Mathematics Class XII Past Paper PDF

**MATHEMATICS** **CLASS XII** Time Allowed: 3 Hours Maximum Marks: 80 General Instructions: 1\. This Question paper contains -- five sections A, B, C, D and E. Each section is compulsory. However, there are internal choices in some questions. 2\. Section A has 18 MCQs and 02 Assertion-Reason...

**MATHEMATICS** **CLASS XII** Time Allowed: 3 Hours Maximum Marks: 80 General Instructions: 1\. This Question paper contains -- five sections A, B, C, D and E. Each section is compulsory. However, there are internal choices in some questions. 2\. Section A has 18 MCQs and 02 Assertion-Reason based questions of 1 mark each. 3\. Section B has 5 Very Short Answer (VSA)-type questions of 2 marks each. 4\. Section C has 6 Short Answer (SA)-type questions of 3 marks each. 5\. Section D has 4 Long Answer (LA)-type questions of 5 marks each. 6\. Section E has 3 source based/case based/passage based/integrated units of assessment (4 marks each) with sub parts. **SECTION -- A** *(Multiple Choice Questions) Each question carries 1 mark.* **1.** Which of the following relations is symmetric but neither reflexive nor transitive for a set A = {1, 2, 3} \(a) R = {(1, 2), (1, 3), (1, 4)} \(b) R = {(1, 2), (2, 1)} \(c) R= {(1, 1), (2, 2), (3, 3)} \(d) R={ (1, 1), (1, 2), (2, 3)} **2.** The matrix A = [\$\\begin{bmatrix} \\& 1\\text{\\;\\;\\;\\;}2 \\\\ \\& 2\\text{\\;\\;\\;\\;}1 \\\\ \\end{bmatrix}\$]{.math.inline}is a: \(a) symmetric matrix \(b) null matrix \(c) skew symmetric matrix \(d) diagonal matrix **3.** Which of the following is not a possible ordered pair for a matrix of 6 elements? \(a) (2, 3) (b) (3, 2) \(c) (1, 6) (d) (3, 1) **4.** Which of the following is the formula for calculating the inverse of a matrix? \(a) [\$\\frac{2}{\|A\|}\$]{.math.inline} adj. A (b) [\$\\frac{1}{\|A\|}\$]{.math.inline} adj. A \(c) [\$\\frac{- 1}{\|A\|}\$]{.math.inline} adj. A (d) [\$\\frac{1}{\|2A\|}\$]{.math.inline} adj. A **5.** The value of [\$\\frac{d\^{2}y}{dx\^{2}},\$]{.math.inline} if *y* = [2sin^ − 1^(cos *x*),]{.math.inline} is: \(a) 0 (b) [\$\\sin\^{- 1}\\left( \\frac{1}{\\cos x} \\right)\$]{.math.inline} \(c) 1 (d) --1 **6.** What is the nature of the function [*f*(*x*) = *x*^3^ − 3*x*^2^ + 4*x*]{.math.inline}on R? \(a) Increasing \(b) Decreasing \(c) Constant \(d) Increasing and decreasing **7.** if f(x)=[\$\\left\\{ \\begin{matrix} \\\\ \\\\ \\frac{\\text{k\\ cosx}}{\\pi - 2x} \\\\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ 3\\ \\ if\\ \\ x = \\frac{\\pi}{2} \\\\ \\end{matrix} \\right.\\ \\ if\\ x \\neq \\frac{\\pi}{2}\$]{.math.inline} is continuous at [\$x = \\frac{\\pi}{2}\$]{.math.inline} ,value of k= \(a) 6 \(b) 3 (c)4 (d)7 **8.** The integral of [\$3e\^{x} + 2\\frac{(\\log x)}{3x}\$]{.math.inline}is: \(a) [\$3e\^{x} + \\frac{1}{3}x\^{2} + c\$]{.math.inline} \(b) [\$e\^{x} - \\frac{8}{3}(\\log x)\^{2} + c\$]{.math.inline} \(c) [\$3e\^{x} - \\frac{1}{3}(\\log x)\^{2} + c\$]{.math.inline} \(d) [\$3e\^{x} + \\frac{1}{3}(\\log x)\^{2} + c\$]{.math.inline} **9.** What is *y* in [∫~*a*~^*b*^*f*(*y*)]{.math.inline}called as? \(a) random variable (b) integral \(c) integrand (d) dummy symbol **10.