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

1. \(If[\begin{matrix}a&c&0\\ b&d&0\\ 0&0&5\end{matrix}]\) is a scalar matrix, then the value of \(a+2b+3c+4d\) is: (1)
   1. 0
   2. 5
   3. 10
   4. 25
2. Given that \(A^{-1}=\frac{1}{7}[\begin{matrix}2&1\\ -3&2\end{matrix}]\) matrix A is (1)
   1. \(7[\begin{matrix}2&-1\\ 3&2\end{matrix}]\)
   2. \([\begin{matrix}2&-1\\ 3&2\end{matrix}]\)
   3. \(\frac{1}{7}[\begin{matrix}2&-1\\ 3&2\end{matrix}]\)
   4. \(\frac{1}{49}[\begin{matrix}2&-1\\ 3&2\end{matrix}]\)
3. \(If~A=[\begin{matrix}2&1\\ -4&-2\end{matrix}].\) then the value of \(I-A+A^{2}-A^{3}+...is\): (1)
   1. \([\begin{matrix}-1&-1\\ 4&3\end{matrix}]\)
   2. \([\begin{matrix}3&1\\ -4&-1\end{matrix}]\)
   3. \([\begin{matrix}0&0\\ 0&0\end{matrix}]\)
   4. \([\begin{matrix}1&0\\ 0&1\end{matrix}]\)
4. \(If~A=[\begin{matrix}-2&0&0\\ 1&2&3\\ 5&1&-1\end{matrix}],\) then the value of | A (adj. A) | is: (1)
   1. 100 I
   2. 10 I
   3. 10
   4. 1000
5. Given that \([\begin{matrix}1&x\end{matrix}][\begin{matrix}4&0\\ -2&0\end{matrix}]=0,\) the value of x is: (1)
   1. -4
   2. -2
   3. 2
   4. 4
6. Derivative of \(e^{2x}\) with respect to \(e^{x}\), is: (1)
   1. \(e^{x}\)
   2. \(2e^{x}\)
   3. \(2e^{2x}\)
   4. \(2e^{3x}\)
7. For what value of k, the function given below is continuous at \(x=0\) ? \(f(x)=\begin{cases}\frac{\sqrt{4+x}-2}{x},&x\ne0\\ k&,&x=0\end{cases}\) (1)
   1. 0
   2. \(\frac{1}{4}\)
   3. 1
   4. 4
8. The value of \(\int_{0}^{3}\frac{dx}{\sqrt{9-x^{2}}}\) is: (1)
   1. \(\frac{\pi}{6}\)
   2. \(\frac{\pi}{4}\)
   3. \(\frac{\pi}{2}\)
   4. \(\frac{\pi}{18}\)
9. The general solution of the differential equation \(x~dy+y~dx=0\) is: (1)
   1. \(xy=c\)
   2. \(x+y=c\)
   3. \(x^{2}+y^{2}=c^{2}\)
   4. \(log~y=log~x+c\)
10. The integrating factor of the differential equation \((x+2y^{2})\frac{dy}{dx}=y(y>0)\) is: (1)
    1. \(\frac{1}{x}\)
    2. x
    3. y
    4. \(\frac{1}{y}\)
11. If \(\vec{a}\) and \(\vec{b}\) are two vectors such that \(|\vec{a}|=1,|\vec{b}|=2~and\vec{a}\cdot\vec{b}=\sqrt{3}\) then the angle between \(2\vec{a}\) and \(-\vec{b}\) is: (1)
    1. \(\frac{\pi}{6}\)
    2. \(\frac{\pi}{3}\)
    3. \(\frac{5\pi}{6}\)
    4. \(\frac{11\pi}{6}\)
12. The vectors \(\vec{a}=2\hat{i}-\hat{j}+\hat{k}\), \(\vec{b}=\hat{i}-3\hat{j}-5\hat{k}\) and \(\vec{c}=-3\hat{i}+4\hat{j}+4\hat{k}\) represents the sides of (1)
    1. an equilateral triangle
    2. an obtuse-angled triangle
    3. an isosceles triangle
    4. a right-angled triangle
13. Let \(\vec{a}\) be any vector such that \(|\vec{a}|=a\) The value of \(|\vec{a}\times\hat{i}|^{2}+|\vec{a}\times\hat{j}|^{2}+|\vec{a}\times\hat{k}|^{2}\) is: (1)
    1. \(a^{2}\)
    2. \(2a^{2}\)
    3. \(3a^{2}\)
    4. 0
14. The vector equation of a line passing through the point (1, -1, 0) and parallel to Y-axis is : (1)
    1. \(\vec{r}=\hat{i}-\hat{j}+\lambda(\hat{i}-\hat{j})\)
    2. \(\vec{r}=\hat{i}-\hat{j}+\lambda\hat{j}\)
    3. \(\vec{r}=\hat{i}-\hat{j}+\lambda\hat{k}\)
    4. \(\vec{r}=\lambda\hat{j}\)
15. The lines \(\frac{1-x}{2}=\frac{y-1}{3}=\frac{z}{1}\) and \(\frac{2x-3}{2p}=\frac{y}{-1}=\frac{z-4}{7}\) are perpendicular to each other for p equal to: (1)
    1. \(-\frac{1}{2}\)
    2. \(\frac{1}{2}\)
    3. 2
    4. 3
16. The maximum value of \(Z=4x+y\) for a L.P.P. whose feasible region is given below is: [Graph image present in source] (1)
    1. 50
    2. 110
    3. 120
    4. 170
17. The probability distribution of a random variable X is:

