Class 10 Question Paper 2023-24 Maths (Half Yearly)

2023-24
HALF YEARLY
EXAM
NCERT BASED
SYLLABUS
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FOR
CBSE BOARD
AND
STATE BOARDS

Half Yearly Exam 2023-24
Question Paper

Class - X Subject – Mathematics
Time – 3 Hours Maximum Marks - 80

General Instructions:

Read the following instructions carefully and follow them

(i) This question paper contains 38 questions. All questions are compulsory.
(ii) This question paper is divided into FIVE Sections - Section A, B, C, D and E.
(iii) In Section-A question number 1 to 18 are Multiple Choice Questions (MCQs) and question number 19
and 20 are Assertion - Reason based questions of 1 mark each.
(iv) In Section –B question number 21 to 25 are Very Short-Answer-I (SA – I) type questions of 2 marks each
(v) In Section – C question number 26 to 31 or Short Answer – II (SA – II) type question carrying 3 marks
each
(vi) In Section –D question number 32 to 35 is Long –Answer (LA) type questions carrying 5 marks each
(vii) In Section –E question number 36 to 38 are case study type questions carrying 4 marks each

SECTION A

1. The product of a non-zero rational number and an irrational number is
(a) always irrational (b) always rational (c) rational or irrational (d) one

2. If the square of difference of the zeroes of the quadratic polynomial x 2 + px + 45 is equal to 144, then
the value of p is
(a) ± 9 (b) ± 12 (c) ± 15 (d) ± 18

3. Let the hypotenuse of an isosceles right angled triangle is 7√2 cm. Then the area of the circle
inscribed in it, is
154 154
(a) 154 cm2 (b) 2 cm2 (c) 2 cm2 (d) 145 cm2
( 2 + √2) ( 2− √2)

4. Point P divides the line segment joining R(-1, 3) and S(9, 8) in the ratio k : 1. If P lies on the line x – y
+ 2 = 0, then the value of k is
(a) 2/3 (b) 1/2 (c) 1/3 (d) 1/4

5. Which of the following pairs of linear equations is inconsistent
(a) x + y = 5; 2x + 2y = 10 (b) x – y = 8; 3x – 3y = 16
(c) 2x + y – 6 = 0; 4x – 2y – 4 = 0 (d) 2x – 2y – 2 = 0; 4x – 4y – 4 = 0

6. In the given figure, ∠ACB = ∠CDA, AC = 8 cm and AD = 3 cm, then BD is

(a) 22/3 cm (b) 26/3 cm
(c) 55/3 cm (d) 64/3 cm

1

7. Which of the following cannot be the probability of an event?
(a) 2/3 (b) - 1.5 (c) 15% (d) 0.7

8. For an event E, P(E) + P(E’) = x, then the value of x3 – 3, is
(a) – 2 (b) 2 (c) 1 (d) – 1

9. If cos2 θ + 2sin2 θ + 3cos2 θ + 4sin2 θ + ⋯200 terms = 10025, where θ is acute, then the value of
sinθ − cos θ is

1− √3 1+2√3 √3−1
(a) (b) (c) (d) 0
2 2 2

7 (1+ sinA)(1−sinA)
10. If cotA = 8, then is
(1+cosA)(1−cosA)
(a) 7/8 (b) 49/64 (c) 1/2 (d) None

11. An arc of a circle is of length 5π cm and the sector it bounds has an area of 20π cm2, then the radius
of the circle is
(a) 4 cm (b) 8 cm (c) 12 cm (d) 16 cm

12. Length of an arc of a sector of angle P (in degrees) of a circle of radius R is
P P P P
(a) 1800 × 2πR (b) 1800 × πR (c) 3600 × 2π𝑅2 (d) 3600 × π𝑅2

13. The sum of the squares of three consecutive integers is 110, then the smallest positive integer is

(a) 6 (b) 5 (c) 7 (d) 4
1
14. If x = √3 is a root of the equation px 2 + (√3 − √2)x − 1 = 0, then the value of p2 + 1 is
(a) √6 (b) 6 (c) 7 (d) 8

15. The values of k for which the roots of the quadratic equation x 2 + 4x + k = 0 are real is
(a) k ≥ 4 (b) k ≤ 4 (c) k ≥ − 4 (d) k ≤ − 4

16. If Sn , the sum of first n terms of an AP is given by Sn = 3n2 − 4n, then the common difference is
(a) 3 (b) - 4 (c) -1 (d) 6

17. If the 3rd and 9th term of an AP are 4 and - 8 respectively, then which term of this AP is zero

(a) 8th term (b) 5th term (c) 10th term (d) 12th term

18. The sum of integers between 100 and 200 which are not divisible by 9 is
(a) 1683 (b) 14850 (c) 13167 (d) None of these

ASSERTION-REASON BASED QUESTIONS
In the following questions, a statement of assertion (A) is followed by a statement of Reason (R).

Choose the correct answer out of the following choices.

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

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

2

4xy
19. Assertion (A): The equation sec 2 θ = (x+y)2 is only possible, when x = y.
Reason(R): sec 2 θ ≥ 1 and therefore (x + y)2 ≤ 0.

20. Assertion (A): The coordinates of the points which divide the line segment joining A(4, -1) and
B (-2, -3) into three equal parts are (2, -5/3) and (0, -7/3).
Reason(R): The points which divide AB in the ratio 1 : 3 and 3 : 1 are called points of trisection of
AB.

