b^2}=1\\ \\text{meets the line}\\ \\frac{x}{7}+\\frac{y}{2\\sqrt{6}} =1\\)\u00a0 on the x-axis and the line \\(\\frac{x}{7}-\\frac{y}{2\\sqrt{6}} =1\\) on the y-axis, then the eccentricity of the ellipse is\n(A) \\(\\frac{5}{7}\\)\n(B) \\(\\frac{2\\sqrt{6}}{7}\\)\n(C) \\(\\frac{3}{7}\\)\n(D) \\(\\frac{2\\sqrt{5}}{7}\\)\n", "ideal": "[\"the correct answer is (A)\", \"the correct answer is \\textbf{(A)}\", \"the correct answer is $\\boxed{\\textbf{(A)}\"]"}, {"prompt": "The tangents at the points A(1, 3) and B(1, \u20131) on the parabola y^2 \u2013 2x \u2013 2y = 1 meet at the point P. Then the area (in unit^2) of the triangle PAB is :\n(A) 4\n(B) 6\n(C) 7\n(D) 8\n", "ideal": "[\"the correct answer is (D)\", \"the correct answer is \\textbf{(D)}\", \"the correct answer is $\\boxed{\\textbf{(D)}\"]"}, {"prompt": "Let the foci of the ellipse \\(\\frac{x^2}{16}+\\frac{y^2}{7}=1\\ \\text{and the hyperbola}\\ \\frac{x^2}{144}-\\frac{y^2}{\\alpha}=\\frac{1}{25}\\) coincide. Then the length of the latus rectum of the hyperbola is :\n(A) \\(\\frac{32}{9}\\)\n(B) \\(\\frac{18}{5}\\)\n(C) \\(\\frac{27}{4}\\)\n(D) \\(\\frac{27}{10}\\)\n", "ideal": "[\"the correct answer is (D)\", \"the correct answer is \\textbf{(D)}\", \"the correct answer is $\\boxed{\\textbf{(D)}\"]"}, {"prompt": "A plane E is perpendicular to the two planes 2x \u2013 2y + z = 0 and x \u2013 y + 2z = 4, and passes through the point P(1, \u20131, 1). If the distance of the plane E from the point Q(a, a, 2) is 3\u221a2, then (PQ)^2 is equal to\n(A) 9\n(B) 12\n(C) 21\n(D) 33\n", "ideal": "[\"the correct answer is (C)\", \"the correct answer is \\textbf{(C)}\", \"the correct answer is $\\boxed{\\textbf{(C)}\"]"}, {"prompt": "The shortest distance between the lines \\(\\frac{x+7}{-6}=\\frac{y-6}{7}=z\\ \\text{and}\\ \\frac{7-x}{2}=y-2=z-6\\) is\n(A) \\(2\\sqrt{29}\\)\n(B) 1\n(C) \\(\\sqrt{\\frac{37}{29}}\\)\n(D) \\(\\sqrt{\\frac{29}{2}}\\)\n", "ideal": "[\"the correct answer is (A)\", \"the correct answer is \\textbf{(A)}\", \"the correct answer is $\\boxed{\\textbf{(A)}\"]"}, {"prompt": "Let \\(\\vec{a}=\\hat{i}-\\hat{j}+2\\hat{k}\\) \\(\\text{and let}\\ \\vec{b}\\ \\text{be a vector such that}\\)\u00a0 \\(\\vec{a}\\times \\vec{b} =2\\hat{i}-\\hat{k}\\ \\text{and}\\ \\vec{a}\\cdot\\vec{b}=3\\)\u00a0 Then the projection of \\(\\vec{b}\\ \\text{on the vector}\\ \\vec{a}- \\vec{b}\\ \\text{is}:\\)\u00a0\u00a0\n(A) \\(\\frac{2}{\\sqrt{21}}\\)\n(B) \\(2\\sqrt{\\frac{3}{7}}\\)\n(C) \\(\\frac{2}{3}\\sqrt{\\frac{7}{3}}\\)\n(D) \\(\\frac{2}{3}\\)\n", "ideal": "[\"the correct answer is (A)\", \"the correct answer is \\textbf{(A)}\", \"the correct answer is $\\boxed{\\textbf{(A)}\"]"}, {"prompt": "If the mean deviation about median for the number 3, 5, 7, 2k, 12, 16, 21, 24 arranged in the ascending order, is 6 then the median is\n(A) 11.5\n(B) 10.5\n(C) 12\n(D) 11\n", "ideal": "[\"the correct answer is (D)\", \"the correct answer is \\textbf{(D)}\", \"the correct answer is $\\boxed{\\textbf{(D)}\"]"}, {"prompt": "\\(2\\sin\\left(\\frac{\\pi}{22}\\right)\\sin\\left(\\frac{3\\pi}{22}\\right)\\sin\\left(\\frac{5\\pi}{22}\\right)\\sin\\left(\\frac{7\\pi}{22}\\right)\\sin\\left(\\frac{9\\pi}{22}\\right)\\) is equal to :\n(A) \\(\\frac{3}{16}\\\\\\)\n(B) \\(\\frac{1}{16}\\\\\\)\n(C) \\(\\frac{1}{32}\\\\\\)\n(D) \\(\\frac{9}{32}\\)\n", "ideal": "[\"the correct answer is (B)\", \"the correct answer is \\textbf{(B)}\", \"the correct answer is $\\boxed{\\textbf{(B)}\"]"}, {"prompt": "Consider the following statements :\nP : Ramu is intelligent.\nQ : Ramu is rich.\nR : Ramu is not honest.\nThe negation of the statement \u201cRamu is intelligent and honest if and only if Ramu is not rich\u201d can be expressed as :\n(A) ((P \u2227 (~ R)) \u2227 Q) \u2227 ((~ Q) \u2227 ((~ P) \u2228 R))\n(B) ((P \u2227 R) \u2227 Q) \u2228 ((~ Q) \u2227 ((~ P) \u2228 (~ R)))\n(C) ((P \u2227 R) \u2227 Q) \u2227 ((~ Q) \u2227 (( ~ P) \u2228 (~ R)))\n(D) ((P \u2227 (~ R)) \u2227 Q) \u2228 ((~ Q) \u2227 ((~ P) \u2227 R))\n", "ideal": "[\"the correct answer is (D)\", \"the correct answer is \\textbf{(D)}\", \"the correct answer is $\\boxed{\\textbf{(D)}\"]"}, {"prompt": "Let f :R\u2192R be a continuous function such that f(3x) \u2013 f(x) = x. If f(8) = 7, then f(14) is equal to\n(A) 4\n(B) 10\n(C) 11\n(D) 16\n", "ideal": "[\"the correct answer is (B)\", \"the correct answer is \\textbf{(B)}\", \"the correct answer is $\\boxed{\\textbf{(B)}\"]"}, {"prompt": "Let O be the origin and A be the point z1 = 1 + 2i. If B is the point z2, Re(z2) < 0, such that OAB is a right angled isosceles triangle with OB as hypotenuse, then which of the following is NOT true?\n(A) \\( argz_2 = \\pi \u2013 tan^{-1}3\\)\n(B) \\((\\text{arg})\\left(z_1-2z_2\\right)=-\\tan^{-1}\\frac{4}{3}\\)\n(C) \\(\\left|z_2\\right|=\\sqrt{10}\\)\n(D) \\(\\left|2z_1-z_2\\right|=5\\)\n", "ideal": "[\"the correct answer is (D)\", \"the correct answer is \\textbf{(D)}\", \"the correct answer is $\\boxed{\\textbf{(D)}\"]"}, {"prompt": "If the system of linear equations.\n8x + y + 4z = \u20132\nx + y + z = 0\n\\(\\lambda x \u2013 3y = \\mu\\)\nhas infinitely many solutions, then the distance of the point (\u03bb, \u03bc, -1/2) from the plane 8x + y + 4z + 2 = 0 is\n(A) \\(3\\sqrt{5}\\)\n(B) \\( 4\\)\n(C) \\(\\frac{26}{9}\\)\n(D) \\(\\frac{10}{3}\\)\n", "ideal": "[\"the correct answer is (D)\", \"the correct answer is \\textbf{(D)}\", \"the correct answer is $\\boxed{\\textbf{(D)}\"]"}, {"prompt": "Let A be a 2 \u00d7 2 matrix with det (A) = \u20131 and det((A + I) (Adj (A) + I)) = 4. Then the sum of the diagonal elements of A can be\n(A) -1\n(B) 2\n(C) 1\n(D) -\u221a2\n", "ideal": "[\"the correct answer is (B)\", \"the correct answer is \\textbf{(B)}\", \"the correct answer is $\\boxed{\\textbf{(B)}\"]"}, {"prompt": "The odd natural number a, such that the area of the region bounded by y = 1, y = 3, x = 0, x = y^a is 364/3, is equal to\n(A) 3\n(B) 5\n(C) 7\n(D) 9\n", "ideal": "[\"the correct answer is (B)\", \"the correct answer is \\textbf{(B)}\", \"the correct answer is $\\boxed{\\textbf{(B)}\"]"}, {"prompt": "Consider two G.Ps. 2, 2^2, 2^3, \u2026. and 4, 4^2, 4^3, \u2026 of 60 and n terms respectively. If the geometric mean of all the 60 + n terms is \\(\\left(2\\right)^\\frac{225}{8}\\ \\text{then}\\ \\sum_{k=1}^{n}k\\left(n-k\\right)\\) is equal to\n(A) 560\n(B) 1540\n(C) 1330\n(D) 2600\n", "ideal": "[\"the correct answer is (C)\", \"the correct answer is \\textbf{(C)}\", \"the correct answer is $\\boxed{\\textbf{(C)}\"]"}, {"prompt": "If the function \\(f\\left(x\\right)=\\left\\{\\frac{\\text{log}_e\\left(1-x+x^2\\right)+\\text{log}_e\\left(1+x+x^2\\right)}{\\underset{k,}{\\sec x}-\\cos x} \\right.,\\underset{x=0}{x\\in\\left(\\frac{-\\pi}{2},\\frac{\\pi}{2}\\right)}-\\left\\{0 \\right\\}\\)\nis continuous at x = 0, then k is equal to\n(A) 1\n(B) \u20131\n(C) e\n(D) 0\n", "ideal": "[\"the correct answer is (A)\", \"the correct answer is \\textbf{(A)}\", \"the correct answer is $\\boxed{\\textbf{(A)}\"]"}, {"prompt": "If\n\\(f\\left(x\\right)=\\left\\{\\begin{matrix}x+a, & x\\leq 0 \\\\\\left|x-4\\right|, & x>0 \\\\\\end{matrix}\\right.\\text{ and }g\\left(x\\right)=\\left\\{\\begin{matrix}x+1, & x<0 \\\\\\left(x-4\\right)^2+b, & x\\geq 0 \\\\\\end{matrix}\\right.\\) are continuous on R, then (gof) (2) + (fog) (\u20132) is equal to\n(A) \u201310\n(B) 10\n(C) 8\n(D) \u20138\n", "ideal": "[\"the correct answer is (D)\", \"the correct answer is \\textbf{(D)}\", \"the correct answer is $\\boxed{\\textbf{(D)}\"]"}, {"prompt": "Let \\(f\\left(x\\right)=\\left\\{\\begin{matrix}x^3-x^2+10x-7, & x\\leq1 \\\\-2x+\\text{log}_2\\left(b^2-4\\right), & x>1 \\\\\\end{matrix}\\right.\\)\nThen the set of all values of b, for which f(x) has maximum value at x = 1, is \n(A) \\( \\left(-6, -2\\right)\\\\ \\)\n(B) \\( \\left(2, 6\\right)\\\\ \\)\n(C) \\( \\left[-6, -2) \\cup (2, 6\\right]\\)\n(D) \\(\\left[-\\sqrt{6},-2\\right]\\cup \\left(2,\\sqrt{6}\\right] \\)\n", "ideal": "[\"the correct answer is (C)\", \"the correct answer is \\textbf{(C)}\", \"the correct answer is $\\boxed{\\textbf{(C)}\"]"}, {"prompt": "\\(If~ a=\\displaystyle \\lim_{n \\to \\infty}\\sum_{k=1}^{n}\\frac{2n}{n^2+k^2}\\text{and}\\) \n\\(f\\left(x\\right)=\\sqrt{\\frac{1-\\cos x}{1+\\cos x}},x\\in\\left(0,1\\right),then\\)\n(A) \\(2\\sqrt{2}f\\left(\\frac{a}{2}\\right)=f\u2019\\left(\\frac{a}{2}\\right)\\)\n(B) \\(f\\left(\\frac{a}{2}\\right)f\u2019\\left(\\frac{a}{2}\\right)=\\sqrt{2}\\)\n(C) \\(\\sqrt{2}f\\left(\\frac{a}{2}\\right)=f\u2019\\left(\\frac{a}{2}\\right)\\)\n(D) \\(f\\left(\\frac{a}{2}\\right)=\\sqrt{2}f\u2019\\left(\\frac{a}{2}\\right)\\)\n", "ideal": "[\"the correct answer is (C)\", \"the correct answer is \\textbf{(C)}\", \"the correct answer is $\\boxed{\\textbf{(C)}\"]"}, {"prompt": "If \\(\\frac{dy}{dx}+2y\\tan x=\\sin x,0<x<\\frac{\\pi}{2}\\ \\text{and}\\ y\\left(\\frac{\\pi}{3}\\right)=0\\)\u00a0 then the maximum value of y(x) is:\n(A) \\(\\frac{1}{8}\\)\n(B) \\(\\frac{3}{4}\\)\n(C) \\(\\frac{1}{4}\\)\n(D) \\(\\frac{3}{8}\\)\n", "ideal": "[\"the correct answer is (A)\", \"the correct answer is \\textbf{(A)}\", \"the correct answer is $\\boxed{\\textbf{(A)}\"]"}, {"prompt": "A point P moves so that the sum of squares of its distances from the points (1, 2) and (\u20132, 1) is 14. Let f(x, y) = 0 be the locus of P, which intersects the x-axis at the points A, B and the y-axis at the points C, D. Then the area of the quadrilateral ACBD is equal to\n(A) \\(\\frac{9}{2} \\)\n(B) \\(\\frac{3\\sqrt{17}}{2} \\)\n(C) \\(\\frac{3\\sqrt{17}}{4} \\)\n(D) \\(9\\)\n", "ideal": "[\"the correct answer is (B)\", \"the correct answer is \\textbf{(B)}\", \"the correct answer is $\\boxed{\\textbf{(B)}\"]"}, {"prompt": "Let the tangent drawn to the parabola y^2 = 24x at the point (\u03b1, \u03b2) is perpendicular to the line 2x + 2y = 5. Then the normal to the hyperbola \\(\\frac{x^2}{\\alpha^2}-\\frac{y^2}{\\beta^2}=1 \\) at the point (\u03b1 + 4, \u03b2 + 4) does NOT pass through the point\n(A) (25, 10)\n(B) (20, 12)\n(C) (30, 8)\n(D) (15, 13)\n", "ideal": "[\"the correct answer is (D)\", \"the correct answer is \\textbf{(D)}\", \"the correct answer is $\\boxed{\\textbf{(D)}\"]"}, {"prompt": "The length of the perpendicular from the point (1, \u20132, 5) on the line passing through (1, 2, 4) and parallel to the line x + y \u2013 z = 0 = x \u2013 2y + 3z \u2013 5 is\n(A) \\(\\sqrt{\\frac{21}{2}}\\)\n(B) \\(\\sqrt{\\frac{9}{2}}\\)\n(C) \\(\\sqrt{\\frac{73}{2}}\\)\n(D) \\(1 \\)\n", "ideal": "[\"the correct answer is (A)\", \"the correct answer is \\textbf{(A)}\", \"the correct answer is $\\boxed{\\textbf{(A)}\"]"}, {"prompt": "Let \\(\\vec{a}=\\alpha\\hat{i}+\\hat{j}-\\hat{k}\\ \\text{and}\\ \\vec{b}=2\\hat{i}+\\hat{j}-\\alpha\\hat{k},\\alpha >0.\\)\u00a0 \\(\\text{If the projection of}\\ \\vec{a}\\times \\vec{b}\\ \\text{on the vector}\\ -\\hat{i}+2\\hat{j}-2\\hat{k}\\) is 30, then \u03b1 is equal to\n(A) \\(\\frac{15}{2} \\)\n(B) \\(8\\)\n(C) \\(\\frac{13}{2} \\)\n(D) \\(7\\)\n", "ideal": "[\"the correct answer is (D)\", \"the correct answer is \\textbf{(D)}\", \"the correct answer is $\\boxed{\\textbf{(D)}\"]"}, {"prompt": "The mean and variance of a binomial distribution are \u03b1 and \u03b1/3, respectively. If P(X = 1) = 4/243 then P(X = 4 or 5) is equal to :\n(A) \\(\\frac{5}{9} \\)\n(B) \\(\\frac{64}{81} \\)\n(C) \\(\\frac{16}{27} \\)\n(D) \\(\\frac{145}{243} \\)\n", "ideal": "[\"the correct answer is (C)\", \"the correct answer is \\textbf{(C)}\", \"the correct answer is $\\boxed{\\textbf{(C)}\"]"}, {"prompt": "Let E1, E2, E3 be three mutually exclusive events such that \\(P\\left(E_1\\right)=\\frac{2+3p}{6},P\\left(E_2\\right)=\\frac{2-p}{8}\\ \\text{and}\\ P\\left(E_3\\right)=\\frac{1-p}{2}.\\) If the maximum and minimum values of p are p1 and p2, then (p1 + p2) is equal to :\n(A) \\(\\frac{2}{3}\\)\n(B) \\(\\frac{5}{3}\\)\n(C) \\(\\frac{5}{4}\\)\n(D) \\(1\\)\n", "ideal": "[\"the correct answer is (B)\", \"the correct answer is \\textbf{(B)}\", \"the correct answer is $\\boxed{\\textbf{(B)}\"]"}, {"prompt": "\\(\\text{Let } S=\\left\\{\\theta\\in\\left[0,2\\pi\\right]:8^{2\\sin^2\\theta}+8^{2\\cos^2\\theta} =16\\right\\} \\). \\(\\text{Then}~ n\\left(S\\right)+\\underset{\\theta\\in S}{\\sum}\\left(\\sec\\left(\\frac{\\pi}{4}+2\\theta\\right)\\text{cosec}\\left(\\frac{\\pi}{4}+2\\theta\\right)\\right)\\text{is equal to }:\\)\n(A) 0\n(B) \u2013 2\n(C) \u2013 4\n(D) 12\n", "ideal": "[\"the correct answer is (C)\", \"the correct answer is \\textbf{(C)}\", \"the correct answer is $\\boxed{\\textbf{(C)}\"]"}, {"prompt": "\\(\\tan\\left(2\\tan^{-1}\\frac{1}{5}+\\sec^{-1}\\frac{\\sqrt{5}}{2}+2\\tan^{-1}\\frac{1}{8}\\right)\\) is equal to :\n(A) \\(1\\)\n(B) \\(2\\)\n(C) \\(\\frac{1}{4} \\)\n(D) \\(\\frac{5}{4} \\)\n", "ideal": "[\"the correct answer is (B)\", \"the correct answer is \\textbf{(B)}\", \"the correct answer is $\\boxed{\\textbf{(B)}\"]"}, {"prompt": "The statement \\(\\left(\\sim\\left(p\\ \\Leftrightarrow \\sim q\\right)\\right)\\wedge q\\) is :\n(A) A tautology\n(B) A contradiction\n(C) \\(\\text{Equivalent to} \\left ( p \\Rightarrow q \\right) \\wedge q\\)\n(D) \\(\\text{Equivalent to} \\left ( p \\Rightarrow q \\right) \\wedge p\\)\n", "ideal": "[\"the correct answer is (D)\", \"the correct answer is \\textbf{(D)}\", \"the correct answer is $\\boxed{\\textbf{(D)}\"]"}, {"prompt": "The minimum value of the sum of the squares of the roots of x^2 + (3 \u2013 a)x + 1 = 2a is\n(A) 4\n(B) 5\n(C) 6\n(D) 8\n", "ideal": "[\"the correct answer is (C)\", \"the correct answer is \\textbf{(C)}\", \"the correct answer is $\\boxed{\\textbf{(C)}\"]"}, {"prompt": "If z = x + iy satisfies | z | \u2013 2 = 0 and |z \u2013 i| \u2013 | z + 5i| = 0, then\n(A) x + 2y \u2013 4 = 0\n(B) x^2 + y \u2013 4 = 0\n(C) x + 2y + 4 = 0\n(D) x^2 \u2013 y + 3 = 0\n", "ideal": "[\"the correct answer is (C)\", \"the correct answer is \\textbf{(C)}\", \"the correct answer is $\\boxed{\\textbf{(C)}\"]"}, {"prompt": "Let \\(A=\\begin{bmatrix}1 \\\\ 1\\\\1\\end{bmatrix}\\text{ and }B=\\begin{bmatrix}9^2 & -10^2 & 11^2 \\\\12^2 & 13^2 & -14^2 \\\\-15^2 & 16^2 & 17^2 \\\\\\end{bmatrix},\\) then the value of A\u2032BA is \n(A) 1224\n(B) 1042\n(C) 540\n(D) 539\n", "ideal": "[\"the correct answer is (D)\", \"the correct answer is \\textbf{(D)}\", \"the correct answer is $\\boxed{\\textbf{(D)}\"]"}, {"prompt": "\\(\\sum_{\\underset{i\\neq j}{i, j=0}}^{n}{^n}C_i\\ ^nC_j\\) is equal to \n(A) \\( 2^{2n} \u2013 ^{2n}C_n\u00a0\\)\n(B) \\( 2^{2n-1} \u2013 ^{2n-1}C_{n-1}\u00a0\\)\n(C) \\(2^{2n}-\\frac{1}{2}\\ ^{2n}C_n\\)\n(D) \\(2^{n-1}+2^{2n-1}C_n\\)\n", "ideal": "[\"the correct answer is (A)\", \"the correct answer is \\textbf{(A)}\", \"the correct answer is $\\boxed{\\textbf{(A)}\"]"}, {"prompt": "Let P and Q be any points on the curves (x \u2013 1)^2 + (y + 1)^2 = 1 and y = x^2, respectively. The distance between P and Q is minimum for some value of the abscissa of P in the interval\n(A) \\(\\left(0,\\frac{1}{4}\\right) \\)\n(B) \\(\\left(\\frac{1}{2},\\frac{3}{4}\\right) \\)\n(C) \\(\\left(\\frac{1}{4},\\frac{1}{2}\\right)\\)\n(D) \\(\\left(\\frac{3}{4},1\\right)\\)\n", "ideal": "[\"the correct answer is (C)\", \"the correct answer is \\textbf{(C)}\", \"the correct answer is $\\boxed{\\textbf{(C)}\"]"}, {"prompt": "Let \\(\\beta=\\displaystyle \\lim_{x \\to 0}\\frac{\\alpha x-\\left(e^{3x}-1\\right)}{\\alpha x\\left(e^{3x}-1\\right)}\\ \\text{for some}\\ \\alpha\\ \\in\\ \\mathbb{R}.\\) Then the value of \u03b1 + \u03b2 is\n(A) \\(\\frac{14}{5} \\)\n(B) \\(\\frac{3}{2} \\)\n(C) \\(\\frac{5}{2} \\)\n(D) \\(\\frac{7}{2} \\)\n", "ideal": "[\"the correct answer is (C)\", \"the correct answer is \\textbf{(C)}\", \"the correct answer is $\\boxed{\\textbf{(C)}\"]"}, {"prompt": "The value of \\(\\text{log}_e2\\frac{d}{dx}\\left(\\text{log}_{\\cos x}\\text{cosec x}\\right) \\textup{ at }x=\\frac{\\pi}{4}\\) is\n(A) \\(-2\\sqrt{2}\\)\n(B) \\(2\\sqrt{2}\\)\n(C) \\(-4\\)\n(D) \\(4\\)\n", "ideal": "[\"the correct answer is (D)\", \"the correct answer is \\textbf{(D)}\", \"the correct answer is $\\boxed{\\textbf{(D)}\"]"}, {"prompt": "\\(\\displaystyle\\int\\limits_0^{20\\pi}\\left(\\left|\\sin x\\right|+\\left|\\cos x\\right|\\right)^2 dx\\) is equal to\n(A) \\(10\\left(\\pi+4\\right)\\)\n(B) \\(10\\left(\\pi+2\\right)\\)\n(C) \\(20\\left(\\pi-2\\right)\\)\n(D) \\(20\\left(\\pi+2\\right)\\)\n", "ideal": "[\"the correct answer is (D)\", \"the correct answer is \\textbf{(D)}\", \"the correct answer is $\\boxed{\\textbf{(D)}\"]"}, {"prompt": "Let the solution curve y = f(x) of the differential equation \\(\\frac{dy}{dx}+\\frac{xy}{x^2-1}=\\frac{x^4+2x}{\\sqrt{1-x^2}},x \\in \\left(-1, 1\\right)\\) pass through the origin. Then \\(\\displaystyle\\int\\limits_{-\\frac{\\sqrt{3}}{2}}^{\\frac{\\sqrt{3}}{2}}f\\left(x\\right)dx\\) is\n(A) \\(\\frac{\\pi}{3}-\\frac{1}{4}\\)\n(B) \\(\\frac{\\pi}{3}-\\frac{\\sqrt{3}}{4}\\)\n(C) \\(\\frac{\\pi}{6}-\\frac{\\sqrt{3}}{4}\\)\n(D) \\(\\frac{\\pi}{6}-\\frac{\\sqrt{3}}{2}\\)\n", "ideal": "[\"the correct answer is (B)\", \"the correct answer is \\textbf{(B)}\", \"the correct answer is $\\boxed{\\textbf{(B)}\"]"}, {"prompt": "The acute angle between the pair of tangents drawn to the ellipse 2x^2 + 3y^2 = 5 from the point (1, 3) is\n(A) \\(\\tan^{-1}\\left(\\frac{16}{7\\sqrt{5}}\\right)\\)\n(B) \\(\\tan^{-1}\\left(\\frac{24}{7\\sqrt{5}}\\right)\\)\n(C) \\(\\tan^{-1}\\left(\\frac{32}{7\\sqrt{5}}\\right)\\)\n(D) \\(\\tan^{-1}\\left(\\frac{3+8\\sqrt{5}}{35}\\right)\\)\n", "ideal": "[\"the correct answer is (B)\", \"the correct answer is \\textbf{(B)}\", \"the correct answer is $\\boxed{\\textbf{(B)}\"]"}, {"prompt": "The equation of a common tangent to the parabolas y = x^2 and y = \u2013(x \u2013 2)^2 is\n(A) y = 4(x \u2013 2)\n(B) y = 4(x \u2013 1)\n(C) y = 4(x + 1)\n(D) y = 4(x + 2)\n", "ideal": "[\"the correct answer is (B)\", \"the correct answer is \\textbf{(B)}\", \"the correct answer is $\\boxed{\\textbf{(B)}\"]"}, {"prompt": "Let the abscissae of the two points P and Q on a circle be the roots of x^2 \u2013 4x \u2013 6 = 0 and the ordinates of P and Q be the roots of y^2 + 2y \u2013 7 = 0. If PQ is a diameter of the circle x^2 + y^2 + 2ax + 2by + c = 0, then the value of (a + b \u2013 c) is\n(A) 12\n(B) 13\n(C) 14\n(D) 16\n", "ideal": "[\"the correct answer is (A)\", \"the correct answer is \\textbf{(A)}\", \"the correct answer is $\\boxed{\\textbf{(A)}\"]"}, {"prompt": "If the line x \u2013 1 = 0 is a directrix of the hyperbola kx^2 \u2013 y^2 = 6, then the hyperbola passes through the point\n(A) \\(\\left(-2\\sqrt{5},6\\right)\\)\n(B) \\(\\left(-\\sqrt{5},3\\right)\\)\n(C) \\(\\left(\\sqrt{5},-2\\right)\\)\n(D) \\(\\left(2\\sqrt{5},3\\sqrt{6}\\right)\\)\n", "ideal": "[\"the correct answer is (C)\", \"the correct answer is \\textbf{(C)}\", \"the correct answer is $\\boxed{\\textbf{(C)}\"]"}, {"prompt": "A vector \\(\\vec{a}\\)is parallel to the line of intersection of the plane determined by the vectors \\(\\hat{i},\\hat{i}+\\hat{j}\\)and the plane determined by the vectors \\(\\hat{i}-\\hat{j},\\hat{i}+\\hat{k}. \\) The obtuse angle between\\(\\vec{a}\\ \\text{and the vector}\\ \\vec{b}=\\hat{i}-2\\hat{j}+2\\hat{k}\\)\u00a0 is\n(A) \\(\\frac{3\\pi}{4} \\)\n(B) \\(\\frac{2\\pi}{3} \\)\n(C) \\(\\frac{4\\pi}{5} \\)\n(D) \\(\\frac{5\\pi}{6}\\)\n", "ideal": "[\"the correct answer is (A)\", \"the correct answer is \\textbf{(A)}\", \"the correct answer is $\\boxed{\\textbf{(A)}\"]"}, {"prompt": "If \\(0<x<\\frac{1}{\\sqrt{2}}\\ \\text{and}\\ \\frac{\\sin^{-1}x}{\\alpha}=\\frac{\\cos^{-1}x}{\\beta}\\)\u00a0 then a value of \\(\\sin\\left(\\frac{2\\pi\\alpha}{\\alpha+\\beta}\\right)\\) is\n(A) \\(4\\sqrt{\\left(1-x^2\\right)}\\left(1-2x^2\\right)\\)\n(B) \\(4x\\sqrt{\\left(1-x^2\\right)}\\left(1-2x^2\\right)\\)\n(C) \\(2x\\sqrt{\\left(1-x^2\\right)}\\left(1-4x^2\\right)\\)\n(D) \\(4\\sqrt{\\left(1-x^2\\right)}\\left(1-4x^2\\right)\\)\n", "ideal": "[\"the correct answer is (B)\", \"the correct answer is \\textbf{(B)}\", \"the correct answer is $\\boxed{\\textbf{(B)}\"]"}, {"prompt": "Negation of the Boolean expression p\u21d4 (q\u21d2p) is\n(A) (~ p) \u2227q\n(B) p\u2227 (~ q)\n(C) (~ p) \u2228 (~ q)\n(D) (~ p) \u2227 (~ q)\n", "ideal": "[\"the correct answer is (D)\", \"the correct answer is \\textbf{(D)}\", \"the correct answer is $\\boxed{\\textbf{(D)}\"]"}, {"prompt": "Let X be a binomially distributed random variable with mean 4 and variance 4/3. Then, 54 P(X\u2264 2) is equal to\n(A) \\(\\frac{73}{27}\\)\n(B) \\(\\frac{146}{27}\\)\n(C) \\(\\frac{146}{81}\\)\n(D) \\(\\frac{126}{81}\\)\n", "ideal": "[\"the correct answer is (B)\", \"the correct answer is \\textbf{(B)}\", \"the correct answer is $\\boxed{\\textbf{(B)}\"]"}, {"prompt": "The integral \\(\\int\\frac{\\left(1-\\frac{1}{\\sqrt{3}}\\right)\\left(\\cos x-\\sin x\\right)}{\\left(1+\\frac{2}{\\sqrt{3}}\\sin 2x\\right)}dx\\) is equal to\n(A) \\(\\frac{1}{2}\\text{log}_e\\left|\\frac{\\tan\\left(\\frac{x}{2}+\\frac{\\pi}{12}\\right)}{\\tan\\left(\\frac{x}{2}+\\frac{\\pi}{6}\\right)}\\right|+C\\)\n(B) \\(\\frac{1}{2}\\text{log}_e\\left|\\frac{\\tan\\left(\\frac{x}{2}+\\frac{x}{6}\\right)}{\\tan\\left(\\frac{x}{2}+\\frac{\\pi}{3}\\right)}\\right|+C\\)\n(C) \\(\\text{log}_e\\left|\\frac{\\tan\\left(\\frac{x}{2}+\\frac{\\pi}{6}\\right)}{\\tan\\left(\\frac{x}{2}+\\frac{\\pi}{12}\\right)}\\right|+C\\)\n(D) \\(\\frac{1}{2}\\text{log}_e\\left|\\frac{\\tan\\left(\\frac{x}{2}-\\frac{\\pi}{12}\\right)}{\\tan\\left(\\frac{x}{2}-\\frac{\\pi}{6}\\right)}\\right|+C\\)\n", "ideal": "[\"the correct answer is (A)\", \"the correct answer is \\textbf{(A)}\", \"the correct answer is $\\boxed{\\textbf{(A)}\"]"}, {"prompt": "The area bounded by the curves y = |x^2 \u2013 1| and y = 1 is\n(A) \\(\\frac{2}{3}\\left(\\sqrt{2}+1\\right)\\)\n(B) \\(\\frac{4}{3}\\left(\\sqrt{2}-1\\right)\\)\n(C) \\(2\\left(\\sqrt{2}-1\\right)\\)\n(D) \\(\\frac{8}{3}\\left(\\sqrt{2}-1\\right)\\)\n", "ideal": "[\"the correct answer is (D)\", \"the correct answer is \\textbf{(D)}\", \"the correct answer is $\\boxed{\\textbf{(D)}\"]"}, {"prompt": "Let R1 and R2 be two relations defined on \u211d by a R1b \u21d4 ab \u2265 0 and aR2b \u21d4 a \u2265 b. Then,\n(A) R1 is an equivalence relation but not R2\n(B) R2 is an equivalence relation but not R1\n(C) Both R1 and R2 are equivalence relations\n(D) Neither R1 nor R2 is an equivalence relation\n", "ideal": "[\"the correct answer is (D)\", \"the correct answer is \\textbf{(D)}\", \"the correct answer is $\\boxed{\\textbf{(D)}\"]"}, {"prompt": "Let \\(f,g : \\mathbb{N} = \\{1\\} \\rightarrow \\mathbb{N}\\ \\text{be functions defined by}\\)\nf(a) = \u03b1, where \u03b1 is the maximum of the powers of those primes p such that p^\u03b1 divides a, and g(a) = a + 1, for all a \u2208 N \u2013 {1}. Then, the function\u00a0 f + g is\n(A) One-one but not onto\n(B) Onto but not one-one\n(C) Both one-one and onto\n(D) Neither one-one nor onto\n", "ideal": "[\"the correct answer is (D)\", \"the correct answer is \\textbf{(D)}\", \"the correct answer is $\\boxed{\\textbf{(D)}\"]"}, {"prompt": "Let the minimum value v0 of \\(v= \\left|z\\right|^2 + \\left|z \u2013 3\\right|^2 + \\left|z \u2013 6i\\right|^2, z \\in \\mathbb {C}\\) is attained at z = z0. Then \\(\\left|2z_0^2 \u2013 \\bar{z}_0^3 + 3 \\right|^2 + v_0^2\\) is equal to\n(A) 1000\n(B) 1024\n(C) 1105\n(D) 1196\n", "ideal": "[\"the correct answer is (A)\", \"the correct answer is \\textbf{(A)}\", \"the correct answer is $\\boxed{\\textbf{(A)}\"]"}, {"prompt": "Let \\(A = \\begin{pmatrix}1 & 2 \\\\-2 & -5 \\\\\\end{pmatrix}\\). Let \u03b1, \u03b2 \u2208 \u211d be such that \u03b1A^2 + \u03b2A = 2I. Then \u03b1 + \u03b2 is equal to\n(A) \u201310\n(B) \u20136\n(C) 6\n(D) 10\n", "ideal": "[\"the correct answer is (D)\", \"the correct answer is \\textbf{(D)}\", \"the correct answer is $\\boxed{\\textbf{(D)}\"]"}, {"prompt": "The remainder when (2021)^2022 + (2022)^2021 is divided by 7 is\n(A) 0\n(B) 1\n(C) 2\n(D) 6\n", "ideal": "[\"the correct answer is (A)\", \"the correct answer is \\textbf{(A)}\", \"the correct answer is $\\boxed{\\textbf{(A)}\"]"}, {"prompt": "Suppose a1, a2, \u2026 an, \u2026 be an arithmetic progression of natural numbers. If the ration of the sum of first five terms to the sum of first nine terms of the progression is 5 : 17 and 110 < a15 < 120, then the sum of the first ten terms of the progression is equal to \n(A) 290\n(B) 380\n(C) 460\n(D) 510\n", "ideal": "[\"the correct answer is (B)\", \"the correct answer is \\textbf{(B)}\", \"the correct answer is $\\boxed{\\textbf{(B)}\"]"}, {"prompt": "Let \u211d \u2192 \u211d be function defined as \\(f(x)=a\\sin \\left(\\frac{\\pi [x]}{2}\\right) + [2-x], a\\in \\mathbb{R}\\) where [t] is the greatest integer less than or equal to t. \\(\\text{If}\\ \\displaystyle \\lim_{ x \\to 1}f(x)\\ \\text{exists, then the value of}\\ \\int_{0}^{4}f(x) dx\\) is equal to\n(A) \u20131\n(B) \u20132\n(C) 1\n(D) 2\n", "ideal": "[\"the correct answer is (B)\", \"the correct answer is \\textbf{(B)}\", \"the correct answer is $\\boxed{\\textbf{(B)}\"]"}, {"prompt": "The area of the smaller region enclosed by the curves y^2 = 8x + 4 and \\(x^2 +y^2 + 4\\sqrt{3}x-4 =0\\) is equal to\n(A) \\(\\frac{1}{3}(2-12\\sqrt{3} + 8\\pi)\\)\n(B) \\(\\frac{1}{3}(2-12\\sqrt{3} + 6\\pi)\\)\n(C) \\(\\frac{1}{3}(4-12\\sqrt{3} + 8\\pi)\\)\n(D) \\(\\frac{1}{3}(4-12\\sqrt{3} + 6\\pi)\\)\n", "ideal": "[\"the correct answer is (C)\", \"the correct answer is \\textbf{(C)}\", \"the correct answer is $\\boxed{\\textbf{(C)}\"]"}, {"prompt": "Let y = y1(x) and y = y2(x) be two distinct solution of the differential equation \\(\\frac{dy}{dx}=x+y,\\) with y1(0) = 0 and y2(0) = 1 respectively. Then, the number of points of intersection of y = y1 (x) and y = y2(x) is\n(A) 0\n(B) 1\n(C) 2\n(D) 3\n", "ideal": "[\"the correct answer is (A)\", \"the correct answer is \\textbf{(A)}\", \"the correct answer is $\\boxed{\\textbf{(A)}\"]"}, {"prompt": "Let P(a, b) be a point on the parabola y^2 = 8x such that the tangent at P passes through the centre of the circle x^2 + y^2 \u2013 10x \u2013 14y + 65 = 0. Let A be the product of all possible values of a and B be the product of all possible values of b. Then the value of A + B is equal to\n(A) 0\n(B) 25\n(C) 40\n(D) 65\n", "ideal": "[\"the correct answer is (D)\", \"the correct answer is \\textbf{(D)}\", \"the correct answer is $\\boxed{\\textbf{(D)}\"]"}, {"prompt": "\\(\\text{Let}~\\vec{a} = \\alpha \\hat{i} + \\hat{j} + \\beta \\hat{k}~\\text{and}~ \\vec{b} = 3 \\hat{i} + 5\\hat{j} + 4 \\hat{k}~ \\text{be two vectors, such that }\\vec{a} \\times \\vec{b} = -\\hat{i} + 9\\hat{j} + 12 \\hat{k}.