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. The given graph illustrates:
   Graph showing an inverse trigonometric function with domain R and range $(-\pi/2, \pi/2)$.
   1. \(y=\tan^{-1}x\)
   2. \(y=\operatorname{cosec}^{-1}x\)
   3. \(y=\cot^{-1}x\)
   4. \(y=\sec^{-1}x\)
2. Domain of \(f(x)=\cos^{-1}x+\sin x\) is :
   1. R
   2. \((-1, 1)\)
   3. \([-1, 1]\)
   4. \([-\pi/2, \pi/2]\)
3. What is the total number of possible matrices of order \(3\times3\) with each entry as \(\sqrt{2}\) or \(\sqrt{3}\)?
   1. 9
   2. 512
   3. 615
   4. 64
4. The matrix \(A=\begin{bmatrix}\sqrt{3}&0&0\\ 0&\sqrt{2}&0\\ 0&0&\sqrt{5}\end{bmatrix}\) is a/an:
   1. scalar matrix
   2. identity matrix
   3. null matrix
   4. symmetric matrix
5. If A and B are two square matrices each of order 3 with \(|A|=3\) and \(|B|=5\), then \(|2AB|\) is:
   1. 30
   2. 120
   3. 15
   4. 225
6. Let A be a square matrix of order 3. If \(|A|=5\), then \(|\operatorname{adj} A|\) is:
   1. 5
   2. 125
   3. 25
   4. -5
7. If \(\begin{bmatrix}2x-1&3x\\ 0&y^{2}-1\end{bmatrix}=\begin{bmatrix}x+3&12\\ 0&35\end{bmatrix},\) then the value of \((x-y)\) is :
   1. 2 or 10
   2. 2 or 10
   3. 2 or - 10
   4. -2 or - 10
8. If \( f(x) = \begin{cases} 1, & \text{if } x \leq 3 \\ ax + b, & \text{if } 3 < x < 5 \\ 7, & \text{if } x \geq 5 \end{cases} \) is continuous for all real numbers, then find the values of \(a\) and \(b\):
   1. \(a=3\), \(b=-8\)
   2. \(a=3\), \(b=8\)
   3. \(a=-3\), \(b=-8\)
   4. \(a=-3\), \(b=8\)
9. If \(f(x)=-2x^{8}\) then the correct statement is :
   1. \(f^{\prime}(\frac{1}{2})=f^{\prime}(-\frac{1}{2})\)
   2. \(f^{\prime}(\frac{1}{2})=-f^{\prime}(-\frac{1}{2})\)
   3. \(-f^{\prime}(\frac{1}{2})=f(-\frac{1}{2})\)
   4. \(f(\frac{1}{2})=-f(-\frac{1}{2})\)
10. A spherical ball has a variable diameter \(\frac{5}{2}(3x+1).\) The rate of change of its volume w.r.t. x, when \(x=1\), is :
    1. \(225\pi\)
    2. \(300\pi\)
    3. \(375\pi\)
    4. \(125\pi\)
11. If \(f:R\rightarrow R\) is defined as \(f(x)=2x-\sin x,\) then f is:
    1. a decreasing function
    2. an increasing function
    3. maximum at \(x=\frac{\pi}{2}\)
    4. maximum at \(x=0\)
12. \(\int\frac{e^{9\log x}-e^{8\log x}}{e^{6\log x}-e^{5\log x}}dx\) is equal to:
    1. \(x+C\)
    2. \(\frac{x^{2}}{2}+C\)
    3. \(\frac{x^{4}}{4}+C\)
    4. \(\frac{x^{3}}{3}+C\)
13. For a function \(f(x)\) which of the following holds true?
    1. \(\int_{a}^{b}f(x)dx=\int_{a}^{b}f(a+b-x)dx\)
    2. \(\int_{-a}^{a}f(x)dx=0,\) if f is an even function
    3. \(\int_{-a}^{a}f(x)dx=2\int_{0}^{a}f(x)dx,\) if f is an odd function
    4. \(\int_{0}^{2a}f(x)dx=\int_{0}^{a}f(x)dx-\int_{0}^{a}f(2a+x)dx\)
14. \(\int\frac{e^{x}}{\sqrt{4-e^{2x}}}dx\) is equal to:
    1. \(\frac{1}{2}\cos^{-1}(e^{x})+C\)
    2. \(\frac{1}{2}\sin^{-1}(e^{x})+C\)
    3. \(\frac{e^{x}}{2}+C\)
    4. \(\sin^{-1}(\frac{e^{x}}{2})+C\)
15. A student tries to tie ropes, parallel to each other from one end of the wall to the other. If one rope is along the vector \(3\hat{i}+15\hat{j}+6\hat{k}\) and the other is along the vector \(2\hat{i}+10\hat{j}+\lambda\hat{k}\), then the value of \(\lambda\) is :
    1. 6
    2. 1
    3. \(\frac{1}{4}\)
    4. 4
16. If \(|\vec{a}+\vec{b}|=|\vec{a}-\vec{b}|\) for any two vectors, then vectors \(\vec{a}\) and \(\vec{b}\) are:
    1. orthogonal vectors
    2. parallel to each other
    3. unit vectors
    4. collinear vectors
17. If \(P(A)=\frac{1}{7}\), \(P(B)=\frac{5}{7}\) and \(P(A\cap B)=\frac{4}{7},\) then \(P(\overline{A}|B)\) is:
    1. \(\frac{6}{7}\)
    2. \(\frac{3}{4}\)
    3. \(\frac{4}{5}\)
    4. \(\frac{1}{5}\)
18. A coin is tossed and a card is selected at random from a well shuffled pack of 52 playing cards. The probability of getting head on the coin and a face card from the pack is :
    1. \(\frac{2}{13}\)
    2. \(\frac{3}{26}\)
    3. \(\frac{19}{26}\)
    4. \(\frac{3}{13}\)