SECTION B

21. Prove that √8 is an irrational number.
OR,
The length, breadth and height of a room are 8 m 25 cm, 6 m 75 cm and 4 m 50 cm respectively. Find
the length of the longest rod that can measure the three dimensions of the room exactly.

22. Prove that the equation x2(a2 + b2) + 2x(ac + bd) + (c2 + d2) = 0 has no real root, if ad ≠ bc.

OR,
If p, q, r are real and p ≠ q, then show that the roots of the equation (p – q)x2 + 5(p + q)x – 2(p – q) =
0 are real and unequal.

23. In triangle ABC, DE is parallel to BC. Find EC if AD = 1.5 cm, DB = 3 cm and AE = 1 cm.
1
24. Find the zeros of the polynomial 𝑥 2 + 6 𝑥 − 2, and verify the relation between the coefficients and
the zeros of the polynomial.

25. The 17th term of an AP exceeds its 10 term by 7. Find the common difference.

SECTION C

26. In a seminar, the number of participants in Hindi, English and Mathematics are 60, 84 and 108
respectively. Find the minimum number of rooms required if in each room the same number of
participants are to be seated and all of them being in the same subject.

27. If 𝛼 and 𝛽 are the roots of the quadratic polynomial 𝑓 (𝑥) = 𝑘𝑥 2 + 4𝑥 + 4 such that 𝛼 2 + 𝛽2 = 24, find
the values of k.
OR,
If 𝛼 and 𝛽 are the zeroes of the quadratic polynomial 𝑝(𝑥) = 𝑎𝑥 2 + 𝑏𝑥 + 𝑐, then evaluate

𝛼2 𝛽2
(i) 𝛼 4 + 𝛽4 (ii) 𝛽2 + 𝛼2 .

28. A die is thrown twice. Find the probability of getting the sum of numbers
(i) an even prime number
(ii) a number lying between 4 and 8
(iii) a perfect square.
OR,
A box contains 12 balls out of which ‘x’ are black. If one ball is drawn at random from the box, what

is the probability that it will be a black colour? If 6 more black balls are put in the box, then the
probability of drawing a black ball is now double of what it was before. Find the value of ‘x’

3

29. In a circle of radius 21 cm, an arc subtends an angle of 600 at the centre. Find
(i) the length of the arc.
(ii) the area of the sector formed by the arc.
(iii) the area of the segment formed by the corresponding chord.

30. In ∆PQR, right angled at Q, PR + QR = 25 cm and PQ = 5 cm. Determine the value of sinP and tanP.

31. If S1 = 3, 7, 11, 15,... upto 125 terms and S2 = 4, 7, 10, 13, 16,… upto 125 terms, then how many terms
are there in S1 that are in S2?

SECTION D

32. The sum of the digits of a two digit number is 9. Also, nine times this number is twice the number
obtained by reversing the order of the digits. Find the number.
OR,
A person is walking with uniform speed and when he has completed half his journey he increased
his speed 20% and arrives at his destination. Last half of his journey he completed 30 minutes
earlier than first half of journey. How long was he walking the first half?

33. In the given figure, altitudes AD and CE of ∆ABC intersect each other
at the point P. Show that

(i) ∆AEP ~∆CDP

(ii) ∆AEP ~ ∆ADB

OR,
State and prove Basic Proportionality theorem.
tanA cotA
34. Prove that: + = 1 + secAcosecA
1−cotA 1−tanA

35. A cottage industry produces a certain number of pottery articles in a day. It was observed on a
particular day that the cost of production of each article (in Rs.) was 3 more than twice the number
of articles produced on the day. If the total cost of the production on that day was Rs 90, then find
the number of articles produced and the cost of each article.

SECTION E

36. In a Diwali occasion, a colorful Rangoli is formed by using different colours, diya, candles and light
etc. in a square PQRS of side 20 cm.

4

(i) Find the area of the sector OCD, where OAB is an equilateral triangle.
(ii) Find the area of the equilateral triangle OAB outside the circle.
(iii) Find the area of the sector OCEDO.
OR,
Find the area of the remaining part of the square PQRS when areas of circle and equilateral triangle
is excluded.

37. There are three friends and they want to play some interesting game. Firstly, they consider some
cards and marked with the numbers 2 to 101 are placed in a box
and mix thoroughly. One card is drawn from the box.

(i) Find the probability that the drawn on the card is a number
which is a perfect square.

(ii) Find the probability that the drawn on the card is a prime
number less than 50.

(iii) Find the probability that the drawn on the card is a number
which is either divisible by 2 or 3.
OR,
Find the probability that the drawn on the card is a number which is not divisible by 3 and 5.

38. In order to conduct a Sports Day activities is in your school, lines have been drawn with chalk
powder at a distance of 1 m each, in a rectangular shaped ground ABCD, 100 flowerpots have been
placed at distance of 1 m from each other along AD, as shown in the figure below. Niharika runs
1/4th the distance AD on the second line and posts a green flag.
Preet runs 1/5th the distance AD on the 8th line and posts a red
flag.

(i) Find the position (coordinates) of the green flag and red flag.

(ii) Find the distance between the green and the red flags.

(iii) If Joy has to post a flag at 1/4 distance from green flag, in the
line segment joining the green and red flags, then where should
he post his flag?

OR,
If Rashmi has to post a blue flag exactly halfway between the line segment joining the two flags,
where should she post her flag? Also find the distance of her flag from the origin (A).

5

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