~\\text{Then the projection of } \\vec{b}-2\\vec{a} ~\\text{on}~ \\vec{b}+ \\vec{a}~\\text {is equal to}\\)\n(A) \\(2\\)\n(B) \\(\\frac{39}{5}\\)\n(C) \\(9\\)\n(D) \\(\\frac{46}{5}\\)\n", "ideal": "[\"the correct answer is (D)\", \"the correct answer is \\textbf{(D)}\", \"the correct answer is $\\boxed{\\textbf{(D)}\"]"}, {"prompt": "\\(\\text{Let}~\\vec{a} = 2\\hat{i}-\\hat{j}+5\\hat{k}~\\text{and}~ \\vec{b} = \\alpha \\hat{i} + \\beta \\hat{j}+2\\hat{k}. \\text{If}~((\\vec{a}\\times \\vec{b}) \\times \\hat{i})\\cdot \\hat{k}=\\frac{23}{2},~ \\text{then} \\left|\\vec{b}\\times 2\\hat{j}\\right| \\text{is equal to} \\)\n(A) \\(4\\)\n(B) \\(5\\)\n(C) \\(\\sqrt{21}\\)\n(D) \\(\\sqrt{17}\\)\n", "ideal": "[\"the correct answer is (B)\", \"the correct answer is \\textbf{(B)}\", \"the correct answer is $\\boxed{\\textbf{(B)}\"]"}, {"prompt": "Let S be the sample space of all five digit numbers. It p is the probability that a randomly selected number from S, is multiple of 7 but not divisible by 5, then 9p is equal to \n(A) 1.0146\n(B) 1.2085\n(C) 1.0285\n(D) 1.1521\n", "ideal": "[\"the correct answer is (C)\", \"the correct answer is \\textbf{(C)}\", \"the correct answer is $\\boxed{\\textbf{(C)}\"]"}, {"prompt": "Let a vertical tower AB of height 2h stands on a horizontal ground. Let from a point P on the ground a man can see upto height h of the tower with an angle of elevation 2\u03b1. \u00a0\\(\\text{When from P, he moves a distance d in the direction of}\\ \\overrightarrow{AP},\\) he can see the top B of the tower with an angle of elevation \u03b1. if d = \u221a7 h, then tan \u03b1 is equal to\n(A) \\(\\sqrt{5}-2\\)\n(B) \\(\\sqrt{3}-1\\)\n(C) \\(\\sqrt{7}-2\\)\n(D) \\(\\sqrt{7}-\\sqrt{3}\\)\n", "ideal": "[\"the correct answer is (C)\", \"the correct answer is \\textbf{(C)}\", \"the correct answer is $\\boxed{\\textbf{(C)}\"]"}, {"prompt": "\\((p \\land r )\\Leftrightarrow ( p \\land (\\sim q))\\) is equivalent to (~ p) when r is\n(A) p\n(B) ~p\n(C) q\n(D) ~q\n", "ideal": "[\"the correct answer is (C)\", \"the correct answer is \\textbf{(C)}\", \"the correct answer is $\\boxed{\\textbf{(C)}\"]"}, {"prompt": "If the plane P passes through the intersection of two mutually perpendicular planes 2x + ky \u2013 5z = 1 and 3kx \u2013 ky + z = 5, k < 3 and intercepts a unit length on positive x-axis, then the intercept made by the plane P on the y-axis is\n(A) \\(\\ \\frac{1}{11} \\)\n(B) \\(\\ \\frac{5}{11} \\)\n(C) \\(\\ 6\\)\n(D) \\(\\ 7 \\)\n", "ideal": "[\"the correct answer is (D)\", \"the correct answer is \\textbf{(D)}\", \"the correct answer is $\\boxed{\\textbf{(D)}\"]"}, {"prompt": "Let A(1, 1), B(-4, 3) and C(-2, -5) be vertices of a triangle ABC, P be a point on side BC, and \u03941 and \u03942 be the areas of triangles APB and ABC, respectively. If \u03941 : \u03942 = 4 : 7, then the area enclosed by the lines AP, AC and the x-axis is\n(A) \\(\\frac{1}{4}\\)\n(B) \\(\\frac{3}{4}\\)\n(C) \\(\\frac{1}{2}\\)\n(D) \\(1\\)\n", "ideal": "[\"the correct answer is (C)\", \"the correct answer is \\textbf{(C)}\", \"the correct answer is $\\boxed{\\textbf{(C)}\"]"}, {"prompt": "If the circle \\(x^2 + y^2 \u2013 2gx + 6y \u2013 19c = 0, g, c \\in \\mathbb {R} \\) passes through the point (6, 1) and its centre lies on the line x \u2013 2cy = 8, then the length of intercept made by the circle on x-axis is\n(A) \\(\\sqrt{11}\\)\n(B) \\(4\\)\n(C) \\(3 \\)\n(D) \\(2\\sqrt{23}\\)\n", "ideal": "[\"the correct answer is (D)\", \"the correct answer is \\textbf{(D)}\", \"the correct answer is $\\boxed{\\textbf{(D)}\"]"}, {"prompt": "Let a function f: \u211d \u2192 \u211d be defined as :\n\\(f(x) = \\left\\{\\begin{matrix}\\int_{0}^{x}(5-|t-3|)dt, & x>4 \\\\x^2 + bx, & x \\le4 \\\\\\end{matrix}\\right.\\) where b \u2208 \u211d. If f is continuous at x = 4 then which of the following statements is NOT true?\n(A) \\(\\text{f is not differentiable at x} = 4 \\)\n(B) \\(f'(3)+f'(5)=\\frac{35}{4}\\)\n(C) \\(f \\text{ is increasing in } \\left(-\\infty, \\frac{1}{8}\\right) \\cup (8, \\infty)\\)\n(D) \\(f \\text{ has a local minima at } x = \\frac{1}{8}\\)\n", "ideal": "[\"the correct answer is (C)\", \"the correct answer is \\textbf{(C)}\", \"the correct answer is $\\boxed{\\textbf{(C)}\"]"}, {"prompt": "The domain of the function \\(f\\left(x\\right)=\\sin^{-1}\\left[2x^2-3\\right]+\\text{log}_2\\left(\\text{log}_\\frac{1}{2}\\left(x^2-5x+5\\right)\\right) \\)\nwhere [t] is the greatest integer function, is\n(A) \\(\\left(-\\sqrt{\\frac{5}{2}},\\frac{5-\\sqrt{5}}{2}\\right)\\)\n(B) \\(\\left(\\frac{5-\\sqrt{5}}{2},\\frac{5+\\sqrt{5}}{2}\\right)\\)\n(C) \\(\\left(1, \\frac{5-\\sqrt{5}}{2}\\right)\\)\n(D) \\(\\left(1, \\frac{5+\\sqrt{5}}{2}\\right)\\)\n", "ideal": "[\"the correct answer is (C)\", \"the correct answer is \\textbf{(C)}\", \"the correct answer is $\\boxed{\\textbf{(C)}\"]"}, {"prompt": "Let S be the set of (\u03b1, \u03b2), \u03c0 < \u03b1, \u03b2 < 2\u03c0, for which the complex number \\(\\frac{1-i\\sin\\alpha}{1+2i\\sin\\alpha}\\ \\text{is purely imaginary and}\\ \\frac{1+i\\cos\\beta}{1-2i\\cos\\beta}\\ \\text{is purely real},\\) \\(\\text{Let}\\ Z_{\\alpha \\beta} = sin 2\\alpha + i cos 2\\beta, \\left(\\alpha, \\beta\\right) \\in S.\\ \\text{Then}\\) \\(\\sum_{\\left(\\alpha,\\beta\\right)\\in S}\\left(iZ_{\\alpha\\beta}+\\frac{1}{i\\overline{Z}_{\\alpha\\beta}}\\right)\\) is equal to\n(A) 3\n(B) 3i\n(C) 1\n(D) 2 \u2013 i\n", "ideal": "[\"the correct answer is (C)\", \"the correct answer is \\textbf{(C)}\", \"the correct answer is $\\boxed{\\textbf{(C)}\"]"}, {"prompt": "If \u03b1, \u03b2 are the roots of the equation \\(x^2-\\left(5+3^{\\sqrt{\\text{log}_35}}-5^{\\sqrt{\\text{log}_53}}\\right)+3\\left(3^{\\left(\\text{log}_35\\right)^{\\frac{1}{3}}}-5^{\\left(\\text{log}_53\\right)^{\\frac{2}{3}}}-1\\right)=0\\)\nthen the equation, whose roots are \u03b1 + 1/\u03b2\u00a0and \u03b2 + 1/\u03b1 , is\n(A) 3x^2 \u2013 20x \u2013 12 = 0\n(B) 3x^2 \u2013 10x \u2013 4 = 0\n(C) 3x^2 \u2013 10x + 2 = 0\n(D) 3x^2 \u2013 20x + 16 = 0\n", "ideal": "[\"the correct answer is (B)\", \"the correct answer is \\textbf{(B)}\", \"the correct answer is $\\boxed{\\textbf{(B)}\"]"}, {"prompt": "Let \\(A=\\begin{pmatrix}4 & -2 \\\\\\alpha & \\beta \\\\\\end{pmatrix}\\)\nIf A^2 + \u03b3A + 18I = 0, then det (A) is equal to ______.\n(A) \u201318\n(B) 18\n(C) \u201350\n(D) 50\n", "ideal": "[\"the correct answer is (B)\", \"the correct answer is \\textbf{(B)}\", \"the correct answer is $\\boxed{\\textbf{(B)}\"]"}, {"prompt": "If for p \u2260 q \u2260 0, the function \\(f\\left(x\\right)=\\frac{\\sqrt[7]{p\\left(729+x\\right)}-3}{\\sqrt[3]{729+qx}-9}\\) is continuous at x = 0, then\n(A) \\( 7pq~f\\left(0\\right) \u2013 1 = 0\\)\n(B) \\( 63q f\\left(0\\right) \u2013 p^2 = 0\\)\n(C) \\( 21qf\\left(0\\right) \u2013 p^2 = 0\\)\n(D) \\( 7pq f\\left(0\\right) \u2013 9 = 0\\)\n", "ideal": "[\"the correct answer is (B)\", \"the correct answer is \\textbf{(B)}\", \"the correct answer is $\\boxed{\\textbf{(B)}\"]"}, {"prompt": "Let \\(f\\left(x\\right)=2+\\left|x\\right|-\\left|x-1\\right|+\\left|x+1\\right| ,x\\in R.\\)\n\\(\\text{Consider}\\left(S1\\right) :f\u2019\\left(-\\frac{3}{2}\\right)+f\u2019\\left(-\\frac{1}{2}\\right)+f\u2019\\left(\\frac{1}{2}\\right)+f\u2019\\left(\\frac{3}{2}\\right)=2\\)\n\\(\\left(S2\\right):\\displaystyle\\int\\limits_{-2}^2f\\left(x\\right)dx=12\\)\nThen, \n(A) Both (S1) and (S2) are correct\n(B) Both (S1) and (S2) are wrong\n(C) Only (S1) is correct\n(D) Only (S2) is correct\n", "ideal": "[\"the correct answer is (D)\", \"the correct answer is \\textbf{(D)}\", \"the correct answer is $\\boxed{\\textbf{(D)}\"]"}, {"prompt": "Let the sum of an infinite G.P., whose first term is a and the common ratio is r, be 5. Let the sum of its first five terms be 98/25. Then the sum of the first 21 terms of an AP, whose first term is 10ar, n^th term is an and the common difference is 10ar^2, is equal to\n(A) 21 a11\n(B) 22 a11\n(C) 15 a16\n(D) 14 a16\n", "ideal": "[\"the correct answer is (A)\", \"the correct answer is \\textbf{(A)}\", \"the correct answer is $\\boxed{\\textbf{(A)}\"]"}, {"prompt": "The area of the region enclosed by \\(y\\leq 4x^2, x^2\\leq 9y\\text{ and }y\\leq 4,\\) is equal to\n(A) \\(\\frac{40}{3}\\)\n(B) \\(\\frac{56}{3}\\)\n(C) \\(\\frac{112}{3}\\)\n(D) \\(\\frac{80}{3}\\)\n", "ideal": "[\"the correct answer is (D)\", \"the correct answer is \\textbf{(D)}\", \"the correct answer is $\\boxed{\\textbf{(D)}\"]"}, {"prompt": "\\(\\displaystyle\\int\\limits_0^2\\left(\\left|2x^2-3x\\right|+\\left[x-\\frac{1}{2}\\right]\\right)dx, \\) where [t] is the greatest integer function, is equal to\n(A) \\(\\frac{7}{6}\\)\n(B) \\(\\frac{19}{12}\\)\n(C) \\(\\frac{31}{12}\\)\n(D) \\(\\frac{3}{2}\\)\n", "ideal": "[\"the correct answer is (B)\", \"the correct answer is \\textbf{(B)}\", \"the correct answer is $\\boxed{\\textbf{(B)}\"]"}, {"prompt": "Consider a curve y = y(x) in the first quadrant as shown in the figure. Let the area A1 is twice the area A2. Then the normal to the curve perpendicular to the line 2x \u2013 12y = 15 does NOT pass through the point. \n(A) \\( \\left(6, 21\\right)\\\\\\)\n(B) \\( \\left(8, 9\\right)\\\\ \\)\n(C) \\( \\left(10, -4\\right)\\\\ \\)\n(D) \\( \\left(12, -15\\right)\\)\n", "ideal": "[\"the correct answer is (C)\", \"the correct answer is \\textbf{(C)}\", \"the correct answer is $\\boxed{\\textbf{(C)}\"]"}, {"prompt": "The equations of the sides AB, BC and CA of a triangle ABC are 2x + y = 0, x + py = 39 and x \u2013 y = 3, respectively and P(2, 3) is its circumcentre. Then which of the following is NOT true?\n(A) \\( \\left(AC\\right)^2 = 9p\\\\ \\)\n(B) \\( \\left(AC\\right)^2 + p^2 = 136\\\\\\)\n(C) \\( 32 < area\\left(\\triangle ABC\\right) < 36\\\\ \\)\n(D) \\(34 < area\\left(\\triangle ABC\\right) < 38\\)\n", "ideal": "[\"the correct answer is (D)\", \"the correct answer is \\textbf{(D)}\", \"the correct answer is $\\boxed{\\textbf{(D)}\"]"}, {"prompt": "A circle C1 passes through the origin O and has diameter 4 on the positive x-axis. The line y = 2x gives a chord OA of circle C1. Let C2 be the circle with OA as a diameter. If the tangent to C2 at the point A meets the x-axis at P and y-axis at Q, then QA :AP is equal to\n(A) 1 : 4\n(B) 1 : 5\n(C) 2 : 5\n(D) 1 : 3\n", "ideal": "[\"the correct answer is (A)\", \"the correct answer is \\textbf{(A)}\", \"the correct answer is $\\boxed{\\textbf{(A)}\"]"}, {"prompt": "If the length of the latus rectum of a parabola, whose focus is (a, a) and the tangent at its vertex is x + y = a, is 16, then |a| is equal to :\n(A) \\(2\\sqrt{2}\\)\n(B) \\(2\\sqrt{3}\\)\n(C) \\(4\\sqrt{2}\\)\n(D) \\(4\\)\n", "ideal": "[\"the correct answer is (C)\", \"the correct answer is \\textbf{(C)}\", \"the correct answer is $\\boxed{\\textbf{(C)}\"]"}, {"prompt": "If the length of the perpendicular drawn from the point P(a, 4, 2), a> 0 on the line \\(\\frac{x+1}{2}=\\frac{y-3}{3} =\\frac{z-1}{-1}\\text{is }2\\sqrt{6}\\ \\text{units}\\ \\text{and}\\ Q\\left(\\alpha_1,\\alpha_2,\\alpha_3\\right)\\) is the image of the point P in this line, then \\(a+\\sum_{i=1}^{3}\\alpha_i \\) is equal to :\n(A) 7\n(B) 8\n(C) 12\n(D) 14\n", "ideal": "[\"the correct answer is (B)\", \"the correct answer is \\textbf{(B)}\", \"the correct answer is $\\boxed{\\textbf{(B)}\"]"}, {"prompt": "If the line of intersection of the planes ax + by = 3 and ax + by + cz = 0, a> 0 makes an angle 30\u00b0 with the plane y \u2013 z + 2 = 0, then the direction cosines of the line are :\n(A) \\(\\frac{1}{\\sqrt{2}},\\frac{1}{\\sqrt{2}},0\\)\n(B) \\(\\frac{1}{\\sqrt{2}},-\\frac{1}{\\sqrt{2}},0\\)\n(C) \\(\\frac{1}{\\sqrt{5}},-\\frac{2}{\\sqrt{5}},0\\)\n(D) \\(\\frac{1}{2},-\\frac{\\sqrt{3}}{2},0\\)\n", "ideal": "[\"the correct answer is (B)\", \"the correct answer is \\textbf{(B)}\", \"the correct answer is $\\boxed{\\textbf{(B)}\"]"}, {"prompt": "Let X have a binomial distribution B(n, p) such that the sum and the product of the mean and variance of X are 24 and 128 respectively. If \\(P\\left(X>n \u2013 3\\right) = \\frac{k}{2^n},\\) then k is equal to :\n(A) 528\n(B) 529\n(C) 629\n(D) 630\n", "ideal": "[\"the correct answer is (B)\", \"the correct answer is \\textbf{(B)}\", \"the correct answer is $\\boxed{\\textbf{(B)}\"]"}, {"prompt": "A six faced die is biased such that\n3 \u00d7 P (a prime number) = 6 \u00d7 P (a composite number) = 2 \u00d7 P (1).\nLet X be a random variable that counts the number of times one gets a perfect square on some throws of this die. If the die is thrown twice, then the mean of X is :\n(A) \\(\\frac{3}{11}\\)\n(B) \\(\\frac{5}{11}\\)\n(C) \\(\\frac{7}{11}\\)\n(D) \\(\\frac{8}{11}\\)\n", "ideal": "[\"the correct answer is (D)\", \"the correct answer is \\textbf{(D)}\", \"the correct answer is $\\boxed{\\textbf{(D)}\"]"}, {"prompt": "The angle of elevation of the top P of a vertical tower PQ of height 10 from a point A on the horizontal ground is 45\u00b0, Let R be a point on AQ and from a point B, vertically above R, the angle of elevation of P is 60\u00b0. If \\(\\angle BAQ = 30^\\circ\\), AB = d and the area of the trapezium PQRB is \u03b1, then the ordered pair (d, \u03b1) is :\n(A) \\(\\left(10\\left(\\sqrt{3}-1\\right),25\\right) \\)\n(B) \\(\\left(10\\left(\\sqrt{3}-1\\right),\\frac{25}{2}\\right) \\)\n(C) \\(\\left(10\\left(\\sqrt{3}+1\\right),25\\right) \\)\n(D) \\(\\left(10\\left(\\sqrt{3}+1\\right),\\frac{25}{2}\\right) \\)\n", "ideal": "[\"the correct answer is (A)\", \"the correct answer is \\textbf{(A)}\", \"the correct answer is $\\boxed{\\textbf{(A)}\"]"}, {"prompt": "\\(\\text{ Let } S=\\left\\{0\\in\\left(0,\\frac{\\pi}{2}\\right):\\sum_{m=1}^{9}\\sec\\left(\\theta+\\left(m-1\\right)\\frac{\\pi}{6}\\right)\\sec\\left(\\theta+\\frac{m\\pi}{6}\\right)=-\\frac{8}{\\sqrt{3}} \\right\\} \\) Then\n(A) \\(S=\\left\\{\\frac{\\pi}{12}\\right\\}\\)\n(B) \\(S=\\left\\{\\frac{2\\pi}{3}\\right\\}\\)\n(C) \\(\\sum_{\\theta\\in S}\\theta= \\frac{\\pi}{2}\\)\n(D) \\(\\sum_{\\theta\\in S}\\theta= \\frac{3\\pi}{4}\\)\n", "ideal": "[\"the correct answer is (C)\", \"the correct answer is \\textbf{(C)}\", \"the correct answer is $\\boxed{\\textbf{(C)}\"]"}, {"prompt": "If the truth value of the statement \\(\\left(P\\wedge\\left(\\sim R\\right)\\right)\\rightarrow\\left(\\left(\\sim R\\right)\\wedge Q\\right)\\) is F, then the truth value of which of the following is F?\n(A) \\(\u00a0P\\vee Q ~\\rightarrow ~\\sim R\\)\n(B) \\( R\\vee Q ~\\rightarrow ~\\sim P\\)\n(C) \\( \\sim \\left(P\\vee Q \\right)~\\rightarrow ~\\sim R\\)\n(D) \\( \\sim \\left(R\\vee Q \\right)~\\rightarrow ~\\sim P\\)\n", "ideal": "[\"the correct answer is (D)\", \"the correct answer is \\textbf{(D)}\", \"the correct answer is $\\boxed{\\textbf{(D)}\"]"}, {"prompt": "Let the solution curve of the differential equation \\(xdy=\\left(\\sqrt{x^2 + y^2 }+y\\right)dx, x>0\\) intersect the line x = 1 at y = 0 and the line x = 2 at y = \u03b1. Then the value of \u03b1 is\n(A) \\(\\frac{1}{2}\\)\n(B) \\(\\frac{3}{2}\\)\n(C) \\(-\\frac{3}{2}\\)\n(D) \\(\\frac{5}{2}\\)\n", "ideal": "[\"the correct answer is (B)\", \"the correct answer is \\textbf{(B)}\", \"the correct answer is $\\boxed{\\textbf{(B)}\"]"}, {"prompt": "Considering only the principal values of the inverse trigonometric functions, the domain of the function \\(f(x)=\\cos^{-1}\\left(\\frac{x^2-4x+2}{x^2+3}\\right)\\) is\n(A) \\(\\left(-\\infty, \\frac{1}{4}\\right]\\)\n(B) \\(\\left[-\\frac{1}{4}, \\infty \\right)\\)\n(C) \\(\\left(\\frac{-1}{3}, \\infty \\right)\\)\n(D) \\(\\left(-\\infty, \\frac{1}{3} \\right]\\)\n", "ideal": "[\"the correct answer is (B)\", \"the correct answer is \\textbf{(B)}\", \"the correct answer is $\\boxed{\\textbf{(B)}\"]"}, {"prompt": "Let the vectors \\(\\vec{a}=(1+t)\\hat{i}+(1-t)\\hat{j}+\\hat{k},\\ \\vec{b}=(1-t)\\hat{i}+(1+t)\\hat{j}+2\\hat{k}\\) and \\(\\vec{c}=t\\hat{i}-t\\hat{j}+\\hat{k}, t\\in \\mathbf{R}\\) be such that for \\(\\alpha, \\beta, \\gamma \\in \\mathbf R,\\ \\alpha \\vec{a}+\\beta \\vec{b}+\\gamma \\vec{c} =\\vec{0}\\ \\Rightarrow \\alpha =\\beta = \\gamma =0.\\) Then, the set of all values of t is\n(A) A non-empty finite set\n(B) Equal to N\n(C) \\( \\text{Equal to}~ \\mathbf{R} \u2013 \\{0\\}\\)\n(D) \\( \\text{Equal to}~ \\mathbf{R}\\)\n", "ideal": "[\"the correct answer is (C)\", \"the correct answer is \\textbf{(C)}\", \"the correct answer is $\\boxed{\\textbf{(C)}\"]"}, {"prompt": "Considering the principal values of the inverse trigonometric functions, the sum of all the solutions of the equation \\(cos^{-1}\\left(x\\right) \u2013 2sin^{-1}\\left(x\\right) = cos^{-1}\\left(2x\\right)\\) is equal to\n(A) \\(0\\)\n(B) \\(1\\)\n(C) \\(\\frac{1}{2}\\)\n(D) \\(-\\frac{1}{2}\\)\n", "ideal": "[\"the correct answer is (A)\", \"the correct answer is \\textbf{(A)}\", \"the correct answer is $\\boxed{\\textbf{(A)}\"]"}, {"prompt": "Let the operations *, \u25c9 \u2208 {\u2227, \u2228}. If (p * q) \u25c9 (p \u25c9 ~q) is a tautology, then the ordered pair (*, \u25c9) is\n(A) (\u2228, \u2227)\n(B) (\u2228, \u2228)\n(C) (\u2227, \u2227)\n(D) (\u2227, \u2228)\n", "ideal": "[\"the correct answer is (B)\", \"the correct answer is \\textbf{(B)}\", \"the correct answer is $\\boxed{\\textbf{(B)}\"]"}, {"prompt": "Let a vector \\(\\vec{a}\\ \\text{has magnitude}\\ 9.\\ \\text{Let a vector}\\ \\vec{b}\\) be such that for every \\(\\left(x, y\\right) \\in \\mathbf R \\times \\mathbf R \u2013 \\{\\left(0, 0\\right)\\},\\ \\text{the vector}\\ (x\\vec{a} + y\\vec{b})\\)\u00a0 \\(\\text{is perpendicular to the vector}\\ (6y\\vec{a} \u2013 18x\\vec{b}).\\) \\(\\text{Then the value of}\\\u00a0 |\\vec{a}\\times \\vec{b}|\\) is equal to\n(A) \\(9\\sqrt{3}\\)\n(B) \\(27\\sqrt{3}\\)\n(C) \\(9\\)\n(D) \\(81\\)\n", "ideal": "[\"the correct answer is (B)\", \"the correct answer is \\textbf{(B)}\", \"the correct answer is $\\boxed{\\textbf{(B)}\"]"}, {"prompt": "For t \u2208 (0, 2\u03c0), if ABC is an equilateral triangle with vertices A(sint, \u2013 cost), B(cost, sint) and C(a, b) such that its orthocentre lies on a circle with centre (1, 1/3), then (a^2 \u2013 b^2) is equal to\n(A) \\(\\frac{8}{3}\\)\n(B) \\(8\\)\n(C) \\(\\frac{77}{9}\\)\n(D) \\(\\frac{80}{9}\\)\n", "ideal": "[\"the correct answer is (B)\", \"the correct answer is \\textbf{(B)}\", \"the correct answer is $\\boxed{\\textbf{(B)}\"]"}, {"prompt": "For \u03b1 \u2208 N, consider a relation R on N given by R = {(x, y) : 3x + \u03b1y is a multiple of 7}. The relation R is an equivalence relation if and only if\n(A) \u03b1 = 14\n(B) \u03b1 is a multiple of 4\n(C) 4 is the remainder when \u03b1 is divided by 10\n(D) 4 is the remainder when \u03b1 is divided by 7\n", "ideal": "[\"the correct answer is (D)\", \"the correct answer is \\textbf{(D)}\", \"the correct answer is $\\boxed{\\textbf{(D)}\"]"}, {"prompt": "If y = y(x), x \u2208 (0, \u03c0/2) be the solution curve of the differential equation \\((\\sin^22x)\\frac{dy}{dx}+ (8\\sin^22x + 2\\sin 4x)y = 2e^{-4x}(2\\sin 2x + \\cos 2x),\\) \\(\\text{with}\\ y\\left(\\frac{\\pi}{4}\\right)=e^{-\\pi},\\ \\text{then}\\ y\\left(\\frac{\\pi}{6}\\right)\\) is equal to\n(A) \\(\\frac{2}{\\sqrt{3}}e^{-2\\pi/3}\\)\n(B) \\(\\frac{2}{\\sqrt{3}}e^{2\\pi/3}\\)\n(C) \\(\\frac{1}{\\sqrt{3}}e^{-2\\pi/3}\\)\n(D) \\(\\frac{1}{\\sqrt{3}}e^{2\\pi/3}\\)\n", "ideal": "[\"the correct answer is (A)\", \"the correct answer is \\textbf{(A)}\", \"the correct answer is $\\boxed{\\textbf{(A)}\"]"}, {"prompt": "If the tangents drawn at the points P and Q on the parabola y^2 = 2x \u2013 3 intersect at the point R(0, 1), then the orthocentre of the triangle PQR is :\n(A) (0, 1)\n(B) (2, \u20131)\n(C) (6, 3)\n(D) (2, 1)\n", "ideal": "[\"the correct answer is (B)\", \"the correct answer is \\textbf{(B)}\", \"the correct answer is $\\boxed{\\textbf{(B)}\"]"}, {"prompt": "Let C be the centre of the circle \\(x^2+y^2-x+2y=\\frac{11}{4}\\) and P be a point on the circle. A line passes through the point C, makes an angle of \u03c0/4with the line CP and intersects the circle at the Q and R. Then the area of the triangle PQR (in unit^2) is :\n(A) \\(2\\)\n(B) \\(2\\sqrt{2}\\)\n(C) \\(8\\sin\\left(\\frac{\\pi}{8}\\right)\\)\n(D) \\(8\\cos\\left(\\frac{\\pi}{8}\\right)\\)\n", "ideal": "[\"the correct answer is (B)\", \"the correct answer is \\textbf{(B)}\", \"the correct answer is $\\boxed{\\textbf{(B)}\"]"}, {"prompt": "The remainder 7^2022 + 3^2022 is divided by 5 is:\n(A) 0\n(B) 2\n(C) 3\n(D) 4\n", "ideal": "[\"the correct answer is (C)\", \"the correct answer is \\textbf{(C)}\", \"the correct answer is $\\boxed{\\textbf{(C)}\"]"}, {"prompt": "Let the matrix \\(A= \\begin{vmatrix}0 & 1 & 0 \\\\0 & 0 & 1 \\\\1 & 0 & 0 \\\\\\end{vmatrix}\\) and the matrix \\(B_0 = A^{49} + 2A^{98}. ~\\text{If} ~B_n = Adj(B_{n \u2013 1})~\\text{for all}~n \\geq 1,\\) then det(B4) is equal to :\n(A) 3^28\n(B) 3^30\n(C) 3^32\n(D) 3^36\n", "ideal": "[\"the correct answer is (C)\", \"the correct answer is \\textbf{(C)}\", \"the correct answer is $\\boxed{\\textbf{(C)}\"]"}, {"prompt": "Let \\(S_1= \\left\\{z_1 \\in C : |z_1 \u2013 3| = \\frac{1}{2}\\right\\}\\ \\text{and}\\ S_2= \\left\\{z_2 \\in C : |z_2 \u2013 |z_2 + 1|| = |z_2 + |z_2 \u2013 1||\\right\\}.