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. Let both \(AB^{\prime}\) and \(B^{\prime}A\) be defined for matrices A and B. If order of A is \(n\times m\), then the order of B is:
   1. \(n\times n\)
   2. \(n\times m\)
   3. \(m\times m\)
   4. \(m\times n\)
2. If \(A=\begin{bmatrix}-1&0&0\\ 0&3&0\\ 0&0&5\end{bmatrix},\) then A is a/an:
   1. scalar matrix
   2. identity matrix
   3. symmetric matrix
   4. skew-symmetric matrix
3. The following graph is a combination of:
   1. \(y=\sin^{-1}x\) and \(y=\cos^{-1}x\)
   2. \(y=\cos^{-1}x\) and \(y=\cos x\)
   3. \(y=\sin^{-1}x\) and \(y=\sin x\)
   4. \(y=\cos^{-1}x\) and \(y=\sin x\)
4. Sum of two skew-symmetric matrices of same order is always a/an:
   1. skew-symmetric matrix
   2. symmetric matrix
   3. null matrix
   4. identity matrix
5. \([\sec^{-1}(-\sqrt{2})-\tan^{-1}(\frac{1}{\sqrt{3}})]\) is equal to:
   1. \(\frac{11\pi}{12}\)
   2. \(\frac{5\pi}{12}\)
   3. \(-\frac{5\pi}{12}\)
   4. \(\frac{7\pi}{12}\)
6. If \(f(x)=\begin{cases}\frac{\log(1+ax)+\log(1-bx)}{x},&for~x\ne0\\ k&,for~x=0\end{cases}\) is continuous at \(x=0\), then the value of k is:
   1. a
   2. \(a+b\)
   3. \(a-b\)
   4. b
7. If \(\tan^{-1}(x^{2}-y^{2})=a\), where 'a' is a constant, then \(\frac{dy}{dx}\) is:
   1. \(\frac{x}{y}\)
   2. \(-\frac{x}{y}\)
   3. \(\frac{a}{x}\)
   4. \(\frac{a}{y}\)
8. If \(y=a\cos(\log x)+b\sin(\log x)\), then \(x^{2}y_{2}+xy_{1}\) is:
   1. \(\cot(\log x)\)
   2. y
   3. -y
   4. \(\tan(\log x)\)
9. Let \(f(x)=|x|\), \(x\in R\). Then, which of the following statements is **incorrect**?
   1. f has a minimum value at \(x=0\).
   2. f has no maximum value in R.
   3. f is continuous at \(x=0\).
   4. f is differentiable at \(x=0\).
10. Let \(f^{\prime}(x)=3(x^{2}+2x)-\frac{4}{x^{3}}+5,\) and \(f(1)=0\). Then, \(f(x)\) is:
    1. \(x^{3}+3x^{2}+\frac{2}{x^{2}}+5x+11\)
    2. \(x^{3}+3x^{2}+\frac{2}{x^{2}}+5x-11\)
    3. \(x^{3}+3x^{2}-\frac{2}{x^{2}}+5x-11\)
    4. \(x^{3}-3x^{2}-\frac{2}{x^{2}}+5x-11\)
11. \(\int\frac{x+5}{(x+6)^{2}}e^{x}dx\) is equal to:
    1. \(\log(x+6)+C\)
    2. \(e^{x}+C\)
    3. \(\frac{e^{x}}{x+6}+C\)
    4. \(\frac{-1}{(x+6)^{2}}+C\)
12. The order and degree of the following differential equation are, respectively: \(-\frac{d^{4}y}{dx^{4}}+2e^{dy/dx}+y^{2}=0\)
    1. 4, 1
    2. 4, not defined
    3. 1, 1
    4. 4, 1
13. The solution for the differential equation \(\log(\frac{dy}{dx})=3x+4y\) is:
    1. \(3e^{4y}+4e^{-3x}+C=0\)
    2. \(e^{3x+4y}+C=0\)
    3. \(3e^{-3y}+4e^{4x}+12C=0\)
    4. \(3e^{-4y}+4e^{3x}+12C=0\)
14. For a Linear Programming Problem (LPP), the given objective function is \(Z=x+2y\). The feasible region PQRS determined by the set of constraints is shown as a shaded region in the graph. \(P\equiv(\frac{3}{13},\frac{24}{13})\) \(Q\equiv(\frac{3}{2},\frac{15}{4})\) \(R\equiv(\frac{7}{2},\frac{3}{4})\) \(S\equiv(\frac{18}{7},\frac{2}{7})\). Which of the following statements is correct?
    1. Z is minimum at \(S(\frac{18}{7},\frac{2}{7})\)
    2. Z is maximum at \(R(\frac{7}{2},\frac{3}{4})\)
    3. (Value of Z at P) > (Value of Z at Q)
    4. (Value of Z at Q) < (Value of Z at R)
15. In a Linear Programming Problem (LPP), the objective function \(Z=2x+5y\) is to be maximised under the following constraints: \(x+y\le4\), \(3x+3y\ge18\), \(x, y\ge0\). Study the graph and select the correct option. The solution of the given LPP:
    1. lies in the shaded unbounded region.
    2. lies in \(\Delta AOB\).
    3. does not exist.
    4. lies in the combined region of \(\Delta AOB\) and unbounded shaded region.
16. Let \(|\vec{a}|=5\) and \(-2\le\lambda\le1\). Then, the range of \(|\lambda\vec{a}|\) is:
    1. [5, 10]
    2. [-2, 5]
    3. [2, 1]
    4. [-10, 5]
17. The area of the region bounded by the curve \(y^{2}=x\) between \(x=0\) and \(x=1\) is:
    1. \(\frac{3}{2}\) sq units
    2. \(\frac{2}{3}\) sq units
    3. 3 sq units
    4. \(\frac{4}{3}\) sq units
18. A box has 4 green, 8 blue and 3 red pens. A student picks up a pen at random, checks its colour and replaces it in the box. He repeats this process 3 times. The probability that at least one pen picked was red is:
    1. \(\frac{124}{125}\)
    2. \(\frac{1}{125}\)
    3. \(\frac{61}{125}\)
    4. \(\frac{64}{125}\)

