Mathematics · Sample Paper 1

General instructions

Section A · Objective

1. Question 11 mark

   The relation $R = \{(1, 1), (2, 2), (3, 3)\}$ on $\{1, 2, 3\}$ is

   1. A.only reflexive
   2. B.an equivalence relation
   3. C.only symmetric
   4. D.not transitive
2. Question 21 mark

   The principal value of $\sin^{-1}\left(-\tfrac12\right)$ is

   1. A.$-\dfrac\pi6$
   2. B.$\dfrac{5\pi}6$
   3. C.$\dfrac{7\pi}6$
   4. D.$\dfrac\pi6$
3. Question 31 mark

   If A is a $2\times3$ matrix and B is a $3\times4$ matrix, the order of AB is

   1. A.$3\times3$
   2. B.$2\times4$
   3. C.$4\times2$
   4. D.not defined
4. Question 41 mark

   If $A$ is skew-symmetric, then its diagonal elements are

   1. A.1
   2. B.0
   3. C.any real numbers
   4. D.equal
5. Question 51 mark

   If $|A| = 4$ for a $3\times3$ matrix A, then $|2A|$ is

   1. A.8
   2. B.16
   3. C.32
   4. D.64
6. Question 61 mark

   If $|A| = 5$ for a $3\times3$ matrix, $|\text{adj}\,A|$ is

   1. A.5
   2. B.25
   3. C.125
   4. D.15
7. Question 71 mark

   The derivative of $e^{\sin x}$ is

   1. A.$e^{\sin x}$
   2. B.$\cos x\,e^{\sin x}$
   3. C.$\sin x\,e^{\cos x}$
   4. D.$e^{\cos x}$
8. Question 81 mark

   $f(x) = |x|$ at $x = 0$ is

   1. A.continuous and differentiable
   2. B.continuous but not differentiable
   3. C.neither
   4. D.differentiable only
9. Question 91 mark

   The function $f(x) = x^3$ is

   1. A.increasing on ℝ
   2. B.decreasing on ℝ
   3. C.constant
   4. D.decreasing for x > 0
10. Question 101 mark

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

    1. A.$\log(1 + x^2) + C$
    2. B.$\tan^{-1}x + C$
    3. C.$\sin^{-1}x + C$
    4. D.$\dfrac{1}{x} + C$
11. Question 111 mark

    $\int_{-1}^{1} x^3\,dx$ equals

    1. A.0
    2. B.1/2
    3. C.1
    4. D.2
12. Question 121 mark

    The order and degree of $\left(\dfrac{d^2y}{dx^2}\right)^3 + \dfrac{dy}{dx} = 0$ are

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

    If $\vec a\cdot\vec b = 0$ for non-zero vectors, the vectors are

    1. A.parallel
    2. B.perpendicular
    3. C.equal
    4. D.collinear
14. Question 141 mark

    $|\hat i\times\hat j|$ equals

    1. A.0
    2. B.1
    3. C.$\sqrt2$
    4. D.−1
15. Question 151 mark

    The direction cosines of the z-axis are

    1. A.(1, 0, 0)
    2. B.(0, 1, 0)
    3. C.(0, 0, 1)
    4. D.(1, 1, 1)
16. Question 161 mark

    In an LPP, the optimal value of the objective function (for a bounded region) occurs at

    1. A.the origin
    2. B.a corner point
    3. C.the centre
    4. D.any interior point
17. Question 171 mark

    If $P(A) = 0.4$, $P(B) = 0.5$ and A, B are independent, $P(A\cap B)$ is

    1. A.0.9
    2. B.0.2
    3. C.0.1
    4. D.0.45
18. Question 181 mark

    If $P(A) = 0.6$, $P(B) = 0.5$ and $P(A\cap B) = 0.3$, then $P(A|B)$ is

    1. A.0.5
    2. B.0.6
    3. C.0.3
    4. D.0.18
19. Question 191 mark

    Assertion (A): $f(x) = x^2 - 4x$ has a local minimum at $x = 2$.

    Reason (R): $f'(2) = 0$ and $f''(2) > 0$.

    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): $\sin^{-1}x + \cos^{-1}x = \dfrac\pi2$ for all $x\in[-1, 1]$.

