Class X Top 10 Sample Papers Mathematics With Solution

Sample
Paper
SET - 1
With a success rate
exceeding 95% in the
2024 boards
Mathematics
Class 10
www.educatorsresource.in
Set - 1
Mathematics-Standard
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 (04 marks each) with sub-parts.
7. All Questions are compulsory. However, an internal choice in 2 Questions of 5 marks, 2 Questions 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 = 22/7 wherever required if not stated.
SECTION A
Section A consists of 20 questions of 1 mark each.

1. If a pair of linear equations in two variables is inconsistent, then the lines represented by two
equations are
(a) intersecting
(b) parallel
(c) always coincident
(d) intersecting or coincident
2. The distance of the point (5, −4) from 𝑥-axis is

(a) 5 units
(b) 4 units
(c) 1 unit
(d) 9 units
3. In the given figure, 𝐷𝐸 ∥ 𝐵𝐶, 𝐴𝐷 = 2 cm, 𝐵𝐷 = 2.5 cm and 𝐴𝐸 = 3.2 cm, then 𝐴𝐶 is equal to
(a) 2.5 cm
(b) 4 cm
(c) 1.3 cm
(d) 7.2 cm
4. (cos 4 ⁡𝑥 − sin4 ⁡𝑥 ) is equal to

(a) 2sin2 ⁡𝑥 − 1
(b) 1 − 2cos 2 ⁡𝑥
(c) sin2 ⁡𝑥 − cos 2 ⁡𝑥
(d) 2cos 2 ⁡𝑥 − 1
5. If probability of success is 0.9%, then probability of failure is

(a) 0.91
(b) 0.091
(c) 99.1
(d) 0.991
6. If 1 is a zero of the polynomial 𝑝(𝑥) = 𝑎𝑥 2 − 3(𝑎 − 1)𝑥 − 1, then find the value of 𝑎.
(a) 1
(b) 2
(c) -1
(d) -2
7. If 𝑃 is a point on 𝑥-axis such that its distance from the origin is 3 units, then the coordinates of a
point 𝑃 are
(a) (0,3)
(b) (3,0)
(c) (0,0)
(d) (0, −3)
8. In a single throw of a pair of dice, the probability of getting the sum as a perfect square is
7
(a) 36
5
(b) 36
8
(c) 36
11
(d)
36
9. Determine 𝑘 for which the system of equations has infinite solutions : 4𝑥 + 𝑦 = 3 and 8𝑥 + 2𝑦 =
5𝑘.
5
(a) 6
6
(b) 5
5
(c) 4
4
(d) 5
10. 5sin2 ⁡30∘ + cos 2 ⁡45∘ − 4tan2 ⁡30∘ is equal to

5
(a) 6
2
(b) 3
5
(c) 8
5
(d) 12
11. The sum (−6) + (0) + (6) + ⋯. upto 13th term =

(a) 390
(b) 1380
(c) 378
(d) 1830
12. From a point 𝑄, the length of the tangent to a circle is 12 cm and the distance of 𝑄 from the centre is
15 cm. The radius of the circle is
(a) 9 cm
(b) 12 cm
(c) 15 cm
(d) 24.5 cm
13. The mean of first ten odd natural numbers is

(a) 5
(b) 10
(c) 20
(d) 19
14. If LCM of 𝑎 and 18 is 36 and HCF of 𝑎 and 18 is 2 , then 𝑎 =

(a) 2
(b) 3
(c) 4
(d) 1
15. In the given figure, 𝑂𝐴 = 4 cm, 𝑂𝐵 = 6 cm, 𝑂𝐷 = 5 cm and 𝑂𝐶 = 7.5 cm, then by which of the
following similarity criterion △ 𝐴𝑂𝐷 ∼△ 𝐵𝑂𝐶 ?
(a) AA
(b) SAS
(c) AAS
(d) SSS
16. A child has a block in the shape of a cube with one letter written on each face as follows:
The cube is thrown once. What is the probability of getting 𝐴 ?

1
(a) 3
1
(b) 6
1
(c) 2
1
(d)
4
17. The zeroes of the polynomial 𝑓(𝑥) = 𝑥 2 − 2√2𝑥 − 16 are

(a) √2, −√2
(b) 4√2, −2√2
(c) −4√2, 2√2
(d) 4√2, 2√2
18. The line 3𝑥 + 𝑦 − 6 = 0 divides the line segment joining 𝐴(1, −1) and 𝐵(3,6) in the ratio
(a) 2: 5
(b) 4: 9
(c) 2: 7
(d) 2: 3
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.
19. Statement A (Assertion) : Both the roots of the equation 𝑥 2 − 𝑥 + 1 = 0 are real.
Statement 𝐑 (Reason) : The roots of the equation 𝑎𝑥 2 + 𝑏𝑥 + 𝑐 = 0 are real if 𝑏2 − 4𝑎𝑐 ≥ 0.
(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.
20. Statement A (Assertion) : The value of each of the trigonometric ratios of an angle do not vary with
the lengths of the sides of the triangle, if the angle remains the same.
𝐵𝐶 𝐴𝐵
Statement 𝐑 (Reason) : In right △ 𝐴𝐵𝐶, ∠𝐵 = 90∘ and ∠𝐴 = 𝜃, sin⁡𝜃 = 𝐴𝐶 < 1 and cos⁡𝜃 = <1
𝐴𝐶
as hypotenuse is the longest side.
(A).
assertion (A).
SECTION B
Section B consists of 5 questions of 2 marks each.

21. If 𝑝, 𝑞, 𝑟 are real and 𝑝 ≠ 𝑞, then show that the roots of the equation (𝑝 − 𝑞)𝑥 2 + 5(𝑝 + 𝑞)𝑥 −
2(𝑝 − 𝑞) = 0 are real and unequal.
OR
The age of father is equal to the square of the age of his son. The sum of the age of father and five
times the age of the son is 66 years. Find their ages.
1
22. If sin⁡𝛼 = 2, then show that (3cos⁡𝛼 − 4cos 3 ⁡𝛼 ) = 0.
23. A toy is in the form of a cone mounted on a hemisphere. The diameter of the base of the cone and
that of hemisphere is 18 cm and the height of cone is 12 cm. Calculate the surface area of the toy.
[Take 𝜋 = 3.14 ]
24. If △ 𝐴𝐵𝐶 ∼△ 𝑃𝑄𝑅, 𝐴𝐵 = 4 cm, 𝑃𝑄 = 10 cm, 𝑄𝑅 = 15 cm, 𝑃𝑅 = 20 cm, then find the perimeter of
△ 𝐴𝐵𝐶.
OR
In △ 𝐷𝐸𝐹, 𝐴𝐵 ∥ 𝐸𝐹 such that 𝐴𝐷 = 6 cm, 𝐴𝐸 = 18 cm and 𝐵𝐹 = 24 cm. Find the length of 𝐷𝐵.
25. In the given figure, 𝑃𝑄 is the common tangent to both the circles. 𝑆𝑅 and 𝑃𝑇 are tangents. If 𝑆𝑅 =
4 cm, 𝑃𝑇 = 7 cm, then find 𝑅𝑃.
SECTION C
Section 𝐶 consists of 6 questions of 3 marks each.
26. A toy is in the shape of a cone mounted on a hemisphere of same base radius. If the volume of the toy
is 231 cm3 and its diameter is 7 cm, then find the height of the toy.
27. If 𝑆𝑛 denotes the sum of first 𝑛 terms of an A.P., prove that 𝑆12 = 3(𝑆8 − 𝑆4 ).
OR
The digits of a positive number of three digits are in A.P. and their sum is 15. The number obtained
by reversing the digits is 594 less than the original number. Find the number.
28. Find the mean of the following data.
10 20 30 40
Class- interval 0 − 10
− 20 − 30 − 40 − 50
Frequency 3 5 9 5 3
29. In the given figure (not drawn to scale), 𝐴𝑀: 𝑀𝐶 = 3: 4, 𝐵𝑃: 𝑃𝑀 = 3: 2 and 𝐵𝑁 = 12 cm. Find 𝐴𝑁.
OR
A boy of height 95 cm is walking away from base of a lamp post at a speed of 1.5 m/s. If the lamp
post is 3.8 m above the ground, find the length of his shadow after 5 seconds.
30. Two ships are anchored on opposite sides of a lighthouse. Their angles of depression as observed
from the top of the lighthouse are 30∘ and 60∘ . The line joining the ships passes through the foot of
the lighthouse. If the height of the lighthouse is 100 m, find the distance between the ships.
(Use √3 = 1.732)
𝑎 2𝑎
31. If 𝑃 and 𝑄 are two points whose coordinates are (𝑎𝑡 2 , 2𝑎𝑡) and (𝑡 2 , − ) respectively and 𝑆 is the
𝑡
1 1
point (𝑎, 0). Show that 𝑆𝑃 + is independent of 𝑡.
𝑆𝑄
SECTION D
Section D consists of 4 questions of 5 marks each.
32. 𝐴, 𝐵 and 𝐶 start cycling around a circular path in the same direction and the same time.
Circumference of the path is 1980 m. If the speed of 𝐴 is 330 m/min, speed of 𝐵 is 198 m/min and
𝐶 is 220 m/min and they start from the same point, then after how much time will they meet again?
33. 𝐴𝐵 is a diameter of a circle. 𝐴𝐻 and 𝐵𝐾 are perpendicular from 𝐴 and 𝐵 respectively to the tangent
at 𝑃. Prove that 𝐴𝐻 + 𝐵𝐾 = 𝐴𝐵.
OR
A triangle 𝑃𝑄𝑅 is drawn to circumscribe a circle of radius 8 cm such that the segment 𝑄𝑇 and 𝑇𝑅,
into which 𝑄𝑅 is divided by the point of contact 𝑇, are of length 14 cm and 16 cm respectively. If
area of △ 𝑃𝑄𝑅 is 336 cm2 , then find the sides 𝑃𝑄 and 𝑃𝑅.
34. In the given figure, if ∠𝐵𝑂𝐶 = 120∘, then find the area of minor segment?
35. Draw the graph of the following pair of linear equations :

𝑥 + 3𝑦 = 6 and 2𝑥 − 3𝑦 = 12
Find the ratio of the areas of the two triangles formed by first line, 𝑥 = 0, 𝑦 = 0 and second line, 𝑥 =
0, 𝑦 = 0.
OR
Mr. Sehgal buys 4 chairs and 5 tables for ₹ 3400. Later he buys another chair and 4 tables more of the
same type for ₹ 2500. Represent this situation algebraically and graphically.
SECTION E
Case study based questions are compulsory.
36. A boy 4 m tall spots a pigeon sitting on the top of a pole of height 54 m from the ground. The angle
of elevation of the pigeon from the eyes of boy at any instant is 60∘. The pigeon flies away
horizontally in such a way that it remained at a constant height from the ground. After 8 seconds, the
angle of elevation of the pigeon from the same point is 45∘.
Use the above information to answer the questions that follow :
(Take √3 = 1.73 )
(i) Find the distance of first position of the pigeon from the eyes of the boy.
(ii) If the distance between the position of pigeon increases, then how will the angle of elevation be
affected ?
(iii) Find the distance between the boy and the pole.
OR
How much distance the pigeon covers in 8 seconds?
37. Rahul goes to a fete in Mussoorie. There he saw a game having prizes - wall clocks, power banks,
puppets and water bottles. The game consists of a box having cards inside it, bearing the numbers 1
to 200, one on each card. A person has to select a card at random. Now, the winning of prizes has the
following conditions:
Wall clock - If the number on the selected card is a perfect square.
Power bank - If the number on the selected card is a multiple of 3.
Puppet - If the number on selected card is divisible by 10.

Water bottle - If the number on the selected card is a prime number more than 100 but less than 150 .
Better luck next time - If the number on the selected card is a perfect cube.
Use the above information to answer the questions that follow:

(i) Find the probability of winning a wall clock.
(ii) Find the probability of winning a puppet.
OR
Find the probability of winning a water bottle.
(iii) Find the probability of getting 'Better Luck next time'.
38. Quadratic equations started around 3000 B.C. with the Babylonians. They were one of the world's
first civilisation, and came up with some great ideas like agriculture, irrigation and writing. There
were many reasons why Babylonians needed to solve quadratic equations. For example to know what
amount of crop you can grow on the square field.
Use the concept of quadratic equation and answer the questions that follow:
(i) The sum of squares of two consecutive integers is 650. Represent it in the form of quadratic
equation.
(ii) The sum of two numbers is 15 and the sum of their reciprocals is 3/10. Find the numbers.
OR
A natural number when increased by 12, equals 160 times its reciprocal. Find the number.
(iii) Two numbers differ by 3 and their product is 504. Represent it in the form of quadratic equation.
p l e P a p e r
Sam
S o lu t i o n
h e m a t i c s
M at
Shop Now
Class 10
Set -1
SOLUTIONS
1. (b): When a pair of linear equations is inconsistent, then the lines represented by two equations are
parallel.
2. (b): Distance of the point (5, −4) from 𝑥-axis =∣ 𝑦-coordinate of the point (5, −4) ∣= 4 units
3. (d): In △ 𝐴𝐵𝐶, 𝐷𝐸 ∥ 𝐵𝐶
𝐴𝐷 𝐴𝐸
∴ By basic proportionality theorem, we have =
𝐷𝐵 𝐸𝐶
2 3.2 2.5 × 3.2

⇒ ⇒ 𝐸𝐶 = = 4 cm
2.5 𝐸𝐶 2
Now, 𝐴𝐶 = 𝐴𝐸 + 𝐸𝐶 = 3.2 + 4 = 7.2 cm
4. (d): cos 4 𝑥 − sin4 𝑥 = (cos 2 𝑥 + sin2 𝑥 )(cos 2 𝑥 − sin2 𝑥 ) = 1 ⋅ (cos 2 𝑥 − (1 − cos 2 𝑥 )) = cos 2 𝑥 −
1 + cos 2 𝑥 = 2cos 2 𝑥 − 1
0.9 9 1000−9 991
5. (d): Probability of failure = 1 - Probability of success = 1 − 100 = 1 − 1000 = = 1000 =
1000
0.991
6. (a): Since it is given that, 1 is a zero of polynomial 𝑝(𝑥) = 𝑎𝑥 2 − 3(𝑎 − 1)𝑥 − 1. ∴ 𝑝(1) = 0.
⇒ 𝑝(1) = 𝑎(1)2 − 3(𝑎 − 1)(1) − 1 = 0 ⇒ 𝑎 − 3𝑎 + 3 − 1 = 0 ⇒ −2𝑎 = −2 ⇒ 𝑎 = 1
7. (b): The coordinate of points on the 𝑥-axis at a distance of 3 units are (−3,0) and (3,0).
8. (a): Total number of possible outcomes = 6 × 6 = 36
Let 𝐸 be the event 'getting the sum as a perfect square'.
∴ Outcomes favorable to 𝐸 are {(1,3), (2,2), (3,1), (3,6), (4,5), (5,4), (6,3)} i.e., 7 in number.
7
∴ 𝑃(𝐸) =
36
9. (b): The given system of equations is
4𝑥 + 𝑦 − 3 = 0 and 8𝑥 + 2𝑦 − 5𝑘 = 0
For infinite solutions,
𝑎1 𝑏1 𝑐1 4 1 −3 1 3
= = ⇒ = = ⇒ =
𝑎2 𝑏2 𝑐2 8 2 −5𝑘 2 5𝑘
6
⇒ 5𝑘 = 6 ⇒ 𝑘 =
5
10. (d): We have, 5sin2 30∘ + cos 2 45∘ − 4tan2 30∘
1 2 1 2 1 2 1 1 1 5
= 5( ) + ( ) − 4( ) = 5 × + − 4× =
2 √2 √3 4 2 3 12
11. (a): Here, 𝑎 = −6 and 𝑑 = 0 − (−6) = 6

13
∴ 𝑆13 = [2 × (−6) + (13 − 1)6]
2
13 13
= [−12 + 72] = × 60 = 390
2 2
12. (a): 𝑄𝑃 is a tangent at 𝑃
∴ ∠𝑃 = 90∘
Given, 𝑂𝑄 = 15 cm and 𝑃𝑄 = 12 cm In △ 𝑂𝑃𝑄
𝑂𝑄2 = 𝑂𝑃2 + 𝑃𝑄2

⇒ (15)2 = 𝑂𝑃2 + 122
⇒ 𝑂𝑃2 = 225 − 144 = 81
⇒ 𝑂𝑃 = 9 cm
13. (b): First ten odd natural numbers are 1,3,5,7,9,11, 13,15,17 and 19.
∴ Required mean
1 + 3 + 5 + 7 + 9 + 11 + 13 + 15 + 17 + 19 100
= = = 10
10 10
14. (c): Product of two numbers = HCF × LCM of two numbers
72
∴ 𝑎 × 18 = 2 × 36 ⇒ 𝑎 = 18 = 4.
15. (b): In △ 𝐴𝑂𝐷 and △ 𝐵𝑂𝐶,

𝑂𝐴 4 2 𝑂𝐷 5 2 𝑂𝐴 𝑂𝐷
= = and = = ⇒ =
𝑂𝐵 6 3 𝑂𝐶 7.5 3 𝑂𝐵 𝑂𝐶
and ∠𝐴𝑂𝐷 = ∠𝐵𝑂𝐶

[Vertically opposite angles]
∴△ 𝐴𝑂𝐷 ∼△ 𝐵𝑂𝐶
[By SAS similarity criterion]
Hence, criterion of similarity used is SAS.
16. (a): Total number of possible outcomes = 6

As there are two 𝐴 's.
∴ Favourable number of outcomes = 2

2 1
∴ Required probability = 6 = 3
17. (b): The zeroes of 𝑓(𝑥) = 𝑥 2 − 2√2𝑥 − 16 are given by

𝑓(𝑥) = 0 i.e., 𝑥 2 − 2√2𝑥 − 16 = 0
⇒ 𝑥 2 − 4√2𝑥 + 2√2𝑥 − 16 = 0
⇒ 𝑥(𝑥 − 4√2) + 2√2(𝑥 − 4√2) = 0
⇒ (𝑥 + 2√2)(𝑥 − 4√2) = 0
⇒ 𝑥 = 4√2 or 𝑥 = −2√2
18. (b): Let the required ratio be 𝑘: 1.

3𝑘+1 6𝑘−1
Then, the point of division is ( 𝑘+1 , ).
𝑘+1
Since, this point lies on 3𝑥 + 𝑦 − 6 = 0.
3(3𝑘 + 1) 6𝑘 − 1
∴ + =6
𝑘+1 𝑘+1
4 4
⇒ 9𝑘 = 4 ⇒ 𝑘 = 9 ∴ Required ratio = 9 : 1 i.e., 4: 9
19. (d): Given quadratic equation is 𝑥 2 − 𝑥 + 1 = 0

∴ 𝑏2 − 4𝑎𝑐 = (−1)2 − 4(1)(1) = −3 < 0
Hence, the given quadratic equation has no real roots.
∴ Assertion (A) is false but Reason (R) is true.
Clearly, Reason (R) is true.

20. (b):
In △ 𝐴𝐵𝐶 and △ 𝑃𝑄𝑅

𝐴𝐵 3 𝑃𝑄 6
sin 𝜃 = 𝐴𝐶 = 5 and sin 𝜃 = 𝑃𝑅 = 10
3 3
⇒ sin 𝜃 = 5 and sin 𝜃 = 5
Similarly, this will also holds for other trigonometric ratios. ∴ Trigonometric ratio does not depend on
the size of the triangle.
∴ Assertion (A) and Reason (R) both are correct statements but Reason (R) is not the correct
explanation of Assertion (A).
21. The given equation is (𝑝 − 𝑞)𝑥 2 + 5(𝑝 + 𝑞)𝑥 − 2(𝑝 − 𝑞) = 0.

