Question Paper

CBSE Class 10 Maths Sample Paper 2027 – Set 1

Time allowed: 3 hours · Maximum marks: 80

General Instructions

1. This question paper contains 38 questions. All questions are compulsory.
2. The question paper is divided into five sections: A, B, C, D and E.
3. In Section A, questions 1 to 18 are multiple choice questions (MCQs) and questions 19 and 20 are Assertion-Reason based questions of 1 mark each.
4. In Section B, questions 21 to 25 are very short answer (VSA) type questions of 2 marks each.
5. In Section C, questions 26 to 31 are short answer (SA) type questions of 3 marks each.
6. In Section D, questions 32 to 35 are long answer (LA) type questions of 5 marks each.
7. In Section E, questions 36 to 38 are case study based questions of 4 marks each, with sub-parts of 1, 1 and 2 marks.
8. There is no overall choice. However, an internal choice has been provided in 2 questions each of Sections B, C and D, and in the 2-mark sub-part of every question in Section E.
9. Draw neat figures wherever required. Take pi = 22/7 wherever required, if not stated.
10. Use of calculators is not allowed.

How to use this paper: set a 3-hour timer and write all your answers on paper, just like the board exam. When you finish, take the quiz at the end of this page to score your objective (MCQ) answers instantly, then open the Detailed Solutions and check the rest step by step, with the marks each step carries.

Section A (20 marks)

This section has 18 multiple choice questions and 2 Assertion-Reason based questions of 1 mark each.

1. The HCF of two numbers is 18 and their product is 3240. Their LCM is

(a) 180
(b) 162
(c) 216
(d) 90

2. If $\alpha$ and $\beta$ are the zeroes of the polynomial $p(x) = x^2 - 5x + k$ and $\alpha - \beta = 1$, then the value of $k$ is

(a) 4
(b) 6
(c) 8
(d) -6

3. The graph of $y = p(x)$ is shown in the figure. The number of zeroes of $p(x)$ is

(a) 0
(b) 1
(c) 2
(d) 3

4. The pair of equations $2x + 3y = 7$ and $6x + ky = 21$ has infinitely many solutions when $k$ equals

(a) -9
(b) 3
(c) 6
(d) 9

5. The values of $k$ for which the quadratic equation $2x^2 + kx + 8 = 0$ has two equal real roots are

(a) $8$ only
(b) $\pm 4$
(c) $\pm 8$
(d) $16$ only

6. The 15th term of the AP $7, 3, -1, -5, \ldots$ is

(a) $-49$
(b) $-53$
(c) $-45$
(d) $63$

7. In the figure, $DE \parallel BC$ in $\triangle ABC$. If $AD = 2$ cm, $DB = 3$ cm and $AE = 3.2$ cm, then $EC$ is

(a) 2.4 cm
(b) 3.6 cm
(c) 4.8 cm
(d) 5.2 cm

8. The point on the x-axis which is equidistant from $A(2, 3)$ and $B(6, -1)$ is

(a) $(2, 0)$
(b) $(3, 0)$
(c) $(4, 0)$
(d) $(0, -3)$

9. In the figure, $PA$ and $PB$ are tangents to the circle with centre $O$. If $\angle APB = 50^\circ$, then $\angle AOB$ is

(a) $40^\circ$
(b) $50^\circ$
(c) $100^\circ$
(d) $130^\circ$

10. A tangent $PQ$ at a point $P$ of a circle of radius 5 cm meets a line through the centre $O$ at $Q$ such that $OQ = 13$ cm. The length of $PQ$ is

(a) 12 cm
(b) 8 cm
(c) $\sqrt{119}$ cm
(d) 18 cm

11. If $5 \tan A = 4$, then the value of $\frac{5 \sin A - 3 \cos A}{5 \sin A + 2 \cos A}$ is

(a) $\frac{1}{6}$
(b) $\frac{1}{3}$
(c) $\frac{7}{6}$
(d) $0$

12. If $\sec\theta + \tan\theta = 3$, then the value of $\sec\theta - \tan\theta$ is

(a) $3$
(b) $\frac{1}{3}$
(c) $-\frac{1}{3}$
(d) $2$

13. A vertical pole 6 m high casts a shadow $2\sqrt{3}$ m long on the level ground. The angle of elevation of the Sun is

(a) $30^\circ$
(b) $45^\circ$
(c) $60^\circ$
(d) $90^\circ$

14. The area of a sector of a circle of radius 14 cm with central angle $45^\circ$ is (use $\pi = \frac{22}{7}$)

(a) $11 \text{ cm}^2$
(b) $38.5 \text{ cm}^2$
(c) $44 \text{ cm}^2$
(d) $77 \text{ cm}^2$