19. Assertion (A): \(f(x)=\begin{cases}x\sin\frac{1}{x},&x\ne0\\ 0&,x=0\end{cases}\) is continuous at \(x=0\).

    Reason (R): When \(x\rightarrow0\), \(\sin\frac{1}{x}\) is a finite value between -1 and 1.

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

20. Assertion (A): Set of values of \(\sec^{-1}(\frac{\sqrt{3}}{2})\) is a null set.

    Reason (R): \(\sec^{-1}x\) is defined for \(x\in R-(-1,1).\)

    (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. Let \(f:A\rightarrow B\) be defined by \(f(x)=\frac{x-2}{x-3}\) ' where \(A=R-\{3\}\) and \(B=R-\{1\}\). Discuss the bijectivity of the function. (2)
22. If \(A=\begin{bmatrix}2&3\\ -1&2\end{bmatrix}\), then show that \(A^{2}-4A+7I=0\). (2)
23. (a) Differentiate \((\frac{5^{x}}{x^{5}})\) with respect to x.
    OR
    (b) If \(-2x^{2}-5xy+y^{3}=76\), then find \(\frac{dy}{dx}\). (2)
24. In a Linear Programming Problem, the objective function \(Z=5x+4y\) needs to be maximised under constraints \(3x+y\le6\), \(x\le1\), \(x, y\ge0\). Express the LPP on the graph and shade the feasible region and mark the corner points. (2)
25. (a) 10 identical blocks are marked with '0' on two of them, '1' on three of them, '2' on four of them and '3' on one of them and put in a box. If X denotes the number written on the block, then write the probability distribution of X and calculate its mean.
    OR
    (b) In a village of 8000 people, 3000 go out of the village to work and 4000... (Question is incomplete in source document) (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) Show that the function \(f:R\rightarrow R\) defined by \(f(x)=4x^{3}-5\), \(\forall x\in R\) is one-one and onto.
    OR
    (b) Let R be a relation defined on a set N of natural numbers such that \(R=\{(x,y)\): xy is a square of a natural number, \(x, y\in N\}\). Determine if the relation R is an equivalence relation. (3)
27. (a) Let \(2x+5y-1=0\) and \(3x+2y-7=0\) represent the equations of two lines on which the ants are moving on the ground. Using matrix method, find a point common to the paths of the ants.
    OR
    (b) A shopkeeper sells 50 Chemistry, 60 Physics and 35 Maths books on day I and sells 40 Chemistry, 45 Physics and 50 Maths books on day II. If the selling price for each such subject book is ₹150 (Chemistry), ₹ 175 (Physics) and ₹ 180 (Maths), then find his total sale in two days, using matrix method. If cost price of all the books together is 35,000, what profit did he earn after the sale of two days? (3)
28. Differentiate \(y=\sqrt{\log\{\sin(\frac{x^{3}}{3}-1)\}}\) with respect to x. (3)
29. Amongst all pairs of positive integers with product as 289, find which of the two numbers add up to the least. (3)
30. In the Linear Programming Problem for objective function \(Z=18x+10y\) subject to constraints \(4x+y\ge20\), \(2x+3y\ge30\), \(x,y\ge0\) find the minimum value of Z. (3)
31. (a) The scalar product of the vector \(\vec{a}=\hat{i}-\hat{j}+2\hat{k}\) with a unit vector along sum of vectors \(\vec{b}=2\hat{i}-4\hat{j}+5\hat{k}\) and \(\vec{c}=\lambda\hat{i}-2\hat{j}-3\hat{k}\) is equal to 1. Find the value of \(\lambda\).
    OR
    (b) Find the shortest distance between the lines: \(\vec{r}=(2\hat{i}-\hat{j}+3\hat{k})+\lambda(\hat{i}-2\hat{j}+3\hat{k})\) and \(\vec{r}=(\hat{i}+4\hat{k})+\mu(3\hat{i}-6\hat{j}+9\hat{k}).\) (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) Find the area of the region bounded by \(x^{2}+y^{2}=25\) and \(x=3\) using integration.
    OR
    (b) Find the area of the region bounded by the curve \(y=|\cos x+\sin x|\), x-axis and the lines \(x=0\) and \(x=\pi\) using integration. (5)
33. (a) Find the equation of the plane passing through the points \((3, 1, 1)\), \((0, 2, 4)\) and \((-2, 1, -1)\).
    OR
    (b) A girl discovered the scratch passing through the origin inclined at an angle \(\frac{\pi}{4}\) anticlockwise along the positive direction of x-axis. Find the area of the region enclosed by the x-axis, the scratch and the circular table top in the first quadrant, using integration. (5)
34. Solve the differential equation \(\frac{dy}{dx}=\cos x-2y.\) (5)
35. (a) Find the point Q on the line \(\frac{2x+4}{6}=\frac{y+1}{2}=\frac{-2z+6}{-4}\) at a distance of \(3\sqrt{2}\) from the point \(P(1,2,3)\).
    OR
    (b) Find the image of the point \((-1,5,2)\) in the line \(\frac{2x-4}{2}=\frac{y}{2}=\frac{2-z}{3}\). Find the length of the line segment joining the points (given point and the image point). (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: Three friends A, B and C (Vector Algebra)