Here, 𝑎 = 𝑝 − 𝑞, 𝑏 = 5(𝑝 + 𝑞) and 𝑐 = −2(𝑝 − 𝑞).
∴ 𝐷 = 𝑏2 − 4𝑎𝑐 = 25(𝑝 + 𝑞)2 − 4(𝑝 − 𝑞)(−2(𝑝 − 𝑞))
= 25(𝑝 + 𝑞)2 + 8(𝑝 − 𝑞)2
We find that, 25(𝑝 + 𝑞)2 > 0 and 8(𝑝 − 𝑞)2 > 0[∵ 𝑝 ≠ 𝑞]
∴ 𝐷 = 25(𝑝 + 𝑞)2 + 8(𝑝 − 𝑞)2 > 0
Hence, roots of the given equation are real and unequal.
OR
Let age of the son be 𝑥 years, then age of his father will be 𝑥 2 years.
According to question, 𝑥 2 + 5(𝑥) = 66
⇒ 𝑥 2 + 5𝑥 − 66 = 0
Here, 𝑎 = 1, 𝑏 = 5 and 𝑐 = −66.
∴ 𝑏2 − 4𝑎𝑐 = (5)2 − 4(1)(−66) = 25 + 264 = 289 > 0
−𝑏 ± √𝑏2 − 4𝑎𝑐 −5 ± √289 −5 ± 17

∴ 𝑥= = =
2𝑎 2(1) 2
−5+17 −5−17
⇒𝑥= or 𝑥 =
2 2
12 −22
⇒ 𝑥= = 6 or 𝑥 = = −11
2 2
Since, age can't be a negative, so 𝑥 = −11 is rejected. Hence, age of the son is 6 years and age of the
father is 36 years.
22. Consider a △ 𝐴𝐵𝐶 in which ∠𝐵 = 90∘ and ∠𝐵𝐴𝐶 = 𝛼.

𝐵𝐶 1
∴ sin 𝛼 = =
𝐴𝐶 2
Let 𝐵𝐶 = 𝑘 units and 𝐴𝐶 = 2𝑘 units, where 𝑘 is a positive number.
By Pythagoras theorem, we have
𝐴𝐶 2 = 𝐴𝐵2 + 𝐵𝐶 2
⇒ 𝐴𝐵2 = 𝐴𝐶 2 − 𝐵𝐶 2 = 4𝑘 2 − 𝑘 2 = 3𝑘 2
⇒ 𝐴𝐵 = √3𝑘 units
𝐴𝐵 √3𝑘 √3
∴ cos 𝛼 = = =
𝐴𝐶 2𝑘 2
3√3 3 √3
Now, L.H.S. = (3cos 𝛼 − 4cos 3 𝛼 ) = −4× = 0 = R.H.S.
2 8
18
23. Radius of the base of the cone and hemisphere (𝑟) = = 9 cm
2
Height of cone (ℎ) = 12 cm
Slant height of cone (𝑙)
= √𝑟 2 + ℎ2 = √92 + 122
= √81 + 144 = √225 = 15 cm
Total surface area of toy
= Curved surface area of hemisphere + Curved surface area of cone

= 2𝜋𝑟 2 + 𝜋𝑟𝑙 = 𝜋𝑟(2𝑟 + 𝑙) = 3.14 × 9(2 × 9 + 15)
= 3.14 × 9 × 33 = 932.58 cm2
24. Given, △ 𝐴𝐵𝐶 ∼△ 𝑃𝑄𝑅 with 𝐴𝐵 = 4 cm and 𝑃𝑄 = 10 cm

Since, △ 𝐴𝐵𝐶 ∼△ 𝑃𝑄𝑅
𝐴𝐵 𝐵𝐶 𝐴𝐶
∴ = =
𝑃𝑄 𝑄𝑅 𝑃𝑅
4 𝐵𝐶 𝐴𝐶
⇒ = =
10 15 20
4 × 15
⇒ 𝐵𝐶 = = 6 cm
10
4×20
and 𝐴𝐶 = = 8 cm
10
∴ Perimeter of △ 𝐴𝐵𝐶 = 𝐴𝐵 + 𝐵𝐶 + 𝐴𝐶
= 4 + 6 + 8 = 18 cm
Hence, perimeter of △ 𝐴𝐵𝐶 is 18 cm.
OR
Given, 𝐴𝐷 = 6 cm, 𝐴𝐸 = 18 cm and 𝐵𝐹 = 24 cm
In △ 𝐷𝐸𝐹, 𝐴𝐵 ∥ 𝐸𝐹
[Given]
∴ By Basic proportionality theorem, we have
𝐷𝐴 𝐷𝐵 6 𝐷𝐵
= ⇒ =
𝐴𝐸 𝐵𝐹 18 24
6 × 24
⇒ 𝐷𝐵 = = 8 cm
18
25. Here, 𝑃𝑇 = 𝑃𝑄
[∵ Tangents drawn from an external point are equal.]
∴ 𝑃𝑄 = 7 cm
Also, 𝑆𝑅 = 𝑄𝑅 ∴ 𝑄𝑅 = 4 cm
Now, 𝑅𝑃 = 𝑃𝑄 − 𝑄𝑅 = 7 − 4 = 3 cm
26. Let ℎ be the height of the cone and 𝑟 be its radius.
Given, 𝑟 = (7/2)cm = 3.5 cm
Volume of the toy = 231 cm3
⇒ Volume of cone + Volume of hemisphere = 231 cm3
1 2
⇒ 𝜋𝑟 2 ℎ + 𝜋𝑟 3 = 231
3 3
1
⇒ 𝜋𝑟 2 (ℎ + 2𝑟) = 231
3
1 22 7 7 7
⇒ × × × (ℎ + 2 × ) = 231
3 7 2 2 2
77
⇒ (ℎ + 7) = 231
6
231 × 6
⇒ℎ+7=
77
⇒ ℎ + 7 = 18 ⇒ ℎ = 11 cm ∴ Height of cone is 11 cm
Total height of toy = Height of cone + radius of hemisphere
= 11 + 3.5 = 14.5 cm
27. Let ' 𝑎 ' be the first term and ' 𝑑 ' be the common difference of the A.P.
12 𝑛
∴ 𝑆12 = {2𝑎 + (12 − 1)𝑑} [∵ 𝑆𝑛 = {2𝑎 + (𝑛 − 1)𝑑}]
2 2
= 6{2𝑎 + 11𝑑} = 12𝑎 + 66𝑑
8
𝑆8 = {2𝑎 + (8 − 1)𝑑} = 4{2𝑎 + 7𝑑} = 8𝑎 + 28𝑑
2
4
𝑆4 = {2𝑎 + (4 − 1)𝑑} = 2{2𝑎 + 3𝑑} = 4𝑎 + 6𝑑
2
Now, R.H.S. = 3(𝑆8 − 𝑆4 ) = 3(8𝑎 + 28𝑑 − 4𝑎 − 6𝑑)
= 3(4𝑎 + 22𝑑) = 12𝑎 + 66𝑑 = 𝑆12 = L.H.S.
OR
Let the required three digit number be 𝑥𝑦𝑧.
Since, the digits, 𝑥, 𝑦, 𝑧 are in A.P.
∴ 𝑥 = 𝑦 − 𝑑 and 𝑧 = 𝑦 + 𝑑, where 𝑑 is common difference.
According to the question, (𝑦 − 𝑑) + 𝑦 + (𝑦 + 𝑑) = 15
⇒ 3𝑦 = 15 ⇒ 𝑦 = 5
Since, number obtained by reversing the digits (𝑧𝑦𝑥) i.e., 100𝑧 + 10𝑦 + 𝑥 is 594 less than original
number.
⇒ (100𝑥 + 10𝑦 + 𝑧) − (100𝑧 + 10𝑦 + 𝑥) = 594
⇒ (100𝑥 − 𝑥) + (𝑧 − 100𝑧) = 594
⇒ 99𝑥 − 99𝑧 = 594 ⇒ 𝑥 − 𝑧 = 6 ⇒ (𝑦 − 𝑑) − (𝑦 + 𝑑) = 6
⇒ −2𝑑 = 6 or 𝑑 = −3
So, 𝑥 = 𝑦 − 𝑑 = 5 − (−3) = 8 and 𝑧 = 𝑦 + 𝑑 = 5 − 3 = 2
∴ The required three digit number is 852.

28. Let the assumed mean 𝑎 = 25.
Now, the frequency distribution table from the given data can be drawn as:
Class- Class mark Frequency 𝒅𝒊 = 𝒙𝒊 − 𝒂 𝒇𝒊 𝒅𝒊

interval ( 𝒙𝒊 ) (𝒇𝒊 )
0 − 10 5 3 -20 -60
10 − 20 15 5 -10 -50
20 − 30 25 9 0 0
30 − 40 35 5 10 50
40 − 50 45 3 20 60
∑𝑓𝑖 = 25 ∑𝑓𝑖 𝑑𝑖 = 0
Σ𝑓𝑖 𝑑𝑖 0
∴ Mean, 𝑥‾ = 𝑎 + = 25 + 25 = 25
Σ𝑓𝑖
29. Given, 𝐴𝑀: 𝑀𝐶 = 3: 4, 𝐵𝑃: 𝑃𝑀 = 3: 2 and 𝐵𝑁 = 12 cm Draw 𝑀𝑅 parallel to 𝐶𝑁 which meets 𝐴𝐵

at the point 𝑅. In △ 𝐵𝑀𝑅, 𝑃𝑁 ∥ 𝑀𝑅
𝐵𝑁 𝐵𝑃
∴ =
𝑁𝑅 𝑃𝑀
[By B.P.T.]
12 3 12 × 2
⇒ = ⇒ 𝑁𝑅 = = 8 cm
𝑁𝑅 2 3
In △ 𝐴𝑁𝐶, 𝑅𝑀 ∥ 𝑁𝐶
𝐴𝑅 𝐴𝑀
∴ =
𝑅𝑁 𝑀𝐶
[By B.P.T.]
𝐴𝑅 3 3×8
⇒ = ⇒ 𝐴𝑅 = = 6 cm
8 4 4
∴ 𝐴𝑁 = 𝐴𝑅 + 𝑅𝑁 = 6 + 8 = 14 cm
OR
Let 𝐴𝐵 be the lamp post and 𝐶𝐷 be the height of the boy. Let 𝐷𝐸 = 𝑥 m be the length of his shadow
and 𝐵𝐷 is the distance covered by boy in seconds = 1.5 × 5 = 7.5 m.
In △ 𝐴𝐵𝐸 and △ 𝐶𝐷𝐸,
∠𝐵 = ∠𝐷 [Each equals 90∘]
∠𝐸 = ∠𝐸 [𝐶𝑜𝑚𝑚𝑜𝑚]
∴ △ 𝐴𝐵𝐸 ∼△ 𝐶𝐷𝐸 [𝐵𝑦 𝐴𝐴 𝑠𝑖𝑚𝑖𝑙𝑎𝑟𝑖𝑡𝑦 𝑐𝑟𝑖𝑡𝑒𝑟𝑖𝑜𝑛]
𝐵𝐸 𝐴𝐵 7.5 + 𝑥 3.8
⇒ = ⇒ =
𝐷𝐸 𝐶𝐷 𝑥 0.95
[∵ 𝐴𝐵 = 3.8 m, 𝐶𝐷 = 95 cm = 0.95 m and 𝐵𝐸
= 𝐵𝐷 + 𝐷𝐸 = (7.5 + 𝑥)m]
⇒ 7.125 + 0.95𝑥 = 3.8𝑥 ⇒ 7.125 = 3.8𝑥 − 0.95𝑥
⇒ 7.125 = 2.85𝑥 ⇒ 𝑥 = 7.125 ÷ 2.85 ⇒ 𝑥 = 2.5
Hence, the length of his shadow after 5 seconds is 2.5 m.
30. Let 𝑂𝐷 be the light house and 𝐴 and 𝐵 be two ships such that 𝐴𝐵 = 𝑑 m. Suppose the distance of one
of the ships from the light house is 𝑥 m, then the distance of the other ship from the light house is
(𝑑 − 𝑥)m.
In right △ 𝐴𝐷𝑂, we have
𝑂𝐷 1 100
tan 30∘ = 𝐴𝐷 ⇒ = ⇒ 𝑥 = 100√3 …..(i)
√3 𝑥
In right △ 𝐵𝐷𝑂, we have
𝑂𝐷 100
tan 60∘ = ⇒ √3 =
𝐵𝐷 𝑑−𝑥
⇒ (𝑑 − 𝑥)√3 = 100 ⇒ √3𝑑 − 𝑥√3 = 100
⇒ √3𝑑 − 100√3 ⋅ √3 = 100 [Using (i)]
400
⇒ √3𝑑 = 400 ⇒ 𝑑 = = 230.94
√3
Thus, the distance between two ships is approximately 230.94 m.

𝑎 −2𝑎
31. We have, points 𝑃(𝑎𝑡 2 , 2𝑎𝑡), 𝑄 ( 2 , ) and 𝑆(𝑎, 0).
𝑡 𝑡
∴ 𝑆𝑃 = √(𝑎𝑡 2 − 𝑎)2 + (2𝑎𝑡 − 0)2 = √𝑎2 𝑡 4 + 𝑎2 − 2𝑎2 𝑡 2 + 4𝑎2 𝑡 2
= √𝑎2 𝑡 4 + 𝑎2 + 2𝑎2 𝑡 2 = √(𝑎𝑡 2 + 𝑎)2 = 𝑎𝑡 2 + 𝑎 ….(i)
2
𝑎 2 −2𝑎 𝑎2 2𝑎2 4𝑎2
𝑆𝑄 = √( − 𝑎) + ( − 0) = √ + 𝑎 2− + 2
𝑡2 𝑡 𝑡4 𝑡2 𝑡
𝑎2 2𝑎 2 𝑎 2 𝑎
= √ 𝑡 4 + 𝑎2 + = √(𝑡 2 + 𝑎) = 𝑡 2 + 𝑎 …(ii)
𝑡2
1 1 1 1
Now, 𝑆𝑃 + 𝑆𝑄 = 𝑎𝑡 2 +𝑎 + 𝑎
+𝑎
𝑡2
[Using (i) and (ii)]
1 𝑡2 1+𝑡 2 1+𝑡 2 1
= + = = = , which is
𝑎𝑡 2 +𝑎 𝑎+𝑎𝑡 2 𝑎+𝑎𝑡 2 𝑎(1+𝑡 2 ) 𝑎
independent of 𝑡.
1980
32. Time taken by 𝐴 to complete 1 round = = 6mins
330
1980
Time taken by 𝐵 to complete 1 round = = 10mins
198
1980
Time taken by 𝐶 to complete 1 round = = 9mins
220
∴ Required number of minutes, when the three cyclists will meet at the starting point again is LCM
(6,10,9) minutes.
∵ 6 = 2 × 3,10 = 2 × 5 and 9 = 3 × 3
∴ LCM(6,10,9) = 2 × 5 × 32 = 90 minutes
So, they will meet after 90 minutes or 1 hour 30 mins.
33. Given, a circle with center 𝑂. 𝐴𝐵 is the diameter of this circle. 𝐻𝐾 is tangent to the circle at 𝑃. 𝐴𝐻
and 𝐵𝐾 are perpendicular to 𝐻𝐾 from 𝐴 and 𝐵 at 𝐻 and 𝐾 respectively. Since, 𝐴𝐻 and 𝐻𝑃 are
tangents from the external point 𝐻.
∴ 𝐴𝐻 = 𝐻𝑃
Also, 𝐾𝐵 and 𝐾𝑃 are tangents from the external point 𝐾.
∴ 𝐵𝐾 = 𝐾𝑃
Adding (i) and (ii), we get
𝐴𝐻 + 𝐵𝐾 = 𝐻𝑃 + 𝑃𝐾 = 𝐻𝐾
𝐴𝐵 ⊥ 𝐴𝐻 and 𝐴𝐵 ⊥ 𝐵𝐾
∵ Tangent is perpendicular to the radius through the point of contact]
∴ ∠1 = ∠2 = 90∘
Also, 𝐴𝐻 ⊥ 𝐻𝐾
⇒ ∠3 = 90∘ and 𝐵𝐾 ⊥ 𝐻𝐾 ⇒ ∠4 = 90∘
[Given]
Thus, ∠1 = ∠2 = ∠3 = ∠4 = 90∘
∴ 𝐴𝐻𝐾𝐵 is a rectangle.
⇒ 𝐴𝐵 = 𝐻𝐾
[∵ Opposite sides of a rectangle are equal ]
From (iii) and (iv), 𝐴𝐻 + 𝐵𝐾 = 𝐴𝐵
OR
Since, length of tangents drawn form an external point to a circle are equal.
∴ 𝑄𝑆 = 𝑄𝑇 = 14 cm,
𝑅𝑈 = 𝑅𝑇 = 16 cm.
Let, 𝑃𝑆 = 𝑃𝑈 = 𝑥 cm
Thus, 𝑃𝑄 = (𝑥 + 14)cm
𝑃𝑅 = (𝑥 + 16)cm
and 𝑄𝑅 = 30 cm
Now, Area of △ 𝑃𝑄𝑅
= Area of △ 𝐼𝑄𝑅 + Area of △ 𝐼𝑄𝑃 + Area of △ 𝐼𝑃𝑅
1 1 1
⇒ 336 = (14 + 16) × 8 + (14 + 𝑥) × 8 + (16 + 𝑥) × 8
2 2 2
⇒ 84 = 30 + 14 + 𝑥 + 16 + 𝑥
⇒ 24 = 2𝑥 ⇒ 𝑥 = 12
Hence, 𝑃𝑄 = 26 cm and 𝑃𝑅 = 28 cm
34. Given ∠𝐵𝑂𝐶 = 120∘

In △ 𝐵𝑂𝐶, 𝑂𝐵 = 𝑂𝐶 = 𝑟
⇒ ∠𝑂𝐵𝐶 = ∠𝑂𝐶𝐵
1
= (180∘ − 120∘) = 30∘
2
Draw 𝑂𝑀 ⊥ 𝐵𝐶
𝑂𝑀 1 1 𝑟
In △ 𝑂𝑀𝐶, 𝑂𝐶 = sin 30∘ = 2 ⇒ 𝑂𝑀 = 2 × 𝑂𝐶 = 2
𝑀𝐶 √3 √3
Also, = cos 30∘ = ⇒ 𝑀𝐶 = 𝑟 ⇒ 2𝑀𝐶 = √3𝑟
𝑂𝐶 2 2
⇒ 𝐵𝐶 = √3𝑟 (∵ 𝐵𝑀 = 𝑀𝐶, by construction )
𝜋𝑟 2𝜃 1
∴ Area of minor segment = − 2 × 𝑂𝑀 × 𝐵𝐶
360∘
120∘ 1 𝑟
= (𝜋𝑟 2 × ) − ( × × √3𝑟)
360∘ 2 2
𝜋𝑟 2 √3 2 𝜋 √3
= − 𝑟 = ( − ) 𝑟2
3 4 3 4
35. Given pair of linear equations is

6𝑥
𝑥 + 3𝑦 = 6 ⇒ 𝑦
3
2𝑥 12
and 2𝑥 − 3𝑦 = 12 ⇒ 𝑦 3
Tables of solutions for (i) and (ii) are given below:

𝑥 6 0
𝑦 0 2
(i)
𝑥 6 0
𝑦 0 -4
(ii)
Now, plotting the points 𝐴(6,0), 𝐵(0,2) and 𝐶(0, −4) on the graph paper and joining them, we get
the graphical representation of the given pair of linear equations, which is as follows:
The line 𝑥 + 3𝑦 = 6 intersects the 𝑦-axis at (0,2) and the line 2𝑥 − 3𝑦 = 12 intersects the 𝑦-axis at
(0, −4) and the two lines intersect at the point (6,0) on 𝑥-axis.
The area of the triangle formed by first line, 𝑥 = 0, 𝑦 = 0 and 𝑥 + 3𝑦 = 6

1 1
= Area of △ 𝐴𝐵𝑂 = 2 (base × height) = 2 × 2 × 6 = 6 sq. units.
Also, the area of the triangle formed by second line, 𝑥 = 0, 𝑦 = 0 and 2𝑥 − 3𝑦 = 12

1
= Area of △ 𝐴𝑂𝐶 = 2 × 4 × 6 = 12 sq. units.
Area of △𝐴𝐵𝑂 6 1
So, the ratio of areas of triangles = = 12 = 2
Area of △𝐴𝑂𝐶
Hence, the ratio of the areas of triangles is 1: 2.

