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 principal value of \(\sin^{-1}(\sin(-\frac{10\pi}{3}))\) is:
   1. \(-\frac{2\pi}{3}\)
   2. \(-\frac{\pi}{3}\)
   3. \(\frac{\pi}{3}\)
   4. \(\frac{2\pi}{3}\)
2. If A and B are square matrices of same order such that \(AB=A\) and \(BA=B\), then \(A^{2}+B^{2}\) is equal to:
   1. \(A+B\)
   2. \(BA\)
   3. \(2(A+B)\)
   4. \(2BA\)
3. For real x, let \(f(x)=x^{3}+5x+1\). Then:
   1. f is one-one but not onto on R
   2. f is onto on R but not one-one
   3. f is one-one and onto on R
   4. f is neither one-one nor onto on R
4. If \(y=\sin^{-1}x,\) then \((1-x^{2})\frac{d^{2}y}{dx^{2}}\) is equal to:
   1. \(x\frac{dy}{dx}\)
   2. \(-x\frac{dy}{dx}\)
   3. \(x^{2}\frac{dy}{dx}\)
   4. \(-x^{2}\frac{dy}{dx}\)
5. The values of \(\lambda\) so that \(f(x)=\sin x-\cos x-\lambda x+C\) decreases for all real values of x are:
   1. \(1<\lambda<\sqrt{2}\)
   2. \(\lambda\ge1\)
   3. \(\lambda\ge\sqrt{2}\)
   4. \(\lambda<1\)
6. If P is a point on the line segment joining (3, 6, -1) and (6, 2, 2) and y-coordinate of P is 4, then its z-coordinate is:
   1. \(-\frac{3}{2}\)
   2. 0
   3. 1
   4. \(\frac{3}{2}\)
7. If M and N are square matrices of order 3 such that det \((M)=m\) and \(MN=mI,\) then det (N) is equal to:
   1. -1
   2. 1
   3. \(-m^{2}\)
   4. \(m^{2}\)
8. If \(f(x)=\begin{cases}3x-2,&0
9. -4
10. \(-\frac{7}{2}\)
11. -2
12. -1

If \(f:N\rightarrow W\) is defined as \(f(n)=\begin{cases}\frac{n}{2},&if~n~is~even\\ 0,&if~n~is~odd\end{cases}\), then f is:

1. injective only
2. surjective only
3. a bijection
4. neither surjective nor injective

The matrix \([\begin{matrix}0&1&-2\\ -1&0&-7\\ 2&7&0\end{matrix}]\) is a:

1. diagonal matrix
2. symmetric matrix
3. skew symmetric matrix
4. scalar matrix

If the sides AB and AC of \(\triangle ABC\) are represented by vectors \(\hat{j}+\hat{k}\) and \(3\hat{i}-\hat{j}+4\hat{k}\) respectively, then the length of the median through A on BC is:

1. \(2\sqrt{2}\) units
2. \(\sqrt{18}\) units
3. \(\frac{\sqrt{34}}{2}\) units
4. \(\frac{\sqrt{48}}{2}\) units

The function f defined by \(f(x)=\begin{cases}x,&if~x\le1\\ 5,&if~x>1\end{cases}\) is not continuous at:

1. \(x=0\)
2. \(x=1\)
3. \(x=2\)
4. \(x=5\)

If \(f(x)=2x+\cos x\), then f(x):

1. has a maxima at \(x=\pi\)
2. has a minima at \(x=\pi\)
3. is an increasing function
4. is a decreasing function

\(\int\frac{\cos 2x-\cos 2\alpha}{\cos x-\cos \alpha}dx\) is equal to:

1. \(2(\sin x+x\cos\alpha)+C\)
2. \(2(\sin x-x\cos\alpha)+C\)
3. \(2(\sin x+2x\cos\alpha)+C\)
4. \(2(\sin x+\sin\alpha)+C\)

