Class X Session 2024-25

Subject - Mathematics (Basic)

Sample Question Paper - 2

Time Allowed: 3 hours Maximum Marks: 80

General Instructions:

1. This Question Paper has 5 Sections A, B, C, D and E.

2. Section A has 20 MCQs carrying 1 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 carrying 04 marks each.

7. All Questions are compulsory. However, an internal choice in 2 Qs of 5 marks, 2 Qs of 3 marks and 2 Questions of
2 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 = wherever required if not stated.
22

Section A
1. If a is rational and √b is irrational, then a + √b is: [1]

a) an irrational number b) an integer

c) a natural number d) a rational number

2. 120 can be expressed as a product of its prime factors as [1]

a) 15 × 23 b) 5 × 23 × 3

c) 5 × 8 × 3 d) 10 × 22 × 3

3. If the equation 9x2 + 6kx + 4 = 0 has equal roots then k = ? [1]

a) -2 or 0 b) 0 only

c) 2 or 0 d) 2 or -2
4. The value of k for which the system of linear equations x + 2y = 3, 5x + ky + 7 = 0 is inconsistent is: [1]

a) − 14

3
b) 5

c) d) 10
2

5. A quadratic equation whose one root is 3 is [1]

a) x2 - 5x + 6 = 0 b) x2 - 6x - 6 = 0

c) x2 - 5x - 6 = 0 d) x2 + 6x - 5 = 0

6. If (a, 0), (0, b) and (x, y) are collinear, then [1]

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 a) ay - bx = 1 b) ax + by = 1

c) ay + bx = ab d) ax - by = ab
7. △ ABC is such that AB = 3 cm, BC = 2 cm and CA = 2.5 cm. If △DEF ∼ △ABC and EF = 4 cm, then [1]
perimeter of △DEF is

a) 30 cm b) 15 cm

c) 22.5 cm d) 7.5 cm
8. In the given figure, DE || BC and all measurements are given in centimetres. The length of AE is: [1]

a) 2.75 cm b) 2.5 cm

c) 2 cm d) 2.25 cm
9. A tangent PQ at point of contact P to a circle of radius 12 cm meets the line through centre O to a point Q such [1]
that OQ = 20 cm, length of tangent PQ is:

a) 15 cm b) 12 cm

c) 13 cm d) 16 cm
–
10. If √3 tan 2θ − 3 = 0 then θ = ? [1]

a) 30o b) 60o

c) 15o d) 45o

11. There is a small island in the middle of a 50 m wide river. A tall tree stands on the island. P and Q are points [1]
directly opposite to each other on the two banks, and in line with the tree. If the angles of elevation of the top of

the tree from P and Q are respectively 60o and 30o, then find the height of the tree.

a) 22.65 m b) 23.56 m

c) 24.69 m d) 21.65 m

12. If cos θ =
2
, then 2 sec2 θ + 2 tan2 θ - 7 is equal to [1]
3

a) 1 b) 4

c) 0 d) 3
13. The area of a quadrant of a circle whose circumference is 616 cm will be [1]

a) 7546 cm2 b) 7500 cm2

c) 7564 cm2 d) 7456 cm2

14. Find the area of the sector if the radius is 5 cm and with an angle of 50o. [1]

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 a) 10.90 cm b) 12.90 cm

c) 13.90 cm d) 11.90 cm
15. One card is drawn at random from a well-shuffled deck of 52 cards. What is the probability of getting a black [1]
face card?

a) b)
3 3

13 14

c) 3

26
d) 1

26

16. In a data, if l = 60, h = 15, f1 = 16, f0 = 6, f2 = 6, then the mode is [1]

a) 67.5 b) 72

c) 60 d) 62
17. A solid is in the shape of a cone standing on a hemisphere with both their radii being equal to 1cm and the height [1]
of the cone is equal to its radius. The volume of the solid is

a) π cm 3
b) 4π cm
3

c) 2π cm 3
d) 3π cm
3

18. The median class for the data given below is: [1]

Class 20 - 40 40 - 60 60 - 80 80 - 100 100 - 120

Frequency 10 12 14 13 17

a) 80 - 100 b) 60 - 80

c) 20 - 40 d) 40 - 60
−− −−−−
19. Assertion (A): Distance of point (a, b) from origin is √b 2
− a
2
[1]
−−−−−−−−−−−−−− −
Reason (R): Distance of point (x, y) from origin is 2
√(x − 0) + (y − 0)
2

a) Both A and R are true and R is the correct b) Both A and R are true but R is not the
explanation of A. correct explanation of A.

c) A is true but R is false. d) A is false but R is true.

20. Assertion (A): L.C.M. and H.C.F. of a and 20 are 100 and 10 respectively, then a = 50. [1]
Reason (R): L.C.M × H.C.F. = First number × Second number

a) Both A and R are true and R is the correct b) Both A and R are true but R is not the
explanation of A. correct explanation of A.

c) A is true but R is false. d) A is false but R is true.

Section B
a1 b1 c1
21. On comparing the ratios a2
, and
c2
, find out whether the lines representing the pair of linear equations [2]
b2

intersect at a point, are parallel or coincide: 6x − 3y + 10 = 0; 2x – y + 9 = 0.