    X	0	1	2	3	4
    P(X)	0.1	k	2k	k	0.1

    where k is some unknown constant. The probability that the random variable X takes the value 2 is: (1)
    1. \(\frac{1}{5}\)
    2. \(\frac{2}{5}\)
    3. \(\frac{4}{5}\)
    4. 1
18. The function \(f(x)=kx-sin~x\) is strictly increasing for (1)
    1. \(k>1\)
    2. \(k<1\)
    3. \(k>-1\)
    4. \(k<-1\)

19. Assertion (A): The relation \(R=\{(x,y):(x+y)\) is a prime number and x, \(y\in N\) is not a reflexive relation.

    Reason (R): The number '2n' is composite for all natural numbers n.

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

    (1)

20. Assertion (A): The corner points of the bounded feasible region of a L.P.P. are shown below. The maximum value of \(Z=x+2y\) occurs at infinite points. [Graph image present in source]

    Reason (R): The optimal solution of a LPP having bounded feasible region must occur at corner points.

    (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)

21. (a) Express \(tan^{-1}(\frac{cos~x}{1-sin~x})\) where \(\frac{-\pi}{2}(2)

    OR

    (b) Find the principal value of \(tan^{-1}(1)+cos^{-1}(-\frac{1}{2})+sin^{-1}(-\frac{1}{\sqrt{2}}).\)

22. (a) If \(y=cos^{3}(sec^{2}2t)\), find \(\frac{dy}{dt}\) . (2)

    OR

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

23. Find the interval in which the function \(f(x)=x^{4}-4x^{3}+10\) is strictly decreasing. (2)
24. The volume of a cube is increasing at the rate of \(6~cm^{3}/s.\) How fast is the surface area of cube increasing, when the length of an edge is 8 cm? (2)
25. Find: \(\int\frac{1}{x(x^{2}-1)}dx.\) (2)

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

26. Given that \(y=(sin~x)^{x}\cdot x^{sin~x}+a^{x},\) find \(\frac{dy}{dx}\) (3)

27. (a) Evaluate \(\int_{0}^{\frac{\pi}{4}}\frac{x~dx}{1+cos~2x+sin~2x}\) (3)

    OR

    (b) Find: \(\int e^{x}[\frac{1}{(1+x^{2})^{\frac{3}{2}}}+\frac{x}{\sqrt{1+x^{2}}}]dx\)