\\) Then, for z1 \u2208 S1 and z2 \u2208 S2, the least value of |z2 \u2013 z1| is :\n(A) \\(0\\)\n(B) \\(\\frac{1}{2}\\)\n(C) \\(\\frac{3}{2}\\)\n(D) \\(\\frac{5}{2}\\)\n", "ideal": "[\"the correct answer is (C)\", \"the correct answer is \\textbf{(C)}\", \"the correct answer is $\\boxed{\\textbf{(C)}\"]"}, {"prompt": "The foot of the perpendicular from a point on the circle x^2 + y^2 = 1, z = 0 to the plane 2x + 3y + z = 6 lies on which one of the following curves?\n(A) \\( \\left(6x + 5y \u2013 12\\right)^2 + 4\\left(3x + 7y \u2013 8\\right)^2 = 1, z = 6 \u2013 2x \u2013 3y\\)\n(B) \\(\\left(5x + 6y \u2013 12\\right)^2 + 4\\left(3x + 5y \u2013 9\\right)^2 = 1, z = 6- 2x \u2013 3y\\)\n(C) \\( \\left(6x + 5y \u2013 14\\right)^2 + 9\\left(3x + 5y \u2013 7\\right)^2 = 1, z = 6 \u2013 2x \u2013 3y\\)\n(D) \\( \\left(5x + 6y \u2013 14\\right)^2 + 9\\left(3x + 7y \u2013 8\\right)^2 = 1, z = 6 \u2013 2x \u2013 3y\\)\n", "ideal": "[\"the correct answer is (B)\", \"the correct answer is \\textbf{(B)}\", \"the correct answer is $\\boxed{\\textbf{(B)}\"]"}, {"prompt": "If the minimum value of \\(f(x)=\\frac{5x^2}{2}+\\frac{\\alpha}{x^5}, x>0\\) is 14, then the value of \u03b1 is equal to\n(A) 32\n(B) 64\n(C) 128\n(D) 256\n", "ideal": "[\"the correct answer is (C)\", \"the correct answer is \\textbf{(C)}\", \"the correct answer is $\\boxed{\\textbf{(C)}\"]"}, {"prompt": "Let \u03b1, \u03b2 and \u03b3 be three positive real numbers. \\(Let f\\left(x\\right) = \\alpha x^5 + \\beta x^3 + \\gamma x,x \\in \\mathbb {R} ~\\text{and } g:\\mathbb{R} \\rightarrow \\mathbb{R}~\\text{be such that}~ g\\left(f(x)\\right) = x~\\text{for all}~ x \\in \\mathbb{R}. \\) If a1, a2, a3, \u2026, an be in arithmetic progression with mean zero, then the value of \\(f\\left( g\\left( \\frac{1}{n} \\sum_{i=1}^{n}f(a_i)\\right)\\right)\\) is equal to\n(A) 0\n(B) 3\n(C) 9\n(D) 27\n", "ideal": "[\"the correct answer is (A)\", \"the correct answer is \\textbf{(A)}\", \"the correct answer is $\\boxed{\\textbf{(A)}\"]"}, {"prompt": "Consider the sequence a1, a2, a3, \u2026 such that a1 = 1, a2 = 2 and \\(a_{n+2}=\\frac{2}{a_{n+1}}+a_n \\text{ for } n = 1,2,3,\u2026\\). If \\(\\left(\\frac{a_1+\\frac{1}{a_2}}{a_3}\\right)\\left(\\frac{a_2+\\frac{1}{a_3}}{a_4}\\right)\\left(\\frac{a_3+\\frac{1}{a_4}}{a_5}\\right)\\cdots\\left(\\frac{a_{30}+\\frac{1}{a_{31}}}{a_{32}}\\right)=2^\\alpha (^{61}C_{31}), \\) then \u03b1 is equal to\n(A) \u201330\n(B) \u201331\n(C) \u201360\n(D) \u201361\n", "ideal": "[\"the correct answer is (C)\", \"the correct answer is \\textbf{(C)}\", \"the correct answer is $\\boxed{\\textbf{(C)}\"]"}, {"prompt": "The minimum value of the twice differentiable function \\(f(x)=\\int_{0}^{x}e^{x-t}f'(t)dt-(x^2-x+1)e^x, x\\in \\mathbb{R}\\) is\n(A) \\(\\ -\\frac{2}{\\sqrt{e}}\\)\n(B) \\(\\ -2\\sqrt{e}\\)\n(C) \\(\\ -\\sqrt{e}\\)\n(D) \\(\\ \\frac{2}{\\sqrt{e}}\\)\n", "ideal": "[\"the correct answer is (A)\", \"the correct answer is \\textbf{(A)}\", \"the correct answer is $\\boxed{\\textbf{(A)}\"]"}, {"prompt": "Let \\(S=\\left\\{x\\in\\left[-6, 3\\right]-\\left\\{-2,2 \\right\\}:\\frac{\\left|x+3\\right|-1}{\\left|x\\right|-2}\\geq 0 \\right\\}\\ \\text{and} T=\\left\\{x\\in \\mathbb{Z}:x^2-7\\left|x\\right|+9\\leq 0\\right\\}.\\) Then the number of elements in S \u22c2 T is \n(A) 7\n(B) 5\n(C) 4\n(D) 3\n", "ideal": "[\"the correct answer is (D)\", \"the correct answer is \\textbf{(D)}\", \"the correct answer is $\\boxed{\\textbf{(D)}\"]"}, {"prompt": "Let \u03b1, \u03b2 be the roots of the equation \\(x^2-\\sqrt{2}x+\\sqrt{6}=0\\ \\text{and}\\ \\frac{1}{\\alpha^2}+1,\\frac{1}{\\beta^2}+1\\) be the roots of the equation x^2 + ax + b = 0 . Then the roots of the equation x^2 \u2013 (a + b \u2013 2)x + (a + b + 2) = 0 are\n(A) Non-real complex number\n(B) Real and both negative\n(C) Real and both positive\n(D) Real and exactly one of them is positive\n", "ideal": "[\"the correct answer is (B)\", \"the correct answer is \\textbf{(B)}\", \"the correct answer is $\\boxed{\\textbf{(B)}\"]"}, {"prompt": "Let A and B be any two 3 \u00d7 3 symmetric and skew symmetric matrices, respectively. Then Which of the following is NOT true? \n(A) A^4 \u2013 B^4 is a symmetric matrix\n(B) AB \u2013 BA is a symmetric matrix\n(C) B^5 \u2013 A^5 is a skew-symmetric matrix\n(D) AB + BA is a skew-symmetric matrix\n", "ideal": "[\"the correct answer is (C)\", \"the correct answer is \\textbf{(C)}\", \"the correct answer is $\\boxed{\\textbf{(C)}\"]"}, {"prompt": "Let f(x) = ax^2 + bx + c be such that f(1) = 3, f(-2) = \u03bb and f(3) = 4. If f(0) + f(1) + f(-2) + f(3) = 14, then \u03bb is equal to\n(A) \\( -4\\\\ \\)\n(B) \\( \\frac{13}{2} \\\\ \\)\n(C) \\(\\frac{23}{2}\\\\ \\)\n(D) \\(4\\)\n", "ideal": "[\"the correct answer is (D)\", \"the correct answer is \\textbf{(D)}\", \"the correct answer is $\\boxed{\\textbf{(D)}\"]"}, {"prompt": "The function f: \u211d \u2192 \u211d defined by \\(f\\left(x\\right)=\\displaystyle \\lim_{n \\to \\infty}\\frac{\\cos\\left(2\\pi x\\right)-x^{2n}\\sin\\left(x-1\\right)}{1+x^{2n+1}-x^{2n}}\\) is continuous for all x in\n(A) \\(\\mathbb{R} \u2013 \\left\\{-1 \\right\\}\\)\n(B) \\(\\mathbb{R} \u2013 \\left\\{-1,1 \\right\\}\\)\n(C) \\(\\mathbb{R} \u2013 \\left\\{1 \\right\\}\\)\n(D) \\(\\mathbb{R} \u2013 \\left\\{0 \\right\\}\\)\n", "ideal": "[\"the correct answer is (B)\", \"the correct answer is \\textbf{(B)}\", \"the correct answer is $\\boxed{\\textbf{(B)}\"]"}, {"prompt": "The function \\(f\\left(x\\right)=xe^{x\\left(1-x\\right)}, x\\in \\mathbb{R} \\) is \n(A) \\(\\text{Increasing in}\\left(-\\frac{1}{2},1\\right)\\)\n(B) \\(\\text{Decreasing in}\\left(\\frac{1}{2},2\\right)\\)\n(C) \\(\\text{Increasing in}\\left(-1,-\\frac{1}{2}\\right)\\)\n(D) \\(\\text{Decreasing in}\\left(-\\frac{1}{2},\\frac{1}{2}\\right)\\)\n", "ideal": "[\"the correct answer is (A)\", \"the correct answer is \\textbf{(A)}\", \"the correct answer is $\\boxed{\\textbf{(A)}\"]"}, {"prompt": "The sum of the absolute maximum and absolute minimum values of the function \\(f\\left(x\\right)=\\tan^{-1}\\left(\\sin x-\\cos x\\right) \\) in the interval [0, \u03c0] is\n(A) \\(0\\)\n(B) \\(\\tan^{-1}\\left(\\frac{1}{\\sqrt{2}}\\right)-\\frac{\\pi}{4}\\)\n(C) \\(\\cos^{-1}\\left(\\frac{1}{\\sqrt{3}}\\right)-\\frac{\\pi}{4}\\)\n(D) \\(\\frac{-\\pi}{12}\\)\n", "ideal": "[\"the correct answer is (C)\", \"the correct answer is \\textbf{(C)}\", \"the correct answer is $\\boxed{\\textbf{(C)}\"]"}, {"prompt": "Let \\(x\\left(t\\right)=2\\sqrt{2}\\cos t\\sqrt{\\sin 2t}\\ \\text{and}\\ y\\left(t\\right)=2\\sqrt{2}\\sin t\\sqrt{\\sin 2t},t\\in \\left(0,\\frac{\\pi}{2}\\right).\\) \\(\\text{Then}\\ \\frac{1+\\left(\\frac{dy}{dx}\\right)^2}{\\frac{d^2y}{dx^2}}\\ \\text{at}\\ t=\\frac{\\pi}{4}\\) is equal to \n(A) \\(\\frac{-2\\sqrt{2}}{3}\\)\n(B) \\(\\frac{2}{3}\\)\n(C) \\(\\frac{1}{3}\\)\n(D) \\(\\frac{-2}{3}\\)\n", "ideal": "[\"the correct answer is (D)\", \"the correct answer is \\textbf{(D)}\", \"the correct answer is $\\boxed{\\textbf{(D)}\"]"}, {"prompt": "Let \\(I_n\\left(x\\right)=\\int_0^x\\frac{1}{\\left(t^2+5\\right)^n}dt, n=1, 2, 3,\\cdots\\) Then \n(A) \\(50I_6-9I_5=x\\overset{\u2018}{I}_5 \\)\n(B) \\(50I_6-11I_5=x\\overset{\u2018}{I}_5 \\)\n(C) \\(50I_6-9I_5=\\overset{\u2018}{I}_5 \\)\n(D) \\(50I_6-11I_5=\\overset{\u2018}{I}_5 \\)\n", "ideal": "[\"the correct answer is (A)\", \"the correct answer is \\textbf{(A)}\", \"the correct answer is $\\boxed{\\textbf{(A)}\"]"}, {"prompt": "The area enclosed by the curves \\(y=\\text{log}_e\\left(x+e^2\\right),\\ x=\\text{log}_e\\left(\\frac{2}{y}\\right)\\text{ and }x=\\text{ log }_e\\ 2,\\) above the line y = 1 is \n(A) \\( 2 + e \u2013 log_e2 \\\\\\)\n(B) \\( 1 + e \u2013 log_e2\\\\ \\)\n(C) \\( e \u2013 log_e2 \\\\\\)\n(D) \\( 1 + log_e2\\)\n", "ideal": "[\"the correct answer is (B)\", \"the correct answer is \\textbf{(B)}\", \"the correct answer is $\\boxed{\\textbf{(B)}\"]"}, {"prompt": "Let y = y(x) be the solution curve of the differential equation \\(\\frac{dy}{dx}+\\frac{1}{x^2-1}y=\\left(\\frac{x-1}{x+1}\\right)^{1/2},x>1 \\) passing through the point (2, \u221a (1/3)). Then \u221a 7 y(8) is\n(A) \\( 11 + 6 log_e3\\\\ \\)\n(B) \\( 19\\\\ \\)\n(C) \\( 12 \u2013 2 log_e3\\\\ \\)\n(D) \\( 19 \u2013 6 log_e3\\)\n", "ideal": "[\"the correct answer is (D)\", \"the correct answer is \\textbf{(D)}\", \"the correct answer is $\\boxed{\\textbf{(D)}\"]"}, {"prompt": "The differential equation of the family of circles passing through the points (0, 2) and (0, \u20132) is\n(A) \\(2xy\\frac{dy}{dx}+\\left(x^2-y^2+4\\right)=0 \\)\n(B) \\(2xy\\frac{dy}{dx}+\\left(x^2+y^2-4\\right)=0\\)\n(C) \\(2xy\\frac{dy}{dx}+\\left(y^2-x^2+4\\right)=0\\)\n(D) \\(2xy\\frac{dy}{dx}-\\left(x^2-y^2+4\\right)=0\\)\n", "ideal": "[\"the correct answer is (A)\", \"the correct answer is \\textbf{(A)}\", \"the correct answer is $\\boxed{\\textbf{(A)}\"]"}, {"prompt": "Let the tangents at two points A and B on the circle x^2 + y^2 \u2013 4x + 3 = 0 meet at origin O(0, 0). Then the area of the triangle OAB is\n(A) \\(\\frac{3\\sqrt{3}}{2} \\)\n(B) \\(\\frac{3\\sqrt{3}}{4} \\)\n(C) \\(\\frac{3}{2\\sqrt{3}} \\)\n(D) \\(\\frac{3}{4\\sqrt{3}}\\)\n", "ideal": "[\"the correct answer is (B)\", \"the correct answer is \\textbf{(B)}\", \"the correct answer is $\\boxed{\\textbf{(B)}\"]"}, {"prompt": "Let the hyperbola \\(H:\\frac{x^2}{a^2}-\\frac{y^2}{b^2}=1\\ \\text{pass through the point}\\ \\left(2\\sqrt{2},-2\\sqrt{2}\\right).\\) A parabola is drawn whose focus is same as the focus of H with positive abscissa and the directrix of the parabola passes through the other focus of H. If the length of the latus rectum of the parabola is e times the length of the latus rectum of H, where e is the eccentricity of H, then which of the following points lies on the parabola?\n(A) \\(\\left(2\\sqrt{3},3\\sqrt{2}\\right)\\)\n(B) \\(\\left(3\\sqrt{3},-6\\sqrt{2}\\right)\\)\n(C) \\(\\left(\\sqrt{3},-\\sqrt{6}\\right)\\)\n(D) \\(\\left(3\\sqrt{6},6\\sqrt{2}\\right)\\)\n", "ideal": "[\"the correct answer is (B)\", \"the correct answer is \\textbf{(B)}\", \"the correct answer is $\\boxed{\\textbf{(B)}\"]"}, {"prompt": "Let the lines \\(\\frac{x-1}{\\lambda}=\\frac{y-2}{1}=\\frac{z-3}{2}\\ \\text{and}\\ \\frac{x+26}{-2}=\\frac{y+18}{3}=\\frac{z+28}{\\lambda}\\) be coplanar and P be the plane containing these two lines. Then which of the following points does NOT lie on P?\n(A) (0, -2, -2)\n(B) (-5, 0, -1)\n(C) (3, -1, 0)\n(D) (0, 4, 5)\n", "ideal": "[\"the correct answer is (D)\", \"the correct answer is \\textbf{(D)}\", \"the correct answer is $\\boxed{\\textbf{(D)}\"]"}, {"prompt": "A plane P is parallel to two lines whose direction rations are \u20132, 1, \u20133 and \u20131, 2, \u20132 and it contains the point (2, 2, \u20132). Let P intersect the co-ordinate axes at the points A, B, C making the intercepts \u03b1, \u03b2, \u03b3. If V is the volume of the tetrahedron OABC, where O is the origin and p = \u03b1 + \u03b2 + \u03b3, then the ordered pair (V, p) is equal to :\n(A) (48, \u201313)\n(B) (24, \u201313)\n(C) (48, 11)\n(D) (24, \u20135)\n", "ideal": "[\"the correct answer is (B)\", \"the correct answer is \\textbf{(B)}\", \"the correct answer is $\\boxed{\\textbf{(B)}\"]"}, {"prompt": "Let S be the set of all a\u2208 R for which the angle between the vectors \\(\\overrightarrow{u}=a\\left(\\text{log}_e b\\right)\\hat{i}-6\\hat{j}+3\\hat{k}\\) and \\(\\overrightarrow{v}=\\left(\\text{log}_e b\\right)\\hat{i}+2\\hat{j}+2a\\left(\\text{log}_e b\\right)\\hat{k},\\left(b>1\\right)\\) is acute. Then S is equal to \n(A) \\(\\left(-\\infty, -\\frac{4}{3}\\right)\\)\n(B) \\(Phi \\)\n(C) \\(\\left(-\\frac{4}{3},0\\right)\\)\n(D) \\(\\left(\\frac{12}{7},\\infty\\right)\\)\n", "ideal": "[\"the correct answer is (B)\", \"the correct answer is \\textbf{(B)}\", \"the correct answer is $\\boxed{\\textbf{(B)}\"]"}, {"prompt": "A horizontal park is in the shape of a triangle OAB with AB = 16. A vertical lamp post OP is erected at the point O such that \\(\\angle PAO = \\angle PBO = 15^\\circ ~\\text{and}~ \\angle PCO = 45^\\circ,\\) where C is the midpoint of AB. Then (OP)^2 is equal to\n(A) \\(\\frac{32}{\\sqrt{3}}\\left(\\sqrt{3}-1\\right) \\)\n(B) \\(\\frac{32}{\\sqrt{3}}\\left(2-\\sqrt{3}\\right) \\)\n(C) \\(\\frac{16}{\\sqrt{3}}\\left(\\sqrt{3}-1\\right) \\)\n(D) \\(\\frac{16}{\\sqrt{3}}\\left(2-\\sqrt{3}\\right) \\)\n", "ideal": "[\"the correct answer is (B)\", \"the correct answer is \\textbf{(B)}\", \"the correct answer is $\\boxed{\\textbf{(B)}\"]"}, {"prompt": "Let A and B be two events such that \\(P\\left(B/A\\right)\\frac{2}{5},\\ P\\left(A/B\\right)=\\frac{1}{7}\\ \\text{and}\\ P\\left(A\\cap B\\right)=\\frac{1}{9}.\\) Consider\n\\(\\left(S1\\right)P\\left(A\u2019\\cup B\\right)=\\frac{5}{6},\\)\n\\(\\left(S2\\right)P\\left(A\u2019\\cap B\u2019\\right)=\\frac{1}{18}.\\) Then\n(A) Both (S1) and (S2) are true\n(B) Both (S1) and (S2) are false\n(C) Only (S1) is true\n(D) Only (S2) is true\n", "ideal": "[\"the correct answer is (A)\", \"the correct answer is \\textbf{(A)}\", \"the correct answer is $\\boxed{\\textbf{(A)}\"]"}, {"prompt": "Let\np : Ramesh listens to music.\nq :Ramesh is out of his village.\nr : It is Sunday.\ns : It is Saturday.\nThen the statement \u201cRamesh listens to music only if he is in his village and it is Sunday or Saturday\u201d can be expressed as\n(A) \\( \\left(\\left(\\sim q\\right) \\wedge \\left(r\\vee s\\right)\\right) \\Rightarrow p\\\\ \\)\n(B) \\( \\left(q\\wedge \\left(r\\vee s \\right)\\right) \\Rightarrow p\\\\ \\)\n(C) \\( p\\Rightarrow \\left(q\\wedge \\left(r\\vee s\\right)\\right)\\\\ \\)\n(D) \\(p\\Rightarrow \\left(\\sim q \\right) \\wedge \\left(r\\vee s\\right)\\)\n", "ideal": "[\"the correct answer is (D)\", \"the correct answer is \\textbf{(D)}\", \"the correct answer is $\\boxed{\\textbf{(D)}\"]"}, {"prompt": "Let R be a relation from the set {1, 2, 3, \u2026.., 60} to itself such that R = {(a, b) : b = pq, where p, q\u2265 3 are prime numbers}. Then, the number of elements in R is :\n(A) 600\n(B) 660\n(C) 540\n(D) 720\n", "ideal": "[\"the correct answer is (B)\", \"the correct answer is \\textbf{(B)}\", \"the correct answer is $\\boxed{\\textbf{(B)}\"]"}, {"prompt": "If z = 2 + 3i, then \\(z^5+(\\bar{z})^5\\) is equal to :\n(A) 244\n(B) 224\n(C) 245\n(D) 265\n", "ideal": "[\"the correct answer is (A)\", \"the correct answer is \\textbf{(A)}\", \"the correct answer is $\\boxed{\\textbf{(A)}\"]"}, {"prompt": "Let A and B be two 3 \u00d7 3 non-zero real matrices such that AB is a zero matrix. Then\n(A) the system of linear equations AX = 0 has a unique solution\n(B) the system of linear equations AX = 0 has infinitely many solutions\n(C) B is an invertible matrix\n(D) adj(A) is an invertible matrix\n", "ideal": "[\"the correct answer is (B)\", \"the correct answer is \\textbf{(B)}\", \"the correct answer is $\\boxed{\\textbf{(B)}\"]"}, {"prompt": "If \\(\\frac{1}{\\left(20-a\\right)\\left(40-a\\right)}+\\frac{1}{\\left(40-a\\right)\\left(60-a\\right)}+\\cdots + \\frac{1}{\\left(180-a\\right)\\left(200-a\\right)}=\\frac{1}{256},\\) then the maximum value of a is :\n(A) 198\n(B) 202\n(C) 212\n(D) 218\n", "ideal": "[\"the correct answer is (C)\", \"the correct answer is \\textbf{(C)}\", \"the correct answer is $\\boxed{\\textbf{(C)}\"]"}, {"prompt": "If \\(\\displaystyle \\lim_{x \\to 0}\\frac{\\alpha e^x+\\beta e^{-x}+\\gamma\\sin x}{x\\sin^2 x}=\\frac{2}{3},\\) where \u03b1, \u03b2, \u03b3\u2208R, then which of the following is NOT correct?\n(A) \u03b1^2 + \u03b2^2 + \u03b3^2 = 6\n(B) \u03b1\u03b2 + \u03b2\u03b3 + \u03b3\u03b1 + 1 = 0\n(C) \u03b1\u03b2^2 + \u03b2\u03b3^2 + \u03b3\u03b1^2 + 3 = 0\n(D) \u03b1^2 \u2013 \u03b2^2 + \u03b3^2 = 4\n", "ideal": "[\"the correct answer is (C)\", \"the correct answer is \\textbf{(C)}\", \"the correct answer is $\\boxed{\\textbf{(C)}\"]"}, {"prompt": "The integral \\(\\displaystyle\\int\\limits_0^\\frac{\\pi}{2}\\frac{1}{3+2\\sin x+\\cos x}dx \\) is equal to\n(A) tan^\u20131(2)\n(B) \\(\\tan^{-1}\\left(2\\right)-\\frac{\\pi}{4}\\)\n(C) \\(\\frac{1}{2}\\tan^{-1}\\left(2\\right)-\\frac{\\pi}{8}\\)\n(D) \\(\\frac{1}{2}\\)\n", "ideal": "[\"the correct answer is (B)\", \"the correct answer is \\textbf{(B)}\", \"the correct answer is $\\boxed{\\textbf{(B)}\"]"}, {"prompt": "Let the solution curve y = y(x) of the differential equation \\(\\left(1+e^{2x}\\right)\\left(\\frac{dy}{dx}+y\\right)=1\\) pass through the point (0, \u03c0/2). Then, \\(\\displaystyle \\lim_{x \\to \\infty}e^xy\\left(x\\right)\\) is equal to\n(A) \\(\\frac{\\pi}{4} \\)\n(B) \\(\\frac{3\\pi}{4} \\)\n(C) \\(\\frac{\\pi}{2} \\)\n(D) \\(\\frac{3\\pi}{2} \\)\n", "ideal": "[\"the correct answer is (B)\", \"the correct answer is \\textbf{(B)}\", \"the correct answer is $\\boxed{\\textbf{(B)}\"]"}, {"prompt": "Let a line L pass through the point intersection of the lines bx + 10y \u2013 8 = 0 and \\(2x \u2013 3y = 0, b \\in R \u2013 \\left\\{ \\frac{4}{3}\\right\\}.\\) If the line L also passes through the point (1, 1) and touches the circle 17(x^2 + y^2) = 16, then the eccentricity of the ellipse (x^2/5) + (y^2/5) = 1 is\n(A) \\(\\frac{2}{\\sqrt{5}} \\)\n(B) \\(\\sqrt{\\frac{3}{5}} \\)\n(C) \\(\\frac{1}{\\sqrt{5}}\\)\n(D) \\(\\sqrt{\\frac{2}{5}}\\)\n", "ideal": "[\"the correct answer is (B)\", \"the correct answer is \\textbf{(B)}\", \"the correct answer is $\\boxed{\\textbf{(B)}\"]"}, {"prompt": "If the foot of the perpendicular from the point A(\u20131, 4, 3) on the plane P : 2x + my + nz = 4, is (-2, 7/2, 3/2), then the distance of the point A from the plane P, measured parallel to a line with direction ratios 3, \u20131, \u20134, is equal to\n(A) 1\n(B) \\(\\sqrt{26}\\)\n(C) \\(2\\sqrt{2}\\)\n(D) \\(\\sqrt{14}\\)\n", "ideal": "[\"the correct answer is (B)\", \"the correct answer is \\textbf{(B)}\", \"the correct answer is $\\boxed{\\textbf{(B)}\"]"}, {"prompt": "Let \\(\\vec{a}=3\\hat{i}+\\hat{j}\\ \\text{and}\\ \\vec{b}=\\hat{i}+2\\hat{j}+\\hat{k}.\\) \\(\\text{Let}\\ \\vec{c}\\ \\text{be a vector satisfying}\\ \\vec{a}\\times\\left(\\vec{b}\\times\\vec{c}\\right)=\\vec{b}+\\lambda\\vec{c}.\\) \\(\\text{If}\\ \\vec{b}\\ \\text{and}\\ \\vec{c}\\ \\text{are non-parallel},\\) then the value of \u03bb is\n(A) \u20135\n(B) 5\n(C) 1\n(D) \u20131\n", "ideal": "[\"the correct answer is (A)\", \"the correct answer is \\textbf{(A)}\", \"the correct answer is $\\boxed{\\textbf{(A)}\"]"}, {"prompt": "The angle of elevation of the top of a tower from a point A due north of it is \u03b1 and from a point B at a distance of 9 units due west of A is \\(\\cos^{-1}\\left(\\frac{3}{\\sqrt{13}}\\right).\\) If the distance of the point B from the tower is 15 units, then cot \u03b1 is equal to :\n(A) \\(\\frac{6}{5}\\)\n(B) \\(\\frac{9}{5}\\)\n(C) \\(\\frac{4}{3}\\)\n(D) \\(\\frac{7}{3}\\)\n", "ideal": "[\"the correct answer is (A)\", \"the correct answer is \\textbf{(A)}\", \"the correct answer is $\\boxed{\\textbf{(A)}\"]"}, {"prompt": "The statement (p\u2227q) \u21d2 (p\u2227r) is equivalent to :\n(A) q\u21d2 (p\u2227r)\n(B) p\u21d2 (p\u2227r)\n(C) (p\u2227r) \u21d2 (p\u2227q)\n(D) (p\u2227q) \u21d2 r\n", "ideal": "[\"the correct answer is (D)\", \"the correct answer is \\textbf{(D)}\", \"the correct answer is $\\boxed{\\textbf{(D)}\"]"}, {"prompt": "Let the circumcentre of a triangle with vertices A(a, 3), B(b, 5) and C(a, b), ab > 0 be P(1, 1). If the line AP intersects the line BC at the point Q(k1, k2), then k1 + k2 is equal to :\n(A) 2\n(B) \\(\\frac{4}{7} \\)\n(C) \\(\\frac{2}{7} \\)\n(D) 4\n", "ideal": "[\"the correct answer is (B)\", \"the correct answer is \\textbf{(B)}\", \"the correct answer is $\\boxed{\\textbf{(B)}\"]"}, {"prompt": "Let \\(\\hat{a}\\ \\text{and}\\ \\hat{b}\\) be two unit vectors such that the angle between them is \u03c0/4. If \u03b8 is the angle between the vectors \\(\\left(\\hat{a}+\\hat{b}\\right)\\ \\text{and}\\ \\left(\\hat{a}+2\\hat{b}+2\\left(\\hat{a}\\times\\hat{b}\\right)\\right),\\) then the value of 164 cos^2\u03b8 is equal to :\n(A) \\(90+27\\sqrt{2}\\)\n(B) \\(45+18\\sqrt{2}\\)\n(C) \\(90+3\\sqrt{2}\\)\n(D) \\(54+90\\sqrt{2}\\)\n", "ideal": "[\"the correct answer is (A)\", \"the correct answer is \\textbf{(A)}\", \"the correct answer is $\\boxed{\\textbf{(A)}\"]"}, {"prompt": "If \\(f\\left(\\alpha\\right)=\\displaystyle\\int\\limits_1^\\alpha\\frac{\\text{log}_{10}t}{1+t}dt, \\alpha > 0,\\) then f(e^3) + f(e^\u20133) is equal to :\n(A) 9\n(B) \\(\\frac{9}{2}\\)\n(C) \\(\\frac{9}{\\text{log}_e\\left(10\\right)}\\)\n(D) \\(\\frac{9}{2\\text{log}_e\\left(10\\right)}\\)\n", "ideal": "[\"the correct answer is (D)\", \"the correct answer is \\textbf{(D)}\", \"the correct answer is $\\boxed{\\textbf{(D)}\"]"}, {"prompt": "The area of the region \\(\\left\\{\\left(x,y\\right);\\left|x-1\\right|\\leq y \\leq \\sqrt{5-x^2} \\right\\}\\) is equal to\n(A) \\(\\frac{5}{2}\\sin^{-1}\\left(\\frac{3}{5}\\right)-\\frac{1}{2}\\)\n(B) \\(\\frac{5\\pi}{4}-\\frac{3}{2}\\)\n(C) \\(\\frac{3\\pi}{4}+\\frac{3}{2}\\)\n(D) \\(\\frac{5\\pi}{4}-\\frac{1}{2}\\)\n", "ideal": "[\"the correct answer is (D)\", \"the correct answer is \\textbf{(D)}\", \"the correct answer is $\\boxed{\\textbf{(D)}\"]"}, {"prompt": "Let the focal chord of the parabola P :y^2 = 4x along the line L : y = mx + c, m> 0 meet the parabola at the points M and N. Let the line L be a tangent to the hyperbola H :x^2 \u2013 y^2 = 4. If O is the vertex of P and F is the focus of H on the positive x-axis, then the area of the quadrilateral OMFN is\n(A) \\(2\\sqrt{6} \\)\n(B) \\(2\\sqrt{14} \\)\n(C) \\(4\\sqrt{6} \\)\n(D) \\(4\\sqrt{14} \\)\n", "ideal": "[\"the correct answer is (B)\", \"the correct answer is \\textbf{(B)}\", \"the correct answer is $\\boxed{\\textbf{(B)}\"]"}, {"prompt": "The number of points, where the function f: \u211d \u2192 \u211d, f(x) = |x \u2013 1|cos|x \u2013 2|sin|x \u2013 1| + (x \u2013 3)|x^2 \u2013 5x + 4|, is NOT differentiable, is\n(A) 1\n(B) 2\n(C) 3\n(D) 4\n", "ideal": "[\"the correct answer is (B)\", \"the correct answer is \\textbf{(B)}\", \"the correct answer is $\\boxed{\\textbf{(B)}\"]"}, {"prompt": "Let S = {1, 2, 3, \u2026, 2022}. Then the probability that a randomly chosen number n from the set S such that HCF (n, 2022) = 1, is\n(A) \\(\\frac{128}{1011} \\)\n(B) \\(\\frac{166}{1011} \\)\n(C) \\(\\frac{127}{337} \\)\n(D) \\(\\frac{112}{337} \\)\n", "ideal": "[\"the correct answer is (D)\", \"the correct answer is \\textbf{(D)}\", \"the correct answer is $\\boxed{\\textbf{(D)}\"]"}, {"prompt": "Let \\(f\\left(x\\right)=3^{\\left(x^2-2\\right)^3+4},x\\in \\mathbb{R}.\\) Then which of the following statements are true?\nP :x = 0 is a point of local minima of f\nQ: x = \u221a2 is a point of inflection of f\nR :f \u2032 is increasing for x > \u221a2\n(A) Only P and Q\n(B) Only P and R\n(C) Only Q and R\n(D) All P, Q and R\n", "ideal": "[\"the correct answer is (D)\", \"the correct answer is \\textbf{(D)}\", \"the correct answer is $\\boxed{\\textbf{(D)}\"]"}, {"prompt": "If z \u2260 0 be a complex number such that \\(\\left|z-\\frac{1}{z}\\right|=2,\\) then the maximum value of |z| is\n(A) \u221a2\n(B) 1\n(C) \u221a2 \u2013 1\n(D) \u221a2 + 1\n", "ideal": "[\"the correct answer is (D)\", \"the correct answer is \\textbf{(D)}\", \"the correct answer is $\\boxed{\\textbf{(D)}\"]"}, {"prompt": "Which of the following matrices can NOT be obtained from the matrix \\(\\begin{bmatrix} -1& 2 \\\\1 & -1 \\\\\\end{bmatrix}\\) by a single elementary row operation?