19. Assertion (A): If \(|\vec{a}\times\vec{b}|^{2}+|\vec{a}.\vec{b}|^{2}=256\) and \(|\vec{b}|=8,\) then \(|\vec{a}|=2\).

    Reason (R): \(\sin^{2}\theta+\cos^{2}\theta=1\) and \(|\vec{a}\times\vec{b}|=|\vec{a}||\vec{b}|\sin\theta\) and \(\vec{a}.\vec{b}=|\vec{a}||\vec{b}|\cos\theta.\)

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

20. Assertion (A): Let \(f(x)=e^{x}\) and \(g(x)=\log x\). Then \((f+g)x=e^{x}+\log x\) where domain of \((f+g)\) is R.

    Reason (R): \(Dom(f+g)=Dom(f)\cap Dom(g)\).

    (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.

21. Find the domain of \(f(x)=\sin^{-1}(-x^{2})\). (2)
22. (a) Differentiate \(\sqrt{e^{\sqrt{2x}}}\) with respect to \(e^{\sqrt{2x}}\) for \(x>0.\)
    OR
    (b) If \((x)^{y}=(y)^{x}\), then find \(\frac{dy}{dx}\). (2)
23. Determine the values of x for which \(f(x)=\frac{x-4}{x+1}\), \(x\ne-1\) is an increasing or a decreasing function. (2)
24. (a) If \(\vec{a}\) and \(\vec{b}\) are position vectors of point A and point B respectively, find the position vector of point C on BA produced such that \(BC=3BA\).
    OR
    (b) Vector \(\vec{r}\) is inclined at equal angles to the three axes x, y and z. If magnitude of \(\vec{r}\) is \(5\sqrt{3}\) units, then find \(\vec{r}\) . (2)
25. Determine if the lines \(\vec{r}=(\hat{i}+\hat{j}-\hat{k})+\lambda(3\hat{i}-\hat{j})\) and \(\vec{r}=(4\hat{i}-\hat{k})+\mu(2\hat{i}+3\hat{k})\) intersect with each other. (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.