28. Find: \(\int\frac{3x+5}{\sqrt{x^{2}+2x+4}}dx\) (3)

29. (a) Find the particular solution of the differential equation \(\frac{dy}{dx}=y~cot~2x,\) given that \(y(\frac{\pi}{4})=2.\) (3)

    OR

    (b) Find the particular solution of the differential equation \((xe^{\frac{y}{x}}+y)dx=x~dy\), given that \(y=1\) when \(x=1\)

30. Solve the following linear programming problem graphically: Maximise \(Z=2x+3y\) subject to the constraints: \(x+y\le6\), \(x\ge2\), \(y\le3\), \(x,y\ge0\). (3)

31. (a) A card from a well shuffled deck of 52 playing cards is lost. From the remaining cards of the pack, a card is drawn at random and is found to be a King. Find the probability of the lost card being a King. (3)

    OR

    (b) A biased die is twice as likely to show an even number as an odd number. If such a die is thrown twice, find the probability distribution of the number of sixes. Also, find the mean of the distribution.

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

32. (a) Sketch the graph of \(y=x|x|\) and hence find the area bounded by this curve, X-axis and the ordinates \(x=-2\) and \(x=2,\) using integration. (5)

    OR

    (b) Using integration, find the area bounded by the ellipse \(9x^{2}+25y^{2}=225\), the lines \(x=-2,\) \(x=2\), and the X-axis.

33. (a) Let \(A=R-\{5\}\) and \(B=R-\{1\}\). Consider the function \(f:A\rightarrow B\), defined by \(f(x)=\frac{x-3}{x-5}\). Show that f is one-one and onto. (5)

    OR

    (b) Check whether the relation S in the set of real numbers R defined by \(S=\{(a,b)\): where \(a-b+\sqrt{2}\) is an irrational number is reflexive, symmetric or transitive.

34. If \(A=[\begin{matrix}2&1&-3\\ 3&2&1\\ 1&2&-1\end{matrix}],\) find \(A^{-1}\) and hence solve the following system of equations: \(2x+y-3z=13\), \(3x+2y+z=4\), \(x+2y-z=8\). (5)

35. (a) Find the distance between the line \(\frac{x}{2}=\frac{2y-6}{4}=\frac{1-z}{-1}\) and another line parallel to it passing through the point (4, 0, -5). (5)

    OR

    (b) If the lines \(\frac{x-1}{-3}=\frac{y-2}{2k}=\frac{z-3}{2}\) and \(\frac{x-1}{3k}=\frac{y-1}{1}=\frac{z-6}{-7}\) are perpendicular to each other, find the value of k and hence write the vector equation of a line perpendicular to these two lines and passing through the point (3, -4, 7).

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

36. Case Study 1: Calculator Sales

    A store has been selling calculators at 350 each. A market survey indicates that a reduction in price (p) of calculator increases the number of units (x) sold. The relation between the price and quantity sold is given by the demand function \(p=450-\frac{1}{2}x.\) Based on the above information, answer the following questions :

    1. (i) Determine the number of units (x) that should be sold to maximise the revenue \(R(x)=xp(x)\) Also, verify the result. (2)
    2. (ii) What rebate in price of calculator should the store give to maximise the revenue? (2)
    (4)
37. Case Study 2: Stars in a Constellation

    An instructor at the astronomical centre shows three among the brightest stars in a particular constellation. Assume that the telescope is located at \(O(0,0,0)\) and the three stars have their locations at the points D, A and V having position vectors \(2\hat{i}+3\hat{j}+4\hat{k}\), \(7\hat{i}+5\hat{j}+8\hat{k}\) and \(-3\hat{i}+7\hat{j}+11\hat{k}\) respectively. Based on the above information, answer the following questions :

    1. (i) (Sub-question missing in the provided text. Likely 2 marks.)
    2. (ii) (Sub-question missing in the provided text. Likely 2 marks or split with OR.)
    (4)
38. Case Study 3: [Title Missing]

    (The content for Case Study 3 (Question 38) was not available in the provided text.)

    1. (i) [Missing sub-question]
    2. (ii) [Missing sub-question]
    (4)