OR
Let the cost price of each chair be ₹ 𝑥 and cost price of each table be ₹ 𝑦.
According to the question, we have
3400 − 4𝑥
4𝑥 + 5𝑦 = 3400 ⇒ 𝑦 =
5
and 𝑥 + 4𝑦 = 2500
2500 − 𝑥
⇒𝑦=
4
Table of solutions for (i) is:
𝑥 400 0 850
𝑦 360 680 0
Table of solutions for (ii) is:
𝑥 900 500 0
𝑦 400 500 625
Plotting the points 𝐴(400,360), 𝐵(0,680), 𝐶(850,0) on the graph paper and joining them we get the
line representing 4𝑥 + 5𝑦 = 3400.
Similarly, plotting the points 𝐷(900,400), 𝐸(500,500), 𝐹(0,625) on the same graph paper and
joining them we get the line representing 𝑥 + 4𝑦 = 2500.
Clearly, pair of linear equations intersect each other at point 𝑃(100,600).
36. (i) Distance of first position of pigeon from the eyes of boy = 𝐴𝐶
𝐵𝐶
In △ 𝐴𝐵𝐶, sin 60∘ = 𝐴𝐶
𝐶𝐻 𝐵𝐻 54 − 4 100
⇒ 𝐴𝐶 = = m
sin 60∘ √3/2 √3
(ii) If the distance increases, then the angle of elevation decreases.
(iii) Distance between boy and pole = 𝐴𝐵

𝐵𝐶
Now, in △ 𝐴𝐵𝐶, tan 60∘ = ⇒ √3𝐴𝐵 = 50
𝐴𝐵
50
⇒ 𝐴𝐵 = m
√3
OR
𝐸𝐷
In △ 𝐴𝐸𝐷, tan 45∘ = 𝐴𝐷 ⇒ 𝐴𝐷 = 𝐵𝐶 = 50 m
(∵ 𝐸𝐷 = 𝐵𝐶)
Now, distance between two positions of pigeon = 𝐸𝐶
= 𝐵𝐷 = 𝐴𝐷 − 𝐴𝐵
50 50(1.73 − 1)
= (50 − )m = = 21.09 m
√3 1.73
37. (i) Total number of possible outcome, 𝑛(𝑆) = 200

Let 𝐷 be the event that the number on the selected card is a perfect square.
∴ 𝐷 = {1,4,9,16,25,36,49,64,81,100,121,144,169,196}
𝑛(𝐷) 14 7
⇒ 𝑛(𝐷) = 14 ∴ 𝑃(𝐷) = = =
𝑛(𝑆) 200 100
(ii) Total number of possible outcome, 𝑛(𝑆) = 200
Let 𝐴 be the event that number on selected card is divisible by 10.
∴ 𝐴 = {10,20,30,40,50,60,70,80,90,100,110,120,130, 140,150,160,170,180,190,200}
𝑛(𝐴) 20 1
⇒ 𝑛(𝐴) = 20 ∴ 𝑃(𝐴) = = =
𝑛(𝑆) 200 10
OR
Let 𝐵 be the event that the number on the selected card is a prime number more than 100 but less than
150.
∴ 𝐵 = {101,103,107,109,113,127,131,137,139,149}
𝑛(𝐵) 10 1
⇒ 𝑛(𝐵) = 10 ∴ 𝑃(𝐵) = = =
𝑛(𝑆) 200 20
(iii) Let 𝐸 be the event that the number on the selected card is a perfect cube.
∴ 𝐸 = {1,8,27,64,125} ⇒ 𝑛(𝐸) = 5
5 1
∴ 𝑃(𝐸) = =
200 40
38. (i) Let two consecutive integers be 𝑥, 𝑥 + 1.
Given, 𝑥 2 + (𝑥 + 1)2 = 650 ⇒ 2𝑥 2 + 2𝑥 + 1 − 650 = 0
⇒ 2𝑥 2 + 2𝑥 − 649 = 0
(ii) Let the two numbers be 𝑥 and 15 − 𝑥.

1 1 3
Given, 𝑥 + 15−𝑥 = 10
⇒ 10(15 − 𝑥 + 𝑥) = 3𝑥(15 − 𝑥) ⇒ 50 = 15𝑥 − 𝑥 2
⇒ 𝑥 2 − 15𝑥 + 50 = 0 ⇒ 𝑥 2 − 10𝑥 − 5𝑥 + 50 = 0
⇒ 𝑥(𝑥 − 10) − 5(𝑥 − 10) = 0 ⇒ (𝑥 − 5)(𝑥 − 10) = 0
⇒ 𝑥 = 5,10
Hence number are 5, 10
OR
Let the number be 𝑥.
160
According to question, 𝑥 + 12 = 𝑥
⇒ 𝑥 2 + 12𝑥 − 160 = 0 ⇒ 𝑥 2 + 20𝑥 − 8𝑥 − 160 = 0
⇒ 𝑥(𝑥 + 20) − 8(𝑥 + 20) = 0 ⇒ (𝑥 + 20)(𝑥 − 8) = 0
⇒ 𝑥 = −20,8
∴ 𝑥 = 8 [ -20 is not a natural number, so it is rejected]
(iii) Let the numbers be 𝑥 and 𝑥 + 3.
Given, 𝑥(𝑥 + 3) = 504
⇒ 𝑥 2 + 3𝑥 − 504 = 0
Sample
Paper
Set - 2
With a success rate
2024 boards
Mathematics
Class 10
Set - 2
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.
marks questions of Section E.
SECTION A

1. It is given that there is no solution to the system of equations 𝑥 + 2𝑦 = 3, 𝑎𝑥 + 𝑏𝑦 = 4. Which one
of the following is true?
(a) 𝑎 has a unique value
(b) 𝑏 has a unique value
(c) a can have more than one value
(d) 𝑎 has exactly two different values
2. If 𝑥 = 𝑎tan⁡𝜃 and 𝑦 = 𝑏sec⁡𝜃, then

𝑦2 𝑥2
(a) 2
− =1
𝑏 𝑎2
𝑥2 𝑦2
(b) + =1
𝑎2 𝑏2
𝑥2 𝑦2
(c) − =1
𝑎2 𝑏2
𝑥2 𝑦2
(d) − =0
𝑎2 𝑏2
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SOLUTIONS
3. The probability of getting an even number, when a die is thrown once, is
(a) 1/2
(b) 1/3
(c) 1/6
(d) 5/6
4. If one of the zeroes of the quadratic polynomial (𝑘 − 1)𝑥 2 + 𝑘𝑥 + 1 is -3 , then the value of 𝑘 is
4
(a)
3
−4
(b)
3
2
(c)
3
−2
(d)
3
5. The three consecutive vertices of a parallelogram are (𝑎 + 𝑏, 𝑎 − 𝑏), (2𝑎 + 𝑏, 2𝑎 − 𝑏), (𝑎 − 𝑏, 𝑎 +

𝑏). The fourth vertex is
(a) (𝑎, 𝑏)
(b) (𝑏, 𝑏)
(c) (−𝑏, 𝑏)
(d) (−𝑎, −𝑏)
6. If tan⁡𝜃 = √3, then find the value of sin⁡𝜃cos⁡𝜃.

√3
(a)
2
(b) √3
√3
(c)
4
1
(d)
√3
7. Find the sixteenth term of the A.P. −10, −6, −2,2, …

(a) 10
(b) 20
(c) 40
(d) 50
8. In a △ 𝑋𝑌𝑍, 𝑃𝑄 ∥ 𝑌𝑍 such that 𝑋𝑃 = 2𝑥 cm, 𝑋𝑄 = (𝑥 + 2)cm, 𝑌𝑃 = (6𝑥 − 5)cm, 𝑍𝑄 = (3𝑥 +

1)cm, then find the value of 𝑥.
(a) 2
(b) 4
(c) 3/2
(d) 3
9. The mean and mode of a frequency distribution are 28 and 16 respectively. The median is
(a) 22
(b) 23.5
(c) 24
(d) 24.5
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SOLUTIONS
10. Find the value of 𝑘 for which the system of equations 𝑘𝑥 − 𝑦 = 4,10𝑥 − 2𝑦 = 3 has no solution.
(a) 6
(b) 4
(c) 5
(d) 2
11. Find the distance of the point (36,15) from the origin.
(a) 39 units
(b) 37 units
(c) 36 units
(d) 35 units
12. In the given figure, 𝐸𝐷 ∥ 𝐶𝐴. Express 𝑏 in terms of 𝑥, 𝑦 and 𝑎.
𝑎𝑥
(a) −𝑎
𝑦
𝑎𝑦
(b) −𝑎
𝑥
𝑎𝑦
(c) 𝑎 −
𝑥
𝑎𝑥
(d) 𝑎 −
𝑦
13. A die is thrown once. Find the probability of getting a number which is not a factor of 36.
1
(a)
3
1
(b)
2
1
(c)
6
5
(d)
6
14. A box contains 100 memory cards out of which 25 are good and 75 are defective. A memory card is
selected at random. The probability that selected memory card is defective is
1
(a)
4
1
(b)
2
3
(c)
4
(d) 1
15. Find the least positive integer divisible by 20 and 24.
(a) 24
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SOLUTIONS
(b) 15
(c) 12
(d) 120
1
16. If the sum of the zeroes of a polynomial is − and product of the zeroes of the polynomial is -2, then
6
the polynomial is
1
(a) 𝑥 2 − 𝑥 + 2
6
1
(b) 𝑥2
+ 𝑥−2
6
2
(c) 6𝑥 − 𝑥 + 12
(d) 6𝑥 2 + 𝑥 − 12
17. In what ratio does the point (−2,3) divides the line segment joining the points (−3,5) and (4, −9) ?
(a) 1: 6
(b) 6: 1
(c) 5: 1
(d) 1: 5
18. It is given that △ 𝐴𝐵𝐶 ∼△ 𝐸𝐷𝐹 such that 𝐴𝐵 = 5 cm, 𝐴𝐶 = 7 cm, 𝐷𝐹 = 15 cm and 𝐷𝐸 = 12 cm.
Find the length of 𝐸𝐹.
(a) 13 cm
(b) 14.8 cm
(c) 15.2 cm
(d) 16.8 cm
19. Statement A (Assertion): 5𝑥 2 + 14𝑥 + 10 = 0 has no real roots. Statement 𝐑 (Reason) : 𝑎𝑥 2 + 𝑏𝑥 +

𝑐 = 0 has no real roots if 𝑏 2 < 4𝑎𝑐.
(A).
assertion (A).

8 15
20. Statement A (Assertion): In a △ 𝐴𝐵𝐶, right angled at 𝐵, if sin⁡𝐴 = , then cos⁡𝐴 = and tan⁡𝐴 =
17 17
8
.
15
Hypotenuse Base
Statement 𝐑 (Reason): For acute angle 𝜃, cos⁡𝜃 = , and tan⁡𝜃 = .
Base Perpendicular
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SOLUTIONS
(A).
(b) Both assertion (𝐴) and reason (𝑅) are true and reason (𝑅) is not the correct explanation of
assertion (𝐴).
SECTION B

𝐴𝑂 𝐵𝑂 1
21. In the given figure, = = and 𝐴𝐵 = 5 cm. Find the length of 𝐷𝐶.
𝑂𝐶 𝑂𝐷 2
cos⁡𝐴+sin⁡𝐴 √3+1
22. For what value of 𝐴, = ?
cos⁡𝐴−sin⁡𝐴 √3−1
OR
If 2sin2 ⁡𝜃 − cos2 ⁡𝜃 = 2, then find the value of 𝜃.
23. In the given figure, the radii of two concentric circles are 7 cm and 8 cm. If 𝑃𝐴 = 15 cm then find
𝑃𝐵.
24. Two cubes each of volume 125 cm3 are joined end to end. Find the surface area of the resulting
cuboid.
25. Find the values of 𝑘 for which the quadratic equation (𝑘 + 4)𝑥 2 + (𝑘 + 1)𝑥 + 1 = 0 has real and
equal roots.
OR
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SOLUTIONS
Find the roots of the quadratic equation 𝑥 2 + 3 = 3√11𝑥 by using quadratic formula.
SECTION C

26. In figure, 𝐶𝐷 ∥ 𝐿𝐴 and 𝐷𝐸 ∥ 𝐴𝐶. Find 𝐶𝐿.
27. A sum of ₹2000 is invested at 7% simple interest per year. Calculate the interest at the end of each
year. Do these interest form an A.P.? If so, then find the interest at the end of 27th year making use
of this fact.
OR
What is the common difference of four terms in an A.P. such that the ratio of the product of the first
and fourth terms to that of the second and third is 2: 3 and the sum of all four terms is 20 ?
28. Find the value of 𝑝 from the following data, if its mode is 48 .
Class- 0 10 20 30 40 50 60 70
interval − 10 − 20 − 30 − 40 − 50 − 60 − 70 − 80
Frequency 7 14 13 12 𝑝 18 15 8
29. From the top of a 60 m high building, the angles of depression of the top and the bottom of a tower
are observed to be 30∘ and 60∘ respectively. Find the height of the tower.
30. A metallic cylinder has radius 3 cm and height 5 cm. To reduce its weight, a conical hole is drilled in
3 8
the cylinder. The conical hole has a radius of cm and its depth is cm. Calculate the ratio of the
2 9
volume of metal left in the cylinder to the volume of metal taken out in conical shape.
31. Let 𝑃 and 𝑄 be the points of trisection of the line segment joining the points 𝐴(2, −2) and 𝐵(−7,4)
such that 𝑃 is nearer to 𝐴. Find the coordinates of 𝑃 and 𝑄.
OR
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SOLUTIONS
The base 𝐵𝐶 of an equilateral triangle 𝐴𝐵𝐶 lies on 𝑦-axis. The coordinates of point 𝐶 are (0, −3).
The origin is the mid-point of the base. Find the coordinates of the points 𝐴 and 𝐵. Also, find the
coordinates of another point 𝐷 such that 𝐵𝐴𝐶𝐷 is a rhombus.
SECTION D

32. 𝐴, 𝐵 and 𝐶 start cycling around a circular path in the same direction and at the same time.
33. A chord 𝑃𝑄 of a circle of radius 10 cm subtends an angle of 90∘ at the centre of circle. Find the area
of major and minor segments of the circle.
OR
In a circle of radius 28 cm, an arc subtends an angle of 120 ∘ at the centre. Find
(i) the length of arc.
(ii) area of the minor sector and major sector formed by the arc.
(iii) area of the segment formed by corresponding chord. (Use √3 = 1.7 )
34. In the adjoining figure, the incircle of △ 𝐴𝐵𝐶 touches the sides 𝐵𝐶, 𝐶𝐴 and 𝐴𝐵 at 𝐷, 𝐸 and 𝐹
1
respectively. Show that 𝐴𝐹 + 𝐵𝐷 + 𝐶𝐸 = 𝐴𝐸 + 𝐵𝐹 + 𝐶𝐷 = (Perimeter of △ 𝐴𝐵𝐶 )
2
35. Find the values of 𝑝 and 𝑞 for which the following system of equations has infinitely many solution
2(𝑝 + 𝑞)𝑥 − 4𝑞𝑦 = 7𝑝 + 4𝑞 + 3; 5𝑥 − 𝑦 = 16
OR
Ravi invested some amount at the rate of 8% simple interest and some other amount at the rate of 9%
simple interest. He received yearly interest of ₹ 163. But if he had interchanged the amounts invested,
he would have received ₹3 less as interest. How much did he invest at different rates?
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SOLUTIONS
SECTION E

36. If 𝛼 and 𝛽 are the roots of the quadratic equation then the quadratic equation is (𝑥 − 𝛼)(𝑥 − 𝛽) = 0
if 𝛼 is the one root of the quadratic equation then its satisfy the given quadratic equation and it is also
known that if one roots of a quadratic equation is irrational, then its other roots is also irrational. If
the quadratic equation is 𝑎𝑥 2 + 𝑏𝑥 + 𝑐 = 0 then the sum of roots is equal to −𝑏/𝑎.
(i) If the roots of the quadratic equation are 2, −3, then find its equation.
(ii) Find the factors of the quadratic equation 𝑥 2 − 5𝑥 + 6.
(iii) If one root of the quadratic equation 2𝑥 2 + 𝑘𝑥 + 1 = 0 is −1/2 and if (𝑥 − 𝑘)2 + 19 = 0

equals to 𝑎𝑥 2 + 𝑏𝑥 + 𝑐 = 0, then find the value of 𝑎 + 𝑏 + 𝑐.
OR
Find the sum of roots of polynomial 𝑥 2 − 5𝑥 + 6. Also, find the value of polynomial at 𝑥 = 0.
37. Two friends Richa and Sohan have some savings in their piggy bank. They decided to count the total
coins they both had. After counting they find that they have fifty ₹ 1 coins, forty eight ₹ 2 coins,
thirty six ₹ 5 coins, twenty eight ₹ 10 coins and eight ₹ 20 coins. Now, they said to Nisha, their
another friends, to choose a coin randomly. Find the probability that the coin chosen is
(i) Find the probability that the chosen coin is ₹ 5 coin.
(ii) Find the probability that the chosen coin is of denomination of atleast ₹ 10.
OR
Find the probability that the chosen coin is of denomination of atmost ₹ 5.
(iii) Find the probability that the chosen coin is ₹ 20 coin.
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SOLUTIONS
38. A man sitting on the bridge across a river observes the angles of depression of the banks on opposite
sides of the river are 30∘ and 45∘ respectively. (Take √3 = 1.73 )
Use the above figure to answer the questions that follow:
(i) If the bridge is at a height of 4 m, then find 𝐴𝐷.
(ii) Find 𝐵𝐷.
OR
If 𝐵𝐷 = 15 m, then find the height of the bridge.
(iii) Find the width of the river.
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SOLUTIONS
Sample
Paper
Set - 3
With a success rate
2024 boards
Mathematics
Class 10
Set - 3
SECTION A
1. If the common difference of an A.P. is 5 , then the value of 𝑎18 − 𝑎13 is
(a) 5
(b) 20
(c) 25
(d) 30
2. The probability expressed as a percentage of a particular occurrence can never be

(a) less than 100
(b) less than 0
(c) greater than 1
(d) anything but a whole number
3. A school has five houses 𝐴, 𝐵, 𝐶, 𝐷 and 𝐸. A class has 23 students, 4 from house 𝐴, 8 from house 𝐵, 5
from house 𝐶, 2 from house 𝐷 and rest from house 𝐸. A single student is selected at random to be the
class monitor. The probability that the selected student is not from 𝐴, 𝐵 and 𝐶 is
4
(a) 23
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SOLUTIONS
6
(b) 23
8
(c) 23
17
(d) 23
4. If HCF⁡(26,169) = 13, then LCM⁡(26,169) equal to

