Question Paper

CBSE Class 10 Maths Sample Paper 2027 – Set 3

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. If $a = 2^3 \times 3^2 \times 5$ and $b = 2^2 \times 3^3 \times 7$, then $\frac{\text{LCM}(a, b)}{\text{HCF}(a, b)}$ is equal to

(a) $36$
(b) $105$
(c) $210$
(d) $7560$

2. If $x = 2$ is a zero of the polynomial $2x^2 - 5x + k$, then the value of $k$ is

(a) $-2$
(b) $2$
(c) $18$
(d) $-18$

3. In a quiz, a team gets 4 marks for every correct answer and loses 1 mark for every wrong answer. A team answered 20 questions and scored 55 marks. If $x$ of its answers were correct and $y$ were wrong, the pair of equations that describes this is

(a) $x + y = 20$, $4x + y = 55$
(b) $x - y = 20$, $4x - y = 55$
(c) $x + y = 55$, $4x - y = 20$
(d) $x + y = 20$, $4x - y = 55$

4. The roots of the quadratic equation $2x^2 - 7x + 6 = 0$ are

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

5. The 8th term from the end of the AP $7, 11, 15, \ldots, 147$ is

(a) 119
(b) 115
(c) 123
(d) 35

6. In $\triangle ABC$, $D$ and $E$ are points on the sides $AB$ and $AC$ respectively. For which of the following sets of lengths is $DE \parallel BC$?

(a) $AD = 2$ cm, $DB = 3$ cm, $AE = 3$ cm, $EC = 4$ cm
(b) $AD = 3$ cm, $DB = 4$ cm, $AE = 4.5$ cm, $EC = 6$ cm
(c) $AD = 4$ cm, $DB = 5$ cm, $AE = 5$ cm, $EC = 6$ cm
(d) $AD = 1.5$ cm, $DB = 2$ cm, $AE = 2$ cm, $EC = 3$ cm

7. In the figure, $\triangle ABC \sim \triangle PQR$, with $AB = 5$ cm, $BC = 7$ cm and $CA = 8$ cm. If the perimeter of $\triangle PQR$ is 30 cm, then the length of $PR$ is

(a) 12 cm
(b) 7.5 cm
(c) 10.5 cm
(d) 18 cm

8. If the point $(3, k)$ is the mid-point of the line segment joining $A(-1, 4)$ and $B(p, 6)$, then

(a) $p = 5, k = 7$
(b) $p = 7, k = 10$
(c) $p = 4, k = 5$
(d) $p = 7, k = 5$

9. If the point $P(k, 0)$ divides the line segment joining $A(2, -2)$ and $B(-7, 4)$ in the ratio $1 : 2$, then the value of $k$ is

(a) $1$
(b) $-\frac{5}{2}$
(c) $-1$
(d) $-4$

10. Three vertices of a parallelogram $ABCD$, taken in order, are $A(1, 2)$, $B(4, 3)$ and $C(6, 6)$. The fourth vertex $D$ is

(a) $(9, 7)$
(b) $(3, 5)$
(c) $(-1, -1)$
(d) $(5, 3)$

11. The value of $(\sin 45^\circ + \cos 45^\circ)^2$ is

(a) $0$
(b) $1$
(c) $\frac{1}{2}$
(d) $2$

12. A guard in a watch-tower sees the entry gate of a factory at an angle of depression of $60^\circ$. If his eyes are 18 m above the level ground, the distance of the gate from the foot of the watch-tower is

(a) $18\sqrt{3}$ m
(b) $6\sqrt{3}$ m
(c) 36 m
(d) $12\sqrt{3}$ m

13. In the figure, $PQ$ is the tangent at the point $P$ to a circle with centre $O$, and $OQ$ cuts the circle at $R$. If $PQ = 8$ cm and $QR = 4$ cm, then the radius of the circle is

(a) 4 cm
(b) $4\sqrt{3}$ cm
(c) 6 cm
(d) 10 cm

14. $PA$ and $PB$ are tangents from an external point $P$ to a circle with centre $O$. If $\angle OAB = 20^\circ$, then $\angle APB$ is

(a) $40^\circ$
(b) $20^\circ$
(c) $70^\circ$
(d) $140^\circ$

15. A pendulum 42 cm long swings through an angle of $30^\circ$. The length of the arc described by its bob is (use $\pi = \frac{22}{7}$)

(a) 11 cm
(b) 44 cm
(c) 264 cm
(d) 22 cm

16. Which of the following statements about a grouped frequency distribution is correct?

(a) The modal class is the class with the highest cumulative frequency.
(b) In the formula for the mode, $f_0$ is the frequency of the class just before the modal class.
(c) The class mark of the class 20-30 is 30.
(d) The median class is the class with the highest frequency.

