DELHI PUBLIC SCHOOL, GURGAON

PRE-BOARD EXAMINATION – I (2023 – 2024)

CLASS : X (15/12/23)
SUBJECT : BASIC MATHEMATICS (241)

Duration: 3 Hours M.M: 80

No. of Printed Pages: 08

GENERAL INSTRUCTIONS :
1. This Question Paper has 5 Sections A to E.
2. Section A has 20 MCQs carrying 01 mark each.
3. Section B has 5 questions carrying 02 marks each.
4. Section C has 6 questions carrying 03 marks each.
5. Section D has 4 questions carrying 05 marks each.
6. Section E has 3 case-based integrated units of assessment (04 marks each) with sub-parts of
the values of 1, 1 and 2 marks each, respectively.
7. All questions are compulsory. However, an internal choice in 2 questions of 2 marks,
2 questions of 3 marks and 2 questions of 5 marks has been provided. An internal choice has
been provided in the 2 marks questions of Section E.
8. Draw neat figures wherever required. Take π =22/7 wherever required, if not stated.

SECTION – A
This section consists of 20 questions of 01 mark each.
Choose the correct option.
1. HCF of 96 and 120 is 1
(a) 24 (b) 480 (c) 42 (d) 8
2. The number of quadratic polynomial(s) which have zeroes – 3 and 5 is/are 1
(a) Only one (b) infinite (c) exactly two (d) at most two
3. The pair of equations ax + 2y = 9 and 3x +by = 18 represent parallel lines, where a, b are 1
integers, if
(a) a = b (b) 3a = b (c) 2a = 5b (d) ab = 6

4. A bag contains 5 pink, 8 blue and 7 yellow balls. One ball is drawn at random from the bag. 1
What is the probability of getting neither a blue nor a pink ball?
1 2 7 13
(a) 4 (b) 5 (c) 20 (d) 20

5. The line represented by 4x – 3y = 9 intersects the y – axis at 1

9 9
(a) (0, –3) (b) (4, 0) (c) ( -3, 0) (d) (0, 4)

6. If mean and median of a data are 12 and 15 respectively, then its mode is 1
(a) 21 (b) 13.5 (c) 6 (d) 14

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7. The radius and the curved surface area of a cone are 7cm and 550 cm2 respectively. It’s slant 1
height is
(a) 25cm (b) 50cm (c) 7cm (d) 55cm

8. If the sum of the first n terms of an A.P. is 3n2 + n and its common difference is 6, then its 1
second term is
(a) 2 (b) 3 (c) 1 (d) 4

9. In the adjoining figure, PQ and PR are tangents from an exterior point P to a circle with centre O. 1
If POR = 55, then QPR is

(a) 35 (b) 70 (c) 55 (d) 80

10. If √3 tan 2 – 3 = 0, then the value of  ( where  is an acute angle) is 1

(a) 15 (b) 30 (c) 45 (d) 60

11. The distance between the points A(cos, sin) and B(sin, – cos) is 1
(a) 2 units (b) √2 units (c) 1 units (d) √3 units

12. If a sector of a circle of diameter 12 cm subtends an angle of 90 at the centre, then length of arc 1
of that sector is
(a) 2π cm (b) 5π cm (c) 4π cm (d) 3π cm
13. Which of the following is not a quadratic equation? 1
(a) 2(x-1)2 = 4x2 – 2x + 1 (b) 2x – x2 = x2 + 5
(c) (x2 + 2x)2 = x4 + 3 + 4x3 (d) (√2 𝑥 + √3)2+ x2 = 3x2 – 5x

14. Number of revolutions made by a circular wheel of radius 0.7m in covering a distance of 176m is 1
(a) 24 (b) 22 (c) 40 (d) 75

15. The radius of a solid wooden sphere is r cm. It is divided into two equal parts. The total surface 1
area of each part is
(a) 6πr2 (b) 4πr2 (c) 3πr2 (d) 8πr2

16. If ∆ABC and ∆PQR are similar such that A = 31 and R = 69, then Q is 1
(a) 70 (b) 100 (c) 90 (d) 80

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17. The range of the data 14, 27, 29, 45, 61, 15, 18, 9 is 1
(a) 61 (b) 52 (c) 47 (d) 53

18. Aman represented four quadratic equations on a graph paper as shown below. Which of these 1
graphical representation show real roots.

(a) only (iii) (b) (i) and (ii) (c) (iii) and (iv) (d) (i), (ii) and (iv)

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 but 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) : The length of tangent drawn from an external point at a distance of 13 cm from 1
the centre of a circle of radius 5 cm is 10 cm.
Reason (R) : A tangent to a circle is perpendicular to the radius through the point of contact.

20. Assertion(A) : For 0 ≤  ≤ 90, cosec – cot and cosec + cot are reciprocal of each other. 1

Reason (R) : cosec2 – cot2 = 1.

SECTION – B
This section consists of 5 questions of 02 marks each.

21. The length of the minute hand of a clock is 14 cm. Find the area swept by the minute hand in 25 2
minutes.

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22. Check whether 6n can end with the digit 0 for any natural number n. 2
OR
Prove that 2 + 3√5 is an irrational number given that √5 is an irrational number.

23. A geologist asked his assistant Annie, “can we find the distance between the points P and Q from 2
the figure shown below”.

Annie replied, “It is possible’’.

Is Annie’s statement correct? Justify your answer.