(a) 26
(b) 52
(c) 338
(d) 13
5. Find the value of 𝑥 for which the distance between the points 𝑃(3,4) and 𝑄(𝑥, 7) is √13 units.
(a) 1, −5
(b) −1,5
(c) 1,5
(d) −1, −5
6. If △ 𝐴𝐵𝐶 ∼△ 𝑃𝑄𝑅, 𝐴𝐵 = 4 cm, 𝑃𝑄 = 10 cm, 𝑄𝑅 = 15 cm, 𝑃𝑅 = 20 cm, then find the perimeter of
△ 𝐴𝐵𝐶.
(a) 18 cm
(b) 16 cm
(c) 45 cm
(d) 20 cm
𝑎 (𝑎sin⁡𝜃−𝑏cos⁡𝜃)
7. If tan⁡𝜃 = 𝑏 , then (𝑎sin⁡𝜃+𝑏cos⁡𝜃) is equal to
(𝑎 2 +𝑏2 )
(a) (𝑎2
−𝑏2 )
(𝑎 2 −𝑏2 )
(b) (𝑎2
+𝑏2 )
𝑎2
(c) (𝑎2 +𝑏2 )
𝑏2
(d) (𝑎2+𝑏2 )
8. Find the conditions to be satisfied by coefficients for which the following pair of equations 𝑎𝑥 +
𝑏𝑦 + 𝑐 = 0, 𝑑𝑥 + 𝑒𝑦 + 𝑓 = 0 represent coincident lines.
(a) 𝑎𝑏 = 𝑒𝑑; 𝑏𝑓 = 𝑐𝑒
(b) 𝑎𝑒 = 𝑏𝑑; 𝑏𝑐 = 𝑒𝑓
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SOLUTIONS
(c) 𝑎𝑑 = 𝑏𝑐; 𝑏𝑓 = 𝑐𝑒
(d) 𝑎𝑒 = 𝑏𝑑; 𝑏𝑓 = 𝑐𝑒
9. A bag contains 4 red, 5 black and 3 yellow balls. A ball is taken out of the bag at random. Find the
probability that the ball taken out is not of yellow colour.
2
(a) 3
1
(b) 3
3
(c) 4
1
(d) 2
10. In the given circle, 𝑂 is a centre, 𝑂𝑃 = 8 cm and 𝑂𝑄 = 17 cm, then the length of the tangent 𝑃𝑄 will
be
(a) 10 cm
(b) 14 cm
(c) 15 cm
(d) 25 cm
11. One of the properties of mode is

(a) not easy to calculate
(c) algebraic
(b) it is not affected by greatest and least values
(d) difference of greatest and least values
12. Which of the following is not the graph of a quadratic polynomial?
(a)
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SOLUTIONS
(b)
(c)
(d)
13. If the distance of the point (4, 𝑎) from 𝑥-axis is double its distance from 𝑦-axis, then find the value of
𝑎.
(a) 5
(b) 8
(c) 16
(d) 4
14. Find the value of ' ℎ ' in the adjoining figure, if △ 𝐴𝐷𝐸 ∼△ 𝐴𝐵𝐶.
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SOLUTIONS
(a) 12 m
(b) 15 m
(c) 10 m
(d) 8 m
15. If 2sin⁡2𝜃 = √3, then 𝜃 is equal to

(a) 30∘
(b) 45∘
(c) 60∘
(d) None of these
16. If the system of equations 2𝑥 + 3𝑦 = 5,4𝑥 + 𝑘𝑦 = 10 has infinitely many solutions, then find 𝑘.
(a) 4
(b) 3
(c) 6
(d) 8
17. Find the coordinates of point 𝐴, where 𝐴𝐵 is the diameter of a circle whose centre is 𝑂(2, −3) and 𝐵
is (1,4).
(a) (2,10)
(b) (3, −10)
(c) (3, −1)
(d) (1, −7)
18. The polynomial having zeroes as -2 and 5, is

(a) 𝑥 2 − 5𝑥 − 10
(b) 𝑥 2 + 3𝑥 + 10
(c) 𝑥 2 + 3𝑥 − 10
(d) 𝑥 2 − 3𝑥 − 10
19. Statement A (Assertion): 4𝑥 2 − 12𝑥 + 9 = 0 has repeated roots.

Statement 𝐑 (Reason): The quadratic equation 𝑎𝑥 2 + 𝑏𝑥 + 𝑐 = 0 have repeated roots if 𝐷 > 0.
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SOLUTIONS
(A).
assertion (A).
𝑥 2−1
20. Statement A (Assertion): If sec⁡𝜃 + tan⁡𝜃 = 𝑥, then the value of sin⁡𝜃 = 𝑥 2+1.
1 1
Statement 𝑅 (Reason): If sec⁡𝜃 + tan⁡𝜃 = 𝑥, then 𝑥 + 𝑥 = 2tan⁡𝜃 and 𝑥 − 𝑥 = 2sec⁡𝜃.
(A).
assertion (A).
SECTION B
Section 𝐵 consists of 5 questions of 2 marks each.

21. A vertical stick 12 m long casts shadow 8 m long on the ground. At the same time, a tower casts the
shadow 40 m long on the ground. Determine the height of the tower.
OR
In the given figure, if 𝐴𝐵 ∥ 𝐷𝐶, find the value of 𝑥.
22. For what values of 𝑛, the equation 2𝑥 2 − 𝑛𝑥 + 𝑛 = 0 has coincident roots?
8 √2
23. Show that cosec 2 ⁡60∘ sec 2 ⁡30∘ cos 2 ⁡0∘ sin⁡45∘ cot 2 ⁡60∘ tan2 ⁡60∘ = .
9
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SOLUTIONS
OR
1
If sin⁡(𝐴 + 𝐵) = 1 and sin⁡(𝐴 − 𝐵) = 2, where 0∘ ≤ 𝐴 + 𝐵 ≤ 90∘ and 𝐴 > 𝐵, then find 𝐴 and 𝐵.
24. From a solid cube of side 7 cm, a conical cavity of height 7 cm and radius 3 cm is hollowed out.
Find the volume of the remaining solid.
25. In the given figure, TAS is a tangent to the circle, with centre 𝑂, at the point 𝐴. If ∠𝑂𝐵𝐴 = 32∘, then
find the values of 𝑥 and 𝑦 respectively.
SECTION C

26. If 𝑄(0,1) is equidistant from 𝑃(5, −3) and 𝑅(𝑥, 6), find the values of 𝑥. Also find the distances 𝑄𝑅
and 𝑃𝑅.
27. 𝐴𝐵𝐶 is an isosceles triangle with 𝐴𝐵 = 𝐴𝐶 and 𝐷 is a point on 𝐴𝐶 such that 𝐵𝐶 2 = 𝐴𝐶 × 𝐶𝐷. Prove
that 𝐵𝐷 = 𝐵𝐶.
OR
In △ 𝐴𝐵𝐶, if 𝐴𝐷 ⊥ 𝐵𝐶 and 𝐴𝐷2 = 𝐵𝐷 × 𝐷𝐶, then prove that ∠𝐵𝐴𝐶 = 90∘ .
28. A tower stands on a horizontal plane and is surmounted by a flagstaff. At a point on the plane, 70
metres away from the tower, an observer notices that the angles of elevation of the top and the bottom
of the flagstaff are respectively 60∘ and 45∘. Find the height of the flagstaff and that of the tower.
[Use √3 = 1.732 ]
29. A toy is in the shape of a cylinder with two equal cones stuck to each of its ends. The length of entire
solid is 30 cm and diameter of cylinder and cones is 10.5 cm and the length of the cylinder is 14 cm.
Find its surface area.
30. Find the middle term(s) of the A.P.: 6,12,18, … ,240.

OR
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SOLUTIONS
The 17th term of an A.P. is 5 more than twice its 8th term. If the 11th term of the A.P. is 43 , then
find its 𝑛th term.
31. On sports day of a school, age wise participation of students is shown in the following distribution:
11 13 15 17
Age (in years) 5−7 7−9 9 − 11
− 13 − 15 − 17 − 19
Number of students 𝑥 15 18 30 50 48 𝑥
Find the mode of the data. Also, find missing frequencies when sum of frequencies is 181.
SECTION D
Section 𝐷 consists of 4 questions of 5 marks each.

32. Mr. Sharma and Mr. Arora are family friends and they decided to go for a trip. For the trip they
reserved their rail tickets. Mr. Arora has not taken a half ticket for his child who is 6 years old
whereas Mr. Sharma has taken half tickets for his two children who are 6.5 years and 8 years old. A
railway half ticket costs half of the full fare but the reservation charges are the same as on a full
ticket. Mr. and Mrs. Arora paid ₹ 1700, while Mr. and Mrs. Sharma paid ₹ 2700. Find the full fare of
one ticket and the reservation charges per ticket.
OR
Rajiv walks and cycles at uniform speeds. When he walks for 2hrs and cycles for 1hr, distance
travelled is 24 km. When he walks for 1hr and cycles for 2hrs, distance travelled is 39 km. Find his
speed of walking and cycling. If he walked and cycled for equal time in 3hrs how much distance
does he cover?
33. Let 𝑎, 𝑏 and 𝑐 be rational numbers such that 𝑝 is not a perfect cube. If 𝑎 + 𝑏𝑝1/3 + 𝑐𝑝2/3 = 0, then
prove that 𝑎 = 𝑏 = 𝑐.
34. If 𝑎, 𝑏, 𝑐 are the sides of a right triangle where 𝑐 is hypotenuse, prove that the radius 𝑟 of the circle
𝑎+𝑏−𝑐
which touches the sides of the triangle is given by 𝑟 = 2 or, 𝑟 = 𝑠 − 𝑐, where 𝑠 is the semi-
perimeter of the
triangle.
OR
Two tangents 𝐴𝐵 and 𝐴𝐶 are drawn to two intersecting circles with centres 𝑂 and 𝑂′ respectively
from a point of intersection 𝐴. Let 𝑃 be a point such that 𝐴𝑂𝑃𝑂′ is a parallelogram. Prove that 𝑃 is
circumcentre of △ 𝐴𝐵𝐶.
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SOLUTIONS
35. Find the area of the segment 𝐴𝑌𝐵 as shown in figure, if radius of the circle is 21 cm and ∠𝐴𝑂𝐵 =
22
60∘. ( Use 𝜋 = 7 )
SECTION E
36. In the month of May, the weather forecast department gives the prediction of weather for the month
of June. The given table shows the probabilities of forecast of different days:
Days Sunny Cloudy Partially cloudy Rainy
1 1
Probability 𝑥 𝑦
2 5
Consider the forecast is 100% correct for June.
(i) Find the number of sunny days in June.
(ii) If the number of cloudy days in June is 5 , then find the value of 𝑥.
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SOLUTIONS
3
(iii) If the number of rainy days in June is 6 and the sum of 𝑥 and 𝑦 is 10, then find the number of
cloudy days in June.
OR
If the number of cloudy days in June is 3, then find the probability that the day is not rainy.
37. Suppose a straight vertical tree is broken at some point due to cyclone and the broken part is inclined
at a certain distance from the foot of the tree.
(i) If the top of broken part of a tree touches the ground at a point whose distance from foot of the
tree is equal to height of remaining part, then find its angle of inclination.
(ii) If the top of upper part of broken tree touches ground at a distance of 6 m (from the foot of the
tree) and makes an angle of inclination 60∘. What will be the height of remaining part of the tree?
OR
If the height of a tree is 3 m, which is broken by wind in such a way that its top touches the ground
and makes an angle 30∘ with the ground. At what height from the bottom of the tree is broken by the
(iii) If 𝐴𝐵 = 6 m, 𝐴𝐷 = 2 m, then find 𝐶𝐷.
38. Jia's father who is a mathematician, was looking into maths answer sheet of Jia's recent semester. At
an instance, he came across the below graph and asked her the following questions to test her
knowledge. Try answering the same to test your knowledge.
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SOLUTIONS
(i) Find the degree of polynomial by considering given graph.
(ii) Find the equation of polynomial represented by given graph.
OR
For what value of 𝑥 the value of polynomial is -3 ?
(iii) Find the zeroes of the polynomial.
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SOLUTIONS
Sample
Paper
Set - 4
With a success rate
2024 boards
Mathematics
Class 10
SET-04
2. Section 𝐴 has 20 MCQs carrying 1 mark each.
SECTION A
1. 𝐴𝐵𝐶𝐷 is a rectangle whose three vertices are 𝐵(4,0), 𝐶(4,3) and 𝐷(0,3). The length of one of its
diagonals is
(a) 5 units
(b) 4 units
(c) 3 units
(d) 25 units
2. Two poles of height 13 m and 7 m respectively stand vertically on a plane ground at a distance of
8 m from each other. The distance between their tops is
(a) 9 m
(b) 10 m
(c) 11 m
(d) 12 m
3. In the given figure, if 𝑃𝑄 ∥ 𝐵𝐶. Then the value of 𝐴𝑄 is
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SOLUTIONS
(a) 4.5 cm
(b) 5.5 cm
(c) 5.4 cm
(d) 3.5 cm
4. From a pack of 52 playing cards, a card is drawn at random. The probability that the drawn card is
not a face card, is
3
(a) 13
9
(b) 13
10
(c) 13
3
(d) 4
5. The least number which when divided by 18,24,30 and 42 will leave in each case the same remainder
1, would be
(a) 2520
(b) 2519
(c) 2521
(d) None of these
6. The ratio in which the 𝑥-axis divides the line segment joining 𝐴(3,6) and 𝐵(12, −3) is
(a) 2: 1
(b) 1: 2
(c) −2: 1
(d) 1: −2
7. In the given figure, 𝑃𝑄 and 𝑃𝑅 are two tangents to a circle with centre 𝑂. If ∠𝑄𝑃𝑅 = 46∘, then
∠𝑄𝑂𝑅 equals
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SOLUTIONS
(a) 67∘
(b) 134∘
(c) 44∘
(d) 46∘
8. The areas of three adjacent faces of a rectangular block are 8,10 and 20sq. cm, then volume of
rectangular block is
(a) 1600 cm3
(b) 20 cm3
(c) 40 cm3
(d) 80 cm3
9. Consider the following frequency distribution.
10 20 30 40 50
Class interval 0 − 10
− 20 − 30 − 40 − 50 − 60
Frequency 3 9 15 30 18 5
The modal class is

(a) 10 − 20
(b) 20 − 30
(c) 30 − 40
(d) 40 − 50
10. The zeroes of the quadratic polynomial 𝑥 2 + 25𝑥 + 156 are

(a) both positive
(b) both negative
(c) one positive and one negative
(d) can't be determined
8 (1+sin⁡𝜃)(1−sin⁡𝜃)
11. If 𝜃 is an acute angle such that tan2 ⁡𝜃 = 7, then the value of (1+cos⁡𝜃)(1−cos⁡𝜃) is
7
(a) 8
8
(b) 7
7
(c) 4
64
(d) 49
12. In the given figure, the area of the shaded region is
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SOLUTIONS
(a) 3𝜋cm2
(b) 6𝜋cm2
(c) 9𝜋cm2
(d) 7𝜋cm2
13. If 17th term of an A.P. is 20 more than the 13th term, then the common difference is
(a) 8
(b) 6
(c) 7
(d) 5
14. In the given figure, if 𝐴𝐵 = 9 cm and 𝐶𝐸 = 3 cm, then 𝐴𝐸 =
(a) 11 cm
(b) 6 cm
(c) 5 cm
(d) 3 cm
15. In a swimming pool, base measuring 90 m × 40 m, 150men take a dip. If the average displacement
of water by a man is 8 m3, then rise in water level is
(a) 27.33 cm
(b) 30 cm
(c) 31.33 cm
(d) 33.33 cm
16. The polynomial having zero as -2 , is

(a) 𝑥 2 + 𝑥 − 2
(b) 𝑥 2 − 3𝑥 + 2
(c) 𝑥 2 − 𝑥 − 2
(d) none of these
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SOLUTIONS
𝑄𝐵
17. In the given figure, 𝐵𝐴 ∥ 𝑄𝑅, and 𝐶𝐴 ∥ 𝑆𝑅. Then, 𝐵𝑃 =
𝐴𝐵
(a) 𝐴𝐶
𝐴𝑃
(b) 𝑅𝐴
𝑆𝐶
(c) 𝐶𝑃
𝑃𝐶
(d) 𝑆𝐶
18. If 𝑃(𝐸) = 0.01, then 𝑃( not 𝐸) is equal to

(a) 0.09
(b) 0.1
(c) 0.9
(d) 0.99
19. Statement A (Assertion): In a game, the entry fee is ₹ 10. The game consists of tossing of 3 coins. If
one or two heads show, Amita win the game and gets entry fee. The probability, that she gets the
3
entry fee is .
4
Statement R (Reason): When three coins are tossed together, all the outcomes are {𝐻𝐻𝐻, 𝐻𝐻𝑇, 𝐻𝑇𝐻,
THH, HTT, THT, TTH and TTT}.
(A).
assertion (A).
20. Statement A (Assertion): The distance of the point (2,11) from the 𝑥-axis is 11 units.
Statement 𝐑 (Reason): The distance of a point (𝑥, 𝑦) from 𝑥-axis is its ordinate, i.e., 𝑦 units.
(A).
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SOLUTIONS
assertion (A).
SECTION B

sec2 ⁡𝜃−sin2 ⁡𝜃
21. Prove that = cosec 2 ⁡𝜃 − cos 2 ⁡𝜃.
tan2 ⁡𝜃
22. If 𝑃𝐴 and 𝑃𝐵 are two tangents drawn from a point 𝑃 to a circle with centre 𝑂 touching it at 𝐴 and 𝐵,
prove that 𝑂𝑃 is perpendicular bisector of 𝐴𝐵.
23. A two digit number is such that the product of the digits is 12. When 36 is added to the number the
digits interchange their places. Formulate the quadratic equation to represent this situation.
OR
Find the value of √30 + √30 + √30 + ⋯ by factorisation.
24. A bag contains lemon flavoured candies only. Malini takes out one candy without looking into the
bag. What is the probability that she takes out
(i) an orange flavoured candy?
(ii) a lemon flavoured candy?
25. If △ 𝐴𝐵𝐶 ∼△ 𝐷𝐹𝐸, ∠𝐴 = 30∘ , ∠𝐶 = 50∘ , 𝐴𝐵 = 5 cm, 𝐴𝐶 = 8 cm and 𝐷𝐹 = 7.5 cm, then find 𝐷𝐸
and ∠𝐹.
OR
What value of 𝑥 will make 𝐷𝐸 ∥ 𝐴𝐵 in the given figure?
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SOLUTIONS
SECTION C

26. If the numerator of a fraction is multiplied by 4 and its denominator is increased by 2, it becomes 2.
However, if the numerator is increased by 5 and denominator is multiplied by 6, then the ratio of the
numerator and denominator is 1: 3. Form a pair of linear equations for the problem and solve by
substitution method and hence find the fraction.
1 1 1 1
27. Find the roots of the quadratic equation 𝑎+𝑏+𝑥 = 𝑎 + 𝑏 + 𝑥 , 𝑥 ≠ 0, −(𝑎 + 𝑏).
OR
In a class test, the sum of the marks obtained by Ankur in Mathematics and Science is 28. If he had
got 3 more marks in Mathematics and 4 marks less in Science, then product of marks obtained in the
two subjects would have been 180. Find the marks obtained in the two subjects separately.
28. A statue, 2.4 m tall, stands on the top of a pedestal. From a point on the ground, the angle of
elevation of the top of the statue is 60∘ and from the same point the angle of elevation of the top of
the pedestal is 45∘. Find the height of the pedestal.
29. If 𝐴(5,2), 𝐵(2, −2) and 𝐶(−2, 𝑡) are the vertices of a right angled triangle with ∠𝐵 = 90∘, then find
the value of 𝑡.
OR
𝐴, 𝐵 and 𝐶 are collinear points. The coordinates of 𝐴 and 𝐵 are (3,4) and (7,7) respectively and
𝐴𝐶 = 10 units. Find the coordinates of 𝐶.
30. In figure, two tangents 𝑇𝑃 and 𝑇𝑄 are drawn to a circle with centre 𝑂 from an external point 𝑇. Prove
that ∠𝑃𝑇𝑄 = 2∠𝑂𝑃𝑄.
31. In the given figure, from a cuboidal solid metallic block, of dimensions 15 cm × 10 cm × 5 cm, a
cylindrical hole of diameter 7 cm is drilled out. Find the surface area of the remaining block.
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SOLUTIONS
SECTION D

