SECTION - A (Multiple Choice Questions & Assertion-Reason - 1 Mark Each)

This section comprises of 18 multiple choice questions (MCQs) (Q1 to Q18) and 2 Assertion-Reason based questions (Q19, Q20) of 1 mark each.

1. If \(A=[a_{ij}]\) is an identity matrix, then which of the following is true ?
   1. \(a_{ij}=\begin{cases}0,&if~i=j\\ 1,&if~i\ne j\end{cases}\)
   2. \(a_{ij}=1,\forall i,j\)
   3. \(a_{ij}=0,\forall i,j\)
   4. \(a_{ij}=\begin{cases}0,&if~i\ne j\\ 1,&if~i=j\end{cases}\)
2. Let \(R_{+}\) denote the set of all non-negative real numbers. Then the function \(f:R_{+}\rightarrow R_{+}\) defined as \(f(x)=x^{2}+1\) is :
   1. one-one but not onto
   2. onto but not one-one
   3. both one-one and onto
   4. neither one-one nor onto
3. Let \(A=[\begin{matrix}a&b\\ c&d\end{matrix}]\) be a square matrix such that adj \(A=A\) Then, \((a+b+c+d)\) is equal to :
   1. 2a
   2. 2b
   3. 2c
   4. 0
4. A function \(f(x)=|1-x+|x||\) is:
   1. discontinuous at \(x=1\) only
   2. discontinuous at \(x=0\) only
   3. discontinuous at \(x=0,1\)
   4. continuous everywhere
5. If the sides of a square are decreasing at the rate of \(1.5~cm/s\) the rate of decrease of its perimeter is:
   1. \(1.5~cm/s\)
   2. \(6~cm/s\)
   3. \(3~cm/s\)
   4. \(2.25~cm/s\)
6. \(\int_{-a}^{a}f(x)dx=0,\) if :
   1. \(f(-x)=f(x)\)
   2. \(f(-x)=-f(x)\)
   3. \(f(a-x)=f(x)\)
   4. \(f(a-x)=-f(x)\)
7. \(x~log~x\frac{dy}{dx}+y=2~log~x\) is an example of a :
   1. variable separable differential equation.
   2. homogeneous differential equation.
   3. first order linear differential equation.
   4. differential equation whose degree is not defined.
8. If \(\vec{a}=2\hat{i}-\hat{j}+\hat{k}\) and \(\vec{b}=\hat{i}+\hat{j}-\hat{k}\), then \(\vec{a}\) and \(\vec{b}\):
   1. collinear vectors which are not parallel
   2. parallel vectors
   3. perpendicular vectors
   4. unit vectors
9. If \(\alpha\), \(\beta\) and \(\gamma\) are the angles which a line makes with positive directions of x, y and z axes respectively, then which of the following is not true?
   1. \(cos^{2}\alpha+cos^{2}\beta+cos^{2}\gamma=1\)
   2. \(sin^{2}\alpha+sin^{2}\beta+sin^{2}\gamma=2\)
   3. \(cos~2\alpha+cos~2\beta+cos~2\gamma=-1\)
   4. \(cos~\alpha+cos~\beta+cos~\gamma=1\)
10. The restrictions imposed on decision variables involved in an objective function of a linear programming problem are called :
    1. feasible solutions
    2. constraints
    3. optimal solutions
    4. infeasible solutions
11. Let E and F be two events such that \(P(E)=0\cdot1\), \(P(F)=0\cdot3,\) \(P(E\cup F)=0\cdot4\) then \(P(F|E)\) is:
    1. 0.6
    2. 0.4
    3. 0.5
    4. 0
12. If A and B are two skew symmetric matrices, then \((AB+BA)\) is :
    1. a skew symmetric matrix
    2. a symmetric matrix
    3. a null matrix
    4. an identity matrix
13. If \(|\begin{matrix}1&3&1\\ k&0&1\\ 0&0&1\end{matrix}|=\pm6,\) then the value of k is:
    1. 2
    2. -2
    3. \(\pm2\)
    4. \(\mp2\)
14. The derivative of \(2^{x}\) w.r.t. \(3^{x}\) is:
    1. \((\frac{3}{2})^{x}\frac{log~2}{log~3}\)
    2. \((\frac{2}{3})^{x}\frac{log~3}{log~2}\)
    3. \((\frac{2}{3})^{x}\frac{log~2}{log~3}\)
    4. \((\frac{3}{2})^{x}\frac{log~3}{log~2}\)
15. If \(|\vec{a}|= 2\) and \(-3\le k\le2\), then \(|\vec{k}\vec{a}|\in\):
    1. [-6, 4]
    2. [0, 4]
    3. [4, 6]
    4. [0, 6]
16. If a line makes an angle of \(\frac{\pi}{4}\) with the positive directions of both x-axis and z-axis, then the angle which it makes with the positive direction of y-axis is:
    1. 0
    2. \(\frac{\pi}{4}\)
    3. \(\frac{\pi}{2}\)
    4. \(\pi\)
17. Of the following, which group of constraints represents the feasible region given below ?
    1. \(x+2y\le76\), \(2x+y\ge104\), \(x, y\ge0\)
    2. \(x+2y\le76\), \(2x+y\le104,\) \(x, y\ge0\)
    3. \(x+2y\ge76\), \(2x+y\le104\), \(x, y\ge0\)
    4. \(x+2y\ge76\), \(2x+y\ge104,\) \(x, y\ge0\)
18. If \(A=[\begin{matrix}2&0&0\\ 0&3&0\\ 0&0&5\end{matrix}],\) then \(A^{-1}\) is:
    1. \([\begin{matrix}\frac{1}{2}&0&0\\ 0&3&0\\ 0&0&\frac{1}{5}\end{matrix}]\)
    2. \(30[\begin{matrix}\frac{1}{2}&0&0\\ 0&\frac{1}{3}&0\\ 0&0&\frac{1}{5}\end{matrix}]\)
    3. \(\frac{1}{30}[\begin{matrix}2&0&0\\ 0&3&0\\ 0&0&5\end{matrix}]\)
    4. \(\frac{1}{30}[\begin{matrix}\frac{1}{2}&0&0\\ 0&\frac{1}{3}&0\\ 0&0&\frac{1}{5}\end{matrix}]\)

