MATHEMATICS TERM 1

Grade: X CBSE (MARKS: 80)

Date: 01.09.2025
Duration: 3 Hours

General Instructions:
• This question paper contains 38 questions.
• This Question Paper is divided into 5 Sections A, B, C, D and E.
• In Section A, Questions no. 1-18 are multiple choice questions (MCQs) and questions no. 19 and
20 are Assertion- Reason based questions of 1 mark each.
• In Section B, Questions no. 21-25 are very short answer (VSA) type questions, carrying 02 marks
each.
• In Section C, Questions no. 26-31 are short answer (SA) type questions, carrying 03 marks each.
• In Section D, Questions no. 32-35 are long answer (LA) type questions, carrying 05 marks each.
• In Section E, Questions no. 36-38 are case study – based questions carrying 4 marks each with
sub parts of the values of 1, 1 and 2 marks each respectively.
• All Questions are compulsory. However, an internal choice in 2 Question of Section B, 2
Questions of Section C and 2 Questions of Section D have been provided. An internal choice has
been provided in all the 2 marks questions of Section E.
• Draw neat and clean figures wherever required.
• Take π =22/7 wherever required if not stated.
• Use of calculators is not allowed.
Section A
Choose the correct alternative.
1. If the sum of the zeroes of the polynomial p(x) = (k 2 − 14)x 2 − 2x − 12 is 1, then k takes
the value(s): 1
a. √14 b. – 14 c. 2 d. ± 4
2. AOBC is a rectangle whose three vertices are A (0, 3), O (0, 0) and B (5, 0). The length of its
diagonal is: 1
a. 5 b. 3 c. √34 d. 4
3. The distance between the points (a cos θ + b sin θ, 0) and (0, a sin θ − b cos θ) is:
1
a. a2 + b2 b. a + b c. a2 − b2 d. √a2 + b 2
4. In the given figure, LM is parallel to AB. If AL = 𝑥 − 3, AC = 2𝑥, BM = 𝑥 − 2 and BC = 2𝑥 +
3, find the value of 𝑥.

a. 9 b. 8 c. 10 d. 5
5. The smallest number by which √27 should be multiplied so as to get a rational number is:
1
a. √27 b. 3√3 c. √3 d. 3
6. The zeroes of the quadratic polynomial x2 + 99x + 127 are:
a. both positive b. both negative 1
c. one positive and one negative d. both equal

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7. The value of the expression sin6θ + cos6θ + 3 sin2θ cos2θ is:
1
a. 1 b. 2 c. 0 d. 3
8. For the following distribution.
C. I. 0 – 10 10 – 20 20 – 30 30 – 40 40 – 50
Frequency 5 9 15 10 6 1
The upper limit of the median class will be:
a. 20 b. 30 c. 40 d. 10
9. The probability of selecting a rotten apple randomly from a heap of 900 apples is 0.18.
What is the number of rotten apples in the heap? 1
a. 162 b. 152 c. 200 d. 182
10. 2
If one zero of the polynomial 𝑝(𝑥) = 6x + 37x − (k − 2) is reciprocal of the other, then
what is the value of k? 1
a. – 4 b. – 6 c. 6 d. 4
11. The ratio in which the line segment joining the points A (- 2, - 3) and B (3, 7) is divided by
y – axis is: 1
a. 2 : 3 b. 1 : 3 c. 1 : 2 d. 3 : 1
12. A vertical pole of length 6 m casts a shadow 4 m long on the ground and at the same time a
tower casts a shadow 28 m long. Find the height of the tower. 1
a. 42 m b. 48 m c. 40 m d. 50 m
13. If sin 𝜃 + cos 𝜃 = √2 cos 𝜃, then tan 𝜃 will be equal to:
1
a. √2 − 1 b. √2 + 1 c. √2 d. −√2
14. An event is very unlikely to happen. It’s probability is closest to:
1
a. 0.00001 b. 0.001 c. 0.01 d. 0.1
15. If the height of a vertical pole is √3 times the length of its shadow on the ground, then the
angle of elevation of the sun at that time is: 1
a. 300 b. 600 c. 450 d. 750
16. The sum of the first 15 multiples of 8 is:
a. 970 b. 980 c. 960 d. 990
17. In the figure, DE is parallel to BC. Which of the following is true?