32. Prove that 2√3 + √5 is an irrational number. Also, check whether (2√3 + √5)(2√3 − √5) is
rational or irrational.
33. Find the median of the following frequency distribution:
Weekly wages (in 60 70 80 90 100 110

₹) − 69 − 79 − 89 − 99 − 109 − 119
Number of worker 5 15 20 30 20 8
OR
Find the unknown entries 𝑎, 𝑏, 𝑐, 𝑑, 𝑒 and 𝑓 in the following distribution and hence find their mode.
Height (in cm) 150 155 160 165 170 175 Total
− 155 − 160 − 165 − 170 − 175 − 180
Frequency 12 𝑏 10 𝑑 𝑒 2 50
Cumulative 𝑎 25 𝑐 43 48 𝑓
frequency
34. Solve graphically, the pair of equations 2𝑥 + 𝑦 = 6 and 2𝑥 − 𝑦 + 2 = 0. Find the ratio of the areas
of the two triangles formed by the lines representing these equations with 𝑥-axis and the lines with 𝑦-
axis.
OR
The area of a rectangle increases by 76 square units, if the length and breadth is increased by 2 units.
However, if the length is increased by 3 units and breadth is decreased by 3 units, the area gets
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SOLUTIONS
decreased by 21 square units. Find the length and breadth of the rectangle and hence area of the
rectangle.
35. The angles of depression of the top and the bottom of an 8 m tall building from the top of a
multistoried building are 30∘ and 45∘, respectively. Find the height of the multistoried building and
the distance between the two buildings.
SECTION E

36. Meena's mother start a new shoe shop. To display the shoes, she put 3 pairs of shoes in 1st row, 5
pairs in 2nd row, 7 pairs in 3rd row and so on.
(i) Find the pairs of shoes in 30th row.
(ii) If she arranges 𝑥 pairs of shoes in 15 rows, then find the value of 𝑥.
(iii) If she puts a total of 120 pairs of shoes, then calculate the number of rows required.
OR
Find the difference of pairs of shoes in 17th row and 10th row.
37. Manav had a piece of paper which on folding appears as shown in the figure, where 𝐴𝐵𝐶𝐷 is a
parallelogram in which 𝐷𝐶 is extended to 𝐹 such that 𝐴𝐹 intersects 𝐵𝐶 at E as shown in the given
figure.
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SOLUTIONS
(i) Which similarity supports the condition, △ 𝐴𝐵𝐸 ∼△ 𝐹𝐶𝐸 ?
(ii) Find the perimeter of △ 𝐴𝐵𝐸.
OR
Find the ratio of perimeter (△ 𝐴𝐵𝐸) and perimeter (△ 𝐴𝐹𝐷).
(iii) Find the length of side 𝐵𝐶.
38. Amit goes to a restaurant for dinner. He sits on a table and observe the restaurant's dining etiquette on
the table. He sees that napkins are put on a plate as shown below. Here, 𝑂 is the centre and 𝐴𝑂𝐶 is a
diameter of the circle. Lengths of chord 𝐴𝐵 and 𝐵𝐶 are 6 cm each.
(i) Find the radius of the circle.
(ii) Find the area of minor segment made by chord 𝑃𝑄.
OR
Find the area of plate not covered by napkins.
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SOLUTIONS
(iii) Find the area of sector 𝑂𝑃𝑄.
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SOLUTIONS
Sample
Paper
Set - 5
With a success rate
2024 boards
Mathematics
Class 10
SET-05
SECTION A

1. The sum and product of zeroes of a quadratic polynomial are 0 and √3 respectively. The quadratic
polynomial is
(a) 𝑥 2 − √3
(b) 𝑥 2 + √3
(c) 𝑥 2 − 3
(d) 𝑥 2 + 3
2. If the point 𝐶(−1,2) divides internally the line segment joining 𝐴(2,5) and 𝐵 in the ratio 3: 4, then
find the coordinates of 𝐵.
(a) (2, −5)
(b) (2,5)
(c) (5,2)
(d) (−5, −2)
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SOLUTIONS
𝐴𝐵 𝐵𝐶
3. If in triangles 𝐴𝐵𝐶 and 𝐷𝐸𝐹, 𝐷𝐸 = 𝐹𝐷, then they will be similar, when
(a) ∠𝐵 = ∠𝐸
(b) ∠𝐴 = ∠𝐷
(c) ∠𝐵 = ∠𝐷
(d) ∠𝐴 = ∠𝐹
4. If 𝑂 is the centre of the circle and 𝐴𝐵 is a tangent to the circle at 𝑃, then value of 𝑥 is
(a) 40∘
(b) 50∘
(c) 60∘
(d) 35∘
5. The radius of a sphere (in cm ) whose volume is 36𝜋cm3 given by,

(a) 3 cm
(b) 4 cm
(c) 6 cm
(d) 1.5 cm
6. If 𝑎𝑚 ≠ 𝑏𝑙, then the pair of equations 𝑎𝑥 + 𝑏𝑦 = 𝑐, 𝑙𝑥 + 𝑚𝑦 = 𝑛

(a) has a unique solution
(b) has no solution
(c) has infinitely many solutions
(d) may or may not have a solution
7. Evaluate : 8√3cosec 2 ⁡30∘ ⋅ sin⁡60∘ ⋅ cos⁡60∘ ⋅ tan⁡30∘

(a) 8
(b) 4√3
(c) 8√3
(d) 16√3
8. The mean of 1,2,3,4, … … . . , 𝑛 is given by

𝑛(𝑛+1)
(a) 2
(𝑛+1)
(b) 4
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SOLUTIONS
𝑛
(c) 2
(𝑛+1)
(d) 2
9. If 𝑃(𝑏, −4) is the midpoint of the line segment joining 𝐴(6,6) and 𝐵(−2,3), then 𝑏 =
(a) 0
(b) -1
(c) 2
(d) -2
10. A die is thrown once. Find the probability of getting an odd prime number.
1
(a) 2
1
(b) 3
1
(c) 6
1
(d) 4
11. Sunita picked a prime number from the integers 1 to 20. The probability that it will be the number 13
is
1
(a) 20
1
(b) 8
2
(c) 7
13
(d) 20
12. The least number which when divided by 18,24,30 and 42 will leave in each case the same remainder
1 , would be
(a) 2520
(b) 2519
(c) 2521
(d) None of these
tan⁡𝐴
13. If tan⁡𝐴 = √2 − 1, then what is the value of 1+tan2 ⁡𝐴 ?
√2
(a) 4
4
(b)
√2
1
(c)
√3
√3
(d) 4
14. The length of the minute hand of a clock is √21 cm. Find the area swept by the minute hand from 9
a.m. to 9.10a. m.
(a) 22 cm2
(b) 11 cm2
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SOLUTIONS
(c) 45 cm2
(d) 31 cm2
15. Three numbers in an A.P. have sum 18. Its middle term is
(a) 6
(b) 8
(c) 3
(d) 2
16. What value of 𝑥 will make 𝐷𝐸 ∥ 𝐴𝐵 in the figure?
(a) 1
(b) 2
(c) 3
(d) 4
17. In the given figure, 𝑂𝐵 = 5 cm and 𝑇𝐵 is the tangent at 𝐵 to the circle with centre 𝑂. Find 𝑂𝑇, if 𝐵𝑇
is 4 cm.
(a) √41 cm
(b) √43 cm
(c) √39 cm
(d) √47 cm
18. Find the total surface area of solid opened at the top in the given figure.
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SOLUTIONS
(a) 2𝜋𝑟ℎ
(b) 𝜋𝑟(ℎ + 𝑟)
(c) 2𝜋𝑟(ℎ + 𝑟)
(d) None of these
19. Statement A (Assertion): Consider a pack of cards that are numbered from 1 to 52. If a card is drawn
7
at random from the pack, then the probability that it will have a prime number is 26.
Statement 𝐑 (Reason): From 1 to 52, there are 15 prime numbers.
(A).
assertion (A).
20. Statement A (Assertion): If a point (−3, 𝑘) divides the line segment joining the points (−5, −4) and
2
(−2,3) in the ratio 2: 1 internally, then the value of 𝑘 is 3.
Statement 𝐑 (Reason): Coordinates of point which divides the line segment joining the points (𝑥1 , 𝑦1 )
𝑚𝑥 +𝑛𝑥 𝑚𝑦 +𝑛𝑦
and (𝑥2 , 𝑦2 ) in the ratio 𝑚: 𝑛 internally are 𝑥 = ( 2 1 ) and 𝑦 = ( 2 1 ).
𝑚+𝑛 𝑚+𝑛
(A).
assertion (A).
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SOLUTIONS
SECTION B

21. Find the value of the height ' ℎ ' in the adjoining figure, at which the tennis ball must be hit, so that it
will just pass over the net and land 16 m away from the base of the net.
OR
State whether the given pair of triangles are similar or not. Also, state the similarity criterion used
symbolically between the two given triangles.
22. In the given figure, 𝑃𝐴 and 𝑃𝐵 are tangents to the circle from an external point 𝑃. 𝐶𝐷 is another
tangent touching the circle at 𝑄. If 𝑃𝐴 = 12 cm, 𝑄𝐶 = 𝑄𝐷 = 3 cm, then find 𝑃𝐶 + 𝑃𝐷.
23. Find the roots of the quadratic equation √3𝑥 2 − 2√2𝑥 − 2√3 = 0.
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SOLUTIONS
OR
Solve the equation √3𝑥 2 + 11𝑥 + 6√3 = 0 by factorisation method.
24sec⁡𝐴
24. If 15cot⁡𝐴 = 8, then find the value of .
25
25. Find the probability of getting a natural number less than 100 that is divisible by 7 .
SECTION C
Section C consists of 6 questions of 3 marks each.

26. 𝑃𝐴𝑄 is a tangent to the circle with center 𝑂 at a point 𝐴 as shown in figure. If ∠𝑂𝐵𝐴 = 35∘ , then find
the value of ∠𝐴𝐶𝐵 and ∠𝑄𝐴𝐵.
27. The angle of elevation of a cloud from a point 60 m above a lake is 30∘ and the angle of depression
of the reflection of the cloud in the lake is 60∘ . Find the height of the cloud from the surface of the
lake.
OR
A man standing on the deck of a ship, which is 10 m above water level, observes the angle of
elevation of the top of a hill as 60∘ and angle of depression of the base of the hill as 30∘ . Find the
horizontal distance of the hill from the ship and height of the hill.
28. In the given figure, 𝐴𝐵𝐶𝐷 is a rectangle, then find the values of 𝑥 and 𝑦.
29. A solid toy is in the form of a hemisphere surmounted by a right circular cone of same radius. The
height of the cone is 10 cm and the radius of the base is 7 cm. Determine the volume of the toy. Also
22
find the area of the coloured sheet required to cover the toy. (Use 𝜋 = 7 and √149 = 12.2 )
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SOLUTIONS
30. Solve for 𝑥 :
2 3 23
+ = , 𝑥 ≠ 0, −1,2
𝑥 + 1 2(𝑥 − 2) 5𝑥
OR
Prove that the equation 𝑥 2 (𝑎2 + 𝑏 2 ) + 2𝑥(𝑎𝑐 + 𝑏𝑑) + (𝑐 2 + 𝑑 2 ) = 0 has no real roots, if 𝑎𝑑 ≠ 𝑏𝑐.
31. The vertices of a △ 𝐴𝐵𝐶 are 𝐴(5,5), 𝐵(1,5) and 𝐶(9,1). A line is drawn to intersect sides 𝐴𝐵 and 𝐴𝐶
𝐴𝑃 𝐴𝑄 3
at 𝑃 and 𝑄 respectively, such that 𝐴𝐵 = 𝐴𝐶 = 4. Find the length of the line segment 𝑃𝑄.
SECTION D

32. Find the value of 𝑝 and 𝑞 for which system of equations has infinitely many solutions:
2𝑥 + 3𝑦 = 7; (𝑝 + 𝑞 + 1)𝑥 + (𝑝 + 2𝑞 + 2)𝑦 = 4(𝑝 + 𝑞) + 1
OR
Vijay had some bananas and he divided them into two lots 𝐴 and 𝐵. He sold the first lot at the rate of
₹ 2 for 3 bananas and the second lot at the rate of ₹ 1 per banana and got a total of ₹ 400. If he had
sold the first lot at the rate of ₹1 per banana and the second lot at the rate of ₹4 for 5 bananas, his
total collection would have been ₹460. Find the total number of bananas, he had.
33. The following table gives weekly wages in rupees of workers in a certain commercial organization.
The frequency of class 49-52 is missing. It is known that the mean of the frequency distribution is
47.2. Find the missing frequency.
40 43 46 49 52
Weekly wages (in ₹)
− 43 − 46 − 49 − 52 − 55
Number of workers 31 58 60 ? 27
34. Two poles of equal heights are standing opposite each other on either side of the road, which is 60 m
wide. From a point between them on the road, the angles of elevation of the top of the poles are 60∘
and 30∘ respectively. Find the height of the poles and the distances of the point from the poles. (Use
√3 = 1.732 )
OR
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SOLUTIONS
A 1.2 m tall girl spots a balloon moving with the wind in a horizontal line at a height of 88.2 m from
the ground. The angle of elevation of the balloon from the eyes of the girl at any instant is 60∘ . After
some time, the angle of elevation reduces to 30∘ (see figure). Find the distance travelled by the
balloon during the interval.
35. A circular track around a sports ground has circumference of 1080 m. Two cyclists Rohan and
Sumeet start together and cycled at constant speed of 6 m/s and 9 m/s respectively around the
circular track. After how many minutes, will they meet again at the starting point?
SECTION E

36. On a circular sheet of paper Purvi, a student of class X made a design as shown in the figure to
prepare a poster on some moral values each child should have. The diameter of the circular sheet is
10 cm.
(i) Find the area of the region containing Peace.
(ii) Find the perimeter of the region containing Honesty.
(iii) Find the total area of region containing Peace, Tolerance and Hardwork.
OR
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SOLUTIONS
Find the perimeter of the region containing Hardwork. If the region containing Peace, Tolerance and
Hardwork are combined and put on a straight line, then what type of figure can be obtained?
37. Sheela was revising her chapter on triangles for her upcoming test. Then she started recalling
statement 'If a line is drawn parallel to one side of a triangle then it divides the other two sides in the
same ratio' and drawn the following figure on the paper. Help Sheela in revising the topic.
(i) If 𝐴𝐶 = 10 cm and 𝐵 divides 𝐴𝐷 in ratio of 5: 3 then find the value of 𝐴𝐸.
(ii) If 𝐴𝐵: 𝐵𝐷 = 6: 10 then find the ratio of medians drawn from 𝐴 to △ 𝐴𝐵𝐶 and △ 𝐴𝐷𝐸.
OR
If 𝐴𝐶: 𝐶𝐸 = 4: 𝑥 and 𝐴𝐵: 𝐵𝐷 = 5: 𝑦, then find the value of 𝑥: 𝑦.
(iii) If 𝐴𝐷: 𝐴𝐵 = 11: 5, then find 𝐵𝐶: 𝐷𝐸.
38. Amit was playing a number card game. In the game, some number cards (having both +ve or -ve
numbers) are arranged in a row such that they are following an arithmetic progression. On his first
turn, Amit picks up 6th and 14th card and finds their sum to be -76. On the second turn he picks up
8th and 16th card and finds their sum to be -96.
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SOLUTIONS
(i) What is the difference between the numbers on any two consecutive cards?
(ii) What is the number on the 19th card?
OR
What is the number on the 23rd card?
(iii) What number is written on the first card.
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SOLUTIONS
Sample
Paper
Set - 6
With a success rate
2024 boards
Mathematics
Class 10
SET 06
2. Section 𝐴 has 20MCQs carrying 1 mark each.
SECTION A
1. The sum of first 10 terms of the A.P. 𝑥 − 8, 𝑥 − 2, 𝑥 + 4, … is
(a) 10𝑥 + 210
(b) 10𝑥 + 190
(c) 5𝑥 + 190
(d) 5𝑥 + 210
2. How many tangents can a circle have from a point lying inside the circle?
(a) 2
(b) infinitely many
(c) 1
(d) none of these
3. If △ 𝐴𝐵𝐶 ∼△ 𝑃𝑄𝑅, perimeter of △ 𝐴𝐵𝐶 = 20 cm, perimeter of △ 𝑃𝑄𝑅 = 40 cm and 𝑃𝑅 = 8 cm,

then the length of 𝐴𝐶 is
(a) 8 cm
(b) 6 cm
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SOLUTIONS
(c) 4 cm
(d) 5 cm
4. A letter is chosen at random from the letters of the word 'PRONUNCIATION'. The probability that
the letter chosen is a vowel is
6
(a)
13
2
(b)
3
1
(c)
8
7
(d)
13
5. A solid is in the shape of a cone mounted on a hemisphere of same base radius. If the curved surface
areas of the hemispherical part is half the conical part, then find the ratio of the radius and the height
of the conical part is
1
(a)
√15
1
(b)
√17
1
(c)
√13
1
(d)
√19
6. The correct formula for finding the mode of a grouped frequency distribution is
𝑓1−𝑓0
(a) ℎ + ( )×𝑙
2𝑓1−𝑓0 −𝑓2
𝑓1 −𝑓0
(b) 𝑓1 + ( )×𝑙
2ℎ−𝑓1 −𝑓2
𝑓1 −𝑓0
(c) 𝑙 − ( )×ℎ
2𝑓1 −𝑓0−𝑓2
𝑓1−𝑓0
(d) 𝑙 + ( )×ℎ
2𝑓1−𝑓0−𝑓2
7. If the zeroes of the quadratic polynomial 𝑥 2 + (𝑎 + 1)𝑥 + 𝑏 are 4 and -3 , then 𝑎 − 𝑏 is

(a) 12
(b) 10
(c) 7
(d) 1
8. The value of 𝑘, if the distance between 𝐴(𝑘, 3) and 𝐵(2,3) is 5 units, is

(a) 5
(b) 6
(c) 7
(d) 8
2 1
9. If 2𝑥 = sec⁡𝜃 and = tan⁡𝜃, then 2 (𝑥 2 )=
𝑥 𝑥2
1
(a)
2
(b) 2
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SOLUTIONS
1
(c)
4
(d) 4
10. A card is drawn from a well - shuffled pack of cards. The probability that it will be a black queen is
1
(a)
13
1
(b)
26
3
(c)
13
4
(d)
13
11. The pair of equations 𝑥 + 2𝑦 + 5 = 0 and −3𝑥 − 6𝑦 + 1 = 0 have