15. The minute hand of a clock is 21 cm long. The distance moved by its tip in 20 minutes is (use $\pi = \frac{22}{7}$)

(a) 22 cm
(b) 44 cm
(c) 66 cm
(d) 462 cm

16. The curved surface area of a solid hemisphere is $77 \text{ cm}^2$. Its total surface area is

(a) $38.5 \text{ cm}^2$
(b) $77 \text{ cm}^2$
(c) $115.5 \text{ cm}^2$
(d) $154 \text{ cm}^2$

17. For a distribution, the mean is 24 and the median is 26. Using the empirical relationship, the mode is

(a) 22
(b) 25
(c) 28
(d) 30

18. A card is drawn at random from a well-shuffled deck of 52 playing cards. The probability that it is a red face card is

(a) $\frac{1}{26}$
(b) $\frac{3}{26}$
(c) $\frac{3}{13}$
(d) $\frac{6}{13}$

Directions: In questions 19 and 20, a statement of Assertion (A) is followed by a statement of Reason (R). Choose the correct option.

19. Assertion (A): The quadratic polynomial whose zeroes are $3$ and $-5$ is $x^2 + 2x - 15$.

Reason (R): A quadratic polynomial whose zeroes are $\alpha$ and $\beta$ is $x^2 - (\alpha + \beta)x + \alpha\beta$.

(a) Both Assertion (A) and Reason (R) are true and Reason (R) is the correct explanation of Assertion (A).
(b) Both Assertion (A) and Reason (R) are true, but Reason (R) is not the correct explanation of Assertion (A).
(c) Assertion (A) is true, but Reason (R) is false.
(d) Assertion (A) is false, but Reason (R) is true.

20. Assertion (A): When a die is thrown once, the probability of getting a prime number is $\frac{1}{2}$.

Reason (R): The faces of a die carry 3 prime numbers and 3 composite numbers.

(a) Both Assertion (A) and Reason (R) are true and Reason (R) is the correct explanation of Assertion (A).
(b) Both Assertion (A) and Reason (R) are true, but Reason (R) is not the correct explanation of Assertion (A).
(c) Assertion (A) is true, but Reason (R) is false.
(d) Assertion (A) is false, but Reason (R) is true.

Section B (10 marks)

This section has 5 very short answer (VSA) type questions of 2 marks each.

21. Given that $\sqrt{3}$ is irrational, prove that $5 - 2\sqrt{3}$ is irrational. (2 marks)

22. In $\triangle PQR$, $S$ and $T$ are points on $PQ$ and $PR$ respectively such that $ST \parallel QR$. If $PS = x$ cm, $SQ = (x - 2)$ cm, $PT = (x + 2)$ cm and $TR = (x - 1)$ cm, find the value of $x$. (2 marks)

23. Find the values of $y$ for which the distance between the points $P(1, 2)$ and $Q(4, y)$ is 5 units. (2 marks)

24. Evaluate: $(\sin^2 30^\circ + 4\cot^2 45^\circ - \sec^2 60^\circ)(\text{cosec}^2 45^\circ \sec^2 30^\circ)$ (2 marks)

OR

If $\sin(A - B) = \frac{1}{2}$ and $\cos(A + B) = 0$, where $0^\circ < A + B \leq 90^\circ$ and $A > B$, find $A$ and $B$.

25. From an external point $P$, two tangents $PA$ and $PB$ are drawn to a circle with centre $O$. If $\angle PAB = 50^\circ$, find $\angle AOB$. (2 marks)

OR

Two concentric circles have radii 13 cm and 5 cm. Find the length of the chord of the larger circle which touches the smaller circle.

Section C (18 marks)

This section has 6 short answer (SA) type questions of 3 marks each.

26. Students of Class X are planting saplings along three straight paths of lengths 120 m, 168 m and 240 m. On every path the saplings are planted at equal distances, with one sapling at each end of the path, and the distance between two consecutive saplings is the same on all three paths. What is the greatest possible distance between two consecutive saplings? How many saplings are needed in all? (3 marks)

27. Find the ratio in which the x-axis divides the line segment joining the points $A(3, -4)$ and $B(-2, 6)$. Also find the coordinates of the point of division. (3 marks)

28. Prove that $\frac{\cos A}{1 - \tan A} + \frac{\sin A}{1 - \cot A} = \sin A + \cos A$. (3 marks)

OR

Prove that $\frac{\sec A - 1}{\sec A + 1} = \left(\frac{\sin A}{1 + \cos A}\right)^2$.

