Mathematics · Sample Paper 1

General instructions

Section A · Objective

1. Question 11 mark

   If $A = \{1, 2, 3\}$, the number of elements in its power set is

   1. A.3
   2. B.6
   3. C.8
   4. D.9
2. Question 21 mark

   If $n(A) = 20$, $n(B) = 15$ and $n(A\cap B) = 5$, then $n(A\cup B)$ is

   1. A.30
   2. B.35
   3. C.40
   4. D.25
3. Question 31 mark

   If $A = \{1, 2\}$ and $B = \{a, b, c\}$, the number of relations from A to B is

   1. A.6
   2. B.8
   3. C.32
   4. D.64
4. Question 41 mark

   The range of $f(x) = |x - 1|$ is

   1. A.$\mathbb{R}$
   2. B.$[0, \infty)$
   3. C.$[1, \infty)$
   4. D.$(0, \infty)$
5. Question 51 mark

   The value of $\sin 75^\circ$ is

   1. A.$\dfrac{\sqrt6 + \sqrt2}{4}$
   2. B.$\dfrac{\sqrt6 - \sqrt2}{4}$
   3. C.$\dfrac{\sqrt3 + 1}{2}$
   4. D.$\dfrac{\sqrt3}{2}$
6. Question 61 mark

   $\pi/6$ radians in degrees is

   1. A.$30^\circ$
   2. B.$60^\circ$
   3. C.$45^\circ$
   4. D.$15^\circ$
7. Question 71 mark

   The value of $i^{35}$ is

   1. A.1
   2. B.−1
   3. C.i
   4. D.−i
8. Question 81 mark

   The modulus of $3 - 4i$ is

   1. A.5
   2. B.7
   3. C.1
   4. D.25
9. Question 91 mark

   The solution of $-2x > 6$ is

   1. A.$x > -3$
   2. B.$x < -3$
   3. C.$x > 3$
   4. D.$x < 3$
10. Question 101 mark

    The value of $\,^{8}C_6$ is

    1. A.28
    2. B.56
    3. C.20
    4. D.336
11. Question 111 mark

    The number of arrangements of the letters of the word APPLE is

    1. A.120
    2. B.60
    3. C.24
    4. D.30
12. Question 121 mark

    The number of terms in the expansion of $(x + y)^{10}$ is

    1. A.10
    2. B.11
    3. C.12
    4. D.9
13. Question 131 mark

    The sum of the infinite GP $1 + \tfrac12 + \tfrac14 + \cdots$ is

    1. A.1
    2. B.2
    3. C.3/2
    4. D.∞
14. Question 141 mark

    The slope of the line $3x - 4y + 7 = 0$ is

    1. A.$\tfrac34$
    2. B.$-\tfrac34$
    3. C.$\tfrac43$
    4. D.$-\tfrac43$
15. Question 151 mark

    The focus of the parabola $y^2 = 12x$ is

    1. A.(3, 0)
    2. B.(0, 3)
    3. C.(12, 0)
    4. D.(−3, 0)
16. Question 161 mark

    The distance between (1, 2, 3) and (4, 6, 3) is

    1. A.5
    2. B.7
    3. C.25
    4. D.$\sqrt{13}$
17. Question 171 mark

    $\lim_{x\to0}\dfrac{\sin 3x}{x}$ equals

    1. A.0
    2. B.1
    3. C.3
    4. D.1/3
18. Question 181 mark

    A die is thrown. The probability of getting a prime number is

    1. A.1/2
    2. B.1/3
    3. C.2/3
    4. D.1/6
19. Question 191 mark

    Assertion (A): If every observation is increased by 5, the variance is unchanged.

    Reason (R): Variance depends only on deviations from the mean.

    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): The derivative of $\cos x$ is $-\sin x$.

    Reason (R): The derivative of $\sin x$ is $\cos x$.

    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

   If $U = \{1, 2, \ldots, 9\}$, $A = \{1, 2, 3, 4\}$ and $B = \{2, 4, 6, 8\}$, find $(A\cup B)'$.

2. Question 222 marks

   Solve $x^2 + x + 1 = 0$.

3. Question 232 marks

   Find the 4th term in the expansion of $(x - 2y)^6$.

4. Question 242 marks

   Find the equation of the line through (2, 3) perpendicular to the line 2x − y + 5 = 0.

5. OR

   Question 242 marks

   Find the distance of the point (3, −5) from the line 3x − 4y − 26 = 0.

6. Question 252 marks

   Find the derivative of $f(x) = x^2\sin x$.

Section C · Short answer

1. Question 263 marks

   Prove that $\dfrac{\sin x + \sin 3x}{\cos x + \cos 3x} = \tan 2x$.

2. Question 273 marks

   A committee of 3 men and 2 women is to be chosen from 7 men and 5 women. In how many ways can this be done?

3. Question 283 marks

   The sum of the first n terms of an AP is 3n² + 5n. Find its 20th term and common difference.

4. Question 293 marks

   Find the foci, eccentricity and length of the latus rectum of the ellipse $\dfrac{x^2}{25} + \dfrac{y^2}{9} = 1$.

5. OR

   Question 293 marks

   Find the centre and radius of the circle $x^2 + y^2 - 4x + 6y - 12 = 0$.

6. Question 303 marks

   Solve the system graphically: $x + y \le 5$, $x \ge 1$, $y \ge 0$. Name the corner points of the solution region.

7. Question 313 marks

   Two dice are thrown. Find the probability that (i) the sum is 8, (ii) the sum is a multiple of 5, (iii) both show the same number.

Section D · Long answer

1. Question 325 marks

   Find the mean and variance of: 6, 7, 10, 12, 13, 4, 8, 12.

2. Question 335 marks

   (a) Find the derivative of $\tan x$ from first principles. (b) Evaluate $\lim_{x\to2}\dfrac{x^3 - 8}{x^2 - 4}$.

3. Question 345 marks

   Find the sum to n terms of the series $5 + 55 + 555 + \cdots$.

4. OR

   Question 345 marks

   Using the binomial theorem, expand $(x + \tfrac1x)^6$ and find the term independent of x.

5. Question 355 marks

   If $\cos x = -\tfrac35$ and $x$ lies in the third quadrant, find the values of $\sin x$, $\tan x$, $\sin 2x$, $\cos 2x$ and $\tan 2x$.

Section E · Case study

1. Question 364 marks

   Case study — Sports survey. In a survey of 200 students, 120 play cricket, 90 play football and 70 play hockey. 40 play cricket and football, 30 football and hockey, 50 cricket and hockey, and 20 play all three.

   (i) How many play at least one game? (1 mark)

   (ii) How many play none? (1 mark)

   (iii) How many play exactly one game? (2 marks)

   OR

   (iii) How many play cricket and football but not hockey? (2 marks)

2. Question 374 marks

   Case study — Number plates. A town issues vehicle codes made of 2 different letters (from 26) followed by 3 different digits (from 0–9).

   (i) How many letter pairs are possible? (1 mark)

   (ii) How many 3-digit blocks are possible? (1 mark)

   (iii) How many codes are possible in all? (2 marks)

   OR

   (iii) How many codes end with an even digit? (2 marks)

3. Question 384 marks

   Case study — Lucky draw. A box has 4 red, 5 blue and 6 green tokens. One token is drawn at random.

   (i) Find the probability that it is red. (1 mark)

   (ii) Find the probability that it is not green. (1 mark)

   (iii) Two tokens are drawn together. Find the probability that both are blue. (2 marks)

   OR

   (iii) Two tokens are drawn together. Find the probability that they are of different colours. (2 marks)