\n(A) \\(\\begin{bmatrix} 0& 1 \\\\1 & -1 \\\\\\end{bmatrix}\\)\n(B) \\(\\begin{bmatrix} 1& -1 \\\\-1 & 2\\\\\\end{bmatrix}\\)\n(C) \\(\\begin{bmatrix} -1& 2 \\\\-2 & 7\\\\\\end{bmatrix}\\)\n(D) \\(\\begin{bmatrix} -1& 2 \\\\-1 & 3\\\\\\end{bmatrix}\\)\n", "ideal": "[\"the correct answer is (C)\", \"the correct answer is \\textbf{(C)}\", \"the correct answer is $\\boxed{\\textbf{(C)}\"]"}, {"prompt": "If the system of equations \\(x + y + z = 6\\\\ 2x + 5y + \\alpha z = \\beta\\\\ x + 2y + 3z = 14\\) has infinitely many solutions, then \u03b1 + \u03b2 is equal to\n(A) 8\n(B) 36\n(C) 44\n(D) 48\n", "ideal": "[\"the correct answer is (C)\", \"the correct answer is \\textbf{(C)}\", \"the correct answer is $\\boxed{\\textbf{(C)}\"]"}, {"prompt": "Let the function \\(f(x)= \\left\\{\\begin{matrix}\\frac{\\log_e(1+5x)-\\log_e(1+\\alpha x)}{x} &; \\text{if } x\\in0 \\\\10 & ; \\text{if } x=0 \\\\\\end{matrix}\\right.\\) be continuous at x = 0. Then \u03b1 is equal to\n(A) 10\n(B) \u201310\n(C) 5\n(D) \u20135\n", "ideal": "[\"the correct answer is (D)\", \"the correct answer is \\textbf{(D)}\", \"the correct answer is $\\boxed{\\textbf{(D)}\"]"}, {"prompt": "If [t] denotes the greatest integer \u2264 t, then the value of \\(\\int_{0}^{1}[2x-|3x^2 -5x + 2| + 1]dx \\) is\n(A) \\(\\frac{\\sqrt{37} + \\sqrt{13}-4}{6}\\)\n(B) \\(\\frac{\\sqrt{37} \u2013 \\sqrt{13}-4}{6}\\)\n(C) \\(\\frac{-\\sqrt{37} \u2013 \\sqrt{13}+4}{6}\\)\n(D) \\(\\frac{-\\sqrt{37} + \\sqrt{13}+4}{6}\\)\n", "ideal": "[\"the correct answer is (A)\", \"the correct answer is \\textbf{(A)}\", \"the correct answer is $\\boxed{\\textbf{(A)}\"]"}, {"prompt": "Let \\(\\{a_n\\}_{n=0}^\\infty\\ \\text{be a sequence such that}\\ a_0 = a_1 = 0\\ \\text{and}\\)\\(a_{n+2}=3a_{n+1}-2a_{n} + 1, \\forall \\ n \\ge 0.\\ \\text{Then}\\ a_{25}a_{23}-2a_{25}a_{22}-2a_{23}a_{24}+4a_{22}a_{24}\\) is equal to\n(A) 483\n(B) 528\n(C) 575\n(D) 624\n", "ideal": "[\"the correct answer is (B)\", \"the correct answer is \\textbf{(B)}\", \"the correct answer is $\\boxed{\\textbf{(B)}\"]"}, {"prompt": "\\(\\sum_{r=1}^{20}(r^2+1)(r!)\\) is equal to\n(A) \\( 22! \u2013 21!\\\\ \\)\n(B) \\( 22! \u2013 2(21!)\\\\ \\)\n(C) \\( 21! \u2013 2(20!)\\\\ \\)\n(D) \\( 21! \u2013 20!\\)\n", "ideal": "[\"the correct answer is (B)\", \"the correct answer is \\textbf{(B)}\", \"the correct answer is $\\boxed{\\textbf{(B)}\"]"}, {"prompt": "For \\(I(x)=\\int\\frac{\\sec^2 x \u2013 2022}{\\sin^{2022}x} dx,\\ \\text{if}\\ I\\left(\\frac{\\pi}{4}\\right)=2^{1011},\\) then\n(A) \\(\\ 3^{1010}I\\left(\\frac{\\pi}{3}\\right)-I\\left(\\frac{\\pi}{6}\\right)=0\\)\n(B) \\(\\ 3^{1010}I\\left(\\frac{\\pi}{6}\\right)-I\\left(\\frac{\\pi}{3}\\right)=0\\)\n(C) \\(\\ 3^{1011}I\\left(\\frac{\\pi}{3}\\right)-I\\left(\\frac{\\pi}{6}\\right)=0\\)\n(D) \\(\\ 3^{1011}I\\left(\\frac{\\pi}{6}\\right)-I\\left(\\frac{\\pi}{3}\\right)=0\\)\n", "ideal": "[\"the correct answer is (A)\", \"the correct answer is \\textbf{(A)}\", \"the correct answer is $\\boxed{\\textbf{(A)}\"]"}, {"prompt": "if the solution curve of the differential equation \\(\\frac{dy}{dx}=\\frac{x+y-2}{x-y} \\) passes through the points (2, 1) and (k + 1, 2), k > 0, then\n(A) \\(2\\tan^{-1}\\left(\\frac{1}{k}\\right)=\\log_e(k^2+1)\\)\n(B) \\(\\tan^{-1}\\left(\\frac{1}{k}\\right)=\\log_e(k^2+1)\\)\n(C) \\(2\\tan^{-1}\\left(\\frac{1}{k+1}\\right)=\\log_e(k^2+2k +2)\\)\n(D) \\(2\\tan^{-1}\\left(\\frac{1}{k}\\right)=\\log_e\\left(\\frac{k^2+1}{k^2}\\right)\\)\n", "ideal": "[\"the correct answer is (A)\", \"the correct answer is \\textbf{(A)}\", \"the correct answer is $\\boxed{\\textbf{(A)}\"]"}, {"prompt": "Let y = y(x) be the solution curve of the differential equation \\(\\frac{dy}{dx}+\\left(\\frac{2x^2+11x+13}{x^3+6x^2+11x+6}\\right)y = \\frac{(x+3)}{x+1}, x > -1\\) which passes through the point (0, 1). Then y(1) is equal to\n(A) 1/2\n(B) 3/2\n(C) 5/2\n(D) 7/2\n", "ideal": "[\"the correct answer is (B)\", \"the correct answer is \\textbf{(B)}\", \"the correct answer is $\\boxed{\\textbf{(B)}\"]"}, {"prompt": "Let m1, m2 be the slopes of two adjacent sides of a square of side a such that \\(a^2 + 11a + 3(m_1^2 + m_2^2) = 220. \\) If one vertex of the square is \\(\\left(10\\left(cos~\\alpha \u2013 sin~\\alpha\\right),10\\left(sin~\\alpha + cos~ \\alpha\\right)\\right),\\ \\text{where}\\ \\alpha \\in \\left(0, \\frac{\\pi}{2}\\right)\\) and the equation of one diagonal is \\((\\cos \\alpha \u2013 \\sin \\alpha)x + (\\sin \\alpha + \\cos \\alpha)y = 10,\\ \\text{then}\\ 72(\\sin^4 \\alpha + \\cos^4 \\alpha) + a^2 -3a + 13 \\) is equal to :\n(A) 119\n(B) 128\n(C) 145\n(D) 155\n", "ideal": "[\"the correct answer is (B)\", \"the correct answer is \\textbf{(B)}\", \"the correct answer is $\\boxed{\\textbf{(B)}\"]"}, {"prompt": "The number of elements in the set \n\\(S=\\left\\{x \\in \\mathbb{R} : 2\\cos \\left(\\frac{x^2 + x}{6}\\right)=4^x + 4^{-x}\\right\\}\\)\n(A) 1\n(B) 3\n(C) 0\n(D) infinite\n", "ideal": "[\"the correct answer is (A)\", \"the correct answer is \\textbf{(A)}\", \"the correct answer is $\\boxed{\\textbf{(A)}\"]"}, {"prompt": "Let A(\u03b1, -2), B(\u03b1, 6) and C(\u03b1/4, -2) be vertices of a \u0394ABC. If (5, \u03b1/4) is the circumcentre of \u0394ABC, then which of the following is NOT correct about \u0394ABC? \n(A) Area is 24\n(B) Perimeter is 25\n(C) Circumradius is 5\n(D) Inradius is 2\n", "ideal": "[\"the correct answer is (B)\", \"the correct answer is \\textbf{(B)}\", \"the correct answer is $\\boxed{\\textbf{(B)}\"]"}, {"prompt": "Let Q be the foot of perpendicular drawn from the point P(1, 2, 3) to the plane x + 2y + z = 14. If R is a point on the plane such that \u2220PRQ = 60\u00b0, then the area of \u0394PQR is equal to :\n(A) \\(\\frac{\\sqrt{3}}{2}\\)\n(B) \\(\\sqrt{3}\\)\n(C) \\(2\\sqrt{3}\\)\n(D) \\(3\\)\n", "ideal": "[\"the correct answer is (B)\", \"the correct answer is \\textbf{(B)}\", \"the correct answer is $\\boxed{\\textbf{(B)}\"]"}, {"prompt": "If (2, 3, 9), (5, 2, 1), (1, \u03bb, 8) and (\u03bb, 2, 3) are coplanar, then the product of all possible values of \u03bb is :\n(A) \\(\\frac{21}{2}\\)\n(B) \\(\\frac{59}{8}\\)\n(C) \\(\\frac{57}{8}\\)\n(D) \\(\\frac{95}{8}\\)\n", "ideal": "[\"the correct answer is (D)\", \"the correct answer is \\textbf{(D)}\", \"the correct answer is $\\boxed{\\textbf{(D)}\"]"}, {"prompt": "Bag I contains 3 red, 4 black and 3 white balls and Bag II contains 2 red, 5 black and 2 white balls. One ball is transferred from Bag I to Bag II and then a ball is drawn from Bag II. The ball so drawn is found to be black in colour. Then the probability, that the transferred ball is red, is :\n(A) \\(\\frac{4}{9}\\)\n(B) \\(\\frac{5}{18}\\)\n(C) \\(\\frac{1}{6}\\)\n(D) \\(\\frac{3}{10}\\)\n", "ideal": "[\"the correct answer is (B)\", \"the correct answer is \\textbf{(B)}\", \"the correct answer is $\\boxed{\\textbf{(B)}\"]"}, {"prompt": "Let \\(S = \\{z = x + iy : \\left|z \u2013 1 + i\\right| \\geq \\left|z\\right|, \\left|z\\right| < 2, \\left|z + i\\right| = \\left|z \u2013 1\\right|\\}.\\) Then the set of all values of x, for which w = 2x + iy \u2208 S for some y \u2208 R is\n(A) \\(\\left(-\\sqrt{2}, \\frac{1}{2\\sqrt{2}}\\right]\\)\n(B) \\(\\left(-\\frac{1}{\\sqrt{2}}, \\frac{1}{4}\\right]\\)\n(C) \\(\\left(-\\sqrt{2}, \\frac{1}{2}\\right]\\)\n(D) \\(\\left(-\\frac{1}{\\sqrt{2}}, \\frac{1}{2\\sqrt{2}}\\right]\\)\n", "ideal": "[\"the correct answer is (B)\", \"the correct answer is \\textbf{(B)}\", \"the correct answer is $\\boxed{\\textbf{(B)}\"]"}, {"prompt": "Let \\(\\vec{a},\\vec{b},\\vec{c} \\) be three coplanar concurrent vectors such that angles between any two of them is same. If the product of their magnitudes is 14 and \\((\\vec{a}\\times \\vec{b}) \\cdot (\\vec{b} \\times \\vec{c}) + (\\vec{b} \\times \\vec{c})\\cdot (\\vec{c} \\times \\vec{a}) + (\\vec{c} \\times \\vec{a})\\cdot (\\vec{a} \\times \\vec{b}) = 168,\\ \\text{then}\\ |\\vec{a}| + |\\vec{b}| + |\\vec{c}|\\) is equal to :\n(A) 10\n(B) 14\n(C) 16\n(D) 18\n", "ideal": "[\"the correct answer is (C)\", \"the correct answer is \\textbf{(C)}\", \"the correct answer is $\\boxed{\\textbf{(C)}\"]"}, {"prompt": "The domain of the function \\(f(x)=\\sin^{-1}\\left(\\frac{x^2-3x+2}{x^2+2x+7}\\right)\\) is :\n(A) \\( \\left[1, \\infty\\right)\\\\ \\)\n(B) \\( \\left[-1, 2\\right]\\\\ \\)\n(C) \\( \\left[-1, \\infty\\right)\\\\ \\)\n(D) \\( \\left(-\\infty , 2\\right]\\)\n", "ideal": "[\"the correct answer is (C)\", \"the correct answer is \\textbf{(C)}\", \"the correct answer is $\\boxed{\\textbf{(C)}\"]"}, {"prompt": "The statement \\(\\left(p \\Rightarrow q\\right) \\vee \\left(p \\Rightarrow r\\right)\\) is NOT equivalent to\n(A) \\((p \\wedge (\\sim r))\\Rightarrow q\\)\n(B) \\((\\sim q)\\Rightarrow ((\\sim r)\\vee p)\\)\n(C) \\(p\\Rightarrow (q\\vee r)\\)\n(D) \\((p\\wedge (\\sim q))\\Rightarrow r\\)\n", "ideal": "[\"the correct answer is (B)\", \"the correct answer is \\textbf{(B)}\", \"the correct answer is $\\boxed{\\textbf{(B)}\"]"}, {"prompt": "Let A = {z \u2208 C : 1 \u2264 |z \u2013 (1 + i)| \u2264 2} and B = {z \u2208 A : |z \u2013 (1 \u2013 i)| = 1}. Then, B :\n(A) Is an empty set\n(B) Contains exactly two elements\n(C) Contains exactly three elements\n(D) Is an infinite set\n", "ideal": "[\"the correct answer is (D)\", \"the correct answer is \\textbf{(D)}\", \"the correct answer is $\\boxed{\\textbf{(D)}\"]"}, {"prompt": "The remainder when 3^2022 is divided by 5 is :\n(A) 1\n(B) 2\n(C) 3\n(D) 4\n", "ideal": "[\"the correct answer is (D)\", \"the correct answer is \\textbf{(D)}\", \"the correct answer is $\\boxed{\\textbf{(D)}\"]"}, {"prompt": "The surface area of a balloon of spherical shape being inflated, increases at a constant rate. If initially, the radius of balloon is 3 units and after 5 seconds, it becomes 7 units, then its radius after 9 seconds is :\n(A) 9\n(B) 10\n(C) 11\n(D) 12\n", "ideal": "[\"the correct answer is (A)\", \"the correct answer is \\textbf{(A)}\", \"the correct answer is $\\boxed{\\textbf{(A)}\"]"}, {"prompt": "Bag A contains 2 white, 1 black and 3 red balls and bas B contains 3 black, 2 red and n white balls. One bag is chosen at random and 2 balls drawn from it at random, are found to be 1 red and 1 black. If the probability that both balls come from Bag A is \\(\\frac{6}{11}\\) then n is equal to _______.\n(A) 13\n(B) 6\n(C) 4\n(D) 3\n", "ideal": "[\"the correct answer is (C)\", \"the correct answer is \\textbf{(C)}\", \"the correct answer is $\\boxed{\\textbf{(C)}\"]"}, {"prompt": "Let \\(x^2+y^2+Ax+By+C=0\\) be a circle passing through (0, 6) and touching the parabola y = x^2 at (2, 4). Then A + C is equal to ________.\n(A) 16\n(B) \\(\\frac{88}{5}\\)\n(C) 72\n(D) \u20138\n", "ideal": "[\"the correct answer is (A)\", \"the correct answer is \\textbf{(A)}\", \"the correct answer is $\\boxed{\\textbf{(A)}\"]"}, {"prompt": "The number of values of \u03b1 for which the system of equations :\nx + y + z = \u03b1\n\u03b1x + 2\u03b1y + 3z = \u20131\nx + 3\u03b1y + 5z = 4\nis inconsistent, is\n(A) 0\n(B) 1\n(C) 2\n(D) 3\n", "ideal": "[\"the correct answer is (B)\", \"the correct answer is \\textbf{(B)}\", \"the correct answer is $\\boxed{\\textbf{(B)}\"]"}, {"prompt": "If the sum of the squares of the reciprocals of the roots \u03b1 and \u03b2 of the equation 3x^2 + \u03bbx \u2013 1 = 0 is 15, then 6(\u03b1^3 + \u03b2^3)^2 is equal to :\n(A) 18\n(B) 24\n(C) 36\n(D) 96\n", "ideal": "[\"the correct answer is (B)\", \"the correct answer is \\textbf{(B)}\", \"the correct answer is $\\boxed{\\textbf{(B)}\"]"}, {"prompt": "The set of all values of k for which \\(\\left ( \\tan^{-1}x \\right )^3+\\left ( \\cot^{-1}x \\right )^3=k\\pi ^3, x \\in R,\\) is the interval:\n(A) \\(\\left [ \\frac{1}{32},\\frac{7}{8} \\right )\\)\n(B) \\(\\left ( \\frac{1}{24},\\frac{13}{16} \\right )\\)\n(C) \\(\\left [ \\frac{1}{48},\\frac{13}{16} \\right ]\\)\n(D) \\(\\left [ \\frac{1}{32},\\frac{9}{8} \\right )\\)\n", "ideal": "[\"the correct answer is (A)\", \"the correct answer is \\textbf{(A)}\", \"the correct answer is $\\boxed{\\textbf{(A)}\"]"}, {"prompt": "Let \\(S=\\left\\{ \\sqrt{n}:1\\le n \\le 50 \\text{ and } n \\text{ is odd} \\right\\}\\).\nLet \\(a\\in S \\text{ and } A=\\begin{bmatrix}1 & 0 & a \\\\-1 & 1 & 0 \\\\-a & 0 & 1 \\\\\\end{bmatrix}\\)\nIf \\(\\sum_{a~\\in~S} \\det \\left ( adj A \\right ) = 100 \\lambda\\), then \u03bb is equal to :\n(A) 218\n(B) 221\n(C) 663\n(D) 1717\n", "ideal": "[\"the correct answer is (B)\", \"the correct answer is \\textbf{(B)}\", \"the correct answer is $\\boxed{\\textbf{(B)}\"]"}, {"prompt": "For the function\nf(x) = 4 loge(x \u2013 1) \u2013 2x^2 + 4x + 5, x > 1, which one of the following is NOT correct?\n(A) f is increasing in (1, 2) and decreasing in (2, )\n(B) f(x) = \u20131 has exactly two solutions\n(C) f\u2032(e) \u2013 f\u2032\u2032(2) < 0\n(D) f(x) = 0 has a root in the interval (e, e + 1)\n", "ideal": "[\"the correct answer is (C)\", \"the correct answer is \\textbf{(C)}\", \"the correct answer is $\\boxed{\\textbf{(C)}\"]"}, {"prompt": "If the tangent at the point (x1, y1) on the curve y = x^3 + 3x^2 + 5 passes through the origin, then (x1, y1) does NOT lie on the curve :\n(A) \\(x^2+\\frac{y^2}{81}=2\\)\n(B) \\(\\frac{y^2}{9}-x^2=8 \\)\n(C) y = 4x^2 + 5\n(D) \\(\\frac{x}{3}-y^2=2\\)\n", "ideal": "[\"the correct answer is (D)\", \"the correct answer is \\textbf{(D)}\", \"the correct answer is $\\boxed{\\textbf{(D)}\"]"}, {"prompt": "The sum of absolute maximum and absolute minimum values of the function f(x) = |2x^2 + 3x \u2013 2| + sinx cosx in the interval [0, 1] is :\n(A) \\(3+\\frac{\\sin(1)\\cos^2\\left ( \\frac{1}{2} \\right )}{2} \\)\n(B) \\(3+\\frac{1}{2}\\left( 1+2\\cos(1)\\right )\\sin(1)\\)\n(C) \\(5+\\frac{1}{2}\\left ( \\sin(1)+sin(2) \\right )\\)\n(D) \\(2+\\sin\\left ( \\frac{1}{2} \\right )\\cos\\left ( \\frac{1}{2} \\right )\\)\n", "ideal": "[\"the correct answer is (B)\", \"the correct answer is \\textbf{(B)}\", \"the correct answer is $\\boxed{\\textbf{(B)}\"]"}, {"prompt": "If \\(\\left\\{ a_i\\right\\}_{i=1}^n\\), where n is an even integer, is an arithmetic progression with common difference 1, and \\(\\sum_{i=1}^{n}a_i = 192,\\sum_{i=1}^{n/2}a_{2i}= 120\\), then n is equal to :\n(A) 48\n(B) 96\n(C) 92\n(D) 104\n", "ideal": "[\"the correct answer is (B)\", \"the correct answer is \\textbf{(B)}\", \"the correct answer is $\\boxed{\\textbf{(B)}\"]"}, {"prompt": "If x = x(y) is the solution of the differential equation \\(y\\frac{dx}{dy}=2x+y^3(y+1)e^y, x(1)=0;\\) then x(e) is equal to :\n(A) e^3(e^e \u2013 1)\n(B) e^e(e^3 \u2013 1)\n(C) e^2(e^e + 1)\n(D) e^e(e^2 \u2013 1)\n", "ideal": "[\"the correct answer is (A)\", \"the correct answer is \\textbf{(A)}\", \"the correct answer is $\\boxed{\\textbf{(A)}\"]"}, {"prompt": "Let \u03bbx \u2013 2y = \u03bc be a tangent to the hyperbola a^2x^2 \u2013 y^2 = b^2. Then \\(\\left ( \\frac{\\lambda}{a} \\right )^2-\\left ( \\frac{\\mu}{b} \\right )^2\\) is equal to :\n(A) \u20132\n(B) \u20134\n(C) 2\n(D) 4\n", "ideal": "[\"the correct answer is (D)\", \"the correct answer is \\textbf{(D)}\", \"the correct answer is $\\boxed{\\textbf{(D)}\"]"}, {"prompt": "Let \\(\\hat{a}, \\hat{b}\\)be unit vectors. If \\(\\vec{c}\\) be a vector such that the angle between \\(\\hat{a}\\) and \\(\\vec{c}\\) is \\(\\frac{\\pi}{12}\\) and \\(\\hat{b}=\\vec{c}+2\\left ( \\vec {c}\\times\\hat{a} \\right )\\) then \\(\\left|6 \\vec{c}\\right|^2\\) is equal to:\n(A) \\(6\\left ( 3-\\sqrt{3} \\right )\\)\n(B) \\(3+\\sqrt{3}\\)\n(C) \\(6\\left ( 3+\\sqrt{3} \\right )\\)\n(D) \\(6\\left ( \\sqrt{3} + 1 \\right )\\)\n", "ideal": "[\"the correct answer is (C)\", \"the correct answer is \\textbf{(C)}\", \"the correct answer is $\\boxed{\\textbf{(C)}\"]"}, {"prompt": "If a random variable X follows the Binomial distribution B(33, p) such that 3P(X = 0) = P(X = 1), then the value of \\(\\frac{P\\left ( X=15 \\right )}{P\\left ( X=18 \\right )}-\\frac{P\\left ( X=16 \\right )}{P\\left ( X=17 \\right )}\\) is equal to:\n(A) 1320\n(B) 1088\n(C) \\(\\frac{120}{1331}\\)\n(D) \\(\\frac{1088}{1089}\\)\n", "ideal": "[\"the correct answer is (A)\", \"the correct answer is \\textbf{(A)}\", \"the correct answer is $\\boxed{\\textbf{(A)}\"]"}, {"prompt": "The domain of the function \\(f(x)=\\frac{\\cos^{-1}\\left ( \\frac{x^2-5x+6}{x^2-9} \\right )}{\\log_e\\left ( x^2-3x+2 \\right )}\\) is:\n(A) \\(\\left ( -\\infty,1 \\right )\\cup\\left ( 2, \\infty \\right )\\)\n(B) (2, \u221e)\n(C) \\(\\left [ -\\frac{1}{2},1 \\right )\\cup\\left ( 2,\\infty \\right )\\)\n(D) \\(\\left [ -\\frac{1}{2},1 \\right )\\cup\\left ( 2,\\infty \\right )-\\left\\{ \\frac{3+\\sqrt{5}}{2},\\frac{3-\\sqrt{5}}{2}\\right\\}\\)\n", "ideal": "[\"the correct answer is (D)\", \"the correct answer is \\textbf{(D)}\", \"the correct answer is $\\boxed{\\textbf{(D)}\"]"}, {"prompt": "Let \\(S=\\left\\{ \\theta \\in \\left [ -\\pi, \\pi \\right ]-\\left\\{ \\pm\\frac{\\pi}{2}\\right\\}: \\sin\\theta\\tan\\theta + \\tan\\theta=\\sin2\\theta\\right\\}\\). If \\(T=\\sum_{\\theta = S}\\cos2\\theta\\) then T + n(S) is equal to:\n(A) \\(7+\\sqrt{3}\\)\n(B) 9\n(C) \\(8+\\sqrt{3}\\)\n(D) 10\n", "ideal": "[\"the correct answer is (B)\", \"the correct answer is \\textbf{(B)}\", \"the correct answer is $\\boxed{\\textbf{(B)}\"]"}, {"prompt": "The number of choices for \\(\\Delta \\in \\left\\{ \\land, \\lor, \\Rightarrow, \\Leftrightarrow \\right\\}\\) such that (p \u0394 q) \u21d2 ((p \u0394 ~ q) \u2228 ((~p) \u0394 q)) is a tautology, is\n(A) 1\n(B) 2\n(C) 3\n(D) 4\n", "ideal": "[\"the correct answer is (B)\", \"the correct answer is \\textbf{(B)}\", \"the correct answer is $\\boxed{\\textbf{(B)}\"]"}, {"prompt": "Let x * y = x^2 + y^3 and (x * 1) * 1 = x * (1 * 1). Then a value of \\(2\\ sin^{-1}\\left(\\frac{x^4+x^2-2}{x^4+x^2+2}\\right)\\) is \n(A) \\(\\frac{\\pi}{4}\\)\n(B) \\(\\frac{\\pi}{3}\\)\n(C) \\(\\frac{\\pi}{2}\\)\n(D) \\(\\frac{\\pi}{6}\\)\n", "ideal": "[\"the correct answer is (B)\", \"the correct answer is \\textbf{(B)}\", \"the correct answer is $\\boxed{\\textbf{(B)}\"]"}, {"prompt": "The sum of all the real roots of the equation (e^2^x \u2013 4)(6e^2^x \u2013 5e^x + 1) = 0 is\n(A) loge3\n(B) \u2013loge3\n(C) loge6\n(D) \u2013loge6\n", "ideal": "[\"the correct answer is (B)\", \"the correct answer is \\textbf{(B)}\", \"the correct answer is $\\boxed{\\textbf{(B)}\"]"}, {"prompt": "Let the system of linear equations \nx + y + az = 2\n3x + y + z = 4 \nx + 2z = 1 \nhave a unique solution (x*, y*, z*). If (\u03b1, x*), (y*, \u03b1) and (x*, \u2013y*) are collinear points, then the sum of absolute values of all possible values of \u03b1 is \n(A) 4\n(B) 3\n(C) 2\n(D) 1\n", "ideal": "[\"the correct answer is (C)\", \"the correct answer is \\textbf{(C)}\", \"the correct answer is $\\boxed{\\textbf{(C)}\"]"}, {"prompt": "Let x, y > 0. If x^3y^2 = 2^15, then the least value of 3x + 2y is \n(A) 30\n(B) 32\n(C) 36\n(D) 40\n", "ideal": "[\"the correct answer is (D)\", \"the correct answer is \\textbf{(D)}\", \"the correct answer is $\\boxed{\\textbf{(D)}\"]"}, {"prompt": "Let \\(f(x)\\left\\{\\begin{matrix} \\frac{\\sin(x-[x])}{x-[x]},& x \\in (-2,-1)\\\\ \\max{\\left\\{2x,3[\\left|x \\right|]\\right\\}},&\\left|x \\right|<1 \\\\ 1& ,\\text{otherwise}& \\\\\\end{matrix}\\right. \\) \nWhere [t] denotes greatest integer t. If m is the number of points where f is not continuous and n is the number of points where f is not differentiable, then the ordered pair (m, n) is \n(A) (3, 3)\n(B) (2, 4)\n(C) (2, 3)\n(D) (3, 4)\n", "ideal": "[\"the correct answer is (C)\", \"the correct answer is \\textbf{(C)}\", \"the correct answer is $\\boxed{\\textbf{(C)}\"]"}, {"prompt": "The value of the integral\n\\(\\int_{\\frac{-\\pi}{2}}^{\\frac{\\pi}{2}}\\frac{dx}{(1+e^x)(sin^6x+cos^6x)}\\) is equal to \n(A) 2\u03c0\n(B) 0\n(C) \u03c0\n(D) \u03c0/2\n", "ideal": "[\"the correct answer is (C)\", \"the correct answer is \\textbf{(C)}\", \"the correct answer is $\\boxed{\\textbf{(C)}\"]"}, {"prompt": "\\(\\displaystyle \\lim_{n\\to \\infty}\\left(\\frac{n^2}{(n^2+1)(n+1)}+\\frac{n^2}{(n^2+4)(n+2)}+\\frac{n^2}{(n^2+9)(n+3)}\u2026\u2026+\\frac{n^2}{(n^2+n^2)(n+n)}\\right)\\) \nis equal to \n(A) \\(\\frac{\\pi}{8}+\\frac{1}{4}log_e2\\)\n(B) \\(\\frac{\\pi}{4}+\\frac{1}{8}log_e2\\)\n(C) \\(\\frac{\\pi}{4}-\\frac{1}{8}log_e2\\)\n(D) \\(\\frac{\\pi}{8}+\\frac{1}{8}log_e\\sqrt{2}\\)\n", "ideal": "[\"the correct answer is (A)\", \"the correct answer is \\textbf{(A)}\", \"the correct answer is $\\boxed{\\textbf{(A)}\"]"}, {"prompt": "A particle is moving in the xy-plane along a curve C passing through the point (3, 3). The tangent to the curve C at the point P meets the x-axis at Q. If the y-axis bisects the segment PQ, then C is a parabola with \n(A) Length of latus rectum 3\n(B) Length of latus rectum 6\n(C) \\(Focus\\left(\\frac{4}{3},0\\right )\\)\n(D) \\(Focus\\left(0,\\frac{3}{4}\\right)\\)\n", "ideal": "[\"the correct answer is (A)\", \"the correct answer is \\textbf{(A)}\", \"the correct answer is $\\boxed{\\textbf{(A)}\"]"}, {"prompt": "Let the maximum area of the triangle that can be inscribed in the ellipse \\(\\frac{x^2}{a^2}+\\frac{y^2}{4}=1, a>2,\\) having one of its vertices at one end of the major axis of the ellipse and one of its sides parallel to the y-axis, be 6\u221a3. Then the eccentricity of the ellipse is \n(A) \\(\\frac{\\sqrt{3}}{2}\\)\n(B) \\(\\frac{1}{2}\\)\n(C) \\(\\frac{1}{\\sqrt{2}}\\)\n(D) \\(\\frac{\\sqrt{3}}{4}\\)\n", "ideal": "[\"the correct answer is (A)\", \"the correct answer is \\textbf{(A)}\", \"the correct answer is $\\boxed{\\textbf{(A)}\"]"}, {"prompt": "Let the area of the triangle with vertices A(1, \u03b1), B(\u03b1, 0) and C(0, \u03b1) be 4 sq. units. If the points (\u03b1, \u2013\u03b1), (\u2013\u03b1, \u03b1) and (\u03b1^2, \u03b2) are collinear, then \u03b2 is equal to\n(A) 64\n(B) \u20138\n(C) \u201364\n(D) 512\n", "ideal": "[\"the correct answer is (C)\", \"the correct answer is \\textbf{(C)}\", \"the correct answer is $\\boxed{\\textbf{(C)}\"]"}, {"prompt": "The number of distinct real roots of the equation x^7 \u2013 7x \u2013 2 = 0 is\n(A) 5\n(B) 7\n(C) 1\n(D) 3\n", "ideal": "[\"the correct answer is (D)\", \"the correct answer is \\textbf{(D)}\", \"the correct answer is $\\boxed{\\textbf{(D)}\"]"}, {"prompt": "A random variable X has the following probability distribution : \nX\n0\n1\n2\n3\n4\nP(X)\nk\n2k\n4k\n6k\n8k\nThe value of P(1 < X < 4 | x \u2264 2) is equal to\n(A) \\(\\frac{4}{7}\\)\n(B) \\(\\frac{2}{3}\\)\n(C) \\(\\frac{3}{7}\\)\n(D) \\(\\frac{4}{5}\\)\n", "ideal": "[\"the correct answer is (A)\", \"the correct answer is \\textbf{(A)}\", \"the correct answer is $\\boxed{\\textbf{(A)}\"]"}, {"prompt": "The number of solutions of the equation \\(cos\\left ( x+\\frac{\\pi}{3} \\right)cos\\left (\\frac{\\pi}{3}-x\\right)=\\frac{1}{4}cos^22x,x\\in[-3\\pi,3\\pi]\\) is: \n(A) 8\n(B) 5\n(C) 6\n(D) 7\n", "ideal": "[\"the correct answer is (D)\", \"the correct answer is \\textbf{(D)}\", \"the correct answer is $\\boxed{\\textbf{(D)}\"]"}, {"prompt": "If the shortest distance between the lines \\(\\frac{x-1}{2}=\\frac{y-2}{3}=\\frac{z-3}{\\lambda}~\\text{and}~ \\frac{x-2}{1}=\\frac{y-4}{4}=\\frac{z-5}{5}\\) is 1/\u221a3, then the sum of all possible values of \u03bb is : \n(A) 16\n(B) 6\n(C) 12\n(D) 15\n", "ideal": "[\"the correct answer is (A)\", \"the correct answer is \\textbf{(A)}\", \"the correct answer is $\\boxed{\\textbf{(A)}\"]"}, {"prompt": "Let the points on the plane P be equidistant from the points (\u20134, 2, 1) and (2, \u20132, 3). Then the acute angle between the plane P and the plane 2x + y + 3z = 1 is \n(A) \\(\\frac{\\pi}{6}\\)\n(B) \\(\\frac{\\pi}{4}\\)\n(C) \\(\\frac{\\pi}{3}\\)\n(D) \\(\\frac{5\\pi}{12}\\)\n", "ideal": "[\"the correct answer is (C)\", \"the correct answer is \\textbf{(C)}\", \"the correct answer is $\\boxed{\\textbf{(C)}\"]"}, {"prompt": "\\(\\text{Let}\\ \\hat{a}\\ \\text{and}\\ \\hat{b}\\ \\text{be two unit vectors such that}\\)\u00a0 \\(\\left|(\\hat{a}+\\hat{b})+2(\\hat{a}\\times\\hat{b}) \\right|=2. \\) If \u03b8 \u2208 (0, \u03c0) is the angle between \\(\\hat{a}\\ \\text{and}\\ \\hat{b},\\ \\text{then among the statements}:\\)\n(S1): \\(2\\left|\\hat{a}\\times\\hat{b}\\right|=\\left|\\hat{a}-\\hat{b} \\right| \\)\n(S2): The projection of \\(\\hat{a}\\ \\text{on}\\ (\\hat{a}+\\hat{b})is \\frac{1}{2}\\)\n(A) Only (S1) is true\n(B) Only (S2) is true\n(C) Both (S1) and (S2) are true\n(D) Both (S1) and (S2) are false\n", "ideal": "[\"the correct answer is (C)\", \"the correct answer is \\textbf{(C)}\", \"the correct answer is $\\boxed{\\textbf{(C)}\"]"}, {"prompt": "If \\(y=tan^{-1}(sec~ x^3-tan~x^3),\\frac{\\pi}{2}<x^3<\\frac{3\\pi}{2},\\) then \n(A) xy\u2032\u2032 + 2y\u2032 = 0\n(B) \\(x^2y\u201d-6y+\\frac{3\\pi}{2}=0\\)\n(C) x^2y\u2033 \u2013 6y + 3\u03c0 = 0\n(D) xy\u2033 \u2013 4y\u2032 = 0\n", "ideal": "[\"the correct answer is (B)\", \"the correct answer is \\textbf{(B)}\", \"the correct answer is $\\boxed{\\textbf{(B)}\"]"}, {"prompt": "Consider the following statements:\nA : Rishi is a judge.