(a) a unique solution
(b) exactly two solutions
(c) infinitely many solutions
(d) no solution
12. If the coordinates of one end of a diameter of a circle are (2,3) and the coordinates of its centre are
(−2,5), then the coordinates of the other end of the diameter are
(a) (−6,7)
(b) (6, −7)
(c) (6,7)
(d) (−6, −7)
4
13. If cos⁡𝐴 = , then the value of tan⁡𝐴 is
5
3
(a)
5
3
(b)
4
4
(c)
3
5
(d)
3
𝐴𝐵
14. In two triangles 𝐴𝐵𝐶 and 𝐷𝐸𝐹, ∠𝐴 = ∠𝐸 and ∠𝐵 = ∠𝐹. Then, is equal to
𝐴𝐶
𝐷𝐸
(a)
𝐷𝐹
𝐸𝐷
(b)
𝐸𝐹
𝐸𝐹
(c)
𝐸𝐷
𝐸𝐹
(d)
𝐷𝐹
15. If an arc subtending an angle of 60∘ at the centre of a circle 𝐴 and another arc subtending an angle of
90∘ at the centre of circle 𝐵, are of same length, then the ratio of area of circle 𝐴 to that of circle 𝐵 is
(a) 11: 15
(b) 11: 25
(c) 9: 4
(d) 36: 16
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SOLUTIONS
16. The smallest number which when increased by 17 is exactly divisible by 520 and 468 is
(a) 4680
(b) 4663
(c) 4860
(d) 4636
17. In the given figure, 𝑄𝑅 is a common tangent to the given circles, touching externally at the point 𝑇.
The tangent at 𝑇 meets 𝑄𝑅 at 𝑃. If 𝑃𝑇 = 3.8 cm, then the length of 𝑄𝑅 (in cm) is
(a) 3.8
(b) 7.6
(c) 5.7
(d) 1.9
18. If curved surface area of cylinder is equal to its volume. What is the radius of cylinder?
(a) 2
(b) 3
(c) 4
(d) 1
5
19. Statement A (Assertion): Point 𝑃 (1, ) is equidistant from the points 𝐴(−5,3) and 𝐵(7,2).
2
Statement 𝐑 (Reason): If a point 𝑃 is equidistant from the points 𝐴 and 𝐵, then 𝐴𝑃 = 𝐵𝑃.
(A).
assertion (A).
20. Statement A (Assertion): Two dice are rolled simultaneously. Then the probability of getting prime
1
number on both dice is .
4
Statement 𝐑 (Reason): Sum of probability of all the elementary events of an experiment is zero.
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SOLUTIONS
(A).
assertion (A).
SECTION B

21. If 1 is a root of the quadratic equation 3𝑥 2 + 𝑎𝑥 − 2 = 0 and the quadratic equation 𝑎(𝑥 2 + 6𝑥 ) −
𝑏 = 0 has equal roots, find the value of 𝑏.
22. If cot2 ⁡𝜃(sec⁡𝜃 − 1)(1 + cos⁡𝜃) = 𝑘cos⁡𝜃, then find the value of 𝑘.
23. 14 cards numbered 5,6,7,8,9,10,11,12,13,14,15,16,17,18 are placed in a box and mixed thoroughly.
If a card is drawn from the box, then find the probability that the number on the card divisible by 3 or
2.
24. In the adjoining figure, if 𝑃𝑄 is diameter and 𝑃𝑇 is tangent, then find 𝑥.
OR
In the given figure, 𝑃𝐵 = 24 cm, 𝑂𝑃 = 25 cm, 𝑃𝐴 and 𝑃𝐵 are tangents of the circle. Find the length
of 𝑃𝐴 and 𝑂𝐵.
25. In figure, ∠𝐿𝑀𝐾 = ∠𝑃𝑁𝐾 = 46∘ . Express 𝑥 in terms of 𝑎, 𝑏 and 𝑐, where 𝑎, 𝑏 and 𝑐 are lengths of
𝐿𝑀, 𝑀𝑁 and 𝑁𝐾 respectively.
OR
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SOLUTIONS
In △ 𝐴𝐵𝐶, ∠𝐶𝐴𝐵 = 90∘ and 𝐴𝐷 ⊥ 𝐵𝐶. If 𝐴𝐶 = 75 cm, 𝐴𝐵 = 1 m and 𝐵𝐷 = 80 cm, find 𝐴𝐷.
SECTION C

6 2 1
26. Solve for 𝑥: − = ; 𝑥 ≠ 0,1,2
𝑥 𝑥−1 𝑥−2
OR
₹ 6500 is divided equally among a certain number of persons. If there are 15 more persons, each will
get ₹ 30 less. Find the original number of persons.
27. For which value(s) of 𝜆, the pair of linear equations 𝜆𝑥 + 𝑦 = 𝜆2 and 𝑥 + 𝜆𝑦 = 1 has
(i) no solution?
(ii) infinitely many solutions?
28. Show that 𝐴(6,4), 𝐵(5, −2) and 𝐶(7, −2) are the vertices of an isosceles triangle. Also, find the
length of the median through 𝐴.
OR
If 𝐶(−2,3) is equidistant from 𝐴(3, −1) and 𝐵(𝑥, 8), then find 𝑥. Also, find the distances 𝐵𝐶 and
𝐴𝐵.
29. A wooden toy was made by scooping out a hemisphere of same radius from each end of a solid
cylinder. If the height of the cylinder is 15 cm, and its base is of radius 2.8 cm, then find the volume
of wood in the toy.
30. Two men on either side of a 75 m high building and in line with base of building observe the angles
of elevation of the top of the building as 30∘ and 60∘ . Find the distance between the two men.
[Use √3 = 1.73 ]
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SOLUTIONS
31. In two concentric circles, a chord of the larger circle touches the smaller circle. If the length of this
chord is 8 cm and the diameter of the smaller circle is 6 cm, then find the diameter of the larger
circle.
SECTION D

32. Prove that √2 + √3 is irrational.
𝑎1 𝑏1 𝑐
33. By comparing the ratios , and 1 , find out whether the lines representing the following pairs of
𝑎2 𝑏2 𝑐2
linear equations intersect at a point, are parallel or coincident.
(i) 3𝑥 + 𝑦 − 14 = 0
2𝑥 + 5𝑦 − 5 = 0
(ii) 5𝑥 + 3𝑦 − 15 = 0
10𝑥 + 6𝑦 − 30 = 0
(iii) 5𝑥 − 𝑦 + 7 = 0
15𝑥 − 3𝑦 + 17 = 0
34. Calculate the mode for the following frequency distribution.

Class- 1 5 9 13 17 21 25 29 33 37
interval −4 −8 − 12 − 16 − 20 − 24 − 28 − 32 − 36 − 40
Frequency 2 5 8 9 12 14 14 15 11 10
OR
If the median of the following distribution is 46, find the missing frequencies 𝑝 and 𝑞.
Class- 10 20 30 40 50 60 70 Total
interval − 20 − 30 − 40 − 50 − 60 − 70 − 80
Frequency 12 30 𝑝 65 𝑞 25 18 230
35. The angle of elevation 𝜃 of the top of a lighthouse as seen by a person on the ground is such that
tan⁡𝜃 = 5/12. When the person moves a distance of 240 m towards the lighthouse, the angle of
elevation becomes 𝜙, such that tan⁡𝜙 = 3/4. Find the height of the lighthouse.
OR
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SOLUTIONS
A vertical tower stands on a horizontal plane and is surmounted by a vertical flagstaff of height ℎ. At
a point on the plane, the angle of elevation of the bottom of the flagstaff is 𝛼 and that of the top of the
flagstaff is 𝛽. Find the height of the tower.
SECTION E
36. Upasana bought a wall clock to gift her friend Pratibha on her birthday. The clock contains a small
pendulum of length 15 cm. The minute hand and hour hand of the clock are 10 cm and 7 cm long
respectively.
(i) If the pendulum covers distance of 44 cm in one complete round, then find the angle described by
pendulum at the centre.
(ii) Find the angle described by hour hand in 30 minutes.
(iii) Find the area swept by the minute hand in 35 minutes.
Find the area swept by the hour hand in 1 hour and the area swept by the hour hand between 9 a.m.
and 5 p.m.
37. Amit starts a new bakery shop. To display the cakes, he puts 3 cakes in 1st row, 5 cakes in 2nd row, 7
cakes in 3rd row and so on.
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SOLUTIONS
(i) Find the difference of number of cakes in 17th row and 10th row.
(ii) If he puts a total of 120 cakes, then how many rows are required?
OR
Find the total number of cakes in 5th and 8th row. Also, find the cakes in 30th row.
(iii) On next day, he arranges 𝑥 cakes in 15 rows, then find the value of 𝑥.
38. In a classroom, students were playing with some pieces of cardboard as shown below.
All of a sudden, teacher entered into classroom. She told students to arrange all pieces. On seeing this
beautiful image, she observed that △ 𝐴𝐷𝐻 is right angled triangle, which contains
(i) right triangles 𝐴𝐵𝐽 and 𝐼𝐺𝐻.
(ii) quadrilateral GFJI
(iii) squares 𝐽𝐾𝐿𝑀 and 𝐿𝐶𝐵𝐾
(iv) rectangles 𝑀𝐿𝐸𝐹 and 𝐿𝐶𝐷𝐸.
Use the above figure to answer the questions that follow :
(i) If ∠𝐴𝐵𝐽 = 90∘ and △ 𝐴𝐵𝐽 ∼△ 𝐴𝐷𝐻, then which similarity criterion is used?
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SOLUTIONS
(ii) If ∠𝐽𝐹𝐻 = 90∘ , 𝐴𝐷 = 12 m, 𝑀𝐹 = 4 m, 𝐴𝐻 = 8 m and 𝐽𝐻 = 4 m, then find the length of 𝐽𝑀.
OR
If 𝐼𝐺 ∥ 𝐴𝐷, 𝐴𝐼 = 2𝑦, 𝐼𝐻 = 𝑦 + 3, 𝐻𝐺 = 𝑦 and 𝐷𝐺 = 2𝑦 −1 , then find the value of 𝑦.

𝐴𝐵
(iii) If ∠𝐴𝐵𝐽 = 90∘ , 𝐴𝐷 = 5 cm, and 𝐴𝐻 = 7 m, then find .
𝐴𝐽
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SOLUTIONS
Sample
Paper Set - 7
With a success rate
2024 boards
Mathematics
Class 10
SET 07
6. Section 𝐸 has 3 case based integrated units of assessment ( 04 marks each) with sub-parts.
SECTION A

1. The next term of the A.P. √18, √50, √95, … is
(a) √146
(b) √128
(c) √162
(d) √200
2. A chord of a circle of radius 10 cm subtends a right angle at its centre. The length of the chord (in cm
) is
(a) 5√2
(b) 10√2
5
(c) 2
√
(d) 10√3
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SOLUTIONS
3. The radius of a circle is 18 cm and the angle subtended by an arc of this circle at the centre is 60∘.
Find the length of this arc.
(a) 𝜋cm
(b) 2𝜋cm
(c) 3𝜋cm
(d) 6𝜋cm
4. HCF of two numbers is 23 and their LCM is 1449. If one of the numbers is 161 , then the other
number is
(a) 207
(b) 307
(c) 1449
(d) 570
5. The distance between the points (𝑎cos⁡𝜃 + 𝑏sin⁡𝜃, 0) and (0, 𝑎sin⁡𝜃 − 𝑏cos⁡𝜃) is
(a) 𝑎2 + 𝑏2
(b) 𝑎 + 𝑏
(c) 𝑎2 − 𝑏2
(d) √𝑎2 𝑏2
6. In triangles 𝐴𝐵𝐶 and 𝐷𝐸𝐹, ∠𝐵 = ∠𝐸, ∠𝐹 = ∠𝐶 and 𝐴𝐵 = 3𝐷𝐸. Then, the two triangles are
(a) congruent but not similar
(b) similar but not congruent
(c) neither congruent nor similar
(d) congruent as well as similar
7. The maximum value of sin⁡𝜃 is

1
(a)
2
√3
(b) 2
(c) 1
1
(d) 2
√
8. The volume of the greatest sphere that can be cut off from a cylindrical log of wood of base radius
1 cm and height 5 cm is
4
(a) 3 𝜋cm3
10
(b) 3 𝜋cm3
(c) 5𝜋cm3
20
(d) 3 𝜋cm3
9. The number of zeroes lying between -4 and 4 of the polynomial 𝑓(𝑥) whose graph is given, is
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SOLUTIONS
(a) 2
(b) 3
(c) 4
(d) 1
10. Which of the following cannot be the probability of an event?

1
(a) 2
(b) 1
(c) 0
(d) 1.5
11. In figure, 𝑃𝑄 is tangent to the circle with centre at 𝑂, at the point 𝐵. If ∠𝑂𝐵𝐴 = 40∘ , then ∠𝐴𝐵𝑃 is
equal to
(a) 50∘
(b) 40∘
(c) 60∘
(d) 80∘
12. The height of a cylinder is 14 cm and its curved surface area is 264 cm2 . The volume of the cylinder
is
(a) 296 cm3
(b) 396 cm3
(c) 369 cm3
(d) 503 cm3
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SOLUTIONS
13. If 𝑑𝑖 = 𝑥𝑖 − 13, ∑𝑓𝑖 𝑑𝑖 = 30 and ∑𝑓𝑖 = 120, then mean, 𝑥‾ is equal to
(a) 13
(b) 12.75
(c) 13.25
(d) 14.25
14. For what value of 𝑘, the pair of linear equations 3𝑥 + 𝑦 = 3 and 6𝑥 + 𝑘𝑦 = 8 does not have a
solution?
(a) -2
(b) 2
(c) +1
(d) -1
15. If (−2, −1), (𝑎, 0), (4, 𝑏) and (1,2) are the vertices of a parallelogram, then the values of 𝑎 and 𝑏 are
(a) 1,3
(b) 1,4
(c) 2,3
(d) 3,1
16. In the given figure, find ∠𝐹.
(a) 60∘
(b) 80∘
(c) 40∘
(d) 100∘
17. In △ 𝐴𝐵𝐶, ∠𝐵 = 90∘. If tan⁡𝐴 = √3, then the value of sin⁡𝐴 ⋅ cos⁡𝐶 − cos⁡𝐴 ⋅ sin⁡𝐶 is
1
(a) 2
(b) -1
(c) 1
(d) 0
18. A card is accidently dropped from a pack of 52 playing cards. The probability that it is a red card is
1
(a) 2
1
(b) 13
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SOLUTIONS
1
(c) 52
12
(d) 13
19. Statement A (Assertion): The distance of the point 𝑃(6, −6) from the origin is 6 units.
Statement 𝐑 (Reason): The distance between two points 𝐴(𝑥1 , 𝑦1 ) and 𝐵(𝑥2 , 𝑦2 ) is given by
𝐴𝐵 = √(𝑥2 − 𝑥1 )2 + (𝑦2 − 𝑦1 )2 .
(A).
assertion (A).
20. Statement A (Assertion): Three unbiased coins are tossed together, then the probability of getting
3
exactly 1 head is 8.
Statement R (Reason): Favourable number of outcomes do not lie in the sample space of total number
of outcomes.
(A).
assertion (A).
SECTION B
21. In the given △ 𝐴𝐵𝐶, 𝐷𝐸 ∥ 𝐵𝐶, then find the value of 𝑥.
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SOLUTIONS
22. Find the roots of the quadratic equation 5(𝑥 − 3)2 = 20.
OR
Find the value of 𝑝, for which one root of the quadratic equation 𝑝𝑥 2 − 14𝑥 + 8 = 0 is 6 times the
other.
23. A tangent 𝑃𝑄 at a point 𝑃 of a circle of radius 8 cm meets a line through the centre 𝑂 at a point 𝑄 so
that 𝑂𝑄 = 17 cm. Find the length of 𝑃𝑄.
sin⁡𝐴+cos⁡𝐴
24. Find the value of sin⁡𝐴−cos⁡𝐴, if 16cot⁡𝐴 = 12.
OR
cos⁡𝜃−2cos3 ⁡𝜃
Simplify : .
2sin3 ⁡𝜃−sin⁡𝜃
25. Jayanti throws a pair of dice and records the product of the numbers appearing on the dice. Pihu
throws one die and records the square of the number that appears on it. Who has the better chance of
getting the number 36 ? Justify?
SECTION C

26. From the top of a 15 m high building, the angle of elevation of the top of a cable tower is 60∘ and the
angle of depression of its foot is 30∘. Determine the height of the tower.
27. Represent the following equations graphically.
3𝑥 + 2𝑦 = 12,3𝑥 + 2𝑦 = 18
OR
The sum of the digits of a two-digit number is 8 and number formed by reversing the digits is less
than the given number by 18. Find the number.
28. Sum of the areas of two squares is 544 m2 . If the difference of their perimeters is 32 m, then find the
sides of the two squares.
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SOLUTIONS
29. A thin hollow metallic hemispherical solid is covered by a conical canvas tent. The radius of the
hemisphere is 14 m and total height of vessel (including the height of tent) is 28 m. Find the area of
metal sheet and the canvas required.
30. Show that the points (4,2), (7,5) and (9,7) do not form a triangle.
OR
The 𝑥-coordinate of a point 𝑃 is twice its 𝑦-coordinate. If 𝑃 is equidistant from 𝑄(2, −5) and
𝑅(−3,6), then find the coordinates of 𝑃.
31. In the figure, if 𝐴𝐵 = 10 cm, 𝐵𝐶 = 8 cm and 𝐶𝐴 = 6 cm, then find 𝐵𝑋.
SECTION D
32. Draw the graphs of the equations 𝑥 − 𝑦 + 1 = 0 and 3𝑥 + 2𝑦 − 12 = 0. Determine the coordinates
of the vertices of the triangle formed by these lines and the 𝑥-axis, and shade the triangular region.
OR
Form a pair of linear equations in two variables using the following information and solve it
graphically. Five years ago, Sagar was twice as old as Tiru. Ten years later, Sagar's age will be ten
years more than Tiru's age. Find their present ages.
33. 𝐴, 𝐵 and 𝐶 start cycling around a circular path in the same direction and the same time.
34. A survey regarding the heights (in cm ) of 50 girls of class X of a school was conducted and the
following data was obtained:
Height (in cm) 120 130 140 150 160 Total

− 130 − 140 − 150 − 160 − 170
Number of girls 2 8 12 20 8 50
Find the mean, median and mode of the above data.
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SOLUTIONS
35. An aeroplane is flying at an altitude of 1200 m above the sea level. Two ships are sailing towards it
in the same direction. The angles of depression of the ships as observed from the aeroplane are 60∘
and 30∘ respectively at a moment. Find the distance between both the ships at that moment.
OR
A ladder rests against a wall at an inclination 𝛼 to the horizontal. Its foot is pulled away from the wall
through a distance 𝑝 so that its upper end slides a distance 𝑞 down the wall and then the ladder makes
𝑝 cos⁡𝛽−cos⁡𝛼
an angle 𝛽 to the horizontal. Show that = .
𝑞 sin⁡𝛼−sin⁡𝛽
SECTION E

36. Deeraj was unable in answering the questions related to the following trapezium. His father helped
him in understanding the various facts related to it and guided him in answering them.
(i) Name the pair of similar triangles from the given figure.
(ii) Write the similarity rule applied in this case.
(iii) Find the value of 𝑥.
OR
For what value of 𝑥, the trapezium 𝑃𝑄𝑆𝑅 is isosceles?
37. Ajay is a Class X student. His class teacher Mrs. Kiran arranged a historical trip to great Stupa of
Sanchi. She explained that Stupa of Sanchi is great example of architecture in India. Its base part is
cylindrical in shape. The dome of this stupa is hemispherical in shape, known as Anda. It also
contains a cubical shape part called Hermika at the top. Path around Anda is known as Pradakshina
Path.
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SOLUTIONS
(i) Find the lateral surface area of the Hermika, if the side of cubical part is 8 m.
(ii) The diameter and height of the cylindrical base part are respectively 42 m and 12 m. If the
volume of each brick used is 0.01 m3, then find the number of bricks used to make the cylindrical
base.
OR
If the diameter of the Anda is 42 m, then find the volume of the Anda.
(iii) The radius of the Pradakshina path is 25 m. If Buddhist priest walks 14 rounds on this path, then
find the distance covered by the priest.
38. In a pathology lab, a culture test has been conducted. In the test, the number of bacteria taken into
consideration in various samples is all 3-digit numbers that are divisible by 7, taken in order.
(i) How many bacteria are considered in the fifth sample?
(ii) How many samples should be taken into consideration?
OR
Find the total number of bacteria in the first 10 samples.
(iii) How many bacteria are there in the 7th sample from the last?
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SOLUTIONS
Sample
Paper
Set - 8
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Mathematics
Class 10
SET 08
SECTION A
1. The point which divides the line segment joining the points (7, −6) and (3,4) in the ratio 1: 2
internally is
15 −4
(a) (13 , )
3
−8 17
(b) ( 3 , 3 )
4 15
(c) (3 , 3 )
17 −8
(d) ( 3 , )
3
2. The median class for the following data is
Class interval 20 40 60 80
− 40 − 60 − 80 − 100
Frequency 10 12 20 22
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SOLUTIONS
(a) 20 − 40
(b) 40 − 60
(c) 60 − 80
(d) 80 − 100
3. When a die is thrown, the probability of getting an odd number less than 3 is
1
(a) 6
1
(b) 3
1
(c) 2
(d) 0
4. The pair of equations 𝑦 = 3 and 𝑦 = 8 has