29. A circle is inscribed in $\triangle ABC$ and touches the sides $AB$, $BC$ and $CA$ at $F$, $D$ and $E$ respectively. If $AB = 13$ cm, $BC = 14$ cm and $CA = 15$ cm, find the lengths of $AF$, $BD$ and $CE$. (3 marks)

30. A chord of a circle of radius 14 cm subtends a right angle at the centre. Find the area of the corresponding minor segment. (Use $\pi = \frac{22}{7}$) (3 marks)

OR

A cow is tied with a 14 m long rope to a peg at one corner of a square field of side 21 m. Find the area of the field over which the cow can graze. Also find the increase in the grazing area if the rope were 21 m long. (Use $\pi = \frac{22}{7}$)

31. A box contains 50 cards numbered 3, 4, 5, …, 52. One card is drawn at random from the box. Find the probability that the number on the card is (i) a perfect square, (ii) a multiple of 7, (iii) divisible by both 2 and 5. (3 marks)

Section D (20 marks)

This section has 4 long answer (LA) type questions of 5 marks each.

32. A parking lot charges a fixed amount per hour for a car and a different fixed amount per hour for a scooter. Rakesh parked his car for 3 hours and his scooter for 5 hours and paid ₹ 190. Nisha parked her car for 4 hours and her scooter for 2 hours and paid ₹ 160. Find the hourly charge for a car and for a scooter. How much will be paid for parking one car and one scooter for 6 hours each? (5 marks)

OR

Solve the following pair of linear equations graphically: $x + y = 5$ and $2x - y = 4$. Also find the coordinates of the vertices of the triangle formed by these lines and the x-axis, and find its area.

33. A shopkeeper buys a certain number of notebooks for ₹ 1200. If he had bought 10 more notebooks for the same amount, each notebook would have cost him ₹ 20 less. Find the number of notebooks he bought and the cost of each notebook. (5 marks)

34. State and prove the Basic Proportionality Theorem. Using it, find $AE$ if in $\triangle ABC$, $D$ and $E$ are points on $AB$ and $AC$ such that $DE \parallel BC$, $\frac{AD}{DB} = \frac{3}{5}$ and $AC = 5.6$ cm. (5 marks)

35. The table shows the time (in minutes) that 50 students of a class spent on a learning app on a particular day.

Find the mean and the mode of the data. (5 marks)

OR

The median of the following distribution of the ages of 100 people who visited a health camp is 32 years. Find the values of $x$ and $y$.

Section E (12 marks)

This section has 3 case study based questions of 4 marks each.

36. The seats in a school auditorium are arranged in 20 rows. The first row has 18 seats and every row has 3 seats more than the row in front of it.

(i) How many seats are there in the 10th row? (1 mark)

(ii) Which row has 63 seats? (1 mark)

(iii) Find the total number of seats in the auditorium. (2 marks)

OR

(iii) How many seats are there in the last 5 rows taken together? (2 marks)

37. A mobile tower stands on the flat roof of a building, at the edge of the roof nearest to $P$. From a point $P$ on the level ground, 30 m away from the foot of the building, the angle of elevation of the top of the building is $30^\circ$ and the angle of elevation of the top of the tower is $60^\circ$. (Use $\sqrt{3} = 1.73$)

(i) Find the height of the building. (1 mark)

(ii) Find the distance of $P$ from the top of the tower. (1 mark)

(iii) Find the height of the tower. (2 marks)

OR

(iii) An observer at $P$ walks towards the building until the angle of elevation of the top of the building becomes $60^\circ$. How far does the observer walk? (2 marks)

38. A tent at a scouts' camp is in the shape of a right circular cylinder surmounted by a right circular cone of the same radius. The radius of the base is 3.5 m, the height of the cylindrical part is 3 m and the height of the conical part is 1.2 m. (Use $\pi = \frac{22}{7}$)

(i) Find the slant height of the conical part. (1 mark)

(ii) Find the curved surface area of the cylindrical part. (1 mark)

(iii) Find the area of canvas used to make the tent (floor not included). Also find its cost at ₹ 100 per $\text{m}^2$. (2 marks)

OR

(iii) Find the volume of air inside the tent. (2 marks)