    Three friends A, B and C move out from the same location O at the same time in three different directions to reach their destinations. They move out on straight paths and decide that A and B after reaching their destinations will meet up with C at his predecided destination, following straight paths from A to C and B to C in such a way that \(\vec{OA}=\vec{a}\), \(\vec{OB}=\vec{b}\) and \(\vec{OC}=5\vec{a}-2\vec{b}\) respectively.

    Diagram of vectors OA, OB, and OC

    Based upon the above information, answer the following questions :

    1. Complete the given figure to explain their entire movement plan along the respective vectors. (1 mark)
    2. Find vectors \(\vec{AC}\) and \(\vec{BC}\). (1 mark)
    3. (a) If \(|\vec{a}|=3\) and the distance that from O to B is 2 km, then find the angle between \(\vec{OA}\) and \(\vec{OB}\). Also, find \(|\vec{a}\times\vec{b}|\). (2 marks)
       OR
       (b) If \(\vec{a}=2\hat{i}-\hat{j}+4\hat{k}\) and \(\vec{b}=\hat{j}-\hat{k}\), then find a unit vector perpendicular to \((\vec{a}+\vec{b})\) and \((\vec{a}-\vec{b}).\) (2 marks)
    (4)
37. Case Study 2: Camphor Tablet (Differential Equation)

    Camphor is a waxy, colourless solid with strong aroma that evaporates through the process of sublimation, if left in the open at room temperature.

    Image of Cylindrical-shaped Camphor tablets
    A cylindrical camphor tablet whose height is equal to its radius (r) evaporates when exposed to air such that the rate of reduction of its volume is proportional to its total surface area. Thus, \(\frac{dV}{dt}=kS\) is the differential equation, where V is the volume, S is the surface area and t is the time in hours.

    Based upon the above information, answer the following questions :

    1. Write the order and degree of the given differential equation. (1 mark)
    2. Substituting \(V=\pi r^{3}\) and \(S=2\pi r^{2}\), we get the differential equation \(\frac{dr}{dt}=\frac{2}{3}k.\) Solve it, given that r(0)... (Question is incomplete in source document) (1 mark)
    3. (a) [Part 3(a) of Q37, 2 marks - Missing content]
       OR
       (b) [Part 3(b) of Q37, 2 marks - Missing content]
    (4)
38. Case Study 3: Probability (Question is mostly missing from source)

    The preamble starts: A doctor has been appointed in a village...

    Based upon the above information, answer the following questions :

    1. Find the probability that the doctor visits: (Question is incomplete in source document) (1 mark)
    2. [Part 2 of Q38, 1 mark - Missing content]
    3. (a) [Part 3(a) of Q38, 2 marks - Missing content]
       OR
       (b) [Part 3(b) of Q38, 2 marks - Missing content]
    (4)