The value of \(\int_{0}^{1}\frac{dx}{e^{x}+e^{-x}}\) is:

1. \(-\frac{\pi}{4}\)
2. \(\frac{\pi}{4}\)
3. \(\tan^{-1}e-\frac{\pi}{4}\)
4. \(\tan^{-1}e\)

The order and degree of the differential equation \((\frac{d^{2}y}{dx^{2}})^{2}+(\frac{dy}{dx})^{2}=x\sin(\frac{dy}{dx})\) are:

1. order 2, degree 2
2. order 2, degree 1
3. order 2, degree not defined
4. order 1, degree not defined

The area of the region enclosed by the curve \(y=\sqrt{x}\) and the lines \(x=0\) and \(x=4\) and x-axis is:

1. \(\frac{16}{9}\) sq. units
2. \(\frac{32}{9}\) sq. units
3. \(\frac{16}{3}\) sq. units
4. \(\frac{32}{3}\) sq. units

The corner points of the feasible region of a Linear Programming Problem are (0, 2), (3, 0), (6, 0), (6, 8) and (0, 5). If \(Z=ax+by;\) (a, \(b>0)\) be the objective function, and maximum value of Z is obtained at (0, 2) and (3, 0), then the relation between a and b is:

1. \(a=b\)
2. \(a=3b\)
3. \(b=6a\)
4. \(3a=2b\)

Assertion (A): If A and B are two events such that \(P(A\cap B)=0\) then A and B are independent events.

Reason (R): Two events are independent if the occurrence of one does not effect the occurrence of the other.

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

Assertion (A): In a Linear Programming Problem, if the feasible region is empty, then the Linear Programming Problem has no solution.

Reason (R): A feasible region is defined as the region that satisfies all the constraints.

