Mathematics · Sample Paper 3

General instructions

Section A · Objective

1. Question 11 mark

   The number of reflexive relations on a set with 2 elements is

   1. A.2
   2. B.4
   3. C.8
   4. D.16
2. Question 21 mark

   The principal value of ${\tan }^{-1}(-1)$ is

   1. A.$-\frac{\pi }{4}$
   2. B.$\frac{3\pi }{4}$
   3. C.$\frac{\pi }{4}$
   4. D.$-\frac{3\pi }{4}$
3. Question 31 mark

   If $A$ is a $3\times 2$ matrix, the order of $A^T$ is

   1. A.$3\times 2$
   2. B.$2\times 3$
   3. C.$3\times 3$
   4. D.$2\times 2$
4. Question 41 mark

   If $A=\left[\begin{array}{cc}1& 0\\ 0& -1\end{array}\right]$, then $A^2$ is

   1. A.A
   2. B.I
   3. C.−I
   4. D.O
5. Question 51 mark

   If two rows of a determinant are identical, its value is

   1. A.1
   2. B.0
   3. C.−1
   4. D.undefined
6. Question 61 mark

   The area of a triangle with vertices (0, 0), (4, 0) and (0, 3) using determinants is

   1. A.6
   2. B.12
   3. C.7
   4. D.3.5
7. Question 71 mark

   The derivative of $\cos (x^2)$ is

   1. A.$-\sin (x^2)$
   2. B.$-2x\sin (x^2)$
   3. C.$2x\cos (x^2)$
   4. D.$\sin (x^2)$
8. Question 81 mark

   If $y=e^{2x}$, then $\frac{d^{2}y}{d{x}^2}$ is

   1. A.$2e^{2x}$
   2. B.$4e^{2x}$
   3. C.$e^{2x}$
   4. D.$8e^{2x}$
9. Question 91 mark

   The rate of change of the area of a circle with respect to its radius at r = 5 cm is

   1. A.$10\pi$
   2. B.$25\pi$
   3. C.$5\pi$
   4. D.$20\pi$
10. Question 101 mark

    $\int e^{3x}dx$ equals

    1. A.$e^{3x}+C$
    2. B.$\frac{1}{3}{e}^{3x}+C$
    3. C.$3e^{3x}+C$
    4. D.$\frac{e^{3x+1}}{3x+1}+C$
11. Question 111 mark

    ${\int }_0^1\frac{dx}{1+x^2}$ equals

    1. A.$\frac{\pi }{4}$
    2. B.$\frac{\pi }{2}$
    3. C.1
    4. D.0
12. Question 121 mark

    The order of the differential equation $\frac{d^{3}y}{d{x}^3}+2{\left(\frac{dy}{dx}\right)}^2=0$ is

    1. A.1
    2. B.2
    3. C.3
    4. D.4
13. Question 131 mark

    If $\vec{a}=2\hat{i}+\hat{j}$, then $|\vec{a}|$ is

    1. A.$\sqrt{5}$
    2. B.3
    3. C.5
    4. D.$\sqrt{3}$
14. Question 141 mark

    $\hat{j}\times \hat{k}$ equals

    1. A.$\hat{i}$
    2. B.$-\hat{i}$
    3. C.$\hat{k}$
    4. D.0
15. Question 151 mark

    The direction cosines l, m, n of a line satisfy

    1. A.$l+m+n=1$
    2. B.$l^2+m^2+n^2=1$
    3. C.$lmn=1$
    4. D.$l^2+m^2+n^2=0$
16. Question 161 mark

    In an LPP, the function to be maximised or minimised is the

    1. A.constraint
    2. B.objective function
    3. C.feasible region
    4. D.corner point
17. Question 171 mark

    A coin is tossed three times. The probability of getting exactly two heads is

    1. A.$\frac{3}{8}$
    2. B.$\frac{1}{4}$
    3. C.$\frac{1}{2}$
    4. D.$\frac{1}{8}$
18. Question 181 mark

    If $P(A)=0.5$, $P(B)=0.4$ and $P(A\cap B)=0.2$, then A and B are

    1. A.mutually exclusive
    2. B.independent
    3. C.exhaustive
    4. D.complementary
19. Question 191 mark

    Assertion (A): Every differentiable function is continuous.

    Reason (R): $|x|$ is continuous at 0 but not differentiable there.

    1. A.Both Assertion (A) and Reason (R) are true, and R is the correct explanation of A.
    2. B.Both Assertion (A) and Reason (R) are true, but R is not the correct explanation of A.
    3. C.Assertion (A) is true, but Reason (R) is false.
    4. D.Assertion (A) is false, but Reason (R) is true.
20. Question 201 mark

    Assertion (A): ${\int }_{-a}^{a}f(x)dx=0$ if f is an odd function.

    Reason (R): For an odd function, $f(-x)=-f(x)$.

