TIME: 3 hours MAX.

MARKS: 80

General Instructions:

Read the following instructions carefully and follow them:

1. This question paper contains 38 questions.
2. This Question Paper is divided into 5 Sections A, B, C, D and E.
3. In Section A, Questions no. 1-18 are multiple choice questions (MCQs) and questions no. 19 and 20 are
Assertion- Reason based questions of 1 mark each.
4. In Section B, Questions no. 21-25 are very short answer (VSA) type questions, carrying 02 marks each.
5. In Section C, Questions no. 26-31 are short answer (SA) type questions, carrying 03 marks each.
6. In Section D, Questions no. 32-35 are long answer (LA) type questions, carrying 05 marks each.
7. In Section E, Questions no. 36-38 are case study based questions carrying 4 marks each with sub parts
of the values of 1, 1 and 2 marks each respectively.
8. All Questions are compulsory. However, an internal choice in 2 Questions of section B, 2 Questions of
section C and 2 Questions of section D has been provided. And internal choice has been provided in all
the 2 marks questions of Section E.
9. Draw neat and clean figures wherever required.
10. Take π =22/7 wherever required if not stated.
11. Use of calculators is not allowed.

Section A

1. (sec A + tan A) (1 – sin A) equals:

(a) sec A (b) sin A (c) cosec A (d) cos A

2. 2 cards of hearts and 4 cards of spades are missing from a pack of 52 cards. A card is drawn at random
from the remaining pack. What is the probability of getting a black card? (a) 22/52 (b) 22/46 (c) 24/52 (d)
24/56

3. Which of the following is a quadratic equation?

2
(A) x2 + 2x + 1 = (4 – x)2 + 3 (B) –2x2 = (5 – x)(2x- 5)

(C) (k + 1) x2 + (3/2) x = 7, where k = –1 (D) x3 – x2 = (x – 1)3

4. If two positive integers p and q can be expressed as p = ab2 and q = a3 b; a, b being prime numbers, then

LCM (p, q) is

(A) ab (B) a2 b2 (C) a3 b2 (D) a3 b3

5. If a pole 6 m high casts a shadow 2 √3m long on the ground, then the Sun’s elevation is (a) 60° (b) 45°
(c) 30° (d) 90°

6. If △ 𝐴𝐵𝐶 ∼△ 𝑃𝑄𝑅 such that 3AB = 2PQ and BC=10 cm, then length QR is equal to (A) 10 cm (B) 15 cm
(C) 6.67 cm (D) 30 cm

7. Which of the following cannot be the probability of an event? (A) 0.4 (B) 4% (C) 0.04% (D) 4

8. In figure, if ∠BAC =90° and AD⊥BC. Then,

(a) BD.CD = BZC² (b) AB.AC = BC² (c) BD.CD=AD² (d) AB.AC =AD²

9. If ΔABC ~ ΔEDF and ΔABC is not similar to ΔDEF, then which of the following is not true?

(a) BC · EF = AC · FD (b) AB · EF = AC · DE (c) BC · DE = AB · EF (d) BC · DE = AB · FD

10. If P(A) denotes the probability of an event A, then (A) P(A) < 0 (B) P(A) > 1 (C) 0 ≤ P(A) ≤ 1 (D) –1 ≤
P(A) ≤ 1

11. A girl calculates that the probability of her winning the first prize in a lottery is 0.08. If 6000 tickets are
sold, how many tickets has she bought? (A) 40 (B) 240 (C) 480 (D) 750

12. The times, in seconds, taken by 150 athletes to run a 110 m hurdle race are tabulated below

Class 13.8-14 14-14.2 14.2-14.4 14.4-14.6 14.6-14.8 14.8-15

Frequency 2 4 5 71 48 20

The number of athletes who completed the race in less than 14.6 seconds is :
(a)11 (b) 71 (c) 82 (d) 130

13. If x tan 45° sin 30° = cos 30° tan 30°, then x is equal to
(a) √3 (b) 0.5 (c) 1/ √2 (d) 0

14. The points (-5, 1), (1, p) and (4, -2) are collinear if
the value of p is (a) 3 (b) 2 (c) 1 (d) -1

15. The coordinates of the centroid of a triangle whose vertices are (0, 6), (8,12) and (8, 0) is
(a) (4, 6) (b) (16, 6) (c) (8, 6) (d) (16/3, 6)
𝐴𝐵 𝐵𝐶
16. If in triangles ABC and DEF, = , then they will be similar, if
𝐷𝐸 𝐹𝐷
(a) ∠B = ∠E (b) ∠A = ∠D (c) ∠B = ∠D (d) ∠A = ∠F

17. If in ΔABC, ∠C = 90°, then sin (A + B) =

(a) 0 (b) ½ (c) 1.73 (d) 1

18. A quadratic polynomial, whose zeores are -4 and -5, is

(a) x²-9x + 20 (b) x² + 9x + 20 (c) x²-9x- 20 (d) x² + 9x- 20

DIRECTION: In the question number 19 and 20, a statement of Assertion (A) is followed by a
statement of Reason (R). Choose the correct option
(a) Both assertion (A) and reason (R) are true and reason (R) is the correct explanation of assertion (A)
(b) Both assertion (A) and reason (R) are true and reason (R) is not the correct explanation of assertion
(A)
(c) Assertion (A) is true but reason (R) is false.
(d) Assertion (A) is false but reason (R) is true.
19. Assertion(A): (2 + √3)√3 is an irrational number.
Reason(R): Product of two irrational numbers is always irrational.
20. Assertion(A): 5√2 is not the root of the quadratic equation 𝑥 2 − 3√2 𝑥 − 20 = 0
Reason(R): The root of a quadratic equation satisfies it.