(Choose the correct option based on 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. Let A and B be two square matrices of order 3 such that det (A) \(=3\) and det (B) \(=-4\). Find the value of det (-6AB). (2)
22. (a) Find the least value of 'a' so that \(f(x)=2x^{2}-ax+3\) is an increasing function on [2, 4].
    OR
    (b) If \(f(x)=x+\frac{1}{x}, x\ge1\), show that f is an increasing function. (2)
23. (a) Simplify \(\sin^{-1}(\frac{x}{\sqrt{1+x^{2}}}).\)
    OR
    (b) Find domain of \(\sin^{-1}\sqrt{x-1}\). (2)
24. Calculate the area of the region bounded by the curve \(\frac{x^{2}}{9}+\frac{y^{2}}{4}=1\) and the x-axis using integration. (2)
25. For the curve \(y=5x-2x^{3}\) if x increases at the rate of \(2~\text{units/s}\), then how fast is the slope of the curve changing when \(x=2\) ? (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. (a) If \(f:R^{+}\rightarrow R\) is defined as \(f(x)=\log_{a} x\) (\(a > 0\) and \(a\ne1)\), prove that f is a bijection. (\(R^{+}\) is a set of all positive real numbers.)
    OR
    (b) Let \(A=\{1,2,3\}\) and \(B=\{4,5,6\}\). A relation R from A to B is defined as \(R=\{(x,y):x+y=6, x \in A, y\in B\}\).
    1. Write all elements of R.
    2. Is R a function? Justify.
    3. Determine domain and range of R.
    (3)
27. (a) Find k so that \(f(x)=\begin{cases}\frac{x^{2}-2x-3}{x+1},&x\ne-1\\ k,&x=-1\end{cases}\) is continuous at \(x=-1.\)
    OR
    (b) Check the differentiability of function \(f(x)=x|x|\) at \(x=0\). (3)
28. Evaluate: \(\int_{\pi/2}^{\pi}e^{x}(\frac{1-\sin x}{1-\cos x})dx\) (3)
29. (a) Find the probability distribution of the number of boys in families having three children, assuming equal probability for a boy and a girl.
    OR
    (b) A coin is tossed twice. Let X be a random variable defined as number of heads minus number of tails. Obtain the probability distribution of X and also find its mean. (3)
30. Find the distance of the point (-1, -5, -10) from the point of intersection of the lines \(\frac{x-1}{2}=\frac{y-2}{3}=\frac{z-3}{4}\) and \(\frac{x-4}{5}=\frac{y-1}{2}=z.\) (3)
31. Solve the following Linear Programming Problem using graphical method: Maximise \(Z=100x+50y\) subject to the constraints \(3x+y\le600\), \(x+y\le300\), \(y\le x+200\), \(x\ge0\), \(y\ge0\). (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. If A is a \(3\times3\) invertible matrix, show that for any scalar \(k\ne0.\), \((kA)^{-1}=\frac{1}{k}A^{-1}.\) Hence calculate \((3A)^{-1}\) where \(A=[\begin{matrix}2&-1&1\\ -1&2&-1\\ 1&-1&2\end{matrix}].\) (5)
33. The relation between the height of the plant (y cm) with respect to exposure to sunlight is governed by the equation \(y=4x-\frac{1}{2}x^{2},\) where x is the number of days exposed to sunlight.
    1. Find the rate of growth of the plant with respect to sunlight. (2)
    2. In how many days will the plant attain its maximum height ? What is the maximum height ? (3)
    (5)
34. (a) Find: \(\int\frac{\cos x}{(4+\sin^{2}x)(5-4\cos^{2}x)}dx\)
    OR
    (b) Evaluate: \(\int_{0}^{\pi}\frac{dx}{a^{2}\cos^{2}x+b^{2}\sin^{2}x}\) (5)
35. (a) Show that the area of a parallelogram whose diagonals are represented by \(\vec{a}\) and \(\vec{b}\) is given by \(\frac{1}{2}|\vec{a}\times\vec{b}|.\) Also find the area of a parallelogram whose diagonals are \(2\hat{i}-\hat{j}+\hat{k}\) and \(\hat{i}+3\hat{j}-\hat{k}\).
    OR
    (b) Find the equation of a line in vector and cartesian form which passes through the point \((1,2,-4)\) and is perpendicular to the lines \(\frac{x-8}{3}=\frac{y+19}{-16}=\frac{z-10}{7}\) and \(\vec{r}=15\hat{i}+29\hat{j}+5\hat{k}+\mu(3\hat{i}+8\hat{j}-5\hat{k}).\) (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: Decimal Comparison Test/Probability 📊

    Some students are having a misconception while comparing decimals... [Context of javelin throw distances and probability given].

    1. What is the probability of a student not having misconception but still answers Bijoy in the test? (1)
    2. What is the probability that a randomly selected student answers Bijoy as his answer in the test? (1)
    3. (a) What is the probability that a student who answered as Bijoy is having misconception ? (2)
       OR
       (b) What is the probability that a student who answered as Bijoy is amongst students who do not have the misconception ? (2)
    (4)
37. Case Study 2: Metro Rail Network/Lines in 3D 🚇

    An engineer is designing a new metro rail network... The track for Line A is \(l_{1}:\frac{x-2}{3}=\frac{y+1}{-2}=\frac{z-3}{4}\) while the track for Line B is \(l_{2}:\frac{x-1}{2}=\frac{y-3}{1}=\frac{z+2}{-3}.\)

    1. Find whether the two metro tracks are parallel. (1)
    2. Solar panels are to be installed... Determine the equation of the line representing the placement of solar panels on the rooftop of Line A's stations, given that panels are to be positioned parallel to Line A's track (\(l_{1}\)) and pass through the point \((1,-2,-3)\). (1)
    3. (a) To connect the stations, a pedestrian pathway perpendicular to the two metro lines is to be constructed which passes through point (3, 2, 1). Determine the equation of the pedestrian walkway. (2)
       OR
       (b) Find the shortest distance between Line A and Line B. (2)
    (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)