    1. A.Both Assertion (A) and Reason (R) are true, and R is the correct explanation of A.
    2. B.Both Assertion (A) and Reason (R) are true, but R is not the correct explanation of A.
    3. C.Assertion (A) is true, but Reason (R) is false.
    4. D.Assertion (A) is false, but Reason (R) is true.

Section B · Very short answer

1. Question 212 marks

   Show that the function $f:\mathbb{N}\to \mathbb{N}$, $f(x)=2x$ is one-one but not onto.

2. Question 222 marks

   Evaluate ${\cos }^{-1}\left(\cos \frac{7\pi }{6}\right)$.

3. OR

   Question 222 marks

   Find the value of $\sin \left(\frac{\pi }{3}-{\sin }^{-1}\left(-\frac{1}{2}\right)\right)$.

4. Question 232 marks

   Find $\frac{dy}{dx}$ if $y={\sin }^{-1}(2x\sqrt{1-x^2})$, $-\frac{1}{\sqrt{2}}<x<\frac{1}{\sqrt{2}}$.

5. Question 242 marks

   Find the projection of $\vec{a}=2\hat{i}+3\hat{j}+2\hat{k}$ on $\vec{b}=\hat{i}+2\hat{j}+\hat{k}$.

6. Question 252 marks

   A die is thrown. If E is 'number is greater than 3' and F is 'number is even', find $P(E|F)$.

Section C · Short answer

1. Question 263 marks

   Evaluate $\int x^2\log xdx$.

2. Question 273 marks

   Evaluate $\int \frac{x}{(x-1)(x-2)}dx$.

3. OR

   Question 273 marks

   Evaluate ${\int }_0^{\pi /2}{\sin }^{3}x\cos xdx$.

4. Question 283 marks

   Solve the differential equation $\frac{dy}{dx}-y=e^x$.

5. Question 293 marks

   Maximise $Z=5x+3y$ subject to $3x+5y\le 15$, $5x+2y\le 10$, $x,y\ge 0$.

6. Question 303 marks

   Find the local maximum and minimum values of $f(x)=x^3-3x$.

7. Question 313 marks

   A random variable X has the distribution: X = 1, 2, 3 with P(X) = k, 2k, 3k. Find k, the mean and P(X ≥ 2).

Section D · Long answer

1. Question 325 marks

   Using matrices, solve: $2x+3y+3z=5$, $x-2y+z=-4$, $3x-y-2z=3$.

2. Question 335 marks

   Find the area enclosed between the parabola $y^2=4x$ and the line $y=x$.

3. Question 345 marks

   Find the foot of the perpendicular from the point (1, 2, 3) to the line $\frac{x-6}{3}=\frac{y-7}{2}=\frac{z-7}{-2}$. Also find the perpendicular distance.

4. OR

   Question 345 marks

   Find the angle between the lines $\vec{r}=3\hat{i}+\hat{j}-2\hat{k}+\lambda (\hat{i}-\hat{j}-2\hat{k})$ and $\vec{r}=2\hat{i}-\hat{j}-56\hat{k}+\mu (3\hat{i}-5\hat{j}-4\hat{k})$.

5. Question 355 marks

   Evaluate ${\int }_0^{\pi /2}\log (\sin x)dx$.

Section E · Case study

1. Question 364 marks

   Case study — An open box. A square sheet of tin of side 18 cm is used to make an open box by cutting equal squares of side x cm from each corner and folding up the flaps.

   (i) Express the volume V of the box in terms of x. (1 mark)

   (ii) Find $\frac{dV}{dx}$. (1 mark)

   (iii) Find x for maximum volume and the maximum volume. (2 marks)

   OR

   (iii) Show, using the second derivative, that the volume is maximum at that x. (2 marks)

2. Question 374 marks

   Case study — Exam preparation. In a class, 40% of students study regularly and 60% don't. The probability of passing is 0.9 for regular students and 0.5 for others. A student is picked at random.

   (i) Find the probability that the student studies regularly and passes. (1 mark)

   (ii) Find the probability that the student passes. (1 mark)

   (iii) If the student passed, find the probability that they studied regularly. (2 marks)

   OR

   (iii) If the student failed, find the probability that they did not study regularly. (2 marks)

3. Question 384 marks

   Case study — Bakery planning. A bakery makes cakes and pastries. A cake needs 200 g of flour and 25 g of fat; a pastry needs 100 g of flour and 50 g of fat. The bakery has 5 kg of flour and 1 kg of fat. Profit is ₹50 per cake and ₹30 per pastry. Let x = cakes and y = pastries.

   (i) Write the flour constraint. (1 mark)

   (ii) Write the fat constraint. (1 mark)

   (iii) Find the corner points of the feasible region and the maximum profit. (2 marks)

   OR

   (iii) Is (20, 10) feasible? What is the profit there? (2 marks)