    Reason (R): $\tan^{-1}x + \cot^{-1}x = \dfrac\pi2$ for all real $x$.

    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 $f:\mathbb{R}\to\mathbb{R}$, $f(x) = 3x + 2$ is one-one and onto.

2. Question 222 marks

   Find the value of $\tan^{-1}(1) + \cos^{-1}\left(-\tfrac12\right)$.

3. Question 232 marks

   Find $\dfrac{dy}{dx}$ if $x = a\cos t$, $y = a\sin t$.

4. OR

   Question 232 marks

   Find $\dfrac{dy}{dx}$ if $y = x^x$.

5. Question 242 marks

   Find the angle between $\vec a = \hat i + \hat j$ and $\vec b = \hat j + \hat k$.

6. Question 252 marks

   The radius of a circle is increasing at 0.5 cm/s. How fast is its area increasing when the radius is 4 cm?

Section C · Short answer

1. Question 263 marks

   Evaluate $\int x e^x\,dx$.

2. Question 273 marks

   Evaluate $\int\dfrac{dx}{(x + 1)(x + 2)}$.

3. Question 283 marks

   Solve $\dfrac{dy}{dx} = \dfrac{1 + y^2}{1 + x^2}$.

4. OR

   Question 283 marks

   Solve $\dfrac{dy}{dx} + \dfrac yx = x^2$.

5. Question 293 marks

   Bag I has 3 red and 4 black balls; bag II has 5 red and 6 black balls. A bag is chosen at random and a ball drawn is red. Find the probability that it came from bag I.

6. Question 303 marks

   Maximise $Z = 3x + 2y$ subject to $x + y \le 4$, $x \le 3$, $x, y \ge 0$.

7. Question 313 marks

   Find the area of the region bounded by $y = x^2$, the x-axis and the lines $x = 1$ and $x = 3$.

Section D · Long answer

1. Question 325 marks

   Solve by the matrix method: $x + y + z = 6$, $y + 3z = 11$, $x - 2y + z = 0$.

2. Question 335 marks

   An open box with a square base is to be made from 108 m² of sheet metal. Find the dimensions that give the maximum volume.

3. OR

   Question 335 marks

   Find the area of the region enclosed by the ellipse $\dfrac{x^2}{16} + \dfrac{y^2}{9} = 1$.

4. Question 345 marks

   Find the shortest distance between the lines $\vec r = \hat i + \hat j + \lambda(2\hat i - \hat j + \hat k)$ and $\vec r = 2\hat i + \hat j - \hat k + \mu(3\hat i - 5\hat j + 2\hat k)$.

5. Question 355 marks

   Evaluate $\int_0^{\pi/2}\dfrac{\sin x}{\sin x + \cos x}\,dx$.

Section E · Case study

1. Question 364 marks

   Case study — Profit from a product. A company's profit (in ₹ lakh) from selling x thousand units is $P(x) = -5x^2 + 125x + 37\,500$.

   (i) Find $P'(x)$. (1 mark)

   (ii) For what x is the profit maximum? (1 mark)

   (iii) Find the maximum profit. (2 marks)

   OR

   (iii) On which interval is the profit increasing? (2 marks)

2. Question 374 marks

   Case study — Quality check. A factory has two machines. Machine A makes 60% of the bolts and machine B 40%. 2% of A's bolts and 3% of B's bolts are defective.

   (i) Find the probability that a randomly chosen bolt is defective. (2 marks)

   (ii) A bolt is found defective. Find the probability that it was made by machine B. (2 marks)

   OR

   (ii) Find the probability that a randomly chosen bolt is from A and not defective. (2 marks)

3. Question 384 marks

   Case study — School fair stalls. Three stalls sell pens, notebooks and files. Stall 1 sells 2 pens, 3 notebooks, 1 file; stall 2 sells 1 pen, 2 notebooks, 3 files; stall 3 sells 3 pens, 1 notebook, 2 files. Prices are ₹10 per pen, ₹40 per notebook and ₹20 per file.

   (i) Write the sales matrix S (3×3) and price matrix P (3×1). (1 mark)

   (ii) Find the takings of each stall using SP. (2 marks)

   (iii) Find the total takings. (1 mark)

   OR

   (iii) Is S symmetric? Justify. (1 mark)