𝑎+𝑏 𝑎𝑥 𝑎𝑦 𝑥 𝑎
a. 𝑥 = b. 𝑦 = 𝑎+𝑏 c. 𝑥 = 𝑎+𝑏 d. 𝑦 = 𝑏
𝑎𝑦
𝑎
18. The probability of guessing the correct answer to a certain question is 𝑏. If the probability of
2
not guessing the correct answer is 3, then 1
a. 𝑏 = 4𝑎 b. 𝑏 = 3𝑎 c. 𝑏 = 2𝑎 d. 𝑏 = 𝑎
DIRECTION: In the question number 19 and 20, a statement of Assertion (A) is followed
by a statement of Reason (R).
Choose the correct option.
a) Both assertion (A) and reason (R) are true and reason (R) is the correct explanation of
assertion (A)
b) Both assertion (A) and reason (R) are true and reason (R) is not the correct explanation
of assertion (A)
c) Assertion (A) is true but reason (R) is false.
d) Assertion (A) is false but reason (R) is true.
19. Assertion (A): The mean of the given data is 12.93.
Class 4–7 8 – 11 12 – 15 16 – 19
1
Frequency 5 4 9 10
Reason (R): Modal class of above data is 16 – 19.

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 20. Assertion (A): For 00 < 𝜃 900, cosec 𝜃 – cot 𝜃 and cosec 𝜃 + cot 𝜃 are reciprocal of each
other.
Reason (R): cosec2 𝜃 – cot2 𝜃 = 1
SECTION B
Solve the following questions for 2 marks each.
21. (A) Find α-1 + β-1, if α and β are zeroes of the polynomial 9x2 – 3x – 2.
OR
2 2
(B) If the sum of zeroes of the polynomial 𝑥 − (𝑘 + 3)𝑥 + (5𝑘 − 3) is equal to one – fourth
of the product of the zeroes, find the value of ‘𝑘’.
22. Show that 5 + 2√7 is irrational, where √7 is irrational. 2
23. (A) D is a point on the side BC of a triangle ABC such that ∠ADC = ∠BAC. Show that
CA2 = CB.CD
OR 2
(B) ABCD is a trapezium in which AB parallel to DC and its diagonals intersect each other at
𝐴𝑂 𝐶𝑂
O. Using basic proportionality theorem, prove that 𝐵𝑂 = 𝐷𝑂.
24. Prove the following identity: √𝑠𝑒𝑐 2 𝜃 + 𝑐𝑜𝑠𝑒𝑐 2 𝜃 = tan 𝜃 + cot 𝜃 2
25. Find the middle term of the A.P.: 6, 13, 20, ……, 216. 2
SECTION C
Solve the following questions for 3 marks each.
26. Find the HCF and LCM of 144, 180 and 192 by prime factorization method. 3
27. If α and β are the zeroes of the polynomial p(x) = 2x2 + 5x + k, satisfying the relation,
21 3
α2 + β2 + αβ = then find the value of k.
4