(a) one solution
(b) two solutions
(c) infinitely many solutions
(d) no solution
5. In the given figure, 𝐷𝐸 ∥ 𝐵𝐶. If 𝐷𝐸 = 4 cm, 𝐵𝐶 = 8 cm and area of △ 𝐴𝐷𝐸 = 25sq. cm. Find the
area of △ 𝐴𝐵𝐶.
(a) 150 cm2

(b) 100 cm2
(c) 200 cm2
(d) 80 cm2
6. A tangent to a circle is a line that touches the circle at exactly

(a) two points
(b) three points
(c) one point
(d) none of these
7. The radius of spherical balloon increases from 8 cm to 12 cm. The ratio of the surface areas of the
balloon in two cases is
(a) 2: 3
(b) 3: 2
(c) 8: 27
(d) 4: 9
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SOLUTIONS
8. What is the HCF of smallest prime number and smallest composite number?
(a) 1
(b) 2
(c) 3
(d) 4
9. In △ 𝐴𝐵𝐶, ∠𝐵 = 90∘ , ∠𝐴 = 30∘ and 𝐴𝐵 = 9 cm. Then, 𝐵𝐶 =
(a) 3 cm
(b) 2√3 cm
(c) 3√3 cm
(d) 6 cm
10. A card is drawn from a pack of 52 cards. The probability of drawing a black face card is
2
(a)
13
3
(b) 26
1
(c) 13
3
(d) 52
11. The 44th term of the given A.P. 3,15,27,39, ….. is

(a) 951
(b) 195
(c) 519
(d) 64
12. 𝐴𝐵𝐶 is a right-angled triangle with 𝐵𝐶 = 6 cm and 𝐴𝐵 = 8 cm. A circle with centre 𝑂 and radius
𝑥 cm has been inscribed in △ 𝐴𝐵𝐶 as shown in figure. The value of 𝑥 is
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SOLUTIONS
(a) 1 cm
(b) 2 cm
(c) 3 cm
(d) 4 cm
13. A vessel is in the form of a hollow hemisphere mounted by a hollow cylinder. The diameter of the
hemisphere is 42 cm and the total height of the vessel is 30 cm. Find the inner surface area of the
vessel.
(a) 3500 cm2
(b) 3800 cm2
(c) 3960 cm2
(d) 3900 cm2
14. If one root of the polynomial 𝑓(𝑥) = 3𝑥 2 + 11𝑥 + 𝑝 is reciprocal of the other, then the value of 𝑝 is
(a) 0
(b) 3
1
(c) 3
(d) -3
15. If 𝑃(−1,1) is the mid point of the line segment joining 𝐴(−3, 𝑏) and 𝐵(1, 𝑏 + 4), then 𝑏 =
(a) 1
(b) -1
(c) 2
(d) 0
𝐴𝐷 𝐵𝐸
16. In figure, if 𝐷𝐶 = and ∠𝐶𝐷𝐸 = ∠𝐶𝐸𝐷, then
𝐸𝐶
(a) 𝐵𝐶 = 𝐴𝐶
(b) 𝐴𝐵 = 𝐴𝐶
(c) 𝐴𝐵 = 𝐵𝐶
(d) 𝐶𝐸 = 𝐷𝐸
17. Evaluate : (sin⁡30∘ + sin⁡45∘ )(cos⁡60∘ − cos⁡45∘ )

5
(a) 8
−5
(b) 8
1
(c) 4
−1
(d) 4
18. In the given figure, a semicircle is drawn with 𝑂 as centre and 𝐴𝐵 as diameter. Semicircles are drawn
with 𝐴𝑂 and 𝑂𝐵 as diameters. If 𝐴𝐵 = 28 m, then find the perimeter of the shaded region. (Use 𝜋 =
22
)
7
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SOLUTIONS
(a) 44 m
(b) 88 m
(c) 22 m
(d) 80 m
19. Statement A (Assertion): If the centre of a circle is at the origin and its radius = 2 units, then a point
on the circle is (0,2).
Statement 𝐑 (Reason): The centre of the circle is the mid point of the line joining the end points of its
diameter.
(A).
assertion (A).
20. Statement A (Assertion): Two players Sania and Deepika play a tennis match. If the probability of
Sania winning the match is 0.68, then the probability of Deepika winning the match is 0.32.
Statement 𝐑 (Reason): The sum of the probability of two complementary events is 1.
(A).
assertion (A).
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SOLUTIONS
SECTION B

21. The perimeters of two similar triangles are 30 cm and 20 cm. If one altitude of the former triangle is
12 cm, then find the length of the corresponding altitude of the latter triangle.
1
22. If tan⁡𝜃 = , then prove that 7sin2 ⁡𝜃 + 3cos 2 ⁡𝜃 = 4.
√3
OR
1+tan⁡𝐴
If 1−tan⁡𝐴 = 1, then find the value of sin⁡𝐴 + cos⁡𝐴.
23. The probability of getting a defective toy in a carton of 200 toys is 0.24. Find the number of defective
toys in the carton.
OR
Two dice are rolled together. Find the probability of not getting same number on both the dice.
24. A train travels 360 km at a uniform speed. If the speed had been 5 km/hr more, then it would have
taken 1 hour less for the same journey. Form the quadratic equation to find the speed of the train.
25. If 𝑂 is the centre of a circle, 𝑃𝑄 is a chord and the tangent 𝑃𝑅 at 𝑃 makes an angle of 50∘ with 𝑃𝑄,
then find ∠𝑃𝑂𝑄.
SECTION C

26. Find the roots of the following quadratic equations, if they exist, using the quadratic formula.
(i) 15𝑥 2 − 28 = 𝑥
(ii) 9𝑥 2 + 15𝑥 + 10 = 0
27. Find the coordinates of a point 𝐶 on the line segment joining the points 𝐴(6,3) and 𝐵(−4,5) such
3
that 𝐴𝐶 = 5 𝐴𝐵.
OR
If 𝐴(−2,1), 𝐵(𝑎, 0), 𝐶(4, 𝑏) and 𝐷(1,2) are the vertices of a parallelogram 𝐴𝐵𝐶𝐷, then find the
values of 𝑎 and 𝑏. Hence, find the lengths of its sides.
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SOLUTIONS
28. If △ 𝐴𝐵𝐶 circumscribes a circle with centre 𝑂 which touches the sides 𝐴𝐵, 𝐵𝐶 and 𝐶𝐴 at points 𝑃, 𝑄
and 𝑅 respectively, then find the perimeter of △ 𝐴𝐵𝐶 when 𝐴𝐵 + 𝐶𝑅 = 12 cm.
29. Solve the following pair of linear equations
89𝑥 + 91𝑦 = 449 and 91𝑥 + 89𝑦 = 451.
OR
A company purchased 4 chairs and 3 tables for ₹ 1850. From the same place and at same rate, another
company purchased 3 chairs and 2 tables for ₹ 1300. Find the cost of a chair and a table.
30. A pulley was made from two big equal cylinders stuck at the ends of a small cylinder to draw water
from the well, as shown in the figure. Find its curved surface area.
31. Ramesh is standing on the ground and flying a kite with a string of 240 m, at an angle of elevation of
30∘ while Mukesh is standing on the roof of a 30 m high building and is flying his kite at an
elevation of 45∘. Both the boys are on opposite sides of both the kites. Find the length of the string in
metres, correct to two decimal places, that the Mukesh must have so that the two kites meet. ( Use
√2 = 1.414)
SECTION D

32. The mean of the following frequency table is 50. But the frequencies 𝑓1 and 𝑓2 in class 20 − 40 and
60 − 80 are missing. Find the missing frequencies.
20 40 60 80
Class - interval 0 − 20 Total
− 40 − 60 − 80 − 100
Frequency 17 𝑓1 32 𝑓2 19 120
33. A number consists of two digits. When the number is divided by the sum of its digits, the quotient is
8. If 45 is subtracted from the number, the digits interchange their places, then find the number.
OR
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SOLUTIONS
In a class, when 6 boys were admitted and 6 girls left, the percentage of boys increased from 60% to
72%. Find the original number of boys and girls in the class.
34. Prove that 7 − 2√3 is an irrational number.
35. At a point on level ground, the angle of elevation of a vertical tower is found to be such that its
tangent is 5/12. On walking 192 metres towards the tower, the tangent of the angle of elevation is 3/4.
Find the height of the tower.
OR
A bird is sitting on the top of a 90 m high tree. From a point on the ground, the angle of elevation of
the bird is 45∘. The bird flies away horizontally in such a way that it remained at a constant height
from the ground. After 3 seconds, the angle of elevation of the bird from the same point is 30∘. Find
the speed of flying of the bird.
[Use √3 = 1.73 ]
SECTION E

36. A farmer has triangular field of sides 72 m, 120 m and 96 m. He want to divide the field into two
parts, by joining any two points on two of its smaller sides such that the smaller sides are divided in
the ratio of 5: 3, so that he can plant samplings of tomato in the bigger region and that of mango in
the other region.
(i) Find the area of the complete field.
(ii) The length of largest side of the part in which samplings of mangoes are planted.
(iii) The ratio of the perimeter of the two parts of the field formed.
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SOLUTIONS
OR
Find the area of part, where samplings of tomatoes are planted.
37. Pratibha grew some rose plants and Jasmine plants on terrace of her house in a semicircular design as
shown below.
198
Circumference of the semi-circular design is m. The area of the region in which rose plants are
7
grown
is 77 m2 .
Use the above information to answer questions that follow:
(i) Find the radius of the design.
(ii) Find the radius of the region in which rose plants are grown.
OR
Find the area of the region in which Jasmine plants are grown.
(iii) If the cost of growing Jasmine plants is ₹ 8 per m2 , then find the total cost to grow Jasmine
plants.
38. In a class the teacher asks every student to write an example of A.P. Two friends Geeta and Madhuri
writes their progressions as −5, −2,1,4, … and 187,184,181, … respectively.
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SOLUTIONS
(i) Find the 34th term of the progression written by Madhuri.
(ii) Find the sum of first 10 terms of the progression written by Geeta and Madhuri.
OR
Which term of the two progressions will have the same value and find the value also of that form.
(iii) Find the 19th term of the progression written by Geeta.
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SOLUTIONS
Sample
Paper
Set - 9
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Mathematics
Class 10
SET-09
6. Section 𝐸 has 3 case based integrated units of assessment ( 04 marks each) with sub-parts.
SECTION A

1. From a point 𝑄, a tangent 𝑃𝑄 is drawn to the circle such that the length of 𝑃𝑄 is 12 cm and the
distance of 𝑄 from the centre is 13 cm. The radius of the circle is
(a) 7 cm
(b) 6.5 cm
(c) 5 cm
(d) 9 cm
2. In the following figure, 𝐷𝐸 ∥ 𝐵𝐶. Find the value of 𝑥.
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SOLUTIONS
(a) √5
(b) √6
(c) √3
(d) √7
3. From a point on the ground, which is 15 m away from the foot of a vertical tower, the angle of
elevation of the top of the tower, is found to be 60∘. The height of the tower (in metres) is
(a) 5√3
(b) 15√3
(c) 15
(d) 7.5
4. The volume of the largest right circular cone that can be cut out from a cube of edge 4.2 cm is
(a) 9.7 cm3
(b) 72.6 cm3
(c) 58.2 cm3
(d) 19.4 cm3
5. In the given figure, the number of zeroes of the polynomial 𝑓(𝑥) are
(a) 1
(b) 2
(c) 3
(d) 4
6. From the figure, the value of 25(sin2 ⁡𝜃 + 2cos 2 ⁡𝜃 − tan⁡𝜃) is
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SOLUTIONS
2
(a) 3
2
(b) −
3
3
(c) 2
3
(d) − 2
7. A letter is chosen at random from the English alphabets. Find the probability that the letter chosen
succeeds 𝑉.
2
(a) 13
5
(b) 26
1
(c) 26
1
(d) 2
8. A car has two wipers which do not overlap. Each wiper has a blade of length 42 cm sweeping
through an angle of 120∘. Find the total area cleaned at each sweep of the blades.
(a) 4224 cm2
(b) 3696 cm2
(c) 1848 cm2
(d) 5544 cm2
9. The mean of 𝑛 observations 𝑥1 , 𝑥2 , 𝑥3 , … , 𝑥𝑛 is 𝑥‾. If each observation is multiplied by 𝑝, then the

mean of the new observations is
𝑥‾
(a) 𝑝
(b) 𝑝𝑥‾
(c) 𝑥‾
(d) 𝑝 + 𝑥‾
10. Which of the following equations has the sum of its roots as 3 ?
(a) 2𝑥 2 − 3𝑥 + 6 = 0
(b) −𝑥 2 + 3𝑥 − 3 = 0
3
(c) √2𝑥 2 − 2 𝑥 + 1 = 0
√
(d) 3𝑥 2 − 3𝑥 + 3 = 0
11. If HCF⁡(39,91) = 13, then LCM⁡(39,91) is

(a) 91
(b) 273
(c) 39
(d) 3549
12. The angle of elevation of a ladder leaning against a wall is 60∘ and the foot of the ladder is 9.5 m
away from the wall. The length of the ladder is
(a) 10 m
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SOLUTIONS
(b) 16 m
(c) 18 m
(d) 19 m
13. If in two triangles 𝐷𝐸𝐹 and 𝑃𝑄𝑅, ∠𝐷 = ∠𝑄 and ∠𝑅 = ∠𝐸, then which of the following is not true?
𝐸𝐹 𝐷𝐹
(a) 𝑃𝑅 = 𝑃𝑄
𝐷𝐸 𝐸𝐹
(b) 𝑃𝑄 = 𝑅𝑃
𝐷𝐸 𝐷𝐹
(c) 𝑄𝑅 = 𝑃𝑄
𝐸𝐹 𝐷𝐸
(d) 𝑅𝑃 = 𝑄𝑅
14. The value of sin⁡60∘ + cot⁡45∘ − cosec⁡30∘ is

√3
(a) − 1
2
√3
(b) −2
3
√3
(c) −2
2
9√3+4
(d) 33
15. If the equation 𝑎𝑥 2 + 2𝑥 + 𝑎 = 0 has two distinct real roots, then

(a) 𝐷 > 0
(b) 𝐷 < 0
(c) 𝐷 = 0
(d) None of these
16. In the given figure, 𝑂 is the centre of a circle. 𝑃𝑄𝐿 and 𝑃𝑅𝑀 are the tangents at the points 𝑄 and 𝑅
respectively and 𝑆 is a point on the circle such that ∠𝑆𝑄𝐿 = 50∘. Find the value of ∠𝑂𝑄𝑆.
(a) 40∘
(b) 50∘
(c) 60∘
(d) 70∘
17. In given figure, there is a circle with centre 𝑂 and radius 3.5 cm. If the central angle is 60∘, then the
perimeter of 𝑂𝐴𝑃𝐵 is (Take 𝜋 = 3.14 )
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SOLUTIONS
(a) 35 cm
(b) 32.08 cm
(c) 10.66 cm
(d) 18.33 cm
18. In a throw of a pair of dice, what is the probability of getting a doublet?

1
(a) 3
1
(b) 6
5
(c) 12
2
(d) 3
19. Statement A (Assertion): 3𝑦 2 + 17𝑦 − 30 = 0 have distinct roots.

Statement 𝐑 (Reason): The quadratic equation 𝑎𝑥 2 + 𝑏𝑥 + 𝑐 = 0 have distinct roots (real roots) if
𝐷 > 0.
(A).
assertion (A).
20. Statement A (Assertion): Consider the following frequency distribution:

Class interval 0−4 4−8 8 − 12 12 − 16 16 − 20
Frequency 6 3 5 20 10
The median class is 12 − 16.
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SOLUTIONS
Statement 𝐑 (Reason) : Let 𝑛 = ∑𝑓𝑖 . Then, the class whose cumulative frequency is just lesser than
𝑛
( 2 ) is the median class.
(A).
assertion (A).
SECTION B

21. Show that 5 + 2√7 is an irrational number, where √7 is given to be an irrational number.
OR
Amita, Suneha and Rajiv start preparing cards. In order to complete one card, they take 10, 16 and 20
minutes respectively. If all of them started together, after what time will they start preparing a new
card together?
22. Show that 𝑥 = 2, 𝑦 = 1 and 𝑥 = 4, 𝑦 = 4 are solutions of the system of equations. 3𝑥 − 2𝑦 = 4

6𝑥 − 4𝑦 = 8
23. Show that the points (7,10), (−2,5) and (3, −4) are vertices of an isosceles right triangle.
Find the ratio in which 𝑥-axis divides the line segment joining the points 𝑃(−5,6) and 𝑄(−1, −3).
24. Find how many two-digit numbers are divisible by 6.
25. A wooden toy is in the form of a cone mounted on a hemisphere with the same radius. The diameter
of the base of the conical portion is 6 cm and its height is 4 cm. Determine the volume of the toy.
SECTION C
26. Prove that 2√3 + √5 is an irrational number. Also, check whether (2√3 + √5)(2√3 − √5) is
rational or irrational.
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SOLUTIONS
27. A tree 12 m high, is broken by the wind in such a way that its top touches the ground and makes an
angle 60∘ with the ground. At what height from the bottom the tree is broken by the wind? (Use
√3 = 1.73 )
OR
Two ships are sailing in the sea on the either side of a light house. The angles of depression of two
ships as observed from the top of the light house are 60∘ and 45∘ respectively. If the distance
1+√3
between the ships is 100 ( ) m, then find the height of the light house.
√3
28. In the given figure, 𝐴𝑃 = 3 cm, 𝐴𝑅 = 4.5 cm, 𝐴𝑄 = 6 cm, 𝐴𝐵 = 5 cm and 𝐴𝐶 = 10 cm, then find
𝐴𝐷.
29. A box contains 5 red marbles, 8 white marbles and 4 green marbles. One marble is taken out of the
box at random. What is the probability that the marble taken out will be (i) red? (ii) white? (iii) not
green?
30. If 𝛼 and 𝛽 are the zeroes of the quadratic polynomial 𝑓(𝑥) = 𝑥 2 − 2𝑥 + 3, then find a polynomial
𝛼−1 𝛽−1
whose roots 𝛼+1 and 𝛽+1.
OR
Find the value of 𝑝, for which one zero of the polynomial, 𝑝𝑥 2 − 144𝑥 + 8 is 11 times the other.
31. In the given figure, a circle is inscribed in a △ 𝐴𝐵𝐶 having sides 𝐵𝐶 = 8 cm, 𝐴𝐵 = 10 cm and 𝐴𝐶 =
12 cm. Find the lengths 𝐵𝐿, 𝐶𝑀 and 𝐴𝑁.
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SOLUTIONS
SECTION D