17. While finding the mean of a grouped frequency distribution by the step-deviation method, a student takes the assumed mean $a = 55$ and the class size $h = 10$, and gets $\sum f_i = 40$ and $\sum f_i u_i = -12$. The mean of the distribution is

(a) $54.7$
(b) $58$
(c) $52$
(d) $43$

18. A jar contains 12 marbles, some of which are red. The probability of drawing a red marble at random is $\frac{1}{3}$. If 6 more red marbles are put into the jar, the probability of drawing a red marble becomes

(a) $\frac{5}{9}$
(b) $\frac{5}{6}$
(c) $\frac{1}{2}$
(d) $\frac{2}{3}$

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): A cone and a cylinder have the same base radius 7 cm and the same height 12 cm. The volume of the cylinder is $1848 \text{ cm}^3$, so the volume of the cone is $616 \text{ cm}^3$.

Reason (R): The volume of a cone is one-third of the volume of a cylinder with the same base radius and the same height.

(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 two coins are tossed together, the probability of getting exactly one head is $\frac{1}{3}$.

Reason (R): When two coins are tossed together, the four outcomes HH, HT, TH and TT are equally likely.

(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. The floor of a school's computer lab is a rectangle 5.4 m long and 4.2 m wide. It is to be covered with identical square tiles without cutting any tile. Find the side of the largest square tile that can be used and the number of such tiles needed. (2 marks)

22. One zero of the quadratic polynomial $x^2 - 9x + k$ is twice the other. Find the zeroes and the value of $k$. (2 marks)

OR

Find a quadratic polynomial the sum and product of whose zeroes are $-\frac{1}{4}$ and $-\frac{3}{4}$ respectively. Also find its zeroes.

23. In the figure, $D$ and $E$ are points on the sides $AB$ and $AC$ of $\triangle ABC$ such that $DE \parallel BC$, and $F$ is a point on $AD$ such that $EF \parallel CD$. If $AF = 4$ cm and $FD = 6$ cm, find $DB$. (2 marks)

24. If $\sec A = \frac{29}{21}$, where $A$ is an acute angle, find the value of $\frac{1 + \sin A}{\cos A}$. (2 marks)

25. In the figure, a circle with centre $O$ is inscribed in $\triangle ABC$, which is right-angled at $B$. The circle touches $AB$, $BC$ and $CA$ at $P$, $Q$ and $R$ respectively. If $AB = 8$ cm and $BC = 15$ cm, find the radius of the circle. (2 marks)

OR

From an external point $P$, two tangents $PA$ and $PB$ are drawn to a circle with centre $O$. If $PA = 12$ cm and $OP = 15$ cm, find the perimeter of the quadrilateral $OAPB$.

Section C (18 marks)

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

26. Prove that $\sqrt{5}$ is an irrational number. (3 marks)

27. Find the coordinates of the points of trisection of the line segment joining $A(-2, -3)$ and $B(7, 9)$. Also verify that the distance from $A$ of the point of trisection nearer to $A$ is one-third of $AB$. (3 marks)

28. Prove that $\sec^4 A - \sec^2 A = \tan^4 A + \tan^2 A$. (3 marks)

OR

Prove that $(\sec A - \cos A)(\cot A + \tan A) = \tan A \sec A$.