19. Assertion (A): Every scalar matrix is a diagonal matrix.

    Reason (R): In a diagonal matrix, all the diagonal elements are 0.

    (Choose the correct option from the standard codes A, B, C, D)

20. Assertion (A): Projection of \(\vec{a}\) on \(\vec{b}\) is same as projection of \(\vec{b}\) on \(\vec{a}\).

    Reason (R): Angle between \(\vec{a}\) and \(\vec{b}\) is same as angle between \(\vec{b}\) and \(\vec{a}\) numerically.

    (Choose the correct option from the standard codes A, B, C, D)

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SECTION - B (Very Short Answer Type Questions - 2 Marks Each)

This section comprises of 5 Very Short Answer (VSA) type questions of 2 marks each. (Internal choice has been provided in 2 questions)

21. Evaluate: \(sec^{2}(tan^{-1}\frac{1}{2})+cosec^{2}(cot^{-1}\frac{1}{3})\) (2)

22. (a) If \(x=e^{x/y}\), prove that \(\frac{dy}{dx}=\frac{log~x-1}{(log~x)^{2}}\)

    OR

    (b) Check the differentiability of \(f(x)=\begin{cases}x^{2}+1,&0\le x<1\\ 3-x,&1\le x\le2\end{cases}\) at \(x=1.\)

    (2)

23. (a) Evaluate: \(\int_{0}^{\pi/2}sin~2x~cos~3x~dx\)

    OR

    (b) Given \(\frac{d}{dx}F(x)=\frac{1}{\sqrt{2x-x^{2}}}\) and \(F(1)=0\), find \(F(x)\).

    (2)
24. Find the position vector of point C which divides the line segment joining points A and B having position vectors \(\hat{i}+2\hat{j}-\hat{k}\) and \(-\hat{i}+\hat{j}+\hat{k}\) respectively in the ratio \(4:1\) externally. Further, find \(|\vec{AB}|:|\vec{BC}|\). (2)
25. Let \(\vec{a}\) and \(\vec{b}\) be two non-zero vectors. Prove that \(|\vec{a}\times\vec{b}|\le|\vec{a}||\vec{b}|\). State the condition under which equality holds, i.e., \(|\vec{a}\times\vec{b}|=|\vec{a}||\vec{b}|\) (2)

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SECTION - C (Short Answer Type Questions - 3 Marks Each)

This section comprises of 6 Short Answer (SA) type questions of 3 marks each. (Internal choice has been provided in 3 questions)

26. (a) If \(x~cos(p+y)+cos~p~sin(p+y)=0\), prove that \(cos~p\frac{dy}{dx}=-cos^{2}(p+y),\) where p is a constant.

    OR

    (b) Find the value of a and b so that function f defined as: \(f(x)=\begin{cases}\frac{x-2}{|x-2|}+a,&if~x<2\\ a+b,&if~x=2\\ \frac{x-2}{|x-2|}+b,&if~x>2\end{cases}\) is a continuous function.

    (3)

27. (a) Find the intervals in which the function \(f(x)=\frac{log~x}{x}\) is strictly increasing or strictly decreasing.

    OR

    (b) Find the absolute maximum and absolute minimum values of the function f given by \(f(x)=\frac{x}{2}+\frac{2}{x}\) on the interval [1, 2].