** [\$\\int\_{3}\^{7}{\\sin t - 2\\cos t\\text{\\;\\;}\\text{dt}}\$]{.math.inline}is \(a) [(cos (7) − 2sin (7)) + (cos (3) + 2sin (3))]{.math.inline} \(b) --17 \(c) 12 \(d) [ − (cos (7) − 2sin (7)) + (cos (3) + 2sin (3))]{.math.inline} **11.** The general solution of the differential equation[\$\\frac{\\text{dy}}{\\text{dx}} = \\frac{3\\ secy}{2\\text{cosec}\\mspace{6mu} x}\$]{.math.inline} is : \(a) 3 cosx -- 2 cosy = c \(b) 3 sin x + 2 sin y = c \(c) 3 cos x + 2 tan x = c \(d) 3 cos x + 2 sin y = c **12.** The degree of the D.E. [\$\\frac{d\^{2}y}{dx\^{2}} + 5\\cot\\left( \\frac{\\text{dy}}{\\text{dx}} \\right)\$]{.math.inline} \(a) 5 (b) 3 \(c) 0 (d) not defined **13.** The magnitude of [\$\\overset{\\rightarrow}{a} = \\overset{\\land}{i}\\mspace{6mu} + \\overset{\\land}{j}\\mspace{6mu} + \\overset{\\land}{k}\$]{.math.inline} is : \(a) [\$\\sqrt{3}\$]{.math.inline} (b) [\$\\sqrt{2}\$]{.math.inline} \(c) 0 (d) [\$\\sqrt{4}\$]{.math.inline} **14.** If *a, b, c* are the direction ratios of the line and *l, m, n* are the direction cosines of the line, then which of the following is incorrect? \(a) [\$\\frac{l}{a} = \\frac{m}{b} = \\frac{n}{c} = k\$]{.math.inline} (b) [*l*^2^ + *m*^2^ + *n*^2^ = 1]{.math.inline} \(c) [\$k = \\mspace{6mu} \\pm \\mspace{6mu}\\frac{1}{\\sqrt{a\^{2} + b\^{2} + c\^{2}}}\$]{.math.inline} (d) [*l*^2^ − *m*^2^ = *n*^2^ − 1]{.math.inline} **15.** The value of P (X = 3), if X is the discrete random variable taking values *x*~1~, *x*~2~, *x*~3~ where P(X = 0) = 0, P(X = 1) = [\$\\frac{1}{4}\$]{.math.inline} and P(X = 2) [\$= \\frac{1}{4}\$]{.math.inline} is: \(a) 1 (b) [\$\\frac{1}{2}\$]{.math.inline} \(c) [\$\\frac{1}{3}\$]{.math.inline} (d) [\$\\frac{1}{4}\$]{.math.inline} **16.** What is the area of the triangle whose vertices are (0, 1), (0, 2), (1, 5) ? \(a) 1 Sq. units (b) 2 Sq. units \(c) [\$\\frac{1}{3}\$]{.math.inline} Sq. units (d) [\$\\frac{1}{2}\$]{.math.inline} Sq. units **17.** Which of the following is the reverse law of transposes? \(a) [(*A* − *B*)^′^ = *B*^′^  − *A*^′^]{.math.inline} \(b) [(AB)^′^= *B*^′^*A*^′^]{.math.inline} \(c) [(AB)^′^ = (BA)^′^]{.math.inline} \(d) [(*A* + *B*)^′^ = *B*^′^  + *A*^′^]{.math.inline} **18.** In an LPP, if the objective function z = ax + by has the same maximum value on two corner points of the feasible region, then the number of points of which Z~max~ occurs is: \(a) 0 (b) 2 \(c) finite (d) infinite **ASSERTION-REASON BASED QUESTIONS** In the following questions, a statement of assertion (A) is followed by a statement of reason (R). \(a) Both A and R are true and R is the correct explanation of A. \(b) Both A and R are true but R is not the correct explanation of A. \(c) A is true but R is false. \(d) A is false but R is true. **19.** **Assertion (A)**: The relation R in set A of human beings in a town at a particular time given by : R = {(x, y) : x is exactly 5 years younger than y} **Reason (R):** A relation R on the set A is not symmetric if (a, b) [∈]{.math.inline} R but (b, a) [∉]{.math.inline} R. **20.