28. (A) If the sum of first 7 terms of an AP is 49 and that of 17 terms is 289, find the sum of first n
terms.
OR 3
(B) The 4th term of an AP is zero. Prove that the 25th term of the AP is three times its 11th
term.
29. The shadow of a tower standing on level ground is found to be 40 m longer when the Sun’s
3
altitude is 30° than when it is 60°. Find the height of the tower. (Take √3 = 1.732)
30. Calculate mean using assumed mean method for the following data.
Class 0–5 5 – 10 10 – 15 15 – 20 20 – 25 3
Frequency 8 7 10 13 12
31. (A) If the mid – point of the line joining (3, 4) and (k, 7) is (x, y) and 2x + 2y + 1 = 0, find
the value of k.
OR
(B) Name the type of triangle formed by the points A (–5, 6), B (–4, –2) and C (7, 5).
SECTION D
Solve the following questions for 5 marks each.
32. Two poles of height 𝑎 metres and 𝑏 metres are 𝑝 metres apart. Prove that the height of the
point of intersection of the lines joining the top of each pole to the foot of the opposite pole is 5
𝑎𝑏
given by 𝑎+𝑏 metres.
33. (A) Prove that: sin6 θ + cos 6 θ = 1 − 3sin2 θ. cos2 𝜃
OR 5
(B) If 𝑐𝑜𝑠𝑒𝑐 𝐴 − sin 𝐴 = 𝑙 and sec 𝐴 − cos 𝐴 = 𝑚, prove that 𝑙 2 𝑚2 (𝑙 2 + 𝑚2 + 3) = 1
34. (A) The probability of selecting a blue marble at random from a jar that contains only blue,
black and green marbles is 1/5. The probability of selecting a black marble at random
from the same jar is 1/4. If the jar contains 11 green marbles, find the total number of
marbles in the jar. 5
(B) An integer is chosen between 0 and 100. What is the probability that it is
1. divisible by 7?
2. not divisible by 7?
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 35. (A) If the mean of the following data is 14.7, find the values of 𝑝 and 𝑞.
Class 0–6 6 – 12 12 – 18 18 – 24 24 – 30 30 – 36 36 – 42 Total
Frequency 10 𝑝 4 7 𝑞 4 1 40
OR
(B) The marks of students in two divisions A and B are given below:
Marks of students in divisions
Marks
A B 5
10 – 20 10 25
20 – 30 25 10
30 – 40 30 35
40 – 50 20 15
50 – 60 15 20
Find the modal marks of students in divisions A and B.
SECTION E
Solve the following case study questions for 4 marks each.
36. An interior designer, Sana, hired two painters, Manan and Bhima to make paintings for her
buildings. Both painters were asked to make 50 different paintings each. The prices quoted
by both the painters are given below:
• Manan asked for Rs 6000 for the first painting, and an increment of Rs 200 for each
following painting.
• Bhima asked for Rs 4000 for the first painting, and an increment of Rs 400 for each
following painting.
1. How much money did Manan get for his 25th painting? Show your work. 1
2. How much money did Bhima get in all? Show your work. 1
3. (A) If both Manan and Bhima make paintings at the same pace, find the first painting for
which Bhima will get more money than Manan. Show your steps
OR
2
(B) Sana's friend, Aarti hired Manan and Bhima to make paintings for her at the same rates
as for Sana. Aarti had both painters make the same number of paintings, and paid them
the exact same amount in total. How many paintings did Aarti get each painter to make?
37. Class X students of a secondary school in Kerala have been allotted a rectangular plot of a
land for gardening activity. Saplings of Gulmohar are planted on the boundary at a distance
of 1 m from each other. There is a triangular grassy lawn in the plot as shown in the figure.
The students are to sow seeds of flowering plants on the remaining area of the plot.
Considering A as the origin, answer the following questions.

1. What are the coordinates of R? 1

2. What are the coordinates of P? 1
3. (A) What are the coordinates of the mid – point of seg QR? 2
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 OR
(B) What are the coordinates of the mid – point of seg PR?
38. One evening, Kaushik was in a park. Children were playing cricket. Birds were singing on a
nearby tree of height 80m. He observed a bird on the tree at an angle of elevation of 45°.
When a sixer was hit, a ball flew through the tree frightening the bird to fly away. In 2

seconds, he observed the bird flying at the same height at an angle of elevation of 30° and
the ball flying towards him at the same height at an angle of elevation of 60°.
1. At what distance from the foot of the tree was he observing the bird sitting on the tree? 1
2. What is the speed of the bird in m/min if we consider it had flown 20(√3 + 1) m? 1
3. (A) How far did the bird fly in the mentioned time?
OR 2
(B) After hitting the tree, how far did the ball travel in the sky when Kaushik saw the ball?

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