Mathematics · Sample Paper 3

General instructions

Section A · Objective

1. Question 11 mark

   The number of subsets of a set with 4 elements is

   1. A.8
   2. B.16
   3. C.4
   4. D.15
2. Question 21 mark

   If $A\subset B$, then $A\cap B$ equals

   1. A.A
   2. B.B
   3. C.$\varphi$
   4. D.$A\cup B$
3. Question 31 mark

   If $f(x)=2x-3$, then $f(4)$ is

   1. A.5
   2. B.8
   3. C.11
   4. D.1
4. Question 41 mark

   The range of $f(x)=x^2$, $x\in \mathbb{R}$, is

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

   $\sin (-30^{\circ })$ equals

   1. A.$\frac{1}{2}$
   2. B.$-\frac{1}{2}$
   3. C.$\frac{\sqrt{3}}{2}$
   4. D.$-\frac{\sqrt{3}}{2}$
6. Question 61 mark

   The value of $\cos 105^{\circ }$ is

   1. A.$\frac{\sqrt{2}-\sqrt{6}}{4}$
   2. B.$\frac{\sqrt{6}-\sqrt{2}}{4}$
   3. C.$\frac{\sqrt{6}+\sqrt{2}}{4}$
   4. D.$-\frac{\sqrt{6}+\sqrt{2}}{4}$
7. Question 71 mark

   $i^4+i^5+i^6+i^7$ equals

   1. A.0
   2. B.1
   3. C.i
   4. D.−1
8. Question 81 mark

   The roots of $x^2+2=0$ are

   1. A.$\pm \sqrt{2}$
   2. B.$\pm i\sqrt{2}$
   3. C.$\pm 2i$
   4. D.$\pm 2$
9. Question 91 mark

   The solution of $|x|<3$ is

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

    ${}^n{C}_r={}^n{C}_{n-r}$ is true for

    1. A.only r = 0
    2. B.all valid r
    3. C.only n even
    4. D.never
11. Question 111 mark

    The number of 3-digit numbers that can be formed from the digits 1–5 without repetition is

    1. A.60
    2. B.125
    3. C.10
    4. D.20
12. Question 121 mark

    The middle term in the expansion of $(x+y{)}^8$ is the

    1. A.4th
    2. B.5th
    3. C.6th
    4. D.9th
13. Question 131 mark

    The 10th term of the AP 2, 5, 8, … is

    1. A.29
    2. B.32
    3. C.27
    4. D.30
14. Question 141 mark

    The lines $2x+3y=5$ and $3x-2y=7$ are

    1. A.parallel
    2. B.perpendicular
    3. C.coincident
    4. D.inclined at $45^{\circ }$
15. Question 151 mark

    The centre of the circle $(x-2{)}^2+(y+3{)}^2=16$ is

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

    The coordinates of a point on the z-axis are of the form

    1. A.(x, 0, 0)
    2. B.(0, y, 0)
    3. C.(0, 0, z)
    4. D.(x, y, 0)
17. Question 171 mark

    The derivative of $x^5$ is

    1. A.$5x^4$
    2. B.$x^4$
    3. C.$5x^5$
    4. D.$4x^5$
18. Question 181 mark

    A coin is tossed twice. The probability of getting at least one tail is

    1. A.$\frac{1}{4}$
    2. B.$\frac{1}{2}$
    3. C.$\frac{3}{4}$
    4. D.1
19. Question 191 mark

    Assertion (A): The arithmetic mean of 4 and 16 is greater than their geometric mean.

    Reason (R): For positive unequal numbers, AM > GM.

    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): Events A and B with $P(A\cap B)=0$ are mutually exclusive.

    Reason (R): If A and B are mutually exclusive, $P(A\cup B)=P(A)+P(B)$.

    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 $A=\{1,3,5,7\}$ and $B=\{3,4,5,6\}$, find $A-B$ and $B-A$.

2. Question 222 marks

   Find the value of $\tan \frac{13\pi }{12}$.