** **Assertion (A)**: A mapping shown in the following figure is not surjective. **Reason (R):** A function [*f* : *A* → *B*]{.math.inline} is said to be Surjective if every element of B has pre-image in A. **SECTION -- B** *The section comprises of very short answer type-questions (VSA) of 2 marks each.* **21**. Differentiate [sin^2^x w.r.t.. *e*^cosx^]{.math.inline}. **22.** Find the area of the region bounded by the curve *y* = *x*^2^ and the line *y* = 2 in first quadrant. **23.** Evaluate [∫ tan^4^*x* dx.]{.math.inline} **24.** Find the area of a parallelogram whose adjacent sides are along the vectors [\$\\overset{\\land}{i}\\mspace{6mu} + 2\\overset{\\land}{j}\\mspace{6mu} - \\overset{\\land}{k}\$]{.math.inline} and [\$2\\overset{\\land}{j}\\mspace{6mu} + \\overset{\\land}{k}\$]{.math.inline} respectively. **25.** Find the value of p so that the lines [\$\\frac{1 - x}{3} = \\frac{7y - 14}{2p} = \\frac{z - 3}{2}\$]{.math.inline}and[\$.\\ \\frac{7 - 7x}{3p} = \\frac{y - 5}{1}\$]{.math.inline}=[\$\\frac{6 - z}{5}\$]{.math.inline}are orthogonal. **SECTION -- C** *The section comprises of short answer type questions (SA) of 3 marks each* **26.** Solve the differential equation, [\$\\text{\\ \\ \\ \\ \\ \\ \\ \\ \\ }\\frac{\\text{dy}}{\\text{dx}} = x\^{5}\\tan\^{- 1}x\^{3}\$]{.math.inline}. **OR** Write the general solution [\$\\frac{\\text{dy}}{\\text{dx}} = \\frac{1 - 2y}{3x + 1}\\mspace{6mu}.\$]{.math.inline} **27.** Evaluate [\$\\int\_{0}\^{\\frac{\\pi}{4}}{\\mspace{6mu} log(1 + tanx)\\mspace{6mu}\\text{dx}}\$]{.math.inline} **OR** Find the inverse in which the function *f* given by [*f*(*x*) = 2*x*^3^ − 9*x*^2^ + 12*x* + 15]{.math.inline} is strictly increasing or decreasing. **28.** Find the area of the region enclosed between the parabola *y*^2^ = 4*ax* and the line *y = mx*. **OR** Using method of integration, find the area bounded by the curve \| *x* \| + \| *y* \| = 1 **29.** Let *Z* be the set of all integers and R be the relation on Z defined as R = {(*a, b*) : a [∈]{.math.inline} Z and (*a -- b*) is divisible by 5}. Prove that R is an equivalence relation. **30.** Find the shortest distance between the lines shoes vector equations are: [\$\\overset{\\rightarrow}{r} = (1 - t)\\overset{\\land}{i}\\mspace{6mu} + (t - 2)\\overset{\\land}{j}\\mspace{6mu} + (3 - 2t)\\overset{\\land}{k}\$]{.math.inline} and [\$\\overset{\\rightarrow}{r} = (s + 1)\\overset{\\land}{i}\\mspace{6mu} + (2s - 1)\\overset{\\land}{j}\\mspace{6mu} + (2s + 1)\\overset{\\land}{k}\$]{.math.inline} **31.** Consider the experiment of tossing a coin. If the coin shows head, toss it again but if it shows tail, then throw a die. Find the conditional probability of the event that 'the die shows a number greater than 4' given that 'there is at least one tail' **SECTION -- D** *The section comprises of answer type questions (LA) of 5 marks each.* **32.** Show that the semi-vertical angle of the cone of the maximum volume and of given slant height is [\$\\tan\^{- 1}\\sqrt{2}\$]{.math.inline}. **33.** If [\$A = \\begin{bmatrix} 2 & 3 & 10 \\\\ 4 & - 6 & 5 \\\\ 6 & 9 & - 20 \\\\ \\end{bmatrix},\\mspace{6mu}\\text{find}\\mspace{6mu} A\^{- 1}.