    (3)
28. Find: \(\int\frac{x^{2}+1}{(x^{2}+2)(x^{2}+4)}dx\) (3)

29. (a) Find: \(\int\frac{2+sin~2x}{1+cos~2x}e^{x}dx\)

    OR

    (b) Evaluate: \(\int_{0}^{\pi/4}\frac{1}{sin~x+cos~x}dx\)

    (3)
30. Solve the following linear programming problem graphically: Maximise \(z=4x+3y.\) subject to the constraints \(x+y\le800\), \(2x+y\le1000\), \(x\le400\), \(x,y\ge0\). (3)
31. The chances of P, Q and R getting selected as CEO of a company are in the ratio 4: 1: 2 respectively. The probabilities for the company to increase its profits from the previous year under the new CEO, P, Q or R are 0.3, 0.8 and 0.5 respectively. If the company increased the profits from the previous year, find the probability that it is due to the appointment of R as CEO. (3)

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SECTION - D (Long Answer Type Questions - 5 Marks Each)

This section comprises of 4 Long Answer (LA) type questions of 5 marks each. (Internal choice has been provided in 2 questions)

32. A relation R on set \(A=\{-4,-3,-2,-1,0,1,2,3,4\}\) be defined as \(R=\{(x,y):x+y\) is an integer divisible by 2). Show that R is an equivalence relation. Also, write the equivalence class [2]. (5)

33. (a) It is given that function \(f(x)=x^{4}-62x^{2}+ax+9\) attains local maximum value at \(x=1\) Find the value of 'a', hence obtain all other points where the given function f(x) attains local maximum or local minimum values.

    OR

    (b) The perimeter of a rectangular metallic sheet is 300 cm. It is rolled along one of its sides to form a cylinder. Find the dimensions of the rectangular sheet so that volume of cylinder so formed is maximum.

    (5)
34. Using integration, find the area of the region enclosed between the circle \(x^{2}+y^{2}=16\) and the lines \(x=-2\) and \(x=2.\) (5)

35. (a) Find the equation of the line passing through the point of intersection of the lines \(\frac{x}{1}=\frac{y-1}{2}=\frac{z-2}{3}\) and \(\frac{x-1}{0}=\frac{y}{-3}=\frac{z-7}{2}\) and perpendicular to these given lines.

    OR

    (b) Two vertices of the parallelogram ABCD are given as \(A(-1,2,1)\) and \(B(1,-2,5)\). If the equation of the line passing through C and D is \(\frac{x-4}{1}=\frac{y+7}{-2}=\frac{z-8}{2}\), then find the distance between sides AB and CD. Hence, find the area of parallelogram ABCD.

    (5)

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SECTION - E (Case Study Based Questions - 4 Marks Each)

This section comprises of 3 case study based questions of 4 marks each. (Internal choice has been provided in 2 questions)

36. Case Study 1: Self-Study

    Self-study helps students to build confidence in learning. It boosts the self-esteem of the learners. Recent surveys suggested that close to 50% learners were self-taught using internet resources and upskilled themselves. A student may spend 1 hour to 6 hours in a day in upskilling self. The probability distribution of the number of hours spent by a student is given below : \(P(X=x)=\begin{cases}kx^{2},&for~x=1,2,3\\ 2kx,&for~x=4,5,6\\ 0,&otherwise\end{cases}\) where x denotes the number of hours. Based on the above information, answer the following questions :

    1. (i) Express the probability distribution given above in the form of a probability distribution table. (1 mark)
    2. (ii) Find the value of k. (1 mark)
    3. (iii) (a) Find the mean number of hours spent by the student. (2 marks)
       OR
       (iii) (b) Find \(P(1
    (4)
37. Case Study 2: Bacteria Growth

    A bacteria sample of certain number of bacteria is observed to grow exponentially in a given amount of time. Using exponential growth model, the rate of growth of this sample of bacteria is calculated. The differential equation representing the growth of bacteria is given as: \(\frac{dP}{dt}=kP,\) where P is the population of bacteria at any time 't'. Based on the above information, answer the following questions :

    1. (i) Obtain the general solution of the given differential equation and express it as an exponential function of 't'. (2 marks)
    2. (ii) If population of bacteria is 1000 at \(t=0.\), and 2000 at \(t=1.\), find the value of k. (2 marks)
    (4)
38. Case Study 3: [Content Missing from Source Document]

    (The preamble and questions for Case Study 3 (Question 38) were not available in the source document.)

    1. (i) [Missing sub-question] (1 mark)
    2. (ii) [Missing sub-question] (1 mark)
    3. (iii) (a) [Missing sub-question] (2 marks)
       OR
       (iii) (b) [Missing sub-question] (2 marks)
    (4)