29. Prove that the lengths of the tangents drawn from an external point to a circle are equal. Using this result, find the perimeter of $\triangle PXY$ in the figure, where $PA$ and $PB$ are tangents from $P$ to a circle with centre $O$, and the tangent at the point $C$ of the circle meets $PA$ at $X$ and $PB$ at $Y$. It is given that $PA = 10$ cm. (3 marks)

30. In the figure, $PQ$ is a chord of a circle with centre $O$ and radius 6 cm, and $\angle POQ = 120^\circ$. Find the area of the shaded minor segment. (Use $\pi = 3.14$ and $\sqrt{3} = 1.73$) (3 marks)

OR

An arc of a circle of radius 14 cm is 22 cm long. Find (i) the angle subtended by the arc at the centre, (ii) the area of the minor sector formed by the arc, (iii) the area of the corresponding major sector. (Use $\pi = \frac{22}{7}$)

31. Two dice, one red and one blue, are thrown together. Find the probability that (i) the sum of the numbers on the two dice is 9, (ii) the product of the numbers on the two dice is a perfect square, (iii) the number on the red die is less than the number on the blue die. (3 marks)

Section D (20 marks)

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

32. The length of a rectangular garden in a housing society is 16 m more than its breadth, and its area is 1700 $\text{m}^2$. A gravel path of uniform width is laid inside the garden along all four sides, leaving a rectangular lawn of area 1092 $\text{m}^2$ in the middle. Find the length and breadth of the garden and the width of the path. (5 marks)

33. Solve the following pair of linear equations graphically: $2x + y = 8$ and $x - y + 2 = 0$. Also find the coordinates of the vertices of the triangle formed by these lines and the y-axis, and find its area. (5 marks)

OR

At a school fete, a food stall sold 45 veg rolls and 60 glasses of lemonade on the first day and collected ₹ 2250. On the second day it sold 60 veg rolls and 40 glasses of lemonade and collected ₹ 2400. Find the price of one veg roll and of one glass of lemonade. How much would a customer pay for 4 veg rolls and 3 glasses of lemonade?

34. A devotee whose eyes are 1.5 m above the ground sees the top of a temple gopuram at an angle of elevation of $30^\circ$. After walking 50 m straight towards the gopuram on level ground, he finds that the angle of elevation of its top is $60^\circ$. Find the height of the gopuram and his distance from the gopuram at the second position. (Use $\sqrt{3} = 1.73$) (5 marks)

35. A wooden top (lattu) is in the shape of a cone mounted on a hemisphere of the same radius. The diameter of the hemisphere is 4.2 cm and the total height of the top is 4.1 cm. Find the total surface area of the top and the volume of wood in it. (Use $\pi = \frac{22}{7}$) (5 marks)

OR

A memento is made of a solid wooden cube of edge 10 cm with a solid hemisphere of radius 4.2 cm fixed on the middle of its top face. The memento stands on its bottom face, which is not polished; every other outer surface, including the curved surface of the hemisphere, is to be polished. Find the area to be polished and the volume of the memento. (Use $\pi = \frac{22}{7}$)

Section E (12 marks)

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

36. At a brick kiln, bricks are stacked in rows, one row on top of another. The bottom row has 42 bricks, the row above it has 39, the next one has 36, and so on: every row has 3 bricks fewer than the row just below it. The top row has 6 bricks.

(i) How many bricks are there in the 6th row from the bottom? (1 mark)

(ii) How many rows are there in the stack? (1 mark)

(iii) Find the total number of bricks in the stack. (2 marks)

OR

(iii) Find the total number of bricks in all the rows that have more than 30 bricks each. (2 marks)

37. For his science project, Kabir made a pinhole camera: a closed box with a tiny hole $O$ in the front face and a screen on the back face, 20 cm behind the hole. He points it at a tree $AB$ which is 12 m tall and stands 40 m away from the hole. Light from the top $A$ of the tree passes through $O$ and reaches the screen at $A'$, so an inverted image $A'B'$ of the tree is formed. Take the foot $B$ of the tree, the hole $O$ and the point $B'$ to lie on one horizontal line, with the tree and the screen both vertical.

(i) By which criterion is $\triangle OAB \sim \triangle OA'B'$? (1 mark)

(ii) Find the height of the image $A'B'$. (1 mark)

(iii) Kabir moves the camera 10 m closer to the tree. Find the new height of the image. (2 marks)

OR

(iii) Kabir's friend, who is 1.5 m tall, stands in front of the camera, and the image of the friend on the screen is 5 cm tall. How far from the hole is the friend standing? (2 marks)

38. An electricity company surveyed 60 households in a colony. The table shows the number of units of electricity they used in a month.

(i) How many households used 200 units or more in the month? (1 mark)

(ii) Write the modal class and the median class. (1 mark)

(iii) Find the median of the data. (2 marks)

OR

(iii) Find the mode of the data. (2 marks)