\$]{.math.inline} How can we use [*A*^ − 1^]{.math.inline} to solve the system of equation [\$\\frac{2}{x} + \\frac{3}{y} + \\frac{10}{z} = 2,\\mspace{6mu}\\frac{4}{x} - \\frac{6}{y} + \\frac{5}{z} = 5,\$]{.math.inline} [\$\\text{\\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ }\\frac{6}{x} + \\frac{9}{y} - \\frac{20}{z} = - 4.\$]{.math.inline} **OR** If [\$A = \\begin{bmatrix} 2 & 3 & 1 \\\\ 1 & 2 & 2 \\\\ - 3 & 1 & - 1 \\\\ \\end{bmatrix},\\mspace{6mu}\\text{find}\\mspace{6mu} A\^{- 1}{}\$]{.math.inline} and hence show that how we can use [*A*^ − 1^]{.math.inline} to solve system of equation 2x +y -- 3z = 13, 3x + 2y + z =4 ;\ x + 2y -- z = 8. **34.** Find the area bounded by the triangle whose vertices are (1,0),(2,2),and (3,1) **35.** Probability of solving specific problem independently by A and B are [\$\\frac{1}{2}\\text{and\\ }\\frac{1\\ \\ }{3}\$]{.math.inline}respectively. If both try to solve the problem independently, find the probability that (i) the problem is solved \(ii) exactly one of them solves the problem OR Find the distance of the point A (1, 8, 9) and its image in the joining the points B (0, --1, 3) and\ C (2, --3, --1). **SECTION -- E** *The section comprises of 3 Case-Study/Passage-based questions of 4 marks each with two Sub-Parts, First two Case-Study question have three Sub-Parts (A), (B), (C) of marks 1, 1, 2 respectively. The third Case-Study question has two Sub-Parts of 2 marks each.* **36.** LIC insurance company done the life insurance of 9000 people. In this 2000 drives scooter, 4000 drives truck and rest of then drives car. The probability of a scooter, a truck and a car driver meeting an accident is 0.0, 0.04 and 0.02 respectively. Based on the above information , answer the following questions: A. What is the probability of a person being selected is a car drives? B. What is the probability that there is an accident of car driver. C. If an insured person meets an accident,what is the probability that he is a scooter driver? **37.** A graph is shown below. This graph shows the feasible solution of an LPP. Based on the above information, answer the following questions: \(A) Write the equations shown in the graph? \(B) What are the coordinates of point E? \(C) If Z = 500x + 150y, than what is max, z. OR If Z = 500 x + 100 y, what is the value of x and y for which z is maximum. **38.** The temperature of a person during an intestinal illness is given by f(x) = -- 0.1[*x*^2^]{.math.inline} + mx + 98.6, 0[≤]{.math.inline}x[≤]{.math.inline}12 , m being a constant, where f(x) is the temperature in "F at x days. \(i) Is the function differentiable in the interval (0, 12)? Justify your answer. \(ii) If 6 is the critical point of the function, then find the value of the constant m. \(iii) Find the intervals in which the function is strictly increasing/strictly decreasing. OR \(iii) Find the points of local maximum/local minimum, if any, in the interval (0, 12) as well as the points of absolute maximum/absolute minimum in the interval [\[0,12\]]{.math.inline}. Also, find the corresponding local maximum/local minimum and the absolute maximum/absolute minimum values of the function.