24. 5 sin θ − 3 cos θ 2

If 3tan  = 4, find the value of .
5 sin θ + 2 cos θ
OR
sin 30°−sin 90°+2cos0°
Evaluate: .
tan 30° tan 60°

25. A circle is inscribed in ∆ABC. If AP = 2cm, CR = 4cm and BQ = 3cm, then find the perimeter of 2
∆ABC.

SECTION – C
This section consists of 6 questions of 03 marks each.

26. There is a circular path around a sports field. Sonia takes 18 minutes to drive one round of the 3
field while Ravi takes 12 minutes for the same. Suppose they both start at the same point and at
the same time and go in the same direction, after how many minutes will they meet again at the
starting point?

27. Find the zeroes of the quadratic polynomial x2 – 11x – 12 and verify the relationship between the 3
zeroes and the coefficients.

28. x 2y y 3
Solve the pair of linear equations: + = –1 and x – = 3
2 3 3
OR
The angles of a triangle are x, y and 40. The difference between two angles x and y ( x > y) is
30. Find the value of x and y.

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29. Prove that ( sec  – tan  )2 (1 + sin ) = 1– sin . 3

30. A card is drawn at random from a well-shuffled deck of playing cards. Find the probability that 3
the card drawn is
(i) a black king.
(ii) neither a jack nor a king.
(iii) a card of spade or an ace.
OR
A box contains cards bearing numbers from 6 to 70. If one card is drawn at random from the
box, find the probability that it bears
(i) a one digit number.
(ii) a number divisible by 5.
(iii) an odd number less than 30.

31. In figure, two tangents PQ and PR are drawn to a circle with centre O from an external point P. 3
Prove that QPR = 2OQR.

SECTION – D
This section consists of 4 questions of 05 marks each.

32. 14, 21, 28, 35,… and 26, 39, 52, 65,… are two arithmetic progressions such that the pth term of 5
the first AP is the same as the qth term of the second AP. Derive a relation between p and q.

OR

The sum of the first seven terms of an AP is 182. If its 4th and the 17th terms are in the ratio 1: 5,
find the AP.

33. A man standing on the deck of a ship, which is 10 m above water level, observes an angle of 5
elevation to the top of a hill as 60° and the angle of depression to the base of hill as 30°. Find the
distance of the hill from the ship and the height of the hill.

34. (i) If a line drawn parallel to one side of a triangle to intersect the other two sides in distinct 5
points, then prove that the other two sides are divided in the same ratio.

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 (ii) In the given figure, if AB ∥ EW, AD = 4cm, DE = 12 cm and DW = 24 cm, then find BD.

35. The monthly expenditure on milk for 200 families of a Housing Society is given below: 5

Monthly Expenditure( in ₹) Number of families

1000 – 1500 24
1500 – 2000 40
2000 – 2500 33
2500 – 3000 x
3000 – 3500 30
3500 – 4000 22
4000 - 4500 16
4500 - 5000 7

Find the value of x and hence find mean expenditure on milk.

OR

The median of the following data is 50. Find the value of p and q, if sum of all the frequencies is
90.
Marks obtained Number of students
20 – 30 p
30 – 40 15
40 – 50 25
50 – 60 20
60 – 70 q
70 – 80 8
80 – 90 10

SECTION – E
This section consists of 3 case - based questions of 04 marks each.

36. A small scale industry produces a certain number of boxes of candles in a day. Number of boxes
prepared by each worker on a particular day was 2 more than thrice the number of workers
working in the industry. The number of boxes produced on that particular day was 85.

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 Based on the above information, answer the following questions ( take π = 22/7)

(i) If the number of workers working in the industry is x, then find the number of boxes of 1
candles prepared by each worker on that particular day?
(ii) Form the quadratic equation representing the given situation. 1
(iii) Find the number of workers working in the industry.
OR 2
(iii) Find the number of boxes prepared by each worker.

37. The following diagram shows the position of three electric light poles A, B and C in a society
park.

Inspite, of 3 poles in the park, the kids playing in the park complained about darkness in the
park. So, the society president decided to place 2 more poles in the park.

Based on the above information, answer the following questions

(i) What is the distance between the points B and C? 1

(ii) The fourth pole D is to be placed such that it divides the line segment joining A and B in
the ratio 2:3. Find the coordinates of point D. 1
(iii) What type of triangle is formed by joining the points A, B and C? Justify your answer.
OR 2
(iii) The fifth pole is placed at a point E such that ABEC forms a parallelogram. Find the
coordinates of the point E.

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38. Educational trips are most appropriate for students to learn and explore. Students of class X
visited Red Fort. The teacher explained that this monument is a perfect example of combination
of two or more solid figures. It comprises of seven smaller hemispherical domes built between
two cylindrical pillars where flag hoisting ceremony on Independence day takes place every
year. Also, two bigger hemispherical domes were constructed at each corner as shown below.

Based on the above information, answer the following questions ( take π = 22/7)

(i) Calculate the area of a big hemispherical dome of radius 2m in terms of π ? 1

(ii) Find the lateral surface area of a cylindrical pillar if height of the pillar is 7m and radius
of the base is 1.4m. 1
(iii) What is the ratio of the volume of two big hemispherical domes each of radius 2 m to the
volume of seven small hemispherical domes each of radius 1m? 2
OR
(iii) Find the cost of painting seven smaller domes at the rate of ₹ 50 per square metre if
radius of each dome is 1m.

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