\nB : Rishi is honest.\nC : Rishi is not arrogant.\nThe negation of the statement \u201cif Rishi is a judge and he is not arrogant, then he is honest\u201d is\n(A) B \u2192 (A \u2228 C)\n(B) (~ B) \u2227 (A \u2227 C)\n(C) B \u2192 ((~ A) \u2228 (~ C))\n(D) B \u2192 (A \u2227 C)\n", "ideal": "[\"the correct answer is (B)\", \"the correct answer is \\textbf{(B)}\", \"the correct answer is $\\boxed{\\textbf{(B)}\"]"}, {"prompt": "The slope of normal at any point (x, y), x > 0, y > 0 on the curve y = y(x) is given by \\(\\frac{x^2}{xy-x^2y^2-1}.\\) If the curve passes through the point (1, 1), then e \u00b7 y(e) is equal to \n(A) \\(\\frac{1-tan(1)}{1+tan(1)}\\)\n(B) tan(1)\n(C) 1\n(D) \\(\\frac{1+tan(1)}{1-tan(1)}\\)\n", "ideal": "[\"the correct answer is (D)\", \"the correct answer is \\textbf{(D)}\", \"the correct answer is $\\boxed{\\textbf{(D)}\"]"}, {"prompt": "Let \u03bb* be the largest value of \u03bb for which the function f\u03bb(x) = 4\u03bbx^3 \u2013 36\u03bbx^2 + 36x + 48 is increasing for all x \u2208 \u211d. Then f\u03bb* (1) + f\u03bb* (\u2013 1) is equal to :\n(A) 36\n(B) 48\n(C) 64\n(D) 72\n", "ideal": "[\"the correct answer is (D)\", \"the correct answer is \\textbf{(D)}\", \"the correct answer is $\\boxed{\\textbf{(D)}\"]"}, {"prompt": "Let a circle C touch the lines L1 : 4x \u2013 3y +K1 = 0 and L2 : 4x \u2013 3y + K2 = 0, K1, K2\u2208R. If a line passing through the centre of the circle C intersects L1 at (\u20131, 2) and L2 at (3, \u20136), then the equation of the circle Cis :\n(A) (x \u2013 1)^2 + (y \u2013 2)^2 = 4\n(B) (x + 1)^2 + (y \u2013 2)^2 = 4\n(C) (x \u2013 1)^2 + (y + 2)^2 = 16\n(D) (x \u2013 1)^2 + (y \u2013 2)^2 = 16\n", "ideal": "[\"the correct answer is (C)\", \"the correct answer is \\textbf{(C)}\", \"the correct answer is $\\boxed{\\textbf{(C)}\"]"}, {"prompt": "The value of \\(\\displaystyle\\int\\limits_0^\\pi\\frac{e^{\\cos x}\\sin x}{\\left ( 1+\\cos^2x \\right )\\left ( e^{\\cos x}+e^{-\\cos x} \\right )}dx\\) is equal to :\n(A) \\(\\frac{\\pi^2}{4}\\)\n(B) \\(\\frac{\\pi^2}{2}\\)\n(C) \\(\\frac{\\pi}{4}\\)\n(D) \\(\\frac{\\pi}{2}\\)\n", "ideal": "[\"the correct answer is (C)\", \"the correct answer is \\textbf{(C)}\", \"the correct answer is $\\boxed{\\textbf{(C)}\"]"}, {"prompt": "Let a, b and c be the length of sides of a triangle ABC such that \\(\\frac{a+b}{7}=\\frac{b+c}{8}=\\frac{c+a}{9}\\). If r and R are the radius of incircle and radius of circumcircle of the triangle ABC, respectively, then the value of \\(\\frac{R}{r}\\) is equal to :\n(A) \\(\\frac{5}{2}\\)\n(B) 2\n(C) \\(\\frac{3}{2}\\)\n(D) 1\n", "ideal": "[\"the correct answer is (A)\", \"the correct answer is \\textbf{(A)}\", \"the correct answer is $\\boxed{\\textbf{(A)}\"]"}, {"prompt": "Let \u0192 : N\u2192R be a function such that \u0192(x + y) = 2\u0192(x) \u0192(y) for natural numbers x and y. If \u0192(1) = 2, then the value of \u03b1 for which\n\\(\\sum_{k=1}^{10}f\\left ( \\alpha+k \\right )=\\frac{512}{3}\\left ( 2^{20}-1 \\right )|\\)\nholds, is :\n(A) 2\n(B) 3\n(C) 4\n(D) 6\n", "ideal": "[\"the correct answer is (C)\", \"the correct answer is \\textbf{(C)}\", \"the correct answer is $\\boxed{\\textbf{(C)}\"]"}, {"prompt": "Let A be a 3 \u00d7 3 real matrix such that\n\\(A\\begin{pmatrix}1 \\\\1 \\\\0\\end{pmatrix}= \\begin{pmatrix}1 \\\\1 \\\\0\\end{pmatrix};~~A \\begin{pmatrix}1 \\\\0 \\\\1\\end{pmatrix}= \\begin{pmatrix}-1 \\\\0 \\\\1\\end{pmatrix}\\) and \\(A\\begin{pmatrix}0 \\\\0 \\\\1\\end{pmatrix}= \\begin{pmatrix}1 \\\\1 \\\\2\\end{pmatrix}\\)\nIf X = (x1, x2, x3)^T and I is an identity matrix of order 3, then the system \\(\\left ( A-2I \\right )X=\\begin{pmatrix} 4\\\\ 1\\\\1\\end{pmatrix}\\) has :\n(A) No solution\n(B) Infinitely many solutions\n(C) Unique solution\n(D) Exactly two solutions\n", "ideal": "[\"the correct answer is (B)\", \"the correct answer is \\textbf{(B)}\", \"the correct answer is $\\boxed{\\textbf{(B)}\"]"}, {"prompt": "Let \u0192 : R\u2192R be defined as\n\u0192(x) = x^3 + x \u2013 5\nIf g(x) is a function such that \u0192(g(x)) = \\(x,\\forall^{\u2018}x^{\u2018}\\epsilon \\textbf{R}\\) then g\u2032(63) is equal to_____.\n(A) \\(\\frac{1}{49}\\)\n(B) \\(\\frac{3}{49}\\)\n(C) \\(\\frac{43}{49}\\)\n(D) \\(\\frac{91}{49}\\)\n", "ideal": "[\"the correct answer is (A)\", \"the correct answer is \\textbf{(A)}\", \"the correct answer is $\\boxed{\\textbf{(A)}\"]"}, {"prompt": "Consider the following two propositions :\nP1 : ~ (p \u2192 ~ q)\nP2: (p \u2227 ~q) \u2227 ((-~p) \u2228 q)\nIf the proposition p \u2192 ((~p) \u2228 q) is evaluated as FALSE, then :\n(A) P1 is TRUE and P2 is FALSE\n(B) P1 is FALSE and P2 is TRUE\n(C) Both P1 and P2 are FALSE\n(D) Both P1 and P2 are TRUE\n", "ideal": "[\"the correct answer is (C)\", \"the correct answer is \\textbf{(C)}\", \"the correct answer is $\\boxed{\\textbf{(C)}\"]"}, {"prompt": "If \\(\\frac{1}{2\\cdot3^{10}}+\\frac{1}{2^2\\cdot3^9}+\\dots+\\frac{1}{2^{10}\\cdot3}=\\frac{K}{2^{10}\\cdot3^{10}}\\) then the remainder when K is divided by 6 is :\n(A) 1\n(B) 2\n(C) 3\n(D) 5\n", "ideal": "[\"the correct answer is (D)\", \"the correct answer is \\textbf{(D)}\", \"the correct answer is $\\boxed{\\textbf{(D)}\"]"}, {"prompt": "Let \u0192(x) be a polynomial function such that \u0192(x) + \u0192\u2032(x) + \u0192\u2032\u2032(x) = x^5 + 64. Then, the value of \\(\\displaystyle \\lim_{ x\\to 1}\\frac{f\\left ( x \\right )}{x-1}\\) is equal to :\n(A) \u201315\n(B) \u201360\n(C) 60\n(D) 15\n", "ideal": "[\"the correct answer is (A)\", \"the correct answer is \\textbf{(A)}\", \"the correct answer is $\\boxed{\\textbf{(A)}\"]"}, {"prompt": "Let E1 and E2 be two events such that the conditional probabilities \\(P\\left ( E_1|E_2 \\right )=\\frac{1}{2},\\) \\(P\\left ( E_2|E_1 \\right )=\\frac{3}{4}\\) and \\(P\\left ( E_1\\cap E_2 \\right )=\\frac{1}{8}\\cdot\\) Then:\n(A) \\(P\\left ( E_1\\cap E_2 \\right )=P\\left ( E_1 \\right )\\cdot P\\left ( E_2 \\right )\\)\n(B) \\(P\\left ( E_1^{\u2018}\\cap E_2^{\u2018} \\right )=P\\left ( E_1^{\u2018} \\right )\\cdot P\\left ( E_2 \\right )\\)\n(C) \\(P\\left ( E_1\\cap E_2^{\u2018} \\right )=P\\left ( E_1 \\right )\\cdot P\\left ( E_2 \\right )\\)\n(D) \\(P\\left ( E_1^{\u2018}\\cap E_2 \\right )=P\\left ( E_1 \\right )\\cdot P\\left ( E_2 \\right )\\)\n", "ideal": "[\"the correct answer is (C)\", \"the correct answer is \\textbf{(C)}\", \"the correct answer is $\\boxed{\\textbf{(C)}\"]"}, {"prompt": "Let \\(A=\\begin{bmatrix}0 & -2 \\\\2 & 0 \\\\\\end{bmatrix}\\) If M and N are two matrices given by \\(M=\\displaystyle\\sum\\limits_{k=1}^{10} A^{2k}\\) and \\(N=\\displaystyle\\sum\\limits_{k=1}^{10}A^{2k-1}\\) then MN^2is :\n(A) a non-identity symmetric matrix\n(B) a skew-symmetric matrix\n(C) neither symmetric nor skew-symmetric matrix\n(D) an identity matrix\n", "ideal": "[\"the correct answer is (A)\", \"the correct answer is \\textbf{(A)}\", \"the correct answer is $\\boxed{\\textbf{(A)}\"]"}, {"prompt": "Let g : (0, \u221e) \u2192R be a differentiable function such that \\(\\int\\left ( \\frac{x\\left ( \\cos x-\\sin x \\right )}{e^x+1}+\\frac{g\\left ( x \\right )\\left ( e^x+1-xe^x \\right )}{\\left ( e^x+1 \\right )^2} \\right ) dx=\\frac{xg\\left ( x \\right )}{e^x+1}+c,\\) for all x> 0, where c is an arbitrary constant. Then :\n(A) g is decreasingin \\(\\left ( 0, \\frac{\\pi}{4} \\right )\\)\n(B) g\u2032 is increasing in \\(\\left ( 0, \\frac{\\pi}{4} \\right )\\)\n(C) g + g\u2032 is increasing in \\(\\left ( 0, \\frac{\\pi}{2} \\right )\\)\n(D) g \u2013 g\u2032 is increasing in \\(\\left ( 0, \\frac{\\pi}{2} \\right )\\)\n", "ideal": "[\"the correct answer is (D)\", \"the correct answer is \\textbf{(D)}\", \"the correct answer is $\\boxed{\\textbf{(D)}\"]"}, {"prompt": "Let f :R\u2192R and g : R\u2192R be two functions defined by f(x) = loge(x^2 + 1) \u2013 e^\u2013^x + 1 and \\(g\\left ( x \\right )=\\frac{1-2e^{2x}}{e^x}\\). Then, for which of the following range of \u03b1, the inequality \\(f\\left ( g\\left ( \\frac{\\left ( \\alpha-1 \\right )^2}{3} \\right ) \\right )>f\\left ( g\\left ( \\alpha-\\frac{5}{3} \\right ) \\right )\\) holds?\n(A) (2, 3)\n(B) (\u20132, \u20131)\n(C) (1, 2)\n(D) (\u20131, 1)\n", "ideal": "[\"the correct answer is (A)\", \"the correct answer is \\textbf{(A)}\", \"the correct answer is $\\boxed{\\textbf{(A)}\"]"}, {"prompt": "Let \\(\\vec{a}=a_1\\hat{i}+a_2\\hat{j}+a_3\\hat{k}a_i>0, i=1,~2,~3\\) be a vector which makes equal angles with the coordinate axes OX, OY and OZ. Also, let the projection of \\(\\vec{a}\\) on the vector \\(3\\hat{i}+4\\hat{j}\\) be 7. Let \\(\\vec{b}\\) be a vector obtained by rotating \\(\\vec{a}\\) with 90\u00b0. If \\(\\vec{a},\\vec{b}\\) and x-axis are coplanar, then projection of a vector \\(\\vec{b}\\) on \\(3\\hat{i}+4\\hat{j}\\) is equal to\u00a0:\n(A) \\(\\sqrt{7}\\)\n(B) \\(\\sqrt{2}\\)\n(C) 2\n(D) 7\n", "ideal": "[\"the correct answer is (B)\", \"the correct answer is \\textbf{(B)}\", \"the correct answer is $\\boxed{\\textbf{(B)}\"]"}, {"prompt": "Let y = y(x) be the solution of the differential equation (x + 1)y\u2032 \u2013 y = e^3^x(x + 1)^2, with \\(y\\left ( 0 \\right )=\\frac{1}{3}\\) Then, the point \\(x=-\\frac{4}{3}\\) for the curve y = y(x)is\u00a0:\n(A) not a critical point\n(B) a point of local minima\n(C) a point of local maxima\n(D) a point of inflection\n", "ideal": "[\"the correct answer is (B)\", \"the correct answer is \\textbf{(B)}\", \"the correct answer is $\\boxed{\\textbf{(B)}\"]"}, {"prompt": "If y = m1x + c1 and y = m2x + c2, m1\u2260m2 are two common tangents of circle x^2 + y^2 = 2 and parabola y^2 = x, then the value of 8|m1m2| is equal to :\n(A) \\(3+4\\sqrt{2}\\)\n(B) \\(-5+6\\sqrt{2}\\)\n(C) \\(-4+3\\sqrt{2}\\)\n(D) \\(7+6\\sqrt{2}\\)\n", "ideal": "[\"the correct answer is (C)\", \"the correct answer is \\textbf{(C)}\", \"the correct answer is $\\boxed{\\textbf{(C)}\"]"}, {"prompt": "Let Q be the mirror image of the point P(1, 0, 1) with respect to the plane S: x + y + z = 5. If a line L passing through (1, \u20131, \u20131), parallel to the line PQ meets the plane S at R, then QR^2 is equal to :\n(A) 2\n(B) 5\n(C) 7\n(D) 11\n", "ideal": "[\"the correct answer is (B)\", \"the correct answer is \\textbf{(B)}\", \"the correct answer is $\\boxed{\\textbf{(B)}\"]"}, {"prompt": "If the solution curve y = y(x) of the differential equation y^2dx + (x^2 \u2013 xy + y^2)dy = 0, which passes through the point (1,1) and intersects the line \\(y=\\sqrt{3}x\\) at the point \\(\\left ( \\alpha,\\sqrt{3}\\alpha \\right )\\), then value of \\(log_e\\left ( \\sqrt{3}\\alpha \\right )\\) is equal to :\n(A) \\(\\frac{\\pi}{3}\\)\n(B) \\(\\frac{\\pi}{2}\\)\n(C) \\(\\frac{\\pi}{12}\\)\n(D) \\(\\frac{\\pi}{6}\\)\n", "ideal": "[\"the correct answer is (C)\", \"the correct answer is \\textbf{(C)}\", \"the correct answer is $\\boxed{\\textbf{(C)}\"]"}, {"prompt": "Let \\(x=2t,y=\\frac{t^2}{3}\\) be a conic. Let S be the focus and B be the point on the axis of the conic such that SA\u22a5BA, where A is any point on the conic. If k is the ordinate of the centroid of the \u0394SAB, then \\(\\displaystyle \\lim_{ t\\to 1}k\\) is equal to\n(A) \\(\\frac{17}{18}\\)\n(B) \\(\\frac{19}{18}\\)\n(C) \\(\\frac{11}{18}\\)\n(D) \\(\\frac{13}{18}\\)\n", "ideal": "[\"the correct answer is (D)\", \"the correct answer is \\textbf{(D)}\", \"the correct answer is $\\boxed{\\textbf{(D)}\"]"}, {"prompt": "Let a circle C in complex plane pass through the points z1 = 3 + 4i, z2 = 4 + 3i and z3 = 5i. If z(\u2260z1) is a point on C such that the line through z and z1 is perpendicular to the line through z2 and z3, then arg(z) is equal to:\n(A) \\(\\tan^{-1}\\left (\\frac{2}{\\sqrt{5}} \\right )-\\pi\\)\n(B) \\(\\tan^{-1}\\left ( \\frac{24}{7} \\right )-\\pi\\)\n(C) tan^\u20131 (3) \u2013 \u03c0\n(D) \\(\\tan^{-1}\\left (\\frac{3}{4} \\right )-\\pi\\)\n", "ideal": "[\"the correct answer is (B)\", \"the correct answer is \\textbf{(B)}\", \"the correct answer is $\\boxed{\\textbf{(B)}\"]"}, {"prompt": "Let A = {x \u2208 R : | x + 1 | < 2} and B = {x \u2208 R : | x \u2013 1| \u2265 2}. Then which one of the following statements is NOT true?\n(A) A \u2013 B = (\u20131, 1)\n(B) B \u2013 A = R \u2013 (\u20133, 1)\n(C) A \u22c2 B = (\u20133, \u20131]\n(D) A U B = R \u2013 [1, 3)\n", "ideal": "[\"the correct answer is (B)\", \"the correct answer is \\textbf{(B)}\", \"the correct answer is $\\boxed{\\textbf{(B)}\"]"}, {"prompt": "Let a, b \u2208 R be such that the equation ax^2 \u2013 2bx + 15 = 0 has a repeated root \u03b1. If \u03b1 and \u03b2 are the roots of the equation x^2 \u2013 2bx + 21 = 0, then \u03b1^2 + \u03b2^2 is equal to\n(A) 37\n(B) 58\n(C) 68\n(D) 92\n", "ideal": "[\"the correct answer is (B)\", \"the correct answer is \\textbf{(B)}\", \"the correct answer is $\\boxed{\\textbf{(B)}\"]"}, {"prompt": "Let z1 and z2 be two complex numbers such that \\(\\overline{z_1}=i\\overline{z_2}\\ \\text{and}\\ arg\\left( \\frac{z_1}{\\overline{z_2}} \\right )=\\pi.\\) Then\n(A) \\(arg~ z_2=\\left (\\frac{\\pi}{4}\\right )\\)\n(B) \\(arg z_2=-\\frac{3\\pi}{4}\\)\n(C) \\(arg~ z_1=\\frac{\\pi}{4}\\)\n(D) \\(arg z_1=-\\frac{3\\pi}{4}\\)\n", "ideal": "[\"the correct answer is (C)\", \"the correct answer is \\textbf{(C)}\", \"the correct answer is $\\boxed{\\textbf{(C)}\"]"}, {"prompt": "The system of equations\n\u2013kx + 3y \u2013 14z = 25\n\u201315x + 4y \u2013 kz = 3\n\u20134x + y + 3z = 4\nis consistent for all k in the set\n(A) R\n(B) R \u2013 {\u201311, 13}\n(C) R \u2013 {13}\n(D) R \u2013 {\u201311, 11}\n", "ideal": "[\"the correct answer is (D)\", \"the correct answer is \\textbf{(D)}\", \"the correct answer is $\\boxed{\\textbf{(D)}\"]"}, {"prompt": "\\(\\displaystyle \\lim_{x\\to \\frac{\\pi}{2}}tan^2x\\left((2sin^2x+3sinx+4)^{\\frac{1}{2}}-(sin^2x+6sinx+2)^{\\frac{1}{2}}\\right)\\) is equal to\n(A) 1/12\n(B) -1/18\n(C) -1/12\n(D) 1/6\n", "ideal": "[\"the correct answer is (A)\", \"the correct answer is \\textbf{(A)}\", \"the correct answer is $\\boxed{\\textbf{(A)}\"]"}, {"prompt": "The area of the region enclosed between the parabolas y^2 = 2x \u2013 1 and y^2 = 4x \u2013 3 is\n(A) \u2153\n(B) \u2159\n(C) \u2154\n(D) \u00be\n", "ideal": "[\"the correct answer is (A)\", \"the correct answer is \\textbf{(A)}\", \"the correct answer is $\\boxed{\\textbf{(A)}\"]"}, {"prompt": "The coefficient of x^101 in the expression (5 + x)^500 + x(5 + x)^499 + x^2(5 + x)^498 + \u2026\u2026+ x^500, x > 0, is\n(A) ^501C101 (5)^399\n(B) ^501C101 (5)^400\n(C) ^501C100 (5)^400\n(D) ^500C101 (5)^399\n", "ideal": "[\"the correct answer is (A)\", \"the correct answer is \\textbf{(A)}\", \"the correct answer is $\\boxed{\\textbf{(A)}\"]"}, {"prompt": "The sum 1 + 2 \u22c5 3 + 3 \u22c5 3^2 + \u2026. + 10 \u22c5 3^9 is equal to\n(A) \\(\\frac{2.3^{12}+10}{4}\\)\n(B) \\(\\frac{19.3^{10}+1}{4}\\)\n(C) \\(5.3^{10}-2\\)\n(D) \\(\\frac{9.3^{10}+1}{2}\\)\n", "ideal": "[\"the correct answer is (B)\", \"the correct answer is \\textbf{(B)}\", \"the correct answer is $\\boxed{\\textbf{(B)}\"]"}, {"prompt": "Let P be the plane passing through the intersection of the planes \\(\\overrightarrow{r}.(\\hat{i}+3\\hat{j}-\\hat{k})=5 ~and~ \\overrightarrow{r}~.(2\\hat{i}-\\hat{j}+\\hat{k})=3,\\) and the point (2, 1, \u20132). Let the position vectors of the points X and Y be \\(\\hat{i}-2\\hat{j}+4\\hat{k}\\ \\text{and}\\ 5\\hat{i}-\\hat{j}+2\\hat{k}\\) respectively. Then the points\n(A) X and X + Y are on the same side of P\n(B) Y and Y \u2013 X are on the opposite sides of P\n(C) X and Y are on the opposite sides of P\n(D) X + Y and X \u2013 Y are on the same side of P\n", "ideal": "[\"the correct answer is (C)\", \"the correct answer is \\textbf{(C)}\", \"the correct answer is $\\boxed{\\textbf{(C)}\"]"}, {"prompt": "A circle touches both the y-axis and the line x + y = 0. Then the locus of its center is\n(A) \\(y=\\sqrt{2}x\\)\n(B) \\(x=\\sqrt{2}y\\)\n(C) \\(y^2-x^2=2xy\\)\n(D) \\(x^2-y^2=2xy\\)\n", "ideal": "[\"the correct answer is (D)\", \"the correct answer is \\textbf{(D)}\", \"the correct answer is $\\boxed{\\textbf{(D)}\"]"}, {"prompt": "Water is being filled at the rate of 1 cm^3/sec in a right circular conical vessel (vertex downwards) of height 35 cm and diameter 14 cm. When the height of the water level is 10 cm, the rate (in cm^2/sec) at which the wet conical surface area of the vessel increase, is\n(A) 5\n(B) \\(\\frac{\\sqrt{21}}{5}\\)\n(C) \\(\\frac{\\sqrt{26}}{5}\\)\n(D) \\(\\frac{\\sqrt{26}}{10}\\)\n", "ideal": "[\"the correct answer is (C)\", \"the correct answer is \\textbf{(C)}\", \"the correct answer is $\\boxed{\\textbf{(C)}\"]"}, {"prompt": "If \\(b_n=\\int_{0}^{\\frac{\\pi}{2}}\\frac{cos^2nx}{sinx}dx,n\\epsilon N,\\) then\n(A) b3 \u2013 b2, b4 \u2013 b3, b5 \u2013 b4 are in an A.P. with a common difference \u20132\n(B) \\(\\frac{1}{b_3-b_2},\\frac{1}{b_4-b_3},\\frac{1}{b_5-b_4}\\ \\text{are in an A. P. with common difference 2}\\)\n(C) b3 \u2013 b2, b4 \u2013 b3, b5 \u2013 b4 are in a G.P.\n(D) \\(\\frac{1}{b_3-b_2},\\frac{1}{b_4-b_3},\\frac{1}{b_5-b_4}\\ \\text{are in an A.P. with common difference \u20132}\\)\n", "ideal": "[\"the correct answer is (D)\", \"the correct answer is \\textbf{(D)}\", \"the correct answer is $\\boxed{\\textbf{(D)}\"]"}, {"prompt": "If y = y(x) is the solution of the differential equation \\(2x^2\\frac{dy}{dx}-2xy+3y^2=0\\ \\text{such that}\\ y(e)=\\frac{e}{3},\\) then y(1) is equal to\n(A) \u2153\n(B) \u2154\n(C) 3/2\n(D) 3\n", "ideal": "[\"the correct answer is (B)\", \"the correct answer is \\textbf{(B)}\", \"the correct answer is $\\boxed{\\textbf{(B)}\"]"}, {"prompt": "If the angle made by the tangent at the point (x0, y0) on the curve x = 12(t + sin t cos t), \\(y=12(1+\\sin t)^2,0<t<\\frac{\\pi}{2},\\) with the positive x-axis is \u03c0/3,\u00a0then y0 is equal to:\n(A) \\(6(3+2\\sqrt{2})\\)\n(B) \\(3(7+4\\sqrt{3})\\)\n(C) 27\n(D) 48\n", "ideal": "[\"the correct answer is (C)\", \"the correct answer is \\textbf{(C)}\", \"the correct answer is $\\boxed{\\textbf{(C)}\"]"}, {"prompt": "The value of 2 sin(12\u00b0) \u2013 sin(72\u00b0) is :\n(A) \\(\\frac{\\sqrt{5}(1-\\sqrt{3})}{4}\\)\n(B) \\(\\frac{1-\\sqrt{5})}{8}\\)\n(C) \\(\\frac{\\sqrt{3}(1-\\sqrt{5})}{2}\\)\n(D) \\(\\frac{\\sqrt{3}(1-\\sqrt{5})}{4}\\)\n", "ideal": "[\"the correct answer is (D)\", \"the correct answer is \\textbf{(D)}\", \"the correct answer is $\\boxed{\\textbf{(D)}\"]"}, {"prompt": "A biased die is marked with numbers 2, 4, 8, 16, 32, 32 on its faces and the probability of getting a face with mark n is 1/n. If the die is thrown thrice, then the probability, that the sum of the numbers obtained is 48, is :\n(A) \\(\\frac{7}{2^{11}}\\)\n(B) \\(\\frac{7}{2^{12}}\\)\n(C) \\(\\frac{3}{2^{10}}\\)\n(D) \\(\\frac{13}{2^{12}}\\)\n", "ideal": "[\"the correct answer is (D)\", \"the correct answer is \\textbf{(D)}\", \"the correct answer is $\\boxed{\\textbf{(D)}\"]"}, {"prompt": "The negation of the Boolean expression ((~ q) \u2227 p) \u21d2 ((~ p) \u2228 q) is logically equivalent to :\n(A) p \u21d2 q\n(B) q \u21d2 p\n(C) ~ (p \u21d2 q)\n(D) ~ (q \u21d2 p)\n", "ideal": "[\"the correct answer is (C)\", \"the correct answer is \\textbf{(C)}\", \"the correct answer is $\\boxed{\\textbf{(C)}\"]"}, {"prompt": "If the line y = 4 + kx, k > 0, is the tangent to the parabola y = x \u2013 x^2 at the point P and V is the vertex of the parabola, then the slope of the line through P and V is :\n(A) \\(\\frac{3}{2}\\)\n(B) \\(\\frac{26}{9}\\)\n(C) \\(\\frac{5}{2}\\)\n(D) \\(\\frac{23}{6}\\)\n", "ideal": "[\"the correct answer is (C)\", \"the correct answer is \\textbf{(C)}\", \"the correct answer is $\\boxed{\\textbf{(C)}\"]"}, {"prompt": "The value of \\(tan^{-1}\\left(\\frac{cos\\frac{15\\pi}{4}-1}{sin\\left(\\frac{\\pi}{4}\\right)}\\right )\\) is equal to :\n(A) \\(-\\frac{\\pi}{4}\\)\n(B) \\(-\\frac{\\pi}{8}\\)\n(C) \\(-\\frac{5\\pi}{12}\\)\n(D) \\(-\\frac{4\\pi}{9}\\)\n", "ideal": "[\"the correct answer is (B)\", \"the correct answer is \\textbf{(B)}\", \"the correct answer is $\\boxed{\\textbf{(B)}\"]"}, {"prompt": "The line y = x + 1 meets the ellipse \\(\\frac{x^2}{4}+\\frac{y^2}{2}=1\\) at two points P and Q. If r is the radius of the circle with PQ as diameter then (3r)^2 is equal to :\n(A) 20\n(B) 12\n(C) 11\n(D) 8\n", "ideal": "[\"the correct answer is (A)\", \"the correct answer is \\textbf{(A)}\", \"the correct answer is $\\boxed{\\textbf{(A)}\"]"}, {"prompt": "Let \\(f\\left ( x \\right )=\\frac{x-1}{x+1},x\\epsilon R-\\left\\{0,-1,1 \\right\\}\\) If \u0192^n^+1(x) = \u0192(\u0192^n(x)) for all n\u2208N, then \u0192^6(6) + \u0192^7(7) is equal to :\n(A) \\(\\frac{7}{6}\\)\n(B) \\(-\\frac{3}{2}\\)\n(C) \\(\\frac{7}{12}\\)\n(D) \\(-\\frac{11}{12}\\)\n", "ideal": "[\"the correct answer is (B)\", \"the correct answer is \\textbf{(B)}\", \"the correct answer is $\\boxed{\\textbf{(B)}\"]"}, {"prompt": "Let \\(A=\\left\\{\\textbf{z}~\\epsilon~\\textbf{C}:\\left|\\frac{z+1}{z-1} \\right| <1\\right\\}\\)\nand \\(B=\\left\\{\\textbf{z}~\\epsilon~\\textbf{C}:arg\\left ( \\frac{z-1}{z+1} \\right ) = \\frac{2\\pi}{3}\\right\\}\\)\nThen A\u2229Bis :\n(A) A portion of a circle centred at \\(\\left ( 0,~-\\frac{1}{\\sqrt{3}} \\right )\\) that lies in the second and third quadrants only\n(B) A portion of a circle centred at \\(\\left ( 0,~-\\frac{1}{\\sqrt{3}} \\right )\\) that lies in the second quadrant only\n(C) An empty set\n(D) A portion of a circle of radius \\(\\frac{2}{\\sqrt{3}}\\) that lies in the third quadrant only\n", "ideal": "[\"the correct answer is (B)\", \"the correct answer is \\textbf{(B)}\", \"the correct answer is $\\boxed{\\textbf{(B)}\"]"}, {"prompt": "Let A be a 3 \u00d7 3 invertible matrix. If |adj (24A)| = |adj (3 adj (2A))|, then |A|^2 is equal to :\n(A) 6^6\n(B) 2^12\n(C) 2^6\n(D) 1\n", "ideal": "[\"the correct answer is (C)\", \"the correct answer is \\textbf{(C)}\", \"the correct answer is $\\boxed{\\textbf{(C)}\"]"}, {"prompt": "The ordered pair (a, b), for which the system of linear equations\n3x \u2013 2y + z = b\n5x \u2013 8y + 9z = 3\n2x + y + az = \u20131\nhas no solution, is :\n(A) \\(\\left ( 3,\\frac{1}{3} \\right )\\)\n(B) \\(\\left (- 3,\\frac{1}{3} \\right )\\)\n(C) \\(\\left (- 3,-\\frac{1}{3} \\right )\\)\n(D) \\(\\left ( 3,-\\frac{1}{3} \\right )\\)\n", "ideal": "[\"the correct answer is (C)\", \"the correct answer is \\textbf{(C)}\", \"the correct answer is $\\boxed{\\textbf{(C)}\"]"}, {"prompt": "The remainder when (2021)^2023 is divided by 7 is :\n(A) 1\n(B) 2\n(C) 5\n(D) 6\n", "ideal": "[\"the correct answer is (C)\", \"the correct answer is \\textbf{(C)}\", \"the correct answer is $\\boxed{\\textbf{(C)}\"]"}, {"prompt": "\\(\\displaystyle \\lim_{x\\rightarrow\\frac{1}{\\sqrt{2}}}\\frac{\\sin\\left ( \\cos^{-1}x \\right )-x}{1-\\tan\\left ( \\cos^{-1}x \\right )}\\) is equal to :\n(A) \\(\\sqrt{2}\\)\n(B) \\(-\\sqrt{2}\\)\n(C) \\(\\frac{1}{\\sqrt{2}}\\)\n(D) \\(-\\frac{1}{\\sqrt{2}}\\)\n", "ideal": "[\"the correct answer is (D)\", \"the correct answer is \\textbf{(D)}\", \"the correct answer is $\\boxed{\\textbf{(D)}\"]"}, {"prompt": "g :R\u2192R be two real valued functions defined as \\(f\\left ( x \\right )=\\left\\{\\begin{matrix}-\\left|x+3 \\right|, & x<0 \\\\e^x, & x\\geq 0 \\\\\\end{matrix}\\right.