Section B

21. P is the LCM of 2,4,6,8,10; Q is the LCM of 1,3,5,7,9 and L is the LCM of P and Q. Then, find the
relation between L and P

22. A and B are respectively the points on the sides PQ and PR of a ΔPQR such that PQ = 12.5 cm, PA = 5
cm, BR = 6 cm and PB = 4 cm. Is AB || QR? Give reason for your answer.

OR

A line intersects the y-axis and x-axis at the points P and Q, respectively. If (2, –5) is the mid-point of PQ,
then the find the coordinates of P and Q

23. If 3 coins are tossed simultaneously then what is the probability of getting atmost 2 heads?

24. Find the value of a, if the distance between the points A (–3, –14) and B (a, –5) is 9 units.
25. if tanθ + cot θ = 2, then find the value of tan2 θ + cot 2 θ.
Section C

26. Prove that √5 is an irrational number.

𝑥 2 𝑦 2
27. If 𝑥 = 𝑏𝑠𝑒𝑐 3 𝜃 𝑎𝑛𝑑 𝑦 = 𝑎 𝑡𝑎𝑛3 𝜃, find the value of ( 𝑏 )3 − ( 𝑎 )3 .
OR
Prove that si𝑛2 𝜃. tan 𝜃 + 𝑐𝑜𝑠 2 𝜃. cot 𝜃 + 2 sin 𝜃. cos 𝜃 = tan 𝜃 + cot 𝜃
28. Find a point P on y- axis which is equidistant from the points A(4, 8) and B(-6, 6). Also find the
distance between AP.

29. Find the value of 𝑘 for such that the polynomial 𝑥 2 − (𝑘 + 6)𝑥 + 2(2𝑘 − 1) has sum of its zeroes equal
to half of its products.

30. The following distribution shows the daily pocket allowance of children of a locality. The mean pocket
allowance is Rs 18. Find the missing frequency f.

Daily Pocket Allowance (in Rs) 11-13 13-15 15-17 17-19 19-21 21-23 23-35

Number of children 7 6 9 13 f 5 4

31. All the jacks, queens and kings are removed from a deck of 52 playing cards. The remaining cards are
well shuffled and then one card is drawn at random. Giving ace a value 1 similar value for other cards,
find the probability that the card has a value (i) 7 (ii) greater than 7 (iii) less than 7

Section D

32. Two poles of height ‘a’ metres and ‘b’ metres are ‘p’ metres apart. Prove that the height of the point of
𝑎𝑏
intersection of the lines joining the top of each pole to the foot of the opposite pole is given by 𝑎+𝑏
metres.
33. Prove that:

1 1 2 2
1 − 𝑠𝑖𝑛2 𝜃𝑐𝑜𝑠 2 𝜃
[ + ] 𝑠𝑖𝑛 𝜃𝑐𝑜𝑠 𝜃 =
𝑐𝑜𝑠𝑒𝑐 2 𝜃 − 𝑠𝑖𝑛2 𝜃 𝑠𝑒𝑐 2 𝜃 − 𝑐𝑜𝑠 2 𝜃 2 + 𝑠𝑖𝑛2 𝜃𝑐𝑜𝑠 2 𝜃
34. The following data gives the information on the observed lifetimes (in hours) of 225 electrical
components:

Lifetime (in hours) 0-20 20-40 40-60 60-80 80-100 100-120

Frequency 10 35 52 X 38 29

If the modal value of the given data is 65.625 hours then find x.
35. The line segment joining the points A (3, 2) and B (5,1) is divided at the point P in the ratio 1:2 and it
lies on the line 3x – 18y + k = 0. Find the value of k.

Section E

36. We all have seen the airplanes flying in the sky but might have not thought of how they actually reach
the correct destination. Air Traffic Control (ATC) is a service provided by ground-based air traffic
controllers who direct aircraft on the ground and through a given section of controlled airspace, and can
provide advisory services to aircraft in non-controlled airspace. Actually, all this air traffic is managed
and regulated by using various concepts based on coordinate geometry and trigonometry.

At a given instance, ATC finds that the angle of elevation of an airplane from a point on the ground is 60o.
After a flight of 30 seconds, it is observed that the angle of elevation changes to 30 o. The height of the
plane remains constantly as 3000√3 m. Use the above information to answer the questions that follows-

(a) Draw a neat labelled figure to show the above situation diagrammatically.

(b) What is the distance travelled by the plane in 30 seconds?

OR

Keeping the height constant, during the above flight, it was observed that after 15(√3 -1) seconds, the
angle of elevation changed to 45o. How much is the distance travelled in that duration.

(c) What is the speed of the plane in km/hr.

37. Gardening in the Backyard: In the backyard of house, Shikha has some empty space in the shape of a
APQR. She decided to make it a garden. She divided the whole space into three parts by making
boundaries AB and CD using bricks to grow flowers and vegetables where AB || CD || QR as shown in
figure.
Based on the given information, answer the following questions.
(i)Find the length of AB
(ii) Find the length of CD.
OR
What is ratio of (PB)/(PR) ?
(iii) Find out the ratio of (BP)/(PD)

38. Ravi travels from Delhi to Bengal by train, which passes through many tunnels. The mathematical
representation of one of the tunnel is shown in the given figure.

Use the above information to answer the questions that follow:

(i) If one of the tunnel is represented by p(x) = −8𝑥 2 + 7𝑥 + 1, then find its zeroes.
(ii) If a tunnel is represented by −𝑥 2 + kx + 6 and one of its zero is 1, then find the value of k and its other
zero.
OR
𝛼 𝛽
If 𝛼 and 𝛽 are zeros of p(x) = −9𝑥 2 - 6x + 1 then find the value of +
𝛽 𝛼
(iii) If a polynomial represents the tunnel, whose zeroes are - 2/3 and 1/3 Then, find its polynomial.