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

This section has 20 multiple choice questions of 1 mark each.

1. A function \(f:\mathbb{R}\rightarrow\mathbb{R}\) defined as \(f(x)=x^{2}-4x+5\) is: (1)
   1. injective but not surjective.
   2. surjective but not injective.
   3. both injective and surjective.
   4. neither injective nor surjective.
2. If \(A=[\begin{matrix}a&c&-1\\ b&0&5\\ 1&-5&0\end{matrix}]\) is a skew-symmetric matrix, then the value of \(2a-(b+c)\) is: (1)
   1. 0
   2. 1
   3. -10
   4. 10
3. If A is a square matrix of order 3 such that the value of \(|adj\cdot A|=8,\) then the value of \(|A^{T}|\) is: (1)
   1. \(\sqrt{2}\)
   2. \(-\sqrt{2}\)
   3. 8
   4. \(2\sqrt{2}\)
4. If inverse of matrix \([\begin{matrix}1&3&3\\ 1&\lambda&3\\ 1&3&4\end{matrix}]\) is the matrix \([\begin{matrix}7&-3&-3\\ -1&1&0\\ -1&0&1\end{matrix}],\) then value of \(\lambda\) is: (1)
   1. -4
   2. 1
   3. 3
   4. 4
5. If \([\begin{matrix}x&2&0\end{matrix}][\begin{matrix}5\\ -1\\ x\end{matrix}]=[\begin{matrix}3&1\end{matrix}][\begin{matrix}-2\\ x\end{matrix}],\) then value of x is: (1)
   1. -1
   2. 0
   3. 1
   4. 2
6. Find the matrix \(A^{2}\), where \(A=[a_{ij}]\) is a \(2\times2\) matrix whose elements are given by \(a_{ij}=\) maximum (i, j) - minimum (i, j): (1)
   1. \([\begin{matrix}0&0\\ 0&0\end{matrix}]\)
   2. \([\begin{matrix}1&0\\ 0&1\end{matrix}]\)
   3. \([\begin{matrix}0&1\\ 1&0\end{matrix}]\)
   4. \([\begin{matrix}1&1\\ 1&1\end{matrix}]\)
7. If \(xe^{y}=1\), then the value of \(\frac{dy}{dx}\) at \(x=1\) is : (1)
   1. -1
   2. 1
   3. -e
   4. \(-\frac{1}{e}\)
8. Derivative of \(e^{sin^{2}x}\) with respect to cos x is: (1)
   1. \(sin~x~e^{sin^{2}x}\)
   2. \(cos~x~e^{sin^{2}x}\)
   3. \(-2~cos~x~e^{sin^{2}x}\)
   4. \(-2~sin^{2}x~cos~x~e^{sin^{2}x}\)
9. The function \(f(x)=\frac{x}{2}+\frac{2}{x}\) has a local minima at x equal to: (1)
   1. 2
   2. 1
   3. 0
   4. -2
10. Given a curve \(y=7x-x^{3}\) and x increases at the rate of 2 units per second. The rate at which the slope of the curve is changing, when \(x=5\) is: (1)
    1. \(-60~units/sec\)
    2. \(60~units/sec\)
    3. \(-70~units/sec\)
    4. \(-140~units/sec\)
11. \(\int\frac{1}{x(log~x)^{2}}dx\) is equal to: (1)
    1. \(2~log(log~x)+c\)
    2. \(-\frac{1}{log~x}+c\)
    3. \(\frac{(log~x)^{3}}{3}+c\)
    4. \(\frac{3}{(log~x)^{3}}+c\)
12. The value of \(\int_{-1}^{1}x|x|dx\) is: (1)
    1. \(\frac{1}{6}\)
    2. \(\frac{1}{3}\)
    3. \(-\frac{1}{6}\)
    4. 0
13. Area of the region bounded by curve \(y^{2}=4x\) and the X-axis between \(x=0\) and \(x=1\) is: (1)
    1. \(\frac{2}{3}\)
    2. \(\frac{8}{3}\)
    3. 3
    4. \(\frac{4}{3}\)
14. The order of the differential equation \(\frac{d^{4}y}{dx^{4}}-sin(\frac{d^{2}y}{dx^{2}})=5\) is: (1)
    1. 4
    2. 3
    3. 2
    4. not defined
15. The position vectors of points P and Q are \(\vec{p}\) and \(\vec{q}\) respectively. The point R divides line segment PQ in the ratio 3:1 and S is the mid-point of line segment PR. The position vector of S is: (1)
    1. \(\frac{\vec{p}+3\vec{q}}{4}\)
    2. \(\frac{\vec{p}+3\vec{q}}{8}\)
    3. \(\frac{5\vec{p}+3\vec{q}}{4}\)
    4. \(\frac{5\vec{p}+3\vec{q}}{8}\)
16. The angle which the line \(\frac{x}{1}=\frac{y}{-1}=\frac{z}{0}\) makes with the positive direction of Y-axis is: (1)
    1. \(\frac{5\pi}{6}\)
    2. \(\frac{3\pi}{4}\)
    3. \(\frac{5\pi}{4}\)
    4. \(\frac{7\pi}{4}\)
17. The Cartesian equation of the line passing through the point (1, -3, 2) and parallel to the line: \(\vec{r}=(2+\lambda)\hat{i}+\lambda\hat{j}+(2\lambda-1)\hat{k}\) is: (1)
    1. \(\frac{x-1}{2}=\frac{y+3}{0}=\frac{z-2}{-1}\)
    2. \(\frac{x+1}{1}=\frac{y-3}{1}=\frac{z+2}{2}\)
    3. \(\frac{x+1}{2}=\frac{y-3}{0}=\frac{z+2}{-1}\)
    4. \(\frac{x-1}{1}=\frac{y+3}{1}=\frac{z-2}{2}\)
18. If A and B are events such that \(P(A/B)=P(B/A)\ne0,\) then : (1)
    1. \(A\subset B\), but \(A\ne B\)
    2. \(A=B\)
    3. \(A\cap B=\phi\)
    4. \(P(A)=P(B)\)