\\) and\n\\(g\\left ( x \\right )=\\left\\{\\begin{matrix}x^2+k_1x, & x<0 \\\\4x+k_2, & x\\geq 0 \\\\\\end{matrix}\\right.\\) where k1 and k2 are real constants. If (go\u0192) is differentiable at x = 0, then (go\u0192) (\u20134) + (go\u0192) (4) is equal to :\n(A) 4(e^4 + 1)\n(B) 2(2e^4 + 1)\n(C) 4e^4\n(D) 2(2e^4 \u2013 1)\n", "ideal": "[\"the correct answer is (D)\", \"the correct answer is \\textbf{(D)}\", \"the correct answer is $\\boxed{\\textbf{(D)}\"]"}, {"prompt": "The sum of the absolute minimum and the absolute maximum values of the function \u0192(x) = |3x \u2013 x^2 + 2| \u2013 x in the interval [\u20131, 2] is :\n(A) \\(\\frac{\\sqrt{17}+3}{2}\\)\n(B) \\(\\frac{\\sqrt{17}+5}{2}\\)\n(C) 5\n(D) \\(\\frac{9-\\sqrt{17}}{2}\\)\n", "ideal": "[\"the correct answer is (A)\", \"the correct answer is \\textbf{(A)}\", \"the correct answer is $\\boxed{\\textbf{(A)}\"]"}, {"prompt": "Let S be the set of all the natural numbers, for which the line \\(\\frac{x}{a}+\\frac{y}{b}=2\\) is a tangent to the curve \\(\\left ( \\frac{x}{a} \\right )^n+\\left ( \\frac{y}{b} \\right )^n=2\\) at the point (a, b), ab \u2260 0. Then :\n(A) S = \u0278\n(B) n(S) = 1\n(C) S = {2k : k \u2208 N }\n(D) S = N\n", "ideal": "[\"the correct answer is (D)\", \"the correct answer is \\textbf{(D)}\", \"the correct answer is $\\boxed{\\textbf{(D)}\"]"}, {"prompt": "Let R be the point (3, 7) and let P and Q be two points on the line x + y = 5 such that PQR is an equilateral triangle, Then the area of \u0394PQRis :\n(A) \\(\\frac{25}{4\\sqrt{3}}\\)\n(B) \\(\\frac{25\\sqrt{3}}{2}\\)\n(C) \\(\\frac{25}{\\sqrt{3}}\\)\n(D) \\(\\frac{25}{2\\sqrt{3}}\\)\n", "ideal": "[\"the correct answer is (D)\", \"the correct answer is \\textbf{(D)}\", \"the correct answer is $\\boxed{\\textbf{(D)}\"]"}, {"prompt": "Let C be a circle passing through the points A(2, \u20131) and B (3, 4). The line segment AB is not a diameter of C. If r is the radius of C and its centre lies on the circle \\(\\left ( x-5 \\right )^2+\\left ( y-1 \\right )^2=\\frac{13}{2}\\) then r^2 is equal to :\n(A) 32\n(B) \\(\\frac{65}{2}\\)\n(C) \\(\\frac{61}{2}\\)\n(D) 30\n", "ideal": "[\"the correct answer is (B)\", \"the correct answer is \\textbf{(B)}\", \"the correct answer is $\\boxed{\\textbf{(B)}\"]"}, {"prompt": "Let the normal at the point P on the parabola y^2 = 6x pass through the point (5, \u20138). If the tangent at P to the parabola intersects its directrix at the point Q, then the ordinate of the point Q is :\n(A) \u20133\n(B) \\(-\\frac{9}{4}\\)\n(C) \\(-\\frac{5}{2}\\)\n(D) \u20132\n", "ideal": "[\"the correct answer is (B)\", \"the correct answer is \\textbf{(B)}\", \"the correct answer is $\\boxed{\\textbf{(B)}\"]"}, {"prompt": "If the two lines \\(l_1:\\frac{x-2}{3}=\\frac{y+1}{-2},z=2\\) and \\(l_2:\\frac{x-1}{1}=\\frac{2y+3}{\\alpha}=\\frac{z+5}{2}\\) are perpendicular, then an angle between the lines l2 and \\(l_3:\\frac{1-x}{3}=\\frac{2y-1}{-4}=\\frac{z}{4}\\)is :\n(A) \\(\\cos^{-1}\\left ( \\frac{29}{4}\\right )\\)\n(B) \\(\\sec^{-1}\\left ( \\frac{29}{4}\\right )\\)\n(C) \\(\\cos^{-1}\\left ( \\frac{2}{29} \\right )\\)\n(D) \\(\\cos^{-1}\\left ( \\frac{2}{\\sqrt{29}} \\right )\\)\n", "ideal": "[\"the correct answer is (B)\", \"the correct answer is \\textbf{(B)}\", \"the correct answer is $\\boxed{\\textbf{(B)}\"]"}, {"prompt": "Let the plane 2x + 3y + z + 20 = 0 be rotated through a right angle about its line of intersection with the plane x \u2013 3y\u00a0+ 5z = 8. If the mirror image of the point \\(\\left ( 2,-\\frac{1}{2},2 \\right )\\) in the rotated plane is B( a, b, c),then :\n(A) \\(\\frac{a}{8}=\\frac{b}{5}=\\frac{c}{-4}\\)\n(B) \\(\\frac{a}{4}=\\frac{b}{5}=\\frac{c}{-2}\\)\n(C) \\(\\frac{a}{8}=\\frac{b}{-5}=\\frac{c}{4}\\)\n(D) \\(\\frac{a}{4}=\\frac{b}{5}=\\frac{c}{2}\\)\n", "ideal": "[\"the correct answer is (A)\", \"the correct answer is \\textbf{(A)}\", \"the correct answer is $\\boxed{\\textbf{(A)}\"]"}, {"prompt": "If \\(\\vec{a}\\cdot\\vec{b}=1,\\vec{b}\\cdot\\vec{c}=2~\\textup{and}~\\vec{c}\\cdot\\vec{a}=3\\), then the value of \\(\\left [ \\vec{a}\\times\\left ( \\vec{b}\\times\\vec{c} \\right ),\\vec{b}\\times\\left ( \\vec{c}\\times\\vec{a} \\right ),\\vec{c}\\times\\left ( \\vec{b}\\times\\vec{a} \\right ) \\right ]\\)is :\n(A) 0\n(B) \\(-6\\vec{a}\\cdot\\left ( \\vec{b}\\times\\vec{c} \\right )\\)\n(C) \\(12\\vec{c}\\cdot\\left ( \\vec{a}\\times\\vec{b} \\right )\\)\n(D) \\(-12\\vec{b}\\cdot\\left ( \\vec{c}\\times\\vec{a} \\right )\\)\n", "ideal": "[\"the correct answer is (A)\", \"the correct answer is \\textbf{(A)}\", \"the correct answer is $\\boxed{\\textbf{(A)}\"]"}, {"prompt": "Let a biased coin be tossed 5 times. If the probability of getting 4 heads is equal to the probability of getting 5 heads, then the probability of getting atmost two heads is:\n(A) \\(\\frac{275}{6^5}\\)\n(B) \\(\\frac{36}{5^4}\\)\n(C) \\(\\frac{181}{5^5}\\)\n(D) \\(\\frac{46}{6^4}\\)\n", "ideal": "[\"the correct answer is (D)\", \"the correct answer is \\textbf{(D)}\", \"the correct answer is $\\boxed{\\textbf{(D)}\"]"}, {"prompt": "The mean of the numbers a, b, 8, 5, 10 is 6 and their variance is 6.8. If M is the mean deviation of the numbers about the mean, then 25 M is equal to:\n(A) 60\n(B) 55\n(C) 50\n(D) 45\n", "ideal": "[\"the correct answer is (A)\", \"the correct answer is \\textbf{(A)}\", \"the correct answer is $\\boxed{\\textbf{(A)}\"]"}, {"prompt": "Let \\(f\\left ( x \\right )=2\\cos^{-1}x+4\\cot^{-1}x-3x^2-2x+10,\\chi\\epsilon\\left [ -1,1 \\right ]\\) If [a, b] is the range of the function,f then 4a \u2013 b is equal to :\n(A) 11\n(B) 11 \u2013 \u03c0\n(C) 11 + \u03c0\n(D) 15 \u2013 \u03c0\n", "ideal": "[\"the correct answer is (B)\", \"the correct answer is \\textbf{(B)}\", \"the correct answer is $\\boxed{\\textbf{(B)}\"]"}, {"prompt": "Let \\(\\Delta,\\triangledown \\epsilon\\left\\{ \\wedge ,\\vee \\right\\}\\) be such that \\(p\\triangledown q\\Rightarrow\\left ( \\left ( p\\Delta q \\right )\\triangledown r \\right ) \\) is a tautology. Then \\(\\left ( p\\triangledown q \\right )\\Delta r\\) is logically equivalent to :\n(A) \\(\\left ( p~\\Delta~r \\right )\\vee q\\)\n(B) \\(\\left ( p~\\Delta~r \\right )\\wedge q\\)\n(C) \\(\\left ( p \\wedge r \\right )\\Delta q\\)\n(D) \\(\\left ( p \\triangledown r \\right )\\wedge q\\)\n", "ideal": "[\"the correct answer is (A)\", \"the correct answer is \\textbf{(A)}\", \"the correct answer is $\\boxed{\\textbf{(A)}\"]"}, {"prompt": "Let \\(f:\\mathbb{R}\\rightarrow \\mathbb{R}\\) be defined as f(x) = x \u2013 1 and \\(g:\\mathbb{R}-\\left\\{1,-1 \\right\\}\\to \\mathbb{R}\\) be defined as\\(g(x)=\\frac{x^2}{x^2-1}.\\)\nThen the function fog is:\n(A) One-one but not onto\n(B) Onto but not one-one\n(C) Both one-one and onto\n(D) Neither one-one nor onto\n", "ideal": "[\"the correct answer is (D)\", \"the correct answer is \\textbf{(D)}\", \"the correct answer is $\\boxed{\\textbf{(D)}\"]"}, {"prompt": "If the system of equations \u03b1x + y + z = 5, x + 2y + 3z = 4, x + 3y +5z = \u03b2 has infinitely many solutions, then the ordered pair (\u03b1, \u03b2) is equal to:\n(A) (1, \u20133)\n(B) (\u20131, 3)\n(C) (1, 3)\n(D) (\u20131, \u20133)\n", "ideal": "[\"the correct answer is (C)\", \"the correct answer is \\textbf{(C)}\", \"the correct answer is $\\boxed{\\textbf{(C)}\"]"}, {"prompt": "\\(\\text{If}\\ A=\\sum_{n=1 }^{\\infty}\\frac{1}{\\left(3+(-1)^n\\right)^n}\\ \\text{and}\\ B=\\sum_{n=1 }^{\\infty}\\frac{(-1)^n}{\\left(3+(-1)^n\\right)^n},\\) then A/B is equal to:\n(A) 11/9\n(B) 1\n(C) -11/9\n(D) -11/3\n", "ideal": "[\"the correct answer is (C)\", \"the correct answer is \\textbf{(C)}\", \"the correct answer is $\\boxed{\\textbf{(C)}\"]"}, {"prompt": "\\(\\displaystyle \\lim_{x\\to 0}\\frac{cos(sin~x)-cos~x}{x^4}\\ \\text{is equal to}:\\)\n(A) 1/3\n(B) 1/4\n(C) 1/6\n(D) 1/12\n", "ideal": "[\"the correct answer is (C)\", \"the correct answer is \\textbf{(C)}\", \"the correct answer is $\\boxed{\\textbf{(C)}\"]"}, {"prompt": "Let f(x) = min {1, 1 + x sin x}, 0 \u2264 x \u2264 2\u03c0. If m is the number of points, where f is not differentiable, and n is the number of points, where f is not continuous, then the ordered pair (m, n) is equal to\n(A) (2, 0)\n(B) (1, 0)\n(C) (1, 1)\n(D) (2, 1)\n", "ideal": "[\"the correct answer is (B)\", \"the correct answer is \\textbf{(B)}\", \"the correct answer is $\\boxed{\\textbf{(B)}\"]"}, {"prompt": "Consider a cuboid of sides 2x, 4x and 5x and a closed hemisphere of radius r. If the sum of their surface areas is a constant k, then the ratio x : r, for which the sum of their volumes is maximum, is\n(A) 2 : 5\n(B) 19 : 45\n(C) 3 : 8\n(D) 19 : 15\n", "ideal": "[\"the correct answer is (B)\", \"the correct answer is \\textbf{(B)}\", \"the correct answer is $\\boxed{\\textbf{(B)}\"]"}, {"prompt": "The area of the region bounded by y^2 = 8x and y^2 = 16(3 \u2013 x) is equal to\n(A) \\(\\frac{32}{3}\\)\n(B) \\(\\frac{40}{3}\\)\n(C) 16\n(D) 19\n", "ideal": "[\"the correct answer is (C)\", \"the correct answer is \\textbf{(C)}\", \"the correct answer is $\\boxed{\\textbf{(C)}\"]"}, {"prompt": "\\(\\text{If}\\ \\int\\frac{1}{x}\\sqrt{\\frac{1-x}{1+x}}dx=g(x)+c,g(1)=0,\\ \\text{then}\\ g\\left(\\frac{1}{2}\\right )\\) is equal to\n(A) \\(log_e\\left(\\frac{\\sqrt{3}-1}{\\sqrt{3}+1}\\right)+\\frac{\\pi}{3}\\)\n(B) \\(log_e\\left(\\frac{\\sqrt{3}+1}{\\sqrt{3}-1}\\right)+\\frac{\\pi}{3}\\)\n(C) \\(log_e\\left(\\frac{\\sqrt{3}+1}{\\sqrt{3}-1}\\right)-\\frac{\\pi}{3}\\)\n(D) \\(\\frac{1}{2}log_e\\left(\\frac{\\sqrt{3}-1}{\\sqrt{3}+1}\\right)-\\frac{\\pi}{6}\\)\n", "ideal": "[\"the correct answer is (A)\", \"the correct answer is \\textbf{(A)}\", \"the correct answer is $\\boxed{\\textbf{(A)}\"]"}, {"prompt": "If y = y(x) is the solution of the differential equation \\(x\\frac{dy}{dx}+2y=xe^x,y(1)=0\\) then the local maximum value of the function \\(z(x)=x^2y(x)-e^x,x\\in R\\) is\n(A) 1 \u2013 e\n(B) 0\n(C) 1/2\n(D) \\(\\frac{4}{e}-e\\)\n", "ideal": "[\"the correct answer is (D)\", \"the correct answer is \\textbf{(D)}\", \"the correct answer is $\\boxed{\\textbf{(D)}\"]"}, {"prompt": "If the solution of the differential equation \\(\\frac{dy}{dx}+e^x(x^2-2)y=(x^2-2x)(x^2-2)e^{2x}\\) satisfies y(0) = 0, then the value of y(2) is ______.\n(A) \u20131\n(B) 1\n(C) 0\n(D) e\n", "ideal": "[\"the correct answer is (C)\", \"the correct answer is \\textbf{(C)}\", \"the correct answer is $\\boxed{\\textbf{(C)}\"]"}, {"prompt": "If m is the slope of a common tangent to the curves \\(\\frac{x^2}{16}+\\frac{y^2}{9}=1\\) and x^2 + y^2 = 12, then 12m^2 is equal to:\n(A) 6\n(B) 9\n(C) 10\n(D) 12\n", "ideal": "[\"the correct answer is (B)\", \"the correct answer is \\textbf{(B)}\", \"the correct answer is $\\boxed{\\textbf{(B)}\"]"}, {"prompt": "The locus of the mid-point of the line segment joining the point (4, 3) and the points on the ellipse x^2 + 2y^2 = 4 is an ellipse with eccentricity:\n(A) \\(\\frac{\\sqrt{3}}{2}\\)\n(B) \\(\\frac{1}{2\\sqrt{2}}\\)\n(C) \\(\\frac{1}{\\sqrt{2}}\\)\n(D) \\(\\frac{1}{2}\\)\n", "ideal": "[\"the correct answer is (C)\", \"the correct answer is \\textbf{(C)}\", \"the correct answer is $\\boxed{\\textbf{(C)}\"]"}, {"prompt": "The normal to the hyperbola \\(\\frac{x^2}{a^2}-\\frac{y^2}{9}=1\\) at the point (8, 3\u221a3)\u00a0on it passes through the point:\n(A) \\((15,-2\\sqrt{3})\\)\n(B) \\((9,2\\sqrt{3})\\)\n(C) \\((-1,9\\sqrt{3})\\)\n(D) \\((-1,6\\sqrt{3})\\)\n", "ideal": "[\"the correct answer is (C)\", \"the correct answer is \\textbf{(C)}\", \"the correct answer is $\\boxed{\\textbf{(C)}\"]"}, {"prompt": "If the plane 2x + y \u2013 5z = 0 is rotated about its line of intersection with the plane 3x \u2013 y + 4z \u2013 7 = 0 by an angle of \u03c0/2, then the plane after the rotation passes through the point:\n(A) (2, \u20132, 0)\n(B) (\u20132, 2, 0)\n(C) (1, 0, 2)\n(D) (\u20131, 0, \u20132)\n", "ideal": "[\"the correct answer is (C)\", \"the correct answer is \\textbf{(C)}\", \"the correct answer is $\\boxed{\\textbf{(C)}\"]"}, {"prompt": "If the lines \\(\\vec{r}=(\\hat{i}-\\hat{j}+\\hat{k})+\\lambda (3\\hat{j}-\\hat{k})\\) and \\(\\vec{r}=(\\alpha \\hat{i}-\\hat{j})+\\mu(2\\hat{j}-3\\hat{k})\\) are co-planar, then the distance of the plane containing these two lines from the point (\u03b1, 0, 0) is :\n(A) \\(\\frac{2}{9}\\)\n(B) \\(\\frac{2}{11}\\)\n(C) \\(\\frac{4}{11}\\)\n(D) 2\n", "ideal": "[\"the correct answer is (B)\", \"the correct answer is \\textbf{(B)}\", \"the correct answer is $\\boxed{\\textbf{(B)}\"]"}, {"prompt": "Let \\(\\vec{a}=\\hat{i}+\\hat{j}+2\\hat{k},\\vec{b}=2\\dot{i}-3\\hat{j}+\\hat{k}\\) and \\(\\vec{c}=\\hat{i}-\\hat{j}+\\hat{k}\\) be three given vectors. \\(\\text{Let}\\ \\vec{v}\\ \\text{be a vector in the plane of}\\ \\vec{a}\\ \\text{and}\\ \\vec{b}\\ \\text{whose projection on}\\ \\vec{c}\\ \\text{is}\\ \\frac{2}{\\sqrt{3}}.\\)\n\\(\\text{If}\\ \\vec{v}.\\hat{j}=7,\\ \\text{then}\\ \\vec{v}.(\\hat{i}+\\hat{k})\\ \\text{is equal to}:\\)\n(A) 6\n(B) 7\n(C) 8\n(D) 9\n", "ideal": "[\"the correct answer is (D)\", \"the correct answer is \\textbf{(D)}\", \"the correct answer is $\\boxed{\\textbf{(D)}\"]"}, {"prompt": "The mean and standard deviation of 50 observations are 15 and 2 respectively. It was found that one incorrect observation was taken such that the sum of correct and incorrect observations is 70. If the correct mean is 16, then the correct variance is equal to :\n(A) 10\n(B) 36\n(C) 43\n(D) 60\n", "ideal": "[\"the correct answer is (C)\", \"the correct answer is \\textbf{(C)}\", \"the correct answer is $\\boxed{\\textbf{(C)}\"]"}, {"prompt": "16 sin(20\u00b0) sin(40\u00b0) sin(80\u00b0) is equal to :\n(A) \u221a3\n(B) 2\u221a3\n(C) 3\n(D) 4\u221a3\n", "ideal": "[\"the correct answer is (B)\", \"the correct answer is \\textbf{(B)}\", \"the correct answer is $\\boxed{\\textbf{(B)}\"]"}, {"prompt": "If the inverse trigonometric functions take principal values, then \\(cos^{-1}\\left(\\frac{3}{10}cos\\left(tan^{-1}\\left(\\frac{4}{3}\\right )\\right )+\\frac{2}{5}sin\\left ( \\tan^{-1}\\left (\\frac{4}{3} \\right ) \\right ) \\right )\\) is equal to :\n(A) 0\n(B) \\(\\frac{\\pi}{4}\\)\n(C) \\(\\frac{\\pi}{3}\\)\n(D) \\(\\frac{\\pi}{6}\\)\n", "ideal": "[\"the correct answer is (C)\", \"the correct answer is \\textbf{(C)}\", \"the correct answer is $\\boxed{\\textbf{(C)}\"]"}, {"prompt": "Let r \u2208 {p, q, ~p, ~q} be such that the logical statement r \u2228 (~p) \u21d2 (p \u2227 q) \u2228 r is a tautology. Then r is equal to :\n(A) p\n(B) q\n(C) ~p\n(D) ~q\n", "ideal": "[\"the correct answer is (C)\", \"the correct answer is \\textbf{(C)}\", \"the correct answer is $\\boxed{\\textbf{(C)}\"]"}, {"prompt": "The area of the polygon, whose vertices are the non-real roots of the equation \\(\\overline{z}=iz^2\\) is : \n(A) \\(\\frac{3\\sqrt{3}}{4}\\)\n(B) \\(\\frac{3\\sqrt{3}}{2}\\)\n(C) \\(\\frac{3}{2}\\)\n(D) \\(\\frac{3}{4}\\)\n", "ideal": "[\"the correct answer is (A)\", \"the correct answer is \\textbf{(A)}\", \"the correct answer is $\\boxed{\\textbf{(A)}\"]"}, {"prompt": "Let the system of linear equations x + 2y + z = 2, \u03b1x + 3y \u2013 z = \u03b1, \u2013\u03b1x + y + 2z = \u2013\u03b1 be inconsistent. Then \u03b1 is equal to : \n(A) \\(\\frac{5}{2}\\)\n(B) \\(-\\frac{5}{2}\\)\n(C) \\(\\frac{7}{2}\\)\n(D) \\(-\\frac{7}{2}\\)\n", "ideal": "[\"the correct answer is (D)\", \"the correct answer is \\textbf{(D)}\", \"the correct answer is $\\boxed{\\textbf{(D)}\"]"}, {"prompt": "If \\(x=\\displaystyle\\sum\\limits_{n=0}^\\infty a^n,y=\\displaystyle\\sum\\limits_{n=0}^\\infty b^n, z=\\displaystyle\\sum\\limits_{n=0}^\\infty c^n,\\) where a, b, c are in A.P. and |a| < 1, |b| < 1, |c| < 1, abc\u2260 0,\nthen : \n(A) x, y, zare in A.P.\n(B) x, y, zare in G.P.\n(C) \\(\\frac{1}{x},\\frac{1}{y},\\frac{1}{z}\\) are in A.P.\n(D) \\(\\frac{1}{x}+\\frac{1}{y}+\\frac{1}{z}=1-\\left ( a+b+c \\right )\\)\n", "ideal": "[\"the correct answer is (C)\", \"the correct answer is \\textbf{(C)}\", \"the correct answer is $\\boxed{\\textbf{(C)}\"]"}, {"prompt": "Let \\(\\frac{dy}{dx}=\\frac{ax-by+a}{bx+cy+a}\\) where a, b, c are constants, represent a circle passing through the point (2, 5). Then the shortest distance of the point (11, 6) from this circle is \n(A) 10\n(B) 8\n(C) 7\n(D) 5\n", "ideal": "[\"the correct answer is (B)\", \"the correct answer is \\textbf{(B)}\", \"the correct answer is $\\boxed{\\textbf{(B)}\"]"}, {"prompt": "Let a be an integer such that \\(\\lim\\limits_{x\\rightarrow7}\\frac{18-\\left [ 1-x \\right ]}{\\left [ x-3a \\right ]}\\) exists, where [t] is greatest integer \u2264 t. Then a is equal\nto : \n(A) \u20136\n(B) \u20132\n(C) 2\n(D) 6\n", "ideal": "[\"the correct answer is (A)\", \"the correct answer is \\textbf{(A)}\", \"the correct answer is $\\boxed{\\textbf{(A)}\"]"}, {"prompt": "The number of distinct real roots of x^4 \u2013 4x + 1 = 0 is : \n(A) 4\n(B) 2\n(C) 1\n(D) 0\n", "ideal": "[\"the correct answer is (B)\", \"the correct answer is \\textbf{(B)}\", \"the correct answer is $\\boxed{\\textbf{(B)}\"]"}, {"prompt": "The lengths of the sides of a triangle are 10 + x^2, 10 + x^2 and 20 \u2013 2x^2. If for x = k, the area of the triangle is maximum, then 3k^2 is equal to : \n(A) 5\n(B) 8\n(C) 10\n(D) 12\n", "ideal": "[\"the correct answer is (C)\", \"the correct answer is \\textbf{(C)}\", \"the correct answer is $\\boxed{\\textbf{(C)}\"]"}, {"prompt": "If \\(\\cos^{-1}\\left ( \\frac{y}{2} \\right )=\\textup{log}_e\\left ( \\frac{x}{5} \\right )^5,\\left|y \\right|<2\\) then : \n(A) x^2y\u2032\u2032 + xy\u2032 \u2013 25y = 0\n(B) x^2y\u2032\u2032 \u2013 xy\u2032 \u2013 25y = 0\n(C) x^2y\u2032\u2032 \u2013 xy\u2032+ 25y = 0\n(D) x^2y\u2032\u2032 + xy\u2032+ 25y = 0\n", "ideal": "[\"the correct answer is (D)\", \"the correct answer is \\textbf{(D)}\", \"the correct answer is $\\boxed{\\textbf{(D)}\"]"}, {"prompt": "If \\(\\int\\frac{\\left ( x^2+1 \\right )e^x}{\\left ( x+1 \\right )^2}dx=f\\left ( x \\right )e^x+C\\) where C is a constant, then \\(\\frac{d^3f}{dx^3}\\) at x = 1 is equal to : \n(A) \\(-\\frac{3}{4} \\)\n(B) \\(\\frac{3}{4} \\)\n(C) \\(-\\frac{3}{2} \\)\n(D) \\(\\frac{3}{2} \\)\n", "ideal": "[\"the correct answer is (B)\", \"the correct answer is \\textbf{(B)}\", \"the correct answer is $\\boxed{\\textbf{(B)}\"]"}, {"prompt": "The value of the integral \\(\\displaystyle\\int\\limits_{-2}^2\\frac{\\left|x^3+x \\right|}{\\left (e^{x\\left|x\\right|}+1 \\right ) }dx\\) is equal to: \n(A) 5e^2\n(B) 3e^\u20132\n(C) 4\n(D) 6\n", "ideal": "[\"the correct answer is (D)\", \"the correct answer is \\textbf{(D)}\", \"the correct answer is $\\boxed{\\textbf{(D)}\"]"}, {"prompt": "If \\(\\frac{dy}{dx}+\\frac{2^{x-y}\\left ( 2^y-1 \\right )}{2^x-1}=0,x,y>0,y\\left ( 1 \\right ) =1\\), then y(2) is equal to : \n(A) 2 + log2 3\n(B) 2 + log3 2\n(C) 2 \u2013 log3 2\n(D) 2 \u2013 log2 3\n", "ideal": "[\"the correct answer is (D)\", \"the correct answer is \\textbf{(D)}\", \"the correct answer is $\\boxed{\\textbf{(D)}\"]"}, {"prompt": "In an isosceles triangle ABC, the vertex A is (6, 1) and the equation of the base BC is 2x + y = 4. Let the point B lie on the line x + 3y = 7. If (\u03b1, \u03b2) is the centroid of \u0394ABC, then 15(\u03b1 + \u03b2) is equal to : \n(A) 39\n(B) 41\n(C) 51\n(D) 63\n", "ideal": "[\"the correct answer is (C)\", \"the correct answer is \\textbf{(C)}\", \"the correct answer is $\\boxed{\\textbf{(C)}\"]"}, {"prompt": "Let the eccentricity of an ellipse \\(\\frac{x^2}{a^2}+\\frac{y^2}{b^2}=1,a>b, \\) be \\(\\frac{1}{4} \\). If this ellipse passes through the point \\(\\left ( -4\\sqrt{\\frac{2}{5}},3 \\right )\\), then a^2 + b^2 is equal to : \n(A) 29\n(B) 31\n(C) 32\n(D) 34\n", "ideal": "[\"the correct answer is (B)\", \"the correct answer is \\textbf{(B)}\", \"the correct answer is $\\boxed{\\textbf{(B)}\"]"}, {"prompt": "If two straight lines whose direction cosines are given by the relations l + m \u2013 n = 0, 3l^2 + m^2 + cnl = 0 are parallel, then the positive value of cis : \n(A) 6\n(B) 4\n(C) 3\n(D) 2\n", "ideal": "[\"the correct answer is (A)\", \"the correct answer is \\textbf{(A)}\", \"the correct answer is $\\boxed{\\textbf{(A)}\"]"}, {"prompt": "Let \\(\\vec{a}=\\hat{i}+\\hat{j}-\\hat{k} \\) and \\(\\vec{c}=2\\hat{i}-3\\hat{j}+2\\hat{k} \\). Then the number of vectors \\(\\vec{b}\\) such that \\(\\vec{b}\\times\\vec{c}=\\vec{a}\\) and \\(\\left|\\vec{b} \\right|\\in\\left\\{1,2,\\dots,10 \\right\\}\\) is : \n(A) 0\n(B) 1\n(C) 2\n(D) 3\n", "ideal": "[\"the correct answer is (A)\", \"the correct answer is \\textbf{(A)}\", \"the correct answer is $\\boxed{\\textbf{(A)}\"]"}, {"prompt": "Five numbers, x1, x2, x3, x4, x5 are randomly selected from the numbers 1, 2, 3,\u2026.., 18 and are arranged in the increasing order (x1<x2<x3<x4<x5). The probability that x2 = 7 and x4 = 11 is: \n(A) \\(\\frac{1}{136} \\)\n(B) \\(\\frac{1}{72} \\)\n(C) \\(\\frac{1}{68} \\)\n(D) \\(\\frac{1}{34} \\)\n", "ideal": "[\"the correct answer is (C)\", \"the correct answer is \\textbf{(C)}\", \"the correct answer is $\\boxed{\\textbf{(C)}\"]"}, {"prompt": "Let X be a random variable having binomial distribution B(7, p). If P(X = 3) = 5P(X = 4), then the sum of the mean and the variance of X is: \n(A) \\(\\frac{105}{16}\\)\n(B) \\(\\frac{7}{16}\\)\n(C) \\(\\frac{77}{36}\\)\n(D) \\(\\frac{49}{16}\\)\n", "ideal": "[\"the correct answer is (C)\", \"the correct answer is \\textbf{(C)}\", \"the correct answer is $\\boxed{\\textbf{(C)}\"]"}, {"prompt": "The value of \\(\\cos\\left ( \\frac{2\\pi}{7} \\right )+\\cos\\left ( \\frac{4\\pi}{7} \\right )+\\cos\\left ( \\frac{6\\pi}{7} \\right )\\) is equal to: \n(A) \u20131\n(B) \\(-\\frac{1}{2} \\)\n(C) \\(-\\frac{1}{3} \\)\n(D) \\(-\\frac{1}{4} \\)\n", "ideal": "[\"the correct answer is (B)\", \"the correct answer is \\textbf{(B)}\", \"the correct answer is $\\boxed{\\textbf{(B)}\"]"}, {"prompt": "\\(\\sin^{-1}\\left ( \\sin\\frac{2\\pi}{3} \\right )+\\cos^{-1}\\left ( \\cos\\frac{7\\pi}{6} \\right )+\\tan^{-1}\\left ( \\tan\\frac{3\\pi}{4} \\right )\\) is equal to: \n(A) \\(\\frac{11\\pi}{12}\\)\n(B) \\(\\frac{17\\pi}{12}\\)\n(C) \\(\\frac{31\\pi}{12}\\)\n(D) \\(-\\frac{3\\pi}{4}\\)\n", "ideal": "[\"the correct answer is (A)\", \"the correct answer is \\textbf{(A)}\", \"the correct answer is $\\boxed{\\textbf{(A)}\"]"}, {"prompt": "The boolean expression (~(p \u2227q)) \u2228q is equivalent to: \n(A) q\u2192 (p \u2227q)\n(B) p\u2192q\n(C) p\u2192 (p\u2192q)\n(D) p\u2192 (p\u2228q)\n", "ideal": "[\"the correct answer is (D)\", \"the correct answer is \\textbf{(D)}\", \"the correct answer is $\\boxed{\\textbf{(D)}\"]"}, {"prompt": "The number of points of intersection of |z \u2013 (4 + 3i)| = 2 and |z| + |z \u2013 4| = 6, z \u2208 C, is\n(A) 0\n(B) 1\n(C) 2\n(D) 3\n", "ideal": "[\"the correct answer is (C)\", \"the correct answer is \\textbf{(C)}\", \"the correct answer is $\\boxed{\\textbf{(C)}\"]"}, {"prompt": "Let \\(f(x) = \\begin{vmatrix}a & -1 & 0 \\\\ax & a & -1 \\\\ax^2 & ax & a \\\\\\end{vmatrix},\\) a \u2208 R. Then the sum of the square of all the values of a, for which 2f\u2032(10) \u2013f\u2032(5) + 100 = 0, is\n(A) 117\n(B) 106\n(C) 125\n(D) 136\n", "ideal": "[\"the correct answer is (C)\", \"the correct answer is \\textbf{(C)}\", \"the correct answer is $\\boxed{\\textbf{(C)}\"]"}, {"prompt": "Let for some real numbers \u03b1 and \u03b2, a = \u03b1 \u2013 i\u03b2. If the system of equations 4ix + (1 + i) y = 0 and \\(8\\left ( cos \\frac{2\\pi}{3} + i~sin\\frac{2\\pi}{3} \\right )x + \\overline{a} y = 0\\) has more than one solution, then \u03b1/\u03b2 is equal to\n(A) -2 + \u221a3\n(B) 2 \u2013 \u221a3\n(C) 2 + \u221a3\n(D) -2 \u2013 \u221a3\n", "ideal": "[\"the correct answer is (B)\", \"the correct answer is \\textbf{(B)}\", \"the correct answer is $\\boxed{\\textbf{(B)}\"]"}, {"prompt": "Let A and B be two 3 \u00d7 3 matrices such that AB = I and |A| = \u215b. Then |adj (B adj(2A))| is equal to\n(A) 16\n(B) 32\n(C) 64\n(D) 128\n", "ideal": "[\"the correct answer is (C)\", \"the correct answer is \\textbf{(C)}\", \"the correct answer is $\\boxed{\\textbf{(C)}\"]"}, {"prompt": "Let \\(S=2+\\frac{6}{7}+\\frac{12}{7^2}+\\frac{20}{7^3}+\\frac{30}{7^4}+\u2026\\) Then 4S is equal to\n(A) \\(\\left ( \\frac{7}{3} \\right )^2\\)\n(B) \\(\\left ( \\frac{7^3}{3^2} \\right )\\)\n(C) \\(\\left ( \\frac{7}{3} \\right )^3\\)\n(D) \\(\\left ( \\frac{7^2}{3^3} \\right )\\)\n", "ideal": "[\"the correct answer is (C)\", \"the correct answer is \\textbf{(C)}\", \"the correct answer is $\\boxed{\\textbf{(C)}\"]"}, {"prompt": "If a1, a2, a3 \u2026.. and b1, b2, b3 \u2026.. are A.P., and a1 = 2, a10 = 3, a1b1 = 1 = a10b10, then a4b4 is equal to\n(A) \\(\\frac{35}{27}\\)\n(B) 1\n(C) \\(\\frac{27}{28}\\)\n(D) \\(\\frac{28}{27}\\)\n", "ideal": "[\"the correct answer is (D)\", \"the correct answer is \\textbf{(D)}\", \"the correct answer is $\\boxed{\\textbf{(D)}\"]"}, {"prompt": "If m and n respectively are the number of local maximum and local minimum points of the function \\(f(x) = \\int_{0}^{x^2} \\frac{t^2 \u2013 5t + 4}{2 + e^t}dt,\\) then the ordered pair (m, n) is equal to\n(A) (3, 2)\n(B) (2, 3)\n(C) (2, 2)\n(D) (3, 4)\n", "ideal": "[\"the correct answer is (B)\", \"the correct answer is \\textbf{(B)}\", \"the correct answer is $\\boxed{\\textbf{(B)}\"]"}, {"prompt": "Let f be a differentiable function in (0, \u03c0/2). \\(\\text{If}\\ \\int_{cos x}^{1}t^2 f(t)dt = sin^3 x + cos x,\\ \\text{then}\\ \\frac{1}{\\sqrt{3}} f\u2019\\left ( \\frac{1}{\\sqrt{3}} \\right )\\) is equal to\n(A) \\(6-9\\sqrt{2}\\)\n(B) \\(6-\\frac{9}{\\sqrt{2}}\\)\n(C) \\(\\frac{9}{2}-6\\sqrt{2}\\)\n(D) \\(\\frac{9}{\\sqrt{2}}-6\\)\n", "ideal": "[\"the correct answer is (B)\", \"the correct answer is \\textbf{(B)}\", \"the correct answer is $\\boxed{\\textbf{(B)}\"]"}, {"prompt": "The integral \\(\\int_{0}^{1} \\frac{1}{7^{\\left [ \\frac{1}{x} \\right ]}}dx\\) where [\u22c5] denotes the greatest integer function, is equal to\n(A) \\(1+6log_e\\left (\\frac{6}{7} \\right )\\)\n(B) \\(1- 6log_e\\left (\\frac{6}{7} \\right )\\)\n(C) \\(log_e\\left (\\frac{7}{6} \\right )\\)\n(D) \\(1 -7log_e\\left (\\frac{6}{7} \\right )\\)\n", "ideal": "[\"the correct answer is (A)\", \"the correct answer is \\textbf{(A)}\", \"the correct answer is $\\boxed{\\textbf{(A)}\"]"}, {"prompt": "If the solution curve of the differential equation \\(\\left ( \\left ( tan^{-1}y \\right )-x \\right )dy = \\left ( 1+ y^2 \\right )dx\\) passes through the point (1, 0), then the abscissa of the point on the curve whose ordinate is tan(1), is\n(A) 2e\n(B) 2/e\n(C) 2\n(D) 1/e\n", "ideal": "[\"the correct answer is (B)\", \"the correct answer is \\textbf{(B)}\", \"the correct answer is $\\boxed{\\textbf{(B)}\"]"}, {"prompt": "If the equation of the parabola, whose vertex is at (5, 4) and the directrix is 3x + y \u2013 29 = 0, is x^2 + ay^2 + bxy + cx + dy + k = 0, then a + b + c + d + k is equal to\n(A) 575\n(B) \u2013575\n(C) 576\n(D) \u2013576\n", "ideal": "[\"the correct answer is (D)\", \"the correct answer is \\textbf{(D)}\", \"the correct answer is $\\boxed{\\textbf{(D)}\"]"}, {"prompt": "The set of values of k, for which the circle C : 4x^2 + 4y^2 \u2013 12x + 8y + k = 0 lies inside the fourth quadrant and the point (1, -1/3) lies on or inside the circle C, is\n(A) An empty set\n(B) \\(\\left(6, \\frac{65}{9}\\right]\\)\n(C) \\(\\left( \\frac{80}{9},10\\right]\\)\n(D) \\(\\left(9, \\frac{92}{9}\\right]\\)\n", "ideal": "[\"the correct answer is (D)\", \"the correct answer is \\textbf{(D)}\", \"the correct answer is $\\boxed{\\textbf{(D)}\"]"}, {"prompt": "Let the foot of the perpendicular from the point (1, 2, 4) on the line \\(\\frac{x+2}{4}=\\frac{y-1}{2}=\\frac{z+1}{3}\\) be P, Then the distance of P from the plane 3x + 4y + 12z + 23 = 0 is\n(A) 5\n(B) \\(\\frac{50}{13}\\)\n(C) 4\n(D) \\(\\frac{63}{13}\\)\n", "ideal": "[\"the correct answer is (A)\", \"the correct answer is \\textbf{(A)}\", \"the correct answer is $\\boxed{\\textbf{(A)}\"]"}, {"prompt": "The shortest distance between the lines \\(\\frac{x-3}{2}=\\frac{y-2}{3}=\\frac{z-1}{-1}\\) and \\(\\frac{x+3}{2}=\\frac{y-6}{1}=\\frac{z-5}{3},\\ \\text{is}\\)\n(A) \\(\\frac{18}{\\sqrt{5}}\\)\n(B) \\(\\frac{22}{3 \\sqrt{5}}\\)\n(C) \\(\\frac{46}{3 \\sqrt{5}}\\)\n(D) \\(6 \\sqrt{3}\\)\n", "ideal": "[\"the correct answer is (A)\", \"the correct answer is \\textbf{(A)}\", \"the correct answer is $\\boxed{\\textbf{(A)}\"]"}, {"prompt": "Let \\(\\vec{a}\\ \\text{and}\\ \\vec{b}\\) be the vectors along the diagonals of a parallelogram having area 2\u221a2. Let the angle between \\(\\vec{a}\\ \\text{and}\\ \\vec{b}\\ \\text{be acute,}\\ |\\vec{a}|=1,\\ \\text{and}\\ |\\vec{a} \\cdot \\vec{b}|=|\\vec{a} \\times \\vec{b}|\\)\u00a0\n\\(\\text{If}\\ \\vec{c}=2 \\sqrt{2}(\\vec{a} \\times \\vec{b})-2 \\vec{b}\\ \\text{then an angle between}\\ \\vec{b}\\ \\text{and}\\ \\vec{c}\\ \\text{is}\\)\u00a0\n(A) \\(\\frac{\\pi}{4}\\)\n(B) \\(-\\frac{\\pi}{4}\\)\n(C) \\(\\frac{5\\pi}{6}\\)\n(D) \\(\\frac{3\\pi}{4}\\)\n", "ideal": "[\"the correct answer is (D)\", \"the correct answer is \\textbf{(D)}\", \"the correct answer is $\\boxed{\\textbf{(D)}\"]"}, {"prompt": "The mean and variance of the data 4, 5, 6, 6, 7, 8, x, y, where x < y, are 6 and 9/4, respectively. Then x^4 + y^2 is equal to\n(A) 162\n(B) 320\n(C) 674\n(D) 420\n", "ideal": "[\"the correct answer is (B)\", \"the correct answer is \\textbf{(B)}\", \"the correct answer is $\\boxed{\\textbf{(B)}\"]"}, {"prompt": "If a point A(x, y) lies in the region bounded by the y-axis, straight lines 2y + x = 6 and 5x \u2013 6y = 30, then the probability that y < 1 is\n(A) \\(\\frac{1}{6}\\)\n(B) \\(\\frac{5}{6}\\)\n(C) \\(\\frac{2}{3}\\)\n(D) \\(\\frac{6}{7}\\)\n", "ideal": "[\"the correct answer is (B)\", \"the correct answer is \\textbf{(B)}\", \"the correct answer is $\\boxed{\\textbf{(B)}\"]"}, {"prompt": "The value of \\(\\cot \\left(\\sum_{n=1}^{50} \\tan ^{-1}\\left(\\frac{1}{1+n+n^{2}}\\right)\\right)\\) is\n(A) \\(\\frac{26}{25}\\)\n(B) \\(\\frac{25}{26}\\)\n(C) \\(\\frac{50}{51}\\)\n(D) \\(\\frac{52}{51}\\)\n", "ideal": "[\"the correct answer is (A)\", \"the correct answer is \\textbf{(A)}\", \"the correct answer is $\\boxed{\\textbf{(A)}\"]"}, {"prompt": "\u03b1 = sin 36\u00ba is a root of which of the following equation?\n(A) 16x^4 \u2013 10x^2 \u2013 5 = 0\n(B) 16x^4 + 20x^2 \u2013 5 = 0\n(C) 16x^4 \u2013 20x^2 + 5 = 0\n(D) 16x^4 \u2013 10x^2 + 5 = 0\n", "ideal": "[\"the correct answer is (C)\", \"the correct answer is \\textbf{(C)}\", \"the correct answer is $\\boxed{\\textbf{(C)}\"]"}, {"prompt": "Which of the following statement is a tautology?\n(A) ((~ q) \u2227 p) \u2227 q\n(B) ((~ q) \u2227 p) \u2227 (p \u2227 (~ p))\n(C) ((~ q) \u2227 p) \u2228 (p \u2228 (~p))\n(D) (p \u2227 q) \u2227 (~ (p \u2227 q))\n", "ideal": "[\"the correct answer is (C)\", \"the correct answer is \\textbf{(C)}\", \"the correct answer is $\\boxed{\\textbf{(C)}\"]"}, {"prompt": "If \\(\\displaystyle\\sum\\limits_{k=1}^{31}\\left ( ^{31}C_k \\right )\\left ( ^{31}C_{k-1} \\right )-\\displaystyle\\sum\\limits_{k=1}^{30}\\left ( ^{30}C_k \\right )\\left ( ^{30}C_{k-1} \\right )=\\frac{\\alpha\\left ( 60! \\right )}{\\left ( 30! \\right )\\left ( 31! \\right )}\\) where \u03b1 \u2208 R, then the value of 16\u03b1 is equal to\n(A) 1411\n(B) 1320\n(C) 1615\n(D) 1855\n", "ideal": "[\"the correct answer is (A)\", \"the correct answer is \\textbf{(A)}\", \"the correct answer is $\\boxed{\\textbf{(A)}\"]"}, {"prompt": "Let a function \u0192 : N \u2192N be defined by \\(f\\left ( n \\right )=\\left[\\begin{matrix}2n & n=2,~4,~6,~8,\\dots\\\\n-1, & n=3,~7,~11,~15,\\dots \\\\\\frac{n+1}{2}, & n=1,~5,~9,~13,\\dots \\\\\\end{matrix}\\right. \\) then, \u0192 is\n(A) One-one but not onto\n(B) Onto but not one-one\n(C) Neither one-one nor onto\n(D) One-one and onto\n", "ideal": "[\"the correct answer is (D)\", \"the correct answer is \\textbf{(D)}\", \"the correct answer is $\\boxed{\\textbf{(D)}\"]"}, {"prompt": "If the system of linear equations\n2x + 3y \u2013 z = \u20132\nx + y + z = 4\nx \u2013 y + |\u03bb|z = 4\u03bb \u2013 4\nwhere \u03bb\u2208 R, has no solution, then\n(A) \u03bb = 7\n(B) \u03bb = \u20137\n(C) \u03bb = 8\n(D) \u03bb^2 = 1\n", "ideal": "[\"the correct answer is (B)\", \"the correct answer is \\textbf{(B)}\", \"the correct answer is $\\boxed{\\textbf{(B)}\"]"}, {"prompt": "Let A be a matrix of order 3 \u00d7 3 and det (A) = 2. Then det (det (A) adj (5 adj (A^3))) is equal to ______.\n(A) 512 \u00d7 10^6\n(B) 256 \u00d7 10^6\n(C) 1024 \u00d7 10^6\n(D) 256 \u00d7 10^11\n", "ideal": "[\"the correct answer is (A)\", \"the correct answer is \\textbf{(A)}\", \"the correct answer is $\\boxed{\\textbf{(A)}\"]"}, {"prompt": "The total number of 5-digit numbers, formed by using the digits 1, 2, 3, 5, 6, 7 without repetition, which are multiple of 6, is\n(A) 36\n(B) 48\n(C) 60\n(D) 72\n", "ideal": "[\"the correct answer is (D)\", \"the correct answer is \\textbf{(D)}\", \"the correct answer is $\\boxed{\\textbf{(D)}\"]"}, {"prompt": "Let A1, A2, A3, \u2026 be an increasing geometric progression of positive real numbers. If A1A3A5A7 = 1/1296 and A2 + A4 = 7/36 then, the value of A6 + A8 + A10 is equal to\n(A) 33\n(B) 37\n(C) 43\n(D) 47\n", "ideal": "[\"the correct answer is (C)\", \"the correct answer is \\textbf{(C)}\", \"the correct answer is $\\boxed{\\textbf{(C)}\"]"}, {"prompt": "Let [t] denote the greatest integer less than or equal to t. Then, the value of the integral \\(\\displaystyle\\int\\limits_0^1\\left [-8x^2+6x-1 \\right ]dx \\) is equal to\n(A) \u20131\n(B) \\(\\frac{-5}{4}\\)\n(C) \\(\\frac{\\sqrt{17}-13}{8} \\)\n(D) \\(\\frac{\\sqrt{17}-16}{8} \\)\n", "ideal": "[\"the correct answer is (C)\", \"the correct answer is \\textbf{(C)}\", \"the correct answer is $\\boxed{\\textbf{(C)}\"]"}, {"prompt": "Let f: \u211d \u2192 \u211d be defined as \\(f\\left( x \\right )=\\left[\\begin{matrix}\\left [ e^x \\right ], & x<0 \\\\ae^x+\\left [ x-1 \\right ], & 0\\leq x<1 \\\\b+\\left [ \\sin\\left ( \\pi x \\right ) \\right ],&1\\leq x<2 \\\\\\left [ e^{-x} \\right ]-c, & x\\geq 2 \\\\\\end{matrix}\\right.\\)\nWhere a, b, c \u2208\u00a0 \u211d\u00a0and [t] denotes greatest integer less than or equal to t. Then, which of the following statements is true?\n(A) There exists a, b, c \u2208\u00a0 \u211d\u00a0 such that \u0192iscontinuous on \u2208\u00a0 \u211d\u00a0.\n(B) If \u0192 is discontinuous at exactly one point, then a + b + c = 1\n(C) If \u0192 is discontinuous at exactly one point, then a + b + c \u2260 1\n(D) \u0192 is discontinuous at atleast two points, for any values of a, b and c\n", "ideal": "[\"the correct answer is (C)\", \"the correct answer is \\textbf{(C)}\", \"the correct answer is $\\boxed{\\textbf{(C)}\"]"}, {"prompt": "The area of the region\\(\\left\\{\\left ( x,y \\right ):y^2\\leq 8x,y\\geq \\sqrt{2}x,x\\geq 1 \\right\\}\\) is\n(A) \\(\\frac{13\\sqrt{2}}{6}\\)\n(B) \\(\\frac{11\\sqrt{2}}{6}\\)\n(C) \\(\\frac{5\\sqrt{2}}{6}\\)\n(D) \\(\\frac{19\\sqrt{2}}{6}\\)\n", "ideal": "[\"the correct answer is (B)\", \"the correct answer is \\textbf{(B)}\", \"the correct answer is $\\boxed{\\textbf{(B)}\"]"}, {"prompt": "Let the solution curve y = y(x) of the differential equation\n\\(\\left [\\frac{x}{\\sqrt{x^2-y^2}}+e^\\frac{y}{x} \\right ]x\\frac{dy}{dx}=x+\\left [\\frac{x}{\\sqrt{x^2-y^2}}+e^\\frac{y}{x} \\right ]y\\) pass through the points (1, 0) and (2\u03b1, \u03b1), \u03b1> 0. Then \u03b1 is equal to\n(A) \\(\\frac{1}{2}\\textup{exp}\\left ( \\frac{\\pi}{6}+\\sqrt{e}-1 \\right )\\)\n(B) \\(\\frac{1}{2}\\textup{exp}\\left ( \\frac{\\pi}{3}+e-1 \\right )\\)\n(C) \\(\\textup{exp}\\left ( \\frac{\\pi}{6}+\\sqrt{e}+1 \\right )\\)\n(D) \\(2~\\textup{exp}\\left ( \\frac{\\pi}{3}+\\sqrt{e}-1 \\right )\\)\n", "ideal": "[\"the correct answer is (A)\", \"the correct answer is \\textbf{(A)}\", \"the correct answer is $\\boxed{\\textbf{(A)}\"]"}, {"prompt": "Let y = y(x) be the solution of the differential equation \\(x\\left ( 1-x^2 \\right )\\frac{dy}{dx}+\\left ( 3x^2y-y-4x^3 \\right )=0,~x>1\\) with y(2) = \u20132. Then y(3) is equal to\n(A) \u201318\n(B) \u201312\n(C) \u20136\n(D) \u20133\n", "ideal": "[\"the correct answer is (A)\", \"the correct answer is \\textbf{(A)}\", \"the correct answer is $\\boxed{\\textbf{(A)}\"]"}, {"prompt": "The number of real solutions of x^7 + 5x^3 + 3x + 1 = 0 is equal to ______.\n(A) 0\n(B) 1\n(C) 3\n(D) 5\n", "ideal": "[\"the correct answer is (B)\", \"the correct answer is \\textbf{(B)}\", \"the correct answer is $\\boxed{\\textbf{(B)}\"]"}, {"prompt": "Let the eccentricity of the hyperbola \\(H:\\frac{x^2}{a^2}-\\frac{y^2}{b^2}=1\\) be \u221a(5/2)\u00a0and length of its latus rectum be 6\u221a2, If y = 2x + c is a tangent to the hyperbola H. then the value of c^2 is equal to\n(A) 18\n(B) 20\n(C) 24\n(D) 32\n", "ideal": "[\"the correct answer is (B)\", \"the correct answer is \\textbf{(B)}\", \"the correct answer is $\\boxed{\\textbf{(B)}\"]"}, {"prompt": "If the tangents drawn at the points O(0, 0) and P(1 + \u221a5, 2) on the circle x^2 + y^2 \u2013 2x \u2013 4y = 0 intersect at the point Q, then the area of the triangle OPQ is equal to\n(A) \\(\\frac{3+\\sqrt{5}}{2}\\)\n(B) \\(\\frac{4+2\\sqrt{5}}{2}\\)\n(C) \\(\\frac{5+3\\sqrt{5}}{2}\\)\n(D) \\(\\frac{7+3\\sqrt{5}}{2}\\)\n", "ideal": "[\"the correct answer is (C)\", \"the correct answer is \\textbf{(C)}\", \"the correct answer is $\\boxed{\\textbf{(C)}\"]"}, {"prompt": "If two distinct points Q, R lie on the line of intersection of the planes \u2013x + 2y \u2013 z = 0 and 3x \u2013 5y + 2z = 0 and \\(PQ=PR=\\sqrt{18}\\) where the point P is (1, \u20132, 3), then the area of the triangle PQR is equal to\n(A) \\(\\frac{2}{3}\\sqrt{38}\\)\n(B) \\(\\frac{4}{3}\\sqrt{38}\\)\n(C) \\(\\frac{8}{3}\\sqrt{38} \\)\n(D) \\(\\sqrt{\\frac{152}{3}} \\)\n", "ideal": "[\"the correct answer is (B)\", \"the correct answer is \\textbf{(B)}\", \"the correct answer is $\\boxed{\\textbf{(B)}\"]"}, {"prompt": "The acute angle between the planes P1 and P2, when P1 and P2 are the planes passing through the intersection of the planes 5x + 8y + 13z \u2013 29 = 0 and 8x \u2013 7y + z \u2013 20 = 0 and the points (2, 1, 3) and (0, 1, 2), respectively, is\n(A) \\(\\frac{\\pi}{3} \\)\n(B) \\(\\frac{\\pi}{4} \\)\n(C) \\(\\frac{\\pi}{6} \\)\n(D) \\(\\frac{\\pi}{12} \\)\n", "ideal": "[\"the correct answer is (A)\", \"the correct answer is \\textbf{(A)}\", \"the correct answer is $\\boxed{\\textbf{(A)}\"]"}, {"prompt": "Let the plane \\(P:\\vec{r}\\cdot\\vec{a}=d \\) contain the line of intersection of two planes \\(\\vec{r}.\\left ( \\hat{i}+3\\hat{j}-\\hat{k} \\right )=6\\) and \\(\\vec{r}\\cdot\\left ( -6\\hat{i}+5\\hat{j}-\\hat{k} \\right )=7\\). If the plane P passes through the point (2, 3, 1/2),\u00a0 \\(\\text{then the value of}\\ \\frac{\\left|13\\vec{a} \\right|^2}{d^2}\\ \\text{is equal to}\\)\n(A) 90\n(B) 93\n(C) 95\n(D) 97\n", "ideal": "[\"the correct answer is (B)\", \"the correct answer is \\textbf{(B)}\", \"the correct answer is $\\boxed{\\textbf{(B)}\"]"}, {"prompt": "The probability, that in a randomly selected 3-digit number at least two digits are odd, is\n(A) \\(\\frac{19}{36} \\)\n(B) \\(\\frac{15}{36} \\)\n(C) \\(\\frac{13}{36} \\)\n(D) \\(\\frac{23}{36} \\)\n", "ideal": "[\"the correct answer is (A)\", \"the correct answer is \\textbf{(A)}\", \"the correct answer is $\\boxed{\\textbf{(A)}\"]"}, {"prompt": "Let AB and PQ be two vertical poles, 160 m apart from each other. Let C be the middle point of B and Q, which are feet of these two poles. Let \u03c0/8 and \u03b8 be the angles of elevation from C to P and A, respectively. If the height of pole PQ is twice the height of pole AB, then tan^2\u03b8 is equal to\n(A) \\(\\frac{3-2\\sqrt{2}}{2}\\)\n(B) \\(\\frac{3+\\sqrt{2}}{2}\\)\n(C) \\(\\frac{3-2\\sqrt{2}}{4}\\)\n(D) \\(\\frac{3-\\sqrt{2}}{4}\\)\n", "ideal": "[\"the correct answer is (C)\", \"the correct answer is \\textbf{(C)}\", \"the correct answer is $\\boxed{\\textbf{(C)}\"]"}, {"prompt": "Let p, q, r be three logical statements. Consider the compound statements\nS1 : ((~p) \u2228q) \u2228 ((~p) \u2228r) and\nS2 :p\u2192 (q\u2228r)\nThen, which of the following is NOT true?\n(A) If S2 is True, then S1 is True\n(B) If S2is False, then S1 is False\n(C) If S2 is False, then S1 is True\n(D) If S1 is False, then S2 is False\n", "ideal": "[\"the correct answer is (C)\", \"the correct answer is \\textbf{(C)}\", \"the correct answer is $\\boxed{\\textbf{(C)}\"]"}, {"prompt": "Let R1 = {(a, b) \u2208 N \u00d7 N : |a \u2013 b| \u2264 13} and R2 = {(a, b) \u2208 N \u00d7 N : |a \u2013 b| \u2260 13}. Then on N:\n(A) Both R1 and R2 are equivalence relations\n(B) Neither R1 nor R2 is an equivalence relation\n(C) R1 is an equivalence relation but R2 is not\n(D) R2 is an equivalence relation but R1 is not\n", "ideal": "[\"the correct answer is (B)\", \"the correct answer is \\textbf{(B)}\", \"the correct answer is $\\boxed{\\textbf{(B)}\"]"}, {"prompt": "Let f(x) be a quadratic polynomial such that f(\u20132) + f(3) = 0. If one of the roots of f(x) = 0 is \u20131, then the sum of the roots of f(x) = 0 is equal to:\n(A) \\(\\frac{11}{3}\\)\n(B) \\(\\frac{7}{3}\\)\n(C) \\(\\frac{13}{3}\\)\n(D) \\(\\frac{14}{3}\\)\n", "ideal": "[\"the correct answer is (A)\", \"the correct answer is \\textbf{(A)}\", \"the correct answer is $\\boxed{\\textbf{(A)}\"]"}, {"prompt": "The number of ways to distribute 30 identical candies among four children C1, C2, C3 and C4 so that C2 receives atleast 4 and atmost 7 candies, C3 receives atleast 2 and atmost 6 candies, is equal to:\n(A) 205\n(B) 615\n(C) 510\n(D) 430\n", "ideal": "[\"the correct answer is (D)\", \"the correct answer is \\textbf{(D)}\", \"the correct answer is $\\boxed{\\textbf{(D)}\"]"}, {"prompt": "The term independent of x in the expansion of \\(\\left(1-x^{2}+3 x^{3}\\right)\\left(\\frac{5}{2} x^{3}-\\frac{1}{5 x^{2}}\\right)^{11}, x \\neq 0\\)is:\n(A) \\(\\frac{7}{40}\\)\n(B) \\(\\frac{33}{200}\\)\n(C) \\(\\frac{39}{200}\\)\n(D) \\(\\frac{11}{50}\\)\n", "ideal": "[\"the correct answer is (B)\", \"the correct answer is \\textbf{(B)}\", \"the correct answer is $\\boxed{\\textbf{(B)}\"]"}, {"prompt": "If n arithmetic means are inserted between a and 100 such that the ratio of the first mean to the last mean is 1 : 7 and a + n = 33, then the value of n is:\n(A) 21\n(B) 22\n(C) 23\n(D) 24\n", "ideal": "[\"the correct answer is (C)\", \"the correct answer is \\textbf{(C)}\", \"the correct answer is $\\boxed{\\textbf{(C)}\"]"}, {"prompt": "Let f,g : R \u2192 R be functions defined by\n\\(f(x)=\\left\\{\\begin{array}{ll}{[x],} & x<0 \\\\ |1-x|, & x \\geq 0\\end{array}\\right.$ and $g(x)= \\begin{cases}e^{x}-x, & x<0 \\\\ (x-1)^{2}-1, & x \\geq 0\\end{cases}\\) \nWhere [x] denotes the greatest integer less than or equal to x. Then, the function fog is discontinuous at exactly :\n(A) one point\n(B) two points\n(C) three points\n(D) four points\n", "ideal": "[\"the correct answer is (B)\", \"the correct answer is \\textbf{(B)}\", \"the correct answer is $\\boxed{\\textbf{(B)}\"]"}, {"prompt": "Let f : R \u2192 R be a differentiable function such that \\(f\\left(\\frac{\\pi}{4}\\right)=\\sqrt{2}, f\\left(\\frac{\\pi}{2}\\right)=0 \\textup{and} f^{\\prime}\\left(\\frac{\\pi}{2}\\right)=1\\) and let \\(g(x)=\\int_{x}^{\\frac{\\pi}{4}}\\left(f^{\\prime}(t) \\sec t+\\tan t \\operatorname{sec~t} f(t)\\right) d t\\) \\(\\text{for}\\ x \\in\\left[\\frac{\\pi}{4}, \\frac{\\pi}{2}\\right)\\ \\text{Then}\\ \\lim _{x \\rightarrow\\left(\\frac{\\pi}{2}\\right)^{-}} g(x)\\ \\text{is equal to}\\)\u00a0\n(A) 2\n(B) 3\n(C) 4\n(D) \u20133\n", "ideal": "[\"the correct answer is (B)\", \"the correct answer is \\textbf{(B)}\", \"the correct answer is $\\boxed{\\textbf{(B)}\"]"}, {"prompt": "Let f : R \u2192 R be a continuous function satisfying f(x) + f(x + k) = n, for all x \u2208 R where k > 0 and n is a positive integer. If \\(l_{1}=\\int_{0}^{4 n k} f(x) d x \\quad and \\quad I_{2}=\\int_{-k}^{3 k} f(x) d x\\), then \n(A) \\(I_{1}+2 I_{2}=4 n k\\)\n(B) \\(I_{1}+2 I_{2}=2 n k\\)\n(C) \\(I_{1}+n I_{2}=4 n^{2} k\\)\n(D) \\(l_{1}+n l_{2}=6 n^{2} k\\)\n", "ideal": "[\"the correct answer is (C)\", \"the correct answer is \\textbf{(C)}\", \"the correct answer is $\\boxed{\\textbf{(C)}\"]"}, {"prompt": "The area of the bounded region enclosed by the curve \\(y=3-\\left|x-\\frac{1}{2}\\right|-|x+1|\\) and the x-axis is\n(A) \\(\\frac{9}{4}\\)\n(B) \\(\\frac{45}{16}\\)\n(C) \\(\\frac{27}{8}\\)\n(D) \\(\\frac{63}{16}\\)\n", "ideal": "[\"the correct answer is (C)\", \"the correct answer is \\textbf{(C)}\", \"the correct answer is $\\boxed{\\textbf{(C)}\"]"}, {"prompt": "Let x = x(y) be the solution of the differential equation \\(2 y e^{\\frac{x}{y^{2}}} d x+\\left(y^{2}-4 x e^{\\frac{x}{y^{2}}}\\right) d y=0\\) such that x(1) = 0. Then, x(e) is equal to\n(A) e loge(2)\n(B) -e loge(2)\n(C) e^2 loge(2)\n(D) -e^2 loge(2)\n", "ideal": "[\"the correct answer is (D)\", \"the correct answer is \\textbf{(D)}\", \"the correct answer is $\\boxed{\\textbf{(D)}\"]"}, {"prompt": "Let the slope of the tangent to a curve y = f(x) at (x, y) be given by 2 tanx(cosx \u2013 y). If the curve passes through the point (\u03c0/4, 0) then the value of \\(\\int_{0}^{\\pi / 2} y d x\\) is equal to :\n(A) \\((2-\\sqrt{2})+\\frac{\\pi}{\\sqrt{2}}\\)\n(B) \\(2-\\frac{\\pi}{\\sqrt{2}}\\)\n(C) \\((2+\\sqrt{2})+\\frac{\\pi}{\\sqrt{2}}\\)\n(D) \\(2+\\frac{\\pi}{\\sqrt{2}}\\)\n", "ideal": "[\"the correct answer is (B)\", \"the correct answer is \\textbf{(B)}\", \"the correct answer is $\\boxed{\\textbf{(B)}\"]"}, {"prompt": "Let a triangle be bounded by the lines L1 : 2x + 5y = 10; L2 : \u20134x + 3y = 12 and the line L3, which passes through the point P(2, 3), intersects L2 at A and L1 at B. If the point P divides the line-segment AB, internally in the ratio 1 : 3, then the area of the triangle is equal to\n(A) \\(\\frac{110}{13}\\)\n(B) \\(\\frac{132}{13}\\)\n(C) \\(\\frac{142}{13}\\)\n(D) \\(\\frac{151}{13}\\)\n", "ideal": "[\"the correct answer is (B)\", \"the correct answer is \\textbf{(B)}\", \"the correct answer is $\\boxed{\\textbf{(B)}\"]"}, {"prompt": "Let a > 0, b > 0. Let e and l respectively be the eccentricity and length of the latus rectum of the hyperbola \\(\\frac{x^{2}}{a^{2}}-\\frac{y^{2}}{b^{2}}=1\\) Let e\u2032 and l\u2032 respectively be the eccentricity and length of the latus rectum of its conjugate hyperbola. If \\(\\mathrm{e}^{2}=\\frac{11}{14} l\\ \\text{and}\\ \\left(\\mathrm{e}^{\\prime}\\right)^{2}=\\frac{11}{8} l^{\\prime}\\) then the value of 77a + 44b is equal to :\n(A) 100\n(B) 110\n(C) 120\n(D) 130\n", "ideal": "[\"the correct answer is (D)\", \"the correct answer is \\textbf{(D)}\", \"the correct answer is $\\boxed{\\textbf{(D)}\"]"}, {"prompt": "Let, \\(\\vec{a}=\\alpha \\hat{i}+2 \\hat{j}-\\hat{k}$ and $\\vec{b}=-2 \\hat{i}+\\alpha \\hat{j}+\\hat{k}\\), where \u03b1 \u2208 R. If the area of the parallelogram whose adjacent sides are represented by the vectors \\(\\vec{a}\\ \\text{and}\\ \\vec{b}\\ is\\ \\sqrt{15\\left(\\alpha^{2}+4\\right)}\\ \\text{, then the value of}\\ 2|\\vec{a}|^{2}+(\\vec{a} \\cdot \\vec{b})|\\vec{b}|^{2}\\) is equal to :\n(A) 10\n(B) 7\n(C) 9\n(D) 14\n", "ideal": "[\"the correct answer is (D)\", \"the correct answer is \\textbf{(D)}\", \"the correct answer is $\\boxed{\\textbf{(D)}\"]"}, {"prompt": "If vertex of a parabola is (2, \u20131) and the equation of its directrix is 4x \u2013 3y = 21, then the length of its latus rectum is :\n(A) 2\n(B) 8\n(C) 12\n(D) 16\n", "ideal": "[\"the correct answer is (B)\", \"the correct answer is \\textbf{(B)}\", \"the correct answer is $\\boxed{\\textbf{(B)}\"]"}, {"prompt": "Let the plane ax + by + cz = d pass through (2, 3, \u20135) and is perpendicular to the planes 2x + y \u2013 5z = 10 and 3x + 5y \u2013 7z = 12. If a, b, c, d are integers d > 0 and gcd (|a|, |b|, |c|, d) = 1, then the value of a + 7b + c + 20d is equal to :\n(A) 18\n(B) 20\n(C) 24\n(D) 22\n", "ideal": "[\"the correct answer is (D)\", \"the correct answer is \\textbf{(D)}\", \"the correct answer is $\\boxed{\\textbf{(D)}\"]"}, {"prompt": "The probability that a randomly chosen one-one function from the set {a, b, c, d} to the set {1, 2, 3, 4, 5} satisfies f(a) + 2f(b) \u2013 f(c) = f(d) is :\n(A) \\(\\frac{1}{24}\\)\n(B) \\(\\frac{1}{40}\\)\n(C) \\(\\frac{1}{30}\\)\n(D) \\(\\frac{1}{20}\\)\n", "ideal": "[\"the correct answer is (D)\", \"the correct answer is \\textbf{(D)}\", \"the correct answer is $\\boxed{\\textbf{(D)}\"]"}, {"prompt": "The value of \\(\\lim _{n \\rightarrow \\infty} 6 \\tan \\left\\{\\sum_{r=1}^{n} \\tan ^{-1}\\left(\\frac{1}{r^{2}+3 r+3}\\right)\\right\\}\\) is equal to :\n(A) 1\n(B) 2\n(C) 3\n(D) 6\n", "ideal": "[\"the correct answer is (C)\", \"the correct answer is \\textbf{(C)}\", \"the correct answer is $\\boxed{\\textbf{(C)}\"]"}, {"prompt": "Let \\(\\vec{a}\\ \\text{be a vector which is perpendicular to the vector}\\)\u00a0 \\(3 \\hat{i}+\\frac{1}{2} \\hat{j}+2 \\hat{k}.