32. 𝐴𝐵 is a diameter of a circle. 𝑃 is a point on the semi-circle 𝐴𝑃𝐵. 𝐴𝐻 and 𝐵𝐾 are perpendiculars from
𝐴 and 𝐵 respectively to the tangent at 𝑃, then find 𝐴𝐻 + 𝐵𝐾.
OR
In the given figure, 𝑂 is the centre of a circle of radius 5 cm, 𝑇 is a point such that 𝑂𝑇 = 13 cm and
𝑂𝑇 intersects the circle at 𝐸. If 𝐴𝐵 is the tangent to the circle at 𝐸, find the length of 𝐴𝐵.
33. Jaspal Singh repays his total loan of ₹ 118000 by paying every month starting with the first
installment of ₹1000. If he increases the installment by ₹ 100 every month, what amount will be paid
by him in the 30th installment? What amount of loan does he still have to pay after the 30th
installment?
34 In △ 𝑃𝑄𝑅, ∠𝑄 = 90∘ , 𝑃𝑄 = 8 cm, 𝑃𝑅 − 𝑄𝑅 = 2 cm. Find the lengths of 𝑃𝑅 and 𝑄𝑅. Also,
1+sin⁡𝑃 tan⁡𝑃
evaluate 1+cos⁡𝑃 and sin⁡𝑃 .
OR
If sin⁡𝐴 = 1/√10, then find the value of
tan⁡𝐴+cot⁡𝐴
(i) sin2 ⁡𝐴 + cos 2 ⁡𝐴, (ii) 3sin⁡𝐴+2cos⁡𝐴.
35. A wooden article was made by scooping out a hemisphere from each end of a solid cylinder, as
shown in figure. If the height of the cylinder is 10 cm, and its base is of radius 3.5 cm, find the total
surface area of the article.
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SOLUTIONS
SECTION E

36. Let 𝐴, 𝐵, 𝐶 and 𝐷 are the vertices of a parallelogram 𝐴𝐵𝐶𝐷, which is a garden of different flowers.
Madhu wants to make a garland by using flowers of orchids, roses, marigold and jasmine plants,
which are placed at points 𝐴(−2,1), 𝐵(𝑎, 0), 𝐶(4, 𝑏) and 𝐷(1,2) respectively.
(i) Find the value of ' 𝑎 '.
(ii) Find the value of ' 𝑏 '.
(iii) Find the length of sides 𝐴𝐵 and 𝐴𝐷.
If 𝐴(2,4), 𝐵(−2,3), 𝐶(3,6), 𝐷(𝑥, 𝑦) are the vertices of parallelogram, then find the sum of 𝑥 and 𝑦.
37. Samarth usually go to fruit seller shop with his mother. He observe the following two situations. On
1st day: The cost of 2 kg of apples and 1 kg of grapes was found to be ₹ 160.
On 2nd day: The cost of 5 kg of apples and 2 kg of grapes was found to be ₹380.
Denoting the cost of 1 kg apples by ₹𝑥 and cost of 1 kg of grapes by ₹𝑦, answer the following
questions.
CLICK HERE FOR

SOLUTIONS
(i) Represent algebraically the situation of 1st &2nd day.
(ii) Find cost of 1 kg apples and 1 kg grapes.
OR
At what point the linear equation represented on day 1 intersect 𝑥-axis and 𝑦-axis.
(iii) Find the total cost of 3 kg of apples & 2 kg of grapes.
38. An inspector in an enforcement squad of electricity department visit to a locality of 100 families and
record their monthly consumption of electricity, on the basis of family members, electronic items in
the house and wastage of electricity, which is summarise in the following table.
(i) Find the value of 𝑥 + 𝑦.
(ii) If the median of the above data is 525, then find the value of 𝑥.
OR
Find the average monthly consumption of a family of this locality.
(iii) What will be the upper limit of the modal class?
CLICK HERE FOR

SOLUTIONS
Sample
Paper
Set - 10
With a success rate
2024 boards
Mathematics
Class 10
SET 10
SECTION A

1. Find the value of 𝑘 for which the equation (𝑘 − 12)𝑥 2 + 2(𝑘 − 12)𝑥 + 2 = 0 equal and real
solutions.
(a) 12
(b) 14
(c) Both (a) and (b)
(d) None of the above
2. The HCF and the LCM of 12,21 and 15 respectively, are

(a) 3,140
(b) 12,420
(c) 3,420
(d) 420,3
3. The perimeter of a triangle with vertices (0,4), (0,0) and (3,0) is
(a) 7 + √5
(b) 5
CLICK HERE FOR

SOLUTIONS
(c) 10
(d) 12
4. If tan2 ⁡45∘ − cos 2 ⁡30∘ = 𝑥sin⁡45∘ cos⁡45∘ , then 𝑥 =

(a) 2
(b) -2
1
(c) − 2
1
(d) 2
5. The radius of a sector of a circle with central angle 90∘ is 7 cm. The area of the minor segment of the
circle is
(a) 77/2 cm2
(b) 14 cm2
(c) 18 cm2
(d) 24 cm2
6. Find the distance between the points, 𝐴(2𝑎, 6𝑎) and 𝐵(2𝑎 + √3𝑎, 5𝑎).
(a) √2𝑎2 = 2𝑎
(b) √4𝑎2 = 2𝑎
(c) √6𝑎2 = 2𝑎
(d) √10𝑎2 = 4
7. A box contains 100 discs, numbered from 1 to 100. If one disc is drawn at random from the box, then
the probability that it bears a prime number less than 30, is
7
(a) 100
1
(b)
10
4
(c) 50
9
(d) 50
8. Solve the quadratic equation for 𝑥: 4√3𝑥 2 + 5𝑥 − 2√3 = 0

√3 −2
(a) ,
4 √3
−√3 −2
(b) ,
4 √3
√3 2
(c) ,
4 √3
− √3 2
(d) ,
4 √3
9. A tangent 𝑃𝑄 at a point 𝑃 of a circle of radius 8 cm meets a line through the centre 𝑂 at a point 𝑄 so
that 𝑂𝑄 = 17 cm. Find the length of 𝑃𝑄.
(a) 10 cm
(b) 20 cm
CLICK HERE FOR

SOLUTIONS
(c) 15 cm
(d) 0.5 cm
10. From the given figure, the angle of depression of point 𝐶 from the point 𝑃 is
(a) 45∘
(b) 90∘
(c) 75∘
(d) 30∘
11. Four observations are 2, 4, 6 and 8 . The frequencies of the first three observations are 3,2 and 1
respectively. If the mean of the observations is 4 , then the frequency of the fourth observation is
(a) 8
(b) 4
(c) 1
(d) 2
12. A month is selected at random from a year. The probability that it is May or July is
1
(a) 12
1
(b) 6
3
(c) 4
1
(d) 3
13. The zeroes of the quadratic polynomial 𝑥 2 + 𝑘𝑥 + 𝑘, where 𝑘 > 0

(a) are both positive
(c) are always equal
(b) are both negative
(d) are always unequal
14. In the given figure, 𝐷 and 𝐸 are two points lying on side 𝐴𝐵, such that 𝐴𝐷 = 𝐵𝐸. If 𝐷𝑃 ∥ 𝐵𝐶 and
𝐸𝑄 ∥ 𝐴𝐶, then
CLICK HERE FOR

SOLUTIONS
(a) 𝑃𝑄 ∥ 𝐴𝐵
(b) 𝑃𝑄 = 𝐴𝐵
(c) 𝑃𝑄 ∥ 𝐶𝐷
(d) 𝑃𝑄 = 𝐴𝐶
2sin2 ⁡𝐴+3cot2 ⁡𝐴
15. If cosec⁡𝐴 = √2, then, the value of 4(tan2 ⁡𝐴−cos2 ⁡𝐴) is
(a) 1
(b) 2
(c) 3
(d) 0
16. A wall 8 m long casts a shadow 5 m long. At the same time, a tower casts a shadow 50 m long, then
the height of tower is
(a) 20 m
(b) 80 m
(c) 40 m
(d) 200 m
17. The ratio of the volumes of two spheres is 8: 27. The ratio between the radius of two spheres is
(a) 2: 3
(b) 4: 27
(c) 8: 9
(d) 4: 9
22
18. Find the area of a quadrant of a circle, where the circumference of circle is 44 cm. [Use 𝜋 = ]
7
(a) 38.5 cm2
(b) 77 cm2
(c) 60 cm2
(d) 154 cm2
19. Statement A (Assertion): The arithmetic mean of the following frequency distribution is 25.
10 20 30 40
Class interval 0 − 10
− 20 − 30 − 40 − 50
Frequency 5 18 15 16 6
Σ𝑓𝑖 𝑥𝑖 1
Statement R (Reason) : Mean (𝑥‾) = , where 𝑥𝑖 = 2 (lower limit + upper limit) of 𝑖 th class
Σ𝑓𝑖
interval and 𝑓𝑖 is its frequency.
CLICK HERE FOR

SOLUTIONS
(A).
assertion (A).
20. Statement A (Assertion): 2√2 is not the root of the quadratic equation 𝑥 2 − 4√2𝑥 + 8 = 0.
Statement 𝐑 (Reason): The root of a quadratic equation satisfies it.
(A)
assertion (A)
SECTION B
Section 𝐵 consists of 5 questions of 2 marks each.

21. Mayank made a bird-bath for his garden in the shape of a cylinder with a hemispherical depression at
one end (see figure). The height of the cylinder is 1.45 m and its radius is 30 cm. Find the total
surface area of the bird-bath.
OR
A toy is in the form of a cylinder with two hemispheres at two ends. If the height of the cylinder is
12 cm, and its base is of diameter 7 cm, find the total surface area of the toy.
CLICK HERE FOR

SOLUTIONS
22. Which term of the A.P. 3,14,25,36, … will be 99 more than its 25th term?
23. Represent the following equations graphically: 𝑥 + 𝑦 − 3 = 0,2𝑥 + 3𝑦 − 7 = 0
24. In the given figure, 𝑃𝐴 and 𝑃𝐵 are tangents to the circle from an external point 𝑃. 𝐶𝐷 is another
tangent touching the circle at 𝑄. If 𝑃𝐴 = 12 cm, 𝑄𝐶 = 𝑄𝐷 = 3 cm, then find 𝑃𝐶 + 𝑃𝐷.
25. 𝐴, 𝐵 and 𝐶 starts at the same time in the same direction to run around a circular stadium. 𝐴 completes
a round in 252 seconds, 𝐵 in 308 seconds and 𝐶 in 198 seconds, all starting at the same point. After
what time will they meet again at the starting point?
OR
Three numbers are in the ratio 1: 2: 3 and their HCF is 12. Find the positive square root of largest
number.
SECTION C
26. Obtain the zeroes of the quadratic polynomial √3𝑥 2 − 8𝑥 + 4√3 and verify the relationship between
its zeroes and coefficients.
27. It is given that △ 𝐴𝐵𝐶 ∼△ 𝐸𝐷𝐹 such that 𝐴𝐵 = 5 cm, 𝐴𝐶 = 7 cm, 𝐷𝐹 = 15 cm and 𝐷𝐸 = 12 cm.
Find the lengths of the remaining sides of the triangles.
OR
In the given figure, 𝐴𝐵 ∥ 𝐷𝐸 and 𝐵𝐷 ∥ 𝐸𝐹. Prove that 𝐷𝐶 2 = 𝐶𝐹 × 𝐴𝐶.
CLICK HERE FOR

SOLUTIONS
28. Show that (√3 + √5)2 is an irrational number.
29. A ladder which is inclined at an angle of 60∘ to the horizontal reaches a 11 m long wall at a point
2.35 m below the top of the wall. Find the length of the ladder. (Use √3 = 1.73 )
30. If from an external point 𝐵, two tangents 𝐵𝐶 and 𝐵𝐷 are drawn to a circle with centre 𝑂 such that
∠𝐷𝐵𝐶 = 120∘, prove that 𝐵𝐶 + 𝐵𝐷 = 𝐵𝑂, i.e., 𝐵𝑂 = 2𝐵𝐶.
OR
If two tangents are inclined at an angle of 60∘ to a circle of radius 3 cm, then find length of each
tangent.
31. A die is thrown once. Find the probability of getting:

(i) a prime number;
(ii) a number lying between 2 and 6;
(iii) an odd number.
SECTION D

cosec⁡𝐴−cot⁡𝐴 cosec⁡𝐴+cot⁡𝐴 1+cos2 ⁡𝐴
32. Prove that: + = 2(2cosec 2 ⁡𝐴 − 1) = 2 ( )
cosec⁡𝐴+cot⁡𝐴 cosec⁡𝐴−cot⁡𝐴 1−cos2 ⁡𝐴
OR
𝑚 𝑛 4sin⁡𝐴cos⁡𝐴 4
If 𝑚 = cos⁡𝐴 − sin⁡𝐴 and 𝑛 = cos⁡𝐴 + sin⁡𝐴, then show that − 𝑚 = − cos2 ⁡𝐴−sin2 ⁡𝐴 = − cot⁡𝐴−tan⁡𝐴.
𝑛
33. In the given figure, 𝐴𝐵 is a chord of length 16 cm of a circle of radius 10 cm. The tangents at 𝐴 and
𝐵 intersect at a point 𝑃. Find the length of 𝑃𝐴.
CLICK HERE FOR

SOLUTIONS
OR
In the given figure, a circle is inscribed in a quadrilateral 𝐴𝐵𝐶𝐷 in which ∠𝐵 = 90∘. If 𝐴𝐷 =

23 cm, 𝐴𝐵 = 29 cm and 𝐷𝑆 = 5 cm, find the radius (𝑟) of the circle.
34. A toy is in the form of a right circular cylinder with cone on one end. The height and radius of base of
the cylindrical part are 13 cm and 5 cm respectively. The base of the conical part are same as that of
the cylinder. Calculate the surface area of the toy, if the height of the cone is 12 cm. [Take 𝜋 = 22/7
]
35. The sum of the first 𝑛 terms of an A.P. whose first term is 8 and the common difference is 20 is equal
to the sum of first 2𝑛 terms of another A.P. whose first term is -30 and the common difference is 8.
Find 𝑛.
SECTION E

36. You are interested in interpreting the distribution of ages across preferred modes of communication
consider the data given below.
Age-group Text Message WhatsApp Letter Email Phone call
1−5 0 16 0 0 2
6 − 10 2 8 2 0 3
11 − 15 3 15 5 0 5
16 − 20 4 21 3 1 8
21 − 25 20 30 0 3 10
26 − 30 23 29 0 2 32
CLICK HERE FOR

SOLUTIONS
(i) Find the average age of people who use text message for communication.
(ii) Find the average age of people who use phone call for communication.
(iii) Find the model age of people who use WhatsApp for communication.
OR
Find the median of people who use Letter for communication.
37. In a Maths class, teacher asked the students to mark the points (4, −1) and (−3,2) on a graph paper.
But Shweta mistakenly located the points (−1,4) and (3, −2).
Based on the given information, answer the questions that follow:
(i) The distance between the points, that teacher asked to mark is
(ii) The mid point of the line joining the points plotted by Shweta is
(iii) The coordinates of the point which divides the join of points plotted by Shweta in the ratio 1: 2 is
OR
If all the four points are plotted on a graph paper, then the mid point of the line joining the points
lying in II Quadrant is
38. Anil went for a walk in the evening near railway track with his father who is an expert in
mathematics. He told Anil that path of train A is given by equation 𝑥 + 2𝑦 = 4 and path of train B is
given by equation 2𝑥 + 4𝑦 = 12. His father put some question to Anil. Help Anil to solve the
questions of Anil's father.
CLICK HERE FOR

SOLUTIONS
(i) Equation 𝑥 + 2𝑦 = 4 intersects the 𝑥-axis and 𝑦-axis at which points?
(ii) Draw the graph of given equation. Find the coordinates of point of intersection of two given
equations.
(iii) Equation 2𝑥 + 4𝑦 = 12 intersects the 𝑥-axis and 𝑦-axis at which points?
OR
Show that the system of linear equation represented by two given lines are inconsistent.
CLICK HERE FOR

SOLUTIONS
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CBSE Curriculum
2023-24
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Time Allowed: 3 Hours Maximum Marks: 80

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.

Section A consists of 20 questions of 1 mark each.

2. The distance of the point (5, −4) from 𝑥-axis is

4. (cos 4 ⁡𝑥 − sin4 ⁡𝑥 ) is equal to

5. If probability of success is 0.9%, then probability of failure is

10. 5sin2 ⁡30∘ + cos 2 ⁡45∘ − 4tan2 ⁡30∘ is equal to

11. The sum (−6) + (0) + (6) + ⋯. upto 13th term =

13. The mean of first ten odd natural numbers is

14. If LCM of 𝑎 and 18 is 36 and HCF of 𝑎 and 18 is 2 , then 𝑎 =

The cube is thrown once. What is the probability of getting 𝐴 ?

17. The zeroes of the polynomial 𝑓(𝑥) = 𝑥 2 − 2√2𝑥 − 16 are

(d) Assertion (A) is false but reason (R) is true.

(c) Assertion (A) is true but reason (R) is false.

(d) Assertion (A) is false but reason (R) is true.

Section B consists of 5 questions of 2 marks each.

28. Find the mean of the following data.

35. Draw the graph of the following pair of linear equations :

Use the above information to answer the questions that follow :

How much distance the pigeon covers in 8 seconds?

Power bank - If the number on the selected card is a multiple of 3.

Puppet - If the number on selected card is divisible by 10.

Use the above information to answer the questions that follow:

(ii) Find the probability of winning a puppet.

Find the probability of winning a water bottle.

(iii) Find the probability of getting 'Better Luck next time'.

2 3.2 2.5 × 3.2

⇒ 𝑝(1) = 𝑎(1)2 − 3(𝑎 − 1)(1) − 1 = 0 ⇒ 𝑎 − 3𝑎 + 3 − 1 = 0 ⇒ −2𝑎 = −2 ⇒ 𝑎 = 1

8. (a): Total number of possible outcomes = 6 × 6 = 36

Let 𝐸 be the event 'getting the sum as a perfect square'.

For infinite solutions,

11. (a): Here, 𝑎 = −6 and 𝑑 = 0 − (−6) = 6

Given, 𝑂𝑄 = 15 cm and 𝑃𝑄 = 12 cm In △ 𝑂𝑃𝑄

𝑂𝑄2 = 𝑂𝑃2 + 𝑃𝑄2

⇒ 𝑂𝑃2 = 225 − 144 = 81

15. (b): In △ 𝐴𝑂𝐷 and △ 𝐵𝑂𝐶,

and ∠𝐴𝑂𝐷 = ∠𝐵𝑂𝐶

[By SAS similarity criterion]

Hence, criterion of similarity used is SAS.

16. (a): Total number of possible outcomes = 6

∴ Favourable number of outcomes = 2

17. (b): The zeroes of 𝑓(𝑥) = 𝑥 2 − 2√2𝑥 − 16 are given by

⇒ 𝑥(𝑥 − 4√2) + 2√2(𝑥 − 4√2) = 0

18. (b): Let the required ratio be 𝑘: 1.

Since, this point lies on 3𝑥 + 𝑦 − 6 = 0.

19. (d): Given quadratic equation is 𝑥 2 − 𝑥 + 1 = 0

Hence, the given quadratic equation has no real roots.

∴ Assertion (A) is false but Reason (R) is true.

Clearly, Reason (R) is true.

In △ 𝐴𝐵𝐶 and △ 𝑃𝑄𝑅

21. The given equation is (𝑝 − 𝑞)𝑥 2 + 5(𝑝 + 𝑞)𝑥 − 2(𝑝 − 𝑞) = 0.

∴ 𝐷 = 𝑏2 − 4𝑎𝑐 = 25(𝑝 + 𝑞)2 − 4(𝑝 − 𝑞)(−2(𝑝 − 𝑞))

= 25(𝑝 + 𝑞)2 + 8(𝑝 − 𝑞)2

∴ 𝐷 = 25(𝑝 + 𝑞)2 + 8(𝑝 − 𝑞)2 > 0

Hence, roots of the given equation are real and unequal.