19. Assertion (A): Domain of \(y=cos^{-1}(x)\) is [-1, 1].

    Reason (R): The range of the principal value branch of \(y=cos^{-1}(x)\) is \([0,\pi]-\{\frac{\pi}{2}\}\).

    (Select the correct answer from the codes A, B, C, D as given in the paper)

    (1)

20. Assertion (A): The vectors \(\vec{a}=6\hat{i}+2\hat{j}-8\hat{k}\), \(\vec{b}=10\hat{i}-2\hat{j}-6\hat{k}\), \(\vec{c}=4\hat{i}-4\hat{j}+2\hat{k}\) represent the sides of a right angled triangle.

    Reason (R): Three non-zero vectors of which none of two are collinear forms a triangle if their resultant is zero vector or sum of any two vectors is equal to the third.

    (Select the correct answer from the codes A, B, C, D as given in the paper)

    (1)

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

This section has 5 Very Short Answer questions of 2 marks each.

21. Find value of k if \(sin^{-1}[k~tan(2~cos^{-1}\frac{\sqrt{3}}{2})]=\frac{\pi}{3}.\) (2)

22. (a) Verify whether the function f defined by \(f(x)=\begin{cases}x~sin(\frac{1}{x}),x\ne0\\ 0&,x=0\end{cases}\) is continuous at \(x=0\) or not.

    OR

    (b) Check for differentiability of the function f defined by \(f(x)=|x-5|\), at the point \(x=5\).

    (2)
23. The area of the circle is increasing at a uniform rate of \(2~cm^{2}/sec\). How fast is the circumference of the circle increasing when the radius \(r=5\) cm? (2)

24. (a) Find: \(\int cos^{3}x~e^{log~sin~x}dx\)

    OR

    (b) Find: \(\int\frac{1}{5+4x-x^{2}}dx\)

    (2)
25. Find the vector equation of the line passing through the point (2, 3, -5) and making equal angles with the co-ordinate axes. (2)

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

There are 6 short answer questions in this section.Each is of 3 marks.

26. (a) Find \(\frac{dy}{dx}\) , if \((cos~x)^{y}=(cos~y)^{x}\)

    OR

    (b) If \(\sqrt{1-x^{2}}+\sqrt{1-y^{2}}=a(x-y)\), prove that \(\frac{dy}{dx}=\sqrt{\frac{1-y^{2}}{1-x^{2}}}.\)

    (3)
27. If \(x=a~sin^{3}\theta\), \(y=b~cos^{3}\theta\) then find \(\frac{d^{2}y}{dx^{2}}\) at \(\theta=\frac{\pi}{4}\) (3)

28. (a) Evaluate : \(\int_{0}^{\pi}\frac{e^{cos~x}}{e^{cos~x}+e^{-cos~x}}d~x\)

    OR

    (b) Find: \(\int\frac{2x+1}{(x+1)^{2}(x-1)}dx\)

    (3)

29. (a) Find the particular solution of the differential equation \(\frac{dy}{dx}-2xy=3x^{2}e^{x^{2}};y(0)=5\) .