26. Let \(A=\begin{bmatrix}1\\ 4\\ -2\end{bmatrix}\) and \(C=\begin{bmatrix}3&4&2\\ 12&16&8\\ -6&-8&-4\end{bmatrix}\). Find the matrix B if \(AB=C\). (3)
27. (a) Differentiate \(y=\sin^{-1}(3x-4x^{3})\) w.r.t. x, if \(x\in[-\frac{1}{2},\frac{1}{2}].\)
    OR
    (b) Differentiate \(y=\cos^{-1}(\frac{1-x^{2}}{1+x^{2}})\) with respect to x, when \(x\in(0,1).\) (3)
28. (a) A student wants to pair up natural numbers in such a way that they satisfy the equation \(2x+y=41\), \(x, y\in N\). Find the domain and range of the relation. Check if the relation thus formed is reflexive, symmetric and transitive. Hence, state whether it is an equivalence relation or not.
    OR
    (b) Show that the function \(f:N\rightarrow N\), where N is a set of natural numbers, given by \(f(n) = \begin{cases}n-1,&if~n~is~even\\ n+1,&if~n~is~odd\end{cases}\) is a bijection. (3)
29. Consider the Linear Programming Problem, where the objective function \(Z=(x+4y)\) needs to be minimized subject to constraints \(2x+y\ge1000\), \(x+2y\ge800\), \(x,y\ge0\). Draw a neat graph of the feasible region and find the minimum value of Z. (3)
30. (a) Find the distance of the point \(P(2,4,-1)\) from the line \(\frac{x+5}{1}=\frac{y+3}{4}=\frac{z-6}{-9}\).
    OR
    (b) Let the position vectors of the points A, B and C be \(3\hat{i}-\hat{j}-2\hat{k}\), \(\hat{i}+2\hat{j}-\hat{k}\) and \(\hat{i}+5\hat{j}+3\hat{k}\) respectively. Find the vector and cartesian equations of the line passing through A and parallel to line BC. (3)
31. A person is Head of two independent selection committees I and II. If the probability of making a wrong selection in committee I is 0.03 and that in committee II is 0.01, then find the probability that the person makes the **correct** decision of selection:
    1. in both committees
    2. in only one committee
    (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.

32. (a) Find: \(\int\frac{x^{2}+1}{(x-1)^{2}(x+3)}dx\)
    OR
    (b) Evaluate: \(\int_{0}^{\pi/2}\frac{x}{\sin x+\cos x}dx\) (5)
33. Draw a rough sketch for the curve \(y=2+|x+1|\). Using integration, find the area of the region bounded by the curve \(y=2+|x+1|\), \(x=-4\), \(x=3\) and \(y=0\). (5)
34. (a) Solve the differential equation: \(x^{2}y~dx-(x^{3}+y^{3})dy=0\).
    OR
    (b) Solve the differential equation \((1+x^{2})\frac{dy}{dx}+2xy-4x^{2}=0\) subject to initial condition \(y(0)=0\). (5)
35. Let the polished side of the mirror be along the line \(\frac{x}{1}=\frac{1-y}{-2}=\frac{2z-4}{6}.\) A point \(P(1,6,3)\), some distance away from the mirror, has its image formed behind the mirror. Find the coordinates of the image point and the distance between the point P and its image. (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.

36. Case Study 1: Stationery Purchases (Matrix Method)

    Three students, Neha, Rani and Sam go to a market to purchase stationery items. Neha buys **4 pens, 3 notepads and 2 erasers and pays ₹60**. Rani buys **2 pens, 4 notepads and 6 erasers for ₹90**. Sam pays **₹70 for 6 pens, 2 notepads and 3 erasers**.

    1. Form the equations required to solve the problem of finding the price of each item, and express it in the matrix form \(AX=B\). (1 mark)
    2. Find \(|A|\) and confirm if it is possible to find \(A^{-1}\). (1 mark)
    3. (a) Find \(A^{-1}\), if possible, and write the formula to find X. (2 marks)
       OR
       (b) Find \(A^{2}-8I\) where I is an identity matrix. (2 marks)
    (4)
37. Case Study 2: Ladder Against a Wall (Application of Derivatives)

    A ladder of fixed length 'h' is to be placed along the wall such that it is free to move along the height of the wall.

    1. Express the distance (y) between the wall and foot of the ladder in terms of 'h' and height (x) on the wall at a certain instant. Also, write an expression in terms of h and x for the area (A) of the right triangle, as seen from the side by an observer. (1 mark)
    2. [Part 2 of Q37, 1 mark - Missing content]
    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: [Question Not Provided in Source]

    The third case study question is typically Q38 and carries 4 marks. (The content for this question was not included in the provided text.)

    (4)