3. Question 232 marks

   Find the multiplicative inverse of $4-3i$.

4. OR

   Question 232 marks

   If ${}^n{C}_8={}^n{C}_2$, find ${}^n{C}_2$.

5. Question 242 marks

   Find the equation of the line passing through (−2, 3) with slope −4.

6. Question 252 marks

   Evaluate ${\lim }_{x\to 1}\frac{x^{15}-1}{x^{10}-1}$.

Section C · Short answer

1. Question 263 marks

   Let $A=\{1,2,3,4,5,6\}$. Define a relation R from A to A by $R=\{(x,y):y=x+1\}$. Write R in roster form, and find its domain and range.

2. Question 273 marks

   Prove that $\frac{\cos 7x+\cos 5x}{\sin 7x-\sin 5x}=\cot x$.

3. Question 283 marks

   Using the binomial theorem, evaluate $(102{)}^5$.

4. OR

   Question 283 marks

   The 5th, 8th and 11th terms of a GP are p, q and s. Show that $q^2=ps$.

5. Question 293 marks

   Solve and represent the solution on a number line: $\frac{2x-1}{3}\ge \frac{3x-2}{4}-\frac{2-x}{5}$.

6. Question 303 marks

   Find the equation of the parabola with vertex at the origin, axis along the x-axis and passing through (2, 3).

7. Question 313 marks

   Find the point on the y-axis which is equidistant from A(3, 1, 2) and B(5, 5, 2).

Section D · Long answer

1. Question 325 marks

   Calculate the mean deviation about the mean for the following data:

   | x | 5 | 10 | 15 | 20 | 25 | |---|---|---|---|---|---| | f | 7 | 4 | 6 | 3 | 5 |

2. Question 335 marks

   In how many ways can a team of 11 players be chosen from 15 players if (i) a particular player is always included (ii) two particular players are always excluded (iii) the captain and the vice-captain must be chosen from the selected 11?

3. OR

   Question 335 marks

   How many words, with or without meaning, can be formed from the letters of the word DAUGHTER so that (i) all vowels occur together (ii) all vowels do not occur together?

4. Question 345 marks

   Find the equations of the lines passing through (1, 2) that make an angle of $45^{\circ }$ with the line $y=3x+7$.

5. Question 355 marks

   (a) Find the derivative of $f(x)=\frac{1}{x}$ from first principles. (b) Differentiate $\frac{\sin x+\cos x}{\sin x-\cos x}$.

Section E · Case study

1. Question 364 marks

   Case study — Savings plan. Arjun saves ₹100 in the first month and doubles his saving every month for 6 months. His sister Meera saves ₹500 in the first month and increases her saving by ₹200 every month.

   (i) How much does Arjun save in the 6th month? (1 mark)

   (ii) How much does Meera save in the 6th month? (1 mark)

   (iii) Who saves more in total over the 6 months, and by how much? (2 marks)

   OR

   (iii) In which month does Arjun's monthly saving first exceed Meera's? (2 marks)

2. Question 374 marks

   Case study — Quiz contest. In a quiz, a contestant draws one card at random from a box containing cards numbered 1 to 20. She wins a prize if the number is a multiple of 3 or 5.

   (i) How many outcomes are there in the sample space? (1 mark)

   (ii) How many numbers are multiples of 3? (1 mark)

   (iii) Find the probability that she wins a prize. (2 marks)

   OR

   (iii) Find the probability that the number is a prime. (2 marks)

3. Question 384 marks

   Case study — The whispering gallery. An elliptical room has its two foci 8 m apart. A person whispering at one focus can be heard clearly at the other. The length of the room (major axis) is 10 m.

   (i) Find the value of a (semi-major axis). (1 mark)

   (ii) Find the value of c (distance from centre to focus). (1 mark)

   (iii) Find the width of the room (minor axis) and the equation of the ellipse with the centre at the origin. (2 marks)

   OR

   (iii) Find the eccentricity and the length of the latus rectum. (2 marks)