    OR

    (b) Solve the following differential equation \(x^{2}dy+y(x+y)dx=0\)

    (3)
30. Find a vector of magnitude 4 units perpendicular to each of the vectors \(2\hat{i}-\hat{j}+\hat{k}\) and \(\hat{i}+\hat{j}-\hat{k}\) and hence verify your answer. (3)
31. The random variable X has the following probability distribution where a and b are some constants:

    X	1	2	3	4	5
    \(P(X)\)	0.2	a	a	0.2	b

    If the mean \(E(X)=3\), then find values of a and b and hence determine \(P(X\ge3)\). (3)

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

There are 4 long answer questions in this section.Each question is of 5 marks.

32. (a) If \(A=[\begin{matrix}1&2&-3\\ 2&0&-3\\ 1&2&0\end{matrix}],\) then find \(A^{-1}\) and hence solve the following system of equations: \(x+2y-3z=1\), \(2x-3z=2\), \(x+2y=3\).

    OR

    (b) Find the product of the matrices \([\begin{matrix}1&2&-3\\ 2&3&2\\ 3&-3&-4\end{matrix}][\begin{matrix}-6&17&13\\ 14&5&-8\\ -15&9&-1\end{matrix}]\) and hence solve the system of linear equations: \(x+2y-3z=-4\), \(2x+3y+2z=2\), \(3x-3y-4z=11\).

    (5)
33. Find the area of the region bounded by the curve \(4x^{2}+y^{2}=36\) using integration. (5)

34. (a) Find the co-ordinates of the foot of the perpendicular drawn from the point (2, 3, -8) to the line \(\frac{4-x}{2}=\frac{y}{6}=\frac{1-z}{3}\).Also, find the perpendicular distance of the given point from the line.

    OR

    (b) Find the shortest distance between the lines \(L_{1}\) & \(L_{2}\) given below: \(L_{1}\): The line passing through (2, -1, 1) and parallel to \(\frac{x}{1}=\frac{y}{1}=\frac{z}{3}\).\(L_{2}:\vec{r}=\hat{i}+(2\mu+1)\hat{j}-(\mu+2)\hat{k}\).

    (5)
35. Solve the following L.P.P. graphically: Maximise \(Z=60x+40y\) Subject to \(x+2y\le12\), \(2x+y\le12\), \(4x+5y\ge20\), \(x,y\ge0\). (5)

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

In this section there are 3 case study questions of 4 marks each.

36. Case Study 1: Rail Lines/Relation (Railway Museum)

    Let L be the set of all rail lines on the railway track and R be the relation on L defined by \(R=\{(l_{1},l_{2}):l_{1}\) is parallel to \(l_{2}\}\).

    1. (i) Find whether the relation R is symmetric or not. (1)
    2. (ii) Find whether the relation R is transitive or not. (1)
    3. (iii) If one of the rail lines on the railway track is represented by the equation \(y=3x+2\) then find the set of rail lines in R related to it. (2)
       OR

       Let S be the relation defined by \(S=\{(l_{1},l_{2}):l_{1}\) is perpendicular to \(l_{2}\}\) check whether the relation S is symmetric and transitive. (2)

    (4)
37. Case Study 2: Minimising Area of Visiting Card

    A rectangular visiting card is to contain 24 sq.cm. of printed matter.The margins at the top and bottom of the card are to be 1 cm and the margins on the left and right are to be 1½ cm. Let \(x\) and \(y\) be the dimensions of the printed matter.

    1. (i) Write the expression for the area of the visiting card in terms of x. (2)
    2. (ii) Obtain the dimensions of the card of minimum area. (2)
    (4)
38. Case Study 3: Probability - Bill Payment

    Past experience shows that 70% of the store's customers pay their first month bill in time.The customer who pays the bill in time has the probability of 0.8 of paying in time next month and the customer who doesn't pay in time has the probability of 0.4 of paying in time the next month.

    1. (i) Let \(E_{1}\) and \(E_{2}\) respectively denote the event of customer paying or not paying the first month bill in time. Find \(P(E_{1})\), \(P(E_{2})\). (1)
    2. (ii) Let A denotes the event of customer paying second month's bill in time, then find \(P(A|E_{1})\) and \(P(A|E_{2})\). (1)
    3. (iii) Find the probability of customer paying second month's bill in time. (2)
       OR

       Find the probability of customer paying first month's bill in time if it is found that customer has paid the second month's bill in time. (2)

    (4)