\\ \\text{If }\\ \\vec{a} \\times(2 \\hat{i}+\\hat{k})=2 \\hat{i}-13 \\hat{j}-4 \\hat{k}\\), then the projection of the vector on the vector \\(2 \\hat{i}+2 \\hat{j}+\\hat{k}\\ \\text{is}:\\)\u00a0\n(A) 1/3\n(B) 1\n(C) 5/3\n(D) 7/3\n", "ideal": "[\"the correct answer is (C)\", \"the correct answer is \\textbf{(C)}\", \"the correct answer is $\\boxed{\\textbf{(C)}\"]"}, {"prompt": "If \\(cot~\\alpha = 1\\ \\text{and}\\ sec ~\\beta = -\\frac{5}{3}\\ \\text{where}\\ \\pi<\\alpha<\\frac{3\\pi}{2}\\ \\text{and}\\ \\frac{\\pi}{2}<\\beta<\\pi,\\) then the value of tan(\u03b1 + \u03b2) and the quadrant in which \u03b1 + \u03b2 lies, respectively are :\n(A) -1/7 and IV^th quadrant\n(B) 7 and I^st quadrant\n(C) \u2013 7 and IV^th quadrant\n(D) 1/7 and I^st quadrant\n", "ideal": "[\"the correct answer is (A)\", \"the correct answer is \\textbf{(A)}\", \"the correct answer is $\\boxed{\\textbf{(A)}\"]"}, {"prompt": "The probability that a randomly chosen 2 \u00d7 2 matrix with all the entries from the set of first 10 primes, is singular, is equal to :\n(A) \\(\\frac{133}{10^4} \\)\n(B) \\(\\frac{18}{10^3} \\)\n(C) \\(\\frac{19}{10^3} \\)\n(D) \\(\\frac{271}{10^4} \\)\n", "ideal": "[\"the correct answer is (C)\", \"the correct answer is \\textbf{(C)}\", \"the correct answer is $\\boxed{\\textbf{(C)}\"]"}, {"prompt": "Let the solution curve of the differential equation \\(x\\frac{dy}{dx}-y=\\sqrt{y^2+16x^2},\\) y(1) = 3 be y = y(x). Then y(2) is equal to :\n(A) 15\n(B) 11\n(C) 13\n(D) 17\n", "ideal": "[\"the correct answer is (A)\", \"the correct answer is \\textbf{(A)}\", \"the correct answer is $\\boxed{\\textbf{(A)}\"]"}, {"prompt": "If the mirror image of the point (2, 4, 7) in the plane 3x \u2013 y + 4z = 2 is (a, b, c), then 2a + b + 2c is equal to:\n(A) 54\n(B) 50\n(C) \u20136\n(D) \u201342\n", "ideal": "[\"the correct answer is (C)\", \"the correct answer is \\textbf{(C)}\", \"the correct answer is $\\boxed{\\textbf{(C)}\"]"}, {"prompt": "Let \u0192 : R \u21d2 R be a function defined by :\n\\(f\\left ( x \\right )=\\left\\{\\begin{matrix}\\underset{t\\leq x}{\\max}\\left\\{t^3-3t \\right\\} & :&x\\leq 2 \\\\x^2+2x-6 &:&2<x<3 \\\\\\left [x-3 \\right ]+9 & :&3\\leq x\\leq 5 \\\\2x+1&:&x>5\\end{matrix}\\right. \\)\nwhere [t] is the greatest integer less than or equal to t. Let m be the number of points where \u0192 is not differentiable and \\(l=\\displaystyle\\int\\limits_{-2}^2f\\left ( x \\right )dx\\) Then the ordered pair (m, I) is equal to :\n(A) \\(\\left ( 3,\\frac{27}{4} \\right ) \\)\n(B) \\(\\left ( 3,\\frac{23}{4} \\right ) \\)\n(C) \\(\\left ( 4,\\frac{27}{4} \\right ) \\)\n(D) \\(\\left ( 4,\\frac{23}{4} \\right ) \\)\n", "ideal": "[\"the correct answer is (C)\", \"the correct answer is \\textbf{(C)}\", \"the correct answer is $\\boxed{\\textbf{(C)}\"]"}, {"prompt": "Let \\(\\vec{a}=\\alpha\\hat{i}+3\\hat{j}-\\hat{k},\\vec{b}=3\\hat{i}-\\beta\\hat{j}+4\\hat{k}\\ \\text{and}\\ \\vec{c}=\\hat{i}+2\\hat{j}-2\\hat{k}\\) where \u03b1, \u03b2 \u2208 R, be three vectors. If the projection of \\(\\vec{a}\\ \\text{on}\\ \\vec{c}\\ is\\ \\frac{10}{3}\\ \\text{and}\\ \\vec{b}\\times\\vec{c}=-6\\hat{i}+10\\hat{j}+7\\hat{k}\\) then the value of \u03b1 + \u03b2 is equal to :\n(A) 3\n(B) 4\n(C) 5\n(D) 6\n", "ideal": "[\"the correct answer is (A)\", \"the correct answer is \\textbf{(A)}\", \"the correct answer is $\\boxed{\\textbf{(A)}\"]"}, {"prompt": "The area enclosed by y^2 = 8x and y = \u221a2x that lies outside the triangle formed by \\(y=\\sqrt{2}x,~x=1,~y=2\\sqrt{2}\\) is equal to :\n(A) \\(\\frac{16\\sqrt{2}}{6}\\)\n(B) \\(\\frac{11\\sqrt{2}}{6}\\)\n(C) \\(\\frac{13\\sqrt{2}}{6}\\)\n(D) \\(\\frac{5\\sqrt{2}}{6}\\)\n", "ideal": "[\"the correct answer is (C)\", \"the correct answer is \\textbf{(C)}\", \"the correct answer is $\\boxed{\\textbf{(C)}\"]"}, {"prompt": "If the system of linear equations\n2x + y \u2013 z = 7\nx \u2013 3y + 2z = 1\nx + 4y + \u03b4z = k, where \u03b4, k \u2208 R\nhas infinitely many solutions, then \u03b4 + k is equal to:\n(A) \u20133\n(B) 3\n(C) 6\n(D) 9\n", "ideal": "[\"the correct answer is (B)\", \"the correct answer is \\textbf{(B)}\", \"the correct answer is $\\boxed{\\textbf{(B)}\"]"}, {"prompt": "Let \u03b1 and \u03b2 be the roots of the equation x^2 + (2i \u2013 1) = 0. Then, the value of |\u03b1^2 + \u03b2^2| is equal to:\n(A) 50\n(B) 250\n(C) 1250\n(D) 1500\n", "ideal": "[\"the correct answer is (A)\", \"the correct answer is \\textbf{(A)}\", \"the correct answer is $\\boxed{\\textbf{(A)}\"]"}, {"prompt": "Let \\(\\Delta\\in \\left\\{\\wedge,\\vee,\\Rightarrow,\\Leftrightarrow \\right\\}\\ \\text{be such that}\\left( p\\wedge q \\right)\\Delta \\left( \\left( p\\vee q \\right)\\Rightarrow q \\right)\\) is a tautology. Then \u0394 is equal to :\n(A) \\(\\wedge \\)\n(B) \\(\\vee\\)\n(C) \\(\\Rightarrow\\)\n(D) \\(\\Leftrightarrow\\)\n", "ideal": "[\"the correct answer is (C)\", \"the correct answer is \\textbf{(C)}\", \"the correct answer is $\\boxed{\\textbf{(C)}\"]"}, {"prompt": "Let A = [aij] be a square matrix of order 3 such that aij = 2^j^\u2013^i, for all i, j = 1, 2, 3. Then, the matrix A^2 + A^3 + \u2026 + A^10 is equal to :\n(A) \\(\\left ( \\frac{3^{10}-3}{2} \\right )A\\)\n(B) \\(\\left ( \\frac{3^{10}-1}{2} \\right )A\\)\n(C) \\(\\left ( \\frac{3^{10}+1}{2} \\right )A\\)\n(D) \\(\\left ( \\frac{3^{10}+3}{2} \\right )A\\)\n", "ideal": "[\"the correct answer is (A)\", \"the correct answer is \\textbf{(A)}\", \"the correct answer is $\\boxed{\\textbf{(A)}\"]"}, {"prompt": "Let a set A = A1 \u22c3 A2 \u22c3 \u2026\u22c3 Ak, where Ai \u22c2 Aj = \u03a6 for i \u2260 j, 1 \u2264 i, j \u2264 k. Define the relation R from A to A by R = {(x, y) : y \u2208 Ai if and only if x \u2208 Ai, 1 \u2264 i \u2264 k}. Then, R is :\n(A) reflexive, symmetric but not transitive\n(B) reflexive, transitive but not symmetric\n(C) reflexive but not symmetric and transitive\n(D) an equivalence relation\n", "ideal": "[\"the correct answer is (D)\", \"the correct answer is \\textbf{(D)}\", \"the correct answer is $\\boxed{\\textbf{(D)}\"]"}, {"prompt": "Let \\(\\left\\{a_n \\right\\}_{n=0}^\\infty\\) be a sequence such that a0 = a1 = 0 and an + 2 = 2an + 1 \u2013 an + 1 for all n \u2265 0. \\(\\text{Then}\\ \\displaystyle\\sum\\limits_{n=2}^\\infty\\frac{a_n}{7^n}\\ \\text{is equal to}:\\)\u00a0\n(A) \\(\\frac{6}{343}\\)\n(B) \\(\\frac{7}{216}\\)\n(C) \\(\\frac{8}{343}\\)\n(D) \\(\\frac{49}{216}\\)\n", "ideal": "[\"the correct answer is (B)\", \"the correct answer is \\textbf{(B)}\", \"the correct answer is $\\boxed{\\textbf{(B)}\"]"}, {"prompt": "The distance between the two points A and A\u2032 which lie on y = 2 such that both the line segments AB and A\u2032B (where B is the point (2, 3)) subtend angle \u03c0/4 at the origin, is equal to\n(A) 10\n(B) 48/5\n(C) 52/5\n(D) 3\n", "ideal": "[\"the correct answer is (C)\", \"the correct answer is \\textbf{(C)}\", \"the correct answer is $\\boxed{\\textbf{(C)}\"]"}, {"prompt": "A wire of length 22 m is to be cut into two pieces. One of the pieces is to be made into a square and the other into an equilateral triangle. Then, the length of the side of the equilateral triangle, so that the combined area of the square and the equilateral triangle is minimum, is\n(A) \\(\\frac{22}{9+4\\sqrt{3}}\\)\n(B) \\(\\frac{66}{9+4\\sqrt{3}}\\)\n(C) \\(\\frac{22}{4+9\\sqrt{3}}\\)\n(D) \\(\\frac{66}{4+9\\sqrt{3}}\\)\n", "ideal": "[\"the correct answer is (B)\", \"the correct answer is \\textbf{(B)}\", \"the correct answer is $\\boxed{\\textbf{(B)}\"]"}, {"prompt": "The domain of the function \\(\\cos^{-1}\\left ( \\frac{2\\sin^{-1}\\left ( \\frac{1}{4x^2-1} \\right )}{\\pi} \\right )\\)is :\n(A) \\(\\textbf{R}-\\left\\{-\\frac{1}{2},\\frac{1}{2} \\right\\}\\)\n(B) \\(\\left (-\\infty,-1 \\left . \\right ]\\cup\\left [1,\\infty \\right . \\right )\\cup\\left\\{ 0\\right\\}\\)\n(C) \\(\\left ( -\\infty,\\frac{-1}{2} \\right )\\cup\\left ( \\frac{1}{2},\\infty \\right ) \\cup\\left\\{0 \\right\\}\\)\n(D) \\(\\left (-\\infty,\\frac{-1}{\\sqrt{2}} \\right ]\\cup\\left [ \\frac{1}{\\sqrt{2}},\\infty \\right )\\cup\\left\\{0 \\right\\}\\)\n", "ideal": "[\"the correct answer is (D)\", \"the correct answer is \\textbf{(D)}\", \"the correct answer is $\\boxed{\\textbf{(D)}\"]"}, {"prompt": "If the constant term in the expansion of \\(\\left ( 3x^3-2x^2+\\frac{5}{x^5} \\right ) ^{10}\\) is 2^k\u00b7l, where l is an odd integer, then the value of k is equal to\n(A) 6\n(B) 7\n(C) 8\n(D) 9\n", "ideal": "[\"the correct answer is (D)\", \"the correct answer is \\textbf{(D)}\", \"the correct answer is $\\boxed{\\textbf{(D)}\"]"}, {"prompt": "\\(\\displaystyle\\int\\limits_0^5\\cos\\left ( \\pi\\left ( x-\\left [ \\frac{x}{2} \\right ] \\right ) \\right )dx\\), where [t] denotes greatest integer less than or equal to t, is equal to\n(A) \u20133\n(B) \u20132\n(C) 2\n(D) 0\n", "ideal": "[\"the correct answer is (D)\", \"the correct answer is \\textbf{(D)}\", \"the correct answer is $\\boxed{\\textbf{(D)}\"]"}, {"prompt": "Let PQ be a focal chord of the parabola y^2 = 4x such that it subtends an angle of \u03c0/2 at the point (3, 0). Let the line segment PQ be also a focal chord of the ellipse \\(E:\\frac{x^2}{a^2}+\\frac{y^2}{b^2}=1,a^2>b^2\\). If e is the eccentricity of the ellipse E, then the value of 1/e^2 is equal to\n(A) 1 + \u221a2\n(B) 3 + 2\u221a2\n(C) 1 + 2\u221a3\n(D) 4 + 5\u221a3\n", "ideal": "[\"the correct answer is (B)\", \"the correct answer is \\textbf{(B)}\", \"the correct answer is $\\boxed{\\textbf{(B)}\"]"}, {"prompt": "Let the tangent to the circle C1: x^2 + y^2 = 2 at the point M(\u20131, 1) intersect the circle C2: (x \u2013 3)^2 + (y \u2013 2)^2 = 5, at two distinct points A and B. If the tangents to C2 at the points A and B intersect at N, then the area of the triangle ANB is equal to\n(A) \\(\\frac{1}{2} \\)\n(B) \\(\\frac{2}{3} \\)\n(C) \\(\\frac{1}{6} \\)\n(D) \\(\\frac{5}{3} \\)\n", "ideal": "[\"the correct answer is (C)\", \"the correct answer is \\textbf{(C)}\", \"the correct answer is $\\boxed{\\textbf{(C)}\"]"}, {"prompt": "Let the mean and the variance of 5 observations x1, x2, x3, x4, x5 be 24/5 and 194/25, respectively. If the mean and variance of the first 4 observations are 7/2 and a, respectively, then (4a + x5) is equal to\n(A) 13\n(B) 15\n(C) 17\n(D) 18\n", "ideal": "[\"the correct answer is (B)\", \"the correct answer is \\textbf{(B)}\", \"the correct answer is $\\boxed{\\textbf{(B)}\"]"}, {"prompt": "Let \u03b1 be a root of the equation 1 + x^2 + x^4 = 0. Then the value of \u03b1^1011 + \u03b1^2022 \u2013 \u03b1^3033 is equal to\n(A) 1\n(B) \u03b1\n(C) 1 + \u03b1\n(D) 1 + 2\u03b1\n", "ideal": "[\"the correct answer is (A)\", \"the correct answer is \\textbf{(A)}\", \"the correct answer is $\\boxed{\\textbf{(A)}\"]"}, {"prompt": "Let arg(z) represent the principal argument of the complex number z. Then, |z| = 3 and arg(z \u2013 1) \u2013 arg(z + 1) = \u03c0/4 intersect\n(A) exactly at one point\n(B) exactly at two points\n(C) nowhere\n(D) at infinitely many points\n", "ideal": "[\"the correct answer is (C)\", \"the correct answer is \\textbf{(C)}\", \"the correct answer is $\\boxed{\\textbf{(C)}\"]"}, {"prompt": "Let \\(A = \\begin{bmatrix}2 & -1 \\\\0 & 2 \\\\\\end{bmatrix}\\)\u00a0If B = I \u2013 ^5C1(adjA) + ^5C2(adjA)^2 \u2013 \u2026. \u2013 ^5C5(adjA)^5, then the sum of all elements of the matrix B is\n(A) \u20135\n(B) \u20136\n(C) \u20137\n(D) \u20138\n", "ideal": "[\"the correct answer is (C)\", \"the correct answer is \\textbf{(C)}\", \"the correct answer is $\\boxed{\\textbf{(C)}\"]"}, {"prompt": "The sum of the infinite series \n\\(1+\\frac{5}{6}+\\frac{12}{6^{2}}+\\frac{22}{6^{3}}+\\frac{35}{6^{4}}+\\frac{51}{6^{5}}+\\frac{70}{6^{6}}+\\ldots . .\\) is equal to\n(A) \\(\\frac{425}{216}\\)\n(B) \\(\\frac{429}{216}\\)\n(C) \\(\\frac{288}{125}\\)\n(D) \\(\\frac{280}{125}\\)\n", "ideal": "[\"the correct answer is (C)\", \"the correct answer is \\textbf{(C)}\", \"the correct answer is $\\boxed{\\textbf{(C)}\"]"}, {"prompt": "The value of \\(\\lim _{x \\rightarrow 1} \\frac{\\left(x^{2}-1\\right) \\sin ^{2}(\\pi x)}{x^{4}-2 x^{3}+2 x-1}\\) is equal to\n(A) \\(\\frac{\\pi^{2}}{6}\\)\n(B) \\(\\frac{\\pi^{2}}{3}\\)\n(C) \\(\\frac{\\pi^{2}}{2}\\)\n(D) \\(\\pi^{2}\\)\n", "ideal": "[\"the correct answer is (D)\", \"the correct answer is \\textbf{(D)}\", \"the correct answer is $\\boxed{\\textbf{(D)}\"]"}, {"prompt": "Let f : R \u2192 R be a function defined by;\n\\(f(x)=(x-3)^{n_{1}}(x-5)^{n_{2}}, n_{1}, n_{2} \\in N\\) Then, which of the following is NOT true?\n(A) For n1 = 3, n2 = 4, there exists \u03b1 \u2208 (3, 5) where f attains local maxima.\n(B) For n1 = 4, n2 = 3, there exists \u03b1 \u2208 (3, 5) where f attains local minima.\n(C) For n1 = 3, n2 = 5, there exists \u03b1 \u2208 (3, 5) where f attains local maxima.\n(D) For n1 = 4, n2 = 6, there exists \u03b1 \u2208 (3, 5) where f attains local maxima.\n", "ideal": "[\"the correct answer is (C)\", \"the correct answer is \\textbf{(C)}\", \"the correct answer is $\\boxed{\\textbf{(C)}\"]"}, {"prompt": "Let f be a real valued continuous function on [0, 1] and \\(f(x)=x+\\int_{0}^{1}(x-t) f(t) d t\\) Then, which of the following points (x, y) lies on the curve y = f(x)?\n(A) (2, 4)\n(B) (1, 2)\n(C) (4, 17)\n(D) (6, 8)\n", "ideal": "[\"the correct answer is (D)\", \"the correct answer is \\textbf{(D)}\", \"the correct answer is $\\boxed{\\textbf{(D)}\"]"}, {"prompt": "If \\(\\int_{0}^{2}\\left(\\sqrt{2 x}-\\sqrt{2 x-x^{2}}\\right) d x=\\int_{0}^{1}\\left(1-\\sqrt{1-y^{2}}-\\frac{y^{2}}{2}\\right) d y+\\int_{1}^{2}\\left(2-\\frac{y^{2}}{2}\\right) d y+I\\) then I equal is\n(A) \\(\\int_{0}^{1}\\left(1+\\sqrt{1-y^{2}}\\right) d y\\)\n(B) \\(\\int_{0}^{1}\\left(\\frac{y^{2}}{2}-\\sqrt{1-y^{2}}+1\\right) d y\\)\n(C) \\(\\int_{0}^{1}\\left(1-\\sqrt{1-y^{2}}\\right) d y\\)\n(D) \\(\\int_{0}^{1}\\left(\\frac{y^{2}}{2}+\\sqrt{1-y^{2}}+1\\right) d y\\)\n", "ideal": "[\"the correct answer is (C)\", \"the correct answer is \\textbf{(C)}\", \"the correct answer is $\\boxed{\\textbf{(C)}\"]"}, {"prompt": "If y = y (x) is the solution of the differential equation \\(\\left(1+e^{2 x}\\right) \\frac{d y}{d x}+2\\left(1+y^{2}\\right) e^{x}=0\\) and y(0) = 0, then \\(6\\left(y^{\\prime}(0)+\\left(y\\left(\\log _{e} \\sqrt{3}\\right)\\right)^{2}\\right)\\) is equal to\n(A) 2\n(B) \u20132\n(C) \u20134\n(D) \u20131\n", "ideal": "[\"the correct answer is (C)\", \"the correct answer is \\textbf{(C)}\", \"the correct answer is $\\boxed{\\textbf{(C)}\"]"}, {"prompt": "Let P : y^2 = 4ax, a > 0 be a parabola with focus S. Let the tangents to the parabola P make an angle of \u03c0/4 with the line y = 3x + 5 touch the parabola P at A and B. Then the value of a for which A, B and S are collinear is\n(A) 8 only\n(B) 2 only\n(C) \\(\\frac{1}{4}~\\text{only}\\)\n(D) any a > 0\n", "ideal": "[\"the correct answer is (D)\", \"the correct answer is \\textbf{(D)}\", \"the correct answer is $\\boxed{\\textbf{(D)}\"]"}, {"prompt": "Let \\(\\frac{x-2}{3}=\\frac{y+1}{-2}=\\frac{z+3}{-1}\\) lie on the plane px \u2013 qy + z = 5, for some p, q \u2208 \u211d.\u00a0The shortest distance of the plane from the origin is :\n(A) \\(\\sqrt{\\frac{3}{109}}\\)\n(B) \\(\\sqrt{\\frac{5}{142}}\\)\n(C) \\(\\frac{5}{\\sqrt{71}}\\)\n(D) \\(\\frac{1}{\\sqrt{142}}\\)\n", "ideal": "[\"the correct answer is (B)\", \"the correct answer is \\textbf{(B)}\", \"the correct answer is $\\boxed{\\textbf{(B)}\"]"}, {"prompt": "The distance of the origin from the centroid of the triangle whose two sides have the equations\nx \u2013 2y + 1 = 0 and 2x \u2013 y \u2013 1 = 0 and whose orthocenter is (7/3, 7/3) is : \n(A) \u221a2\n(B) 2\n(C) 2\u221a2\n(D) 4\n", "ideal": "[\"the correct answer is (C)\", \"the correct answer is \\textbf{(C)}\", \"the correct answer is $\\boxed{\\textbf{(C)}\"]"}, {"prompt": "Let Q be the mirror image of the point P(1, 2, 1) with respect to the plane x + 2y + 2z = 16. Let T be a plane passing through the point Q and contains the line \\(\\vec{r}=-\\hat{k}+\\lambda(\\hat{i}+\\hat{j}+2 \\hat{k}), \\lambda \\in \\mathbb{R}\\). Then, which of the following points lies on T?\n(A) (2, 1, 0)\n(B) (1, 2, 1)\n(C) (1, 2, 2)\n(D) (1, 3, 2)\n", "ideal": "[\"the correct answer is (B)\", \"the correct answer is \\textbf{(B)}\", \"the correct answer is $\\boxed{\\textbf{(B)}\"]"}, {"prompt": "Let A, B, C be three points whose position vectors respectively are \n\\(\\vec{a}=\\hat{i}+4 \\hat{j}+3 \\hat{k}\\)\n\\(\\vec{b}=2 \\hat{i}+\\alpha \\hat{j}+4 \\hat{k}, \\alpha \\in \\mathbb{R}\\)\n\\(\\vec{c}=3 \\hat{i}-2 \\hat{j}+5 \\hat{k}\\)\n\\(\\text{If}\\ \\alpha\\ \\text{ is the smallest positive integer for which}\\ \\vec{a}, \\vec{b}, \\vec{c}\\) are non collinear, then the length of the median, in \u0394ABC, through A is:\n(A) \\(\\frac{\\sqrt{82}}{2}\\)\n(B) \\(\\frac{\\sqrt{62}}{2}\\)\n(C) \\(\\frac{\\sqrt{69}}{2}\\)\n(D) \\(\\frac{\\sqrt{66}}{2}\\)\n", "ideal": "[\"the correct answer is (A)\", \"the correct answer is \\textbf{(A)}\", \"the correct answer is $\\boxed{\\textbf{(A)}\"]"}, {"prompt": "The probability that a relation R from {x, y} to {x, y} is both symmetric and transitive, is equal to \n(A) 5/16\n(B) 9/16\n(C) 11/16\n(D) 13/16\n", "ideal": "[\"the correct answer is (A)\", \"the correct answer is \\textbf{(A)}\", \"the correct answer is $\\boxed{\\textbf{(A)}\"]"}, {"prompt": "The number of values of a \u2208 N such that the variance of 3, 7, 12, a, 43 \u2013 a is a natural number is :\n(A) 0\n(B) 2\n(C) 5\n(D) Infinite\n", "ideal": "[\"the correct answer is (A)\", \"the correct answer is \\textbf{(A)}\", \"the correct answer is $\\boxed{\\textbf{(A)}\"]"}, {"prompt": "From the base of a pole of height 20 meter, the angle of elevation of the top of a tower is 60\u00b0. The pole subtends an angle 30\u00b0 at the top of the tower. Then the height of the tower is :\n(A) 15\u221a3\n(B) 20\u221a3\n(C) 20 + 10\u221a3\n(D) 30\n", "ideal": "[\"the correct answer is (D)\", \"the correct answer is \\textbf{(D)}\", \"the correct answer is $\\boxed{\\textbf{(D)}\"]"}, {"prompt": "Negation of the Boolean statement (p \u2228 q) \u21d2 ((~ r) \u2228 p) is equivalent to\n(A) p \u2227 (~ q) \u2227 r\n(B) (~ p) \u2227 (~ q) \u2227 r\n(C) (~p) \u2227 q \u2227 r\n(D) p \u2227 q \u2227 (~ r)\n", "ideal": "[\"the correct answer is (C)\", \"the correct answer is \\textbf{(C)}\", \"the correct answer is $\\boxed{\\textbf{(C)}\"]"}, {"prompt": "Let n \u2265 5 be an integer. If 9^n \u2013 8n \u2013 1 = 64\u03b1 and 6^n \u2013 5n \u2013 1 = 25\u03b2, then \u03b1 \u2013 \u03b2 is equal to\n(A) \\(1+{ }^{n} C_{2}(8-5)+{ }^{n} C_{3}\\left(8^{2}-5^{2}\\right)+\\ldots+{ }^{n} C_{n}\\left(8^{n-1}-5^{n-1}\\right)\\)\n(B) \\(1+{ }^{n} C_{3}(8-5)+{ }^{n} C_{4}\\left(8^{2}-5^{2}\\right)+\\ldots+{ }^{n} C_{n}\\left(8^{n-2}-5^{n-2}\\right)\\)\n(C) \\({ }^{n} C_{3}(8-5)+{ }^{n} C_{4}\\left(8^{2}-5^{2}\\right)+\\ldots+{ }^{n} C_{n}\\left(8^{n-2}-5^{n-2}\\right)\\)\n(D) \\({ }^{n} C_{4}(8-5)+{ }^{n} C_{5}\\left(8^{2}-5^{2}\\right)+\\ldots+{ }^{n} C_{n}\\left(8^{n-3}-5^{n-3}\\right)\\)\n", "ideal": "[\"the correct answer is (C)\", \"the correct answer is \\textbf{(C)}\", \"the correct answer is $\\boxed{\\textbf{(C)}\"]"}], "columns": [{"key": "prompt", "header": "Prompt"}, {"key": "ideal", "header": "Ideal"}]}, "position": {"x": -16, "y": 160}, "selected": false, "positionAbsolute": {"x": -16, "y": 160}, "dragging": false}], "edges": [{"source": "prompt-jee-math", "sourceHandle": "prompt", "target": "eval-jee-math", "targetHandle": "responseBatch", "interactionWidth": 100, "markerEnd": {"type": "arrow", "width": "22px", "height": "22px"}, "id": "reactflow__edge-prompt-1686756357355prompt-eval-1686756357355responseBatch"}, {"source": "prompt-jee-math", "sourceHandle": "prompt", "target": "inspect-jee-math", "targetHandle": "input", "interactionWidth": 100, "markerEnd": {"type": "arrow", "width": "22px", "height": "22px"}, "id": "reactflow__edge-prompt-1686756357355prompt-inspect-1686756357355input"}, {"source": "eval-jee-math", "sourceHandle": "output", "target": "vis-jee-math", "targetHandle": "input", "interactionWidth": 100, "markerEnd": {"type": "arrow", "width": "22px", "height": "22px"}, "id": "reactflow__edge-eval-1686756357355output-vis-1686756357355input"}, {"source": "table-jee-math", "sourceHandle": "Prompt", "target": "prompt-jee-math", "targetHandle": "prompt", "interactionWidth": 100, "markerEnd": {"type": "arrow", "width": "22px", "height": "22px"}, "id": "reactflow__edge-table-1686756385002Prompt-prompt-1686756357355prompt"}], "viewport": {"x": 144, "y": 37, "zoom": 1}}, "cache": {"eval-1686756357355.json": {}, "inspect-1686756357355.json": {}, "prompt-1686756357355.json": {}, "table-1686756385002.json": {}, "vis-1686756357355.json": {}}} |