22. In △ABC, D and E are the points on the sides AB and AC respectively such that DE||BC. If AD = 6x - 7, DB = [2]
4x - 3, AE = 3x - 3 and EC = 2x - 1, find the value of x.
OR
In Fig. check whether AD is the bisector of ∠A of ΔABC if AB =6 cm, AC = 8 cm, BD = 1.5 cm and CD = 2 cm

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23. A quadrilateral ABCD is drawn to circumscribe a circle. Prove that AB + CD = AD + BC [2]

–
24. If sin α = 1
and cot β = √3 , then find the value of cosec α + cosec β. [2]
√2

25. An umbrella has 8 ribs which are equally spaced (see figure). Assuming umbrella to be a flat circle of radius 45 [2]
cm, Find the area between the two consecutive ribs of the umbrella.

OR
Find the area of the segment of a circle of radius 14 cm, if the length of the corresponding arc APB is 22 cm.
Section C
26. Explain why 7 × 11 × 13 + 13 and 7 × 6 × 5 × 4 × 3 × 2 × 1 + 5 are composite numbers. [3]
27. Find the zeroes of the given quadratic polynomials and verify the relationship between the zeroes and the [3]
coefficients.6x 2
− 3 − 7x

28. The sum of the digits of a two-digit number is 9. Also, nine times this number is twice the number obtained by [3]
reversing the order of the number. Find the number. Solve the pair of the linear equation obtained by the
elimination method.
OR
The sum of a two-digit number and the number obtained by reversing the order of its digits is 165. If the digits differ
by 3, find the number.
29. ABCD is a quadrilateral such that ∠D = 90°. A circle C (O, r) touches the sides AB, BC, CD and DA at P, Q, R [3]
and S respectively. If BC = 38 cm, CD = 25 cm and BP = 27 cm, Find r.
sin θ−cos θ+1
30. Prove that , using identity sec . [3]
1 2 2
= θ = 1 + tan θ
sin θ+cos θ−1 sec θ−tan θ

OR
Prove: = cosec A −
1 1
− cosec A
(cot A)(sec A)−cot A (cot A)(sec A)+cot A

31. Two different dice are rolled together. Find the probability of getting (i) the sum of numbers on two dice to be 5, [3]
(ii) even number on both dice, (iii) a doublet.
Section D
32. A rectangular field is 20 m long and 14 m wide. There is a path of equal width all around it, having an area of [5]
111 sq m. Find the width of the path.
OR

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 If the price of a book is reduced by ₹5, a person can buy 5 more books for ₹ 300. Find the original list price of the
book.
33. If BD and QM are medians of triangles ABC and PQR, respectively, where △ABC ∼ △ PQR, prove that [5]
AB

PQ
=
BD

QM
.
34. A solid is in the shape of a cone surmounted on a hemisphere with both their diameters being equal to 7 cm and [5]
the height of the cone is equal to its radius. Find the volume of the solid.
OR
A solid consisting of a right cone standing on a hemisphere is placed upright in a right circular cylinder full of water
and touches the bottom. Find the volume of water left in the cylinder, if the radius of the cylinder is 60 cm and its
height is 180 cm, the radius of the hemisphere is 60 cm and height of the cone is 120 cm, assuming that the
hemisphere and the cone have common base.
35. The following table gives the distribution of the life time of 400 neon lamps: [5]

Lite time (in hours) Number of lamps

1500-2000 14

2000-2500 56

2500-3000 60

3000-3500 86

3500-4000 74

4000-4500 62

4500-5000 48

Find the median life time of a lamp.

Section E
36. Read the following text carefully and answer the questions that follow: [4]
Saving money is a good habit and it should be inculcated in children right from the beginning. Rehan’s mother
brought a piggy bank for Rehan and puts one ₹ 5 coin of her savings in the piggy bank on the first day. She
increases his savings by one ₹ 5 coin daily.

Based on the above information, answer the following questions:

i. How many coins were added to the piggy bank on 8th day?
ii. How much money will be there in the piggy bank after 8 days?
iii. a. If the piggy bank can hold one hundred twenty ₹ 5 coins in all find the number of days she can contribute
to put ₹ 5 coins into it.
OR

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 b. Find the total money saved, when the piggy bank is full.
37. Read the following text carefully and answer the questions that follow: [4]
Using Cartesian Coordinates we mark a point on a graph by how far along and how far up it is.
The left-right (horizontal) direction is commonly called X-axis.
The up-down (vertical) direction is commonly called Y-axis.
In Green Park, New Delhi Suresh is having a rectangular plot ABCD as shown in the following figure. Sapling
of Gulmohar is planted on the boundary at a distance of 1 m from each other. In the plot, Suresh builds his house
in the rectangular area PQRS. In the remaining part of plot, Suresh wants to plant grass.

i. Find the coordinates of the midpoints of the diagonal QS. (1)

ii. Find the length and breadth of rectangle PQRS? (1)
iii. Find Area of rectangle PQRS. (2)
OR
Find the diagonal of rectangle. (2)
38. Read the following text carefully and answer the questions that follow: [4]
Two trees are standing on flat ground. The angle of elevation of the top of Both the trees from a point X on the

ground is 60o. If the horizontal distance between X and the smaller tree is 8 m and the distance of the top of the
two trees is 20 m.

i. Calculate the distance between the point X and the top of the smaller tree. (1)
ii. Calculate the horizontal distance between the two trees. (1)
iii. Find the height of big tree. (2)
OR
Find the height of small tree. (2)

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