Question Paper

CBSE Class 10 Maths Sample Paper 2027 – Set 4

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 smallest number which, when divided by 18, 24 and 30, leaves a remainder of 4 in each case, is

(a) 364
(b) 356
(c) 360
(d) 724

2. If one zero of the polynomial $3x^2 + 8x + k$ is the reciprocal of the other, then the value of $k$ is

(a) $8$
(b) $-3$
(c) $3$
(d) $\frac{1}{3}$

3. Which of the following points lies on both the lines $2x + y = 7$ and $x - y = 2$?

(a) $(1, 5)$
(b) $(4, 2)$
(c) $(5, 3)$
(d) $(3, 1)$

4. The sum of a positive number and its reciprocal is $\frac{10}{3}$. The quadratic equation that represents this situation is

(a) $3x^2 + 10x + 3 = 0$
(b) $3x^2 - 10x + 3 = 0$
(c) $10x^2 - 3x + 10 = 0$
(d) $3x^2 - 10x - 3 = 0$

5. The sum of the first 20 odd natural numbers is

(a) 400
(b) 420
(c) 441
(d) 380

6. The perimeter of the triangle with vertices $O(0, 0)$, $A(8, 0)$ and $B(0, 15)$ is

(a) 23 units
(b) 60 units
(c) 34 units
(d) 40 units

7. In the figure, $\triangle ABC$ is right-angled at $A$ and $AD \perp BC$. If $BD = 4$ cm and $DC = 9$ cm, then $AD$ is

(a) 6.5 cm
(b) 6 cm
(c) 13 cm
(d) 5 cm

8. In the figure, $XY$ and $X'Y'$ are two parallel tangents to a circle with centre $O$. Another tangent $AB$, touching the circle at $C$, meets $XY$ at $A$ and $X'Y'$ at $B$. The measure of $\angle AOB$ is

(a) $60^\circ$
(b) $80^\circ$
(c) $90^\circ$
(d) $120^\circ$

9. If $\cos A = \frac{7}{25}$, where $A$ is an acute angle, then the value of $\tan A + \cot A$ is

(a) $\frac{625}{168}$
(b) $1$
(c) $\frac{168}{625}$
(d) $\frac{527}{168}$

10. The value of $\sin^2 60^\circ - \cos^2 45^\circ + \tan^2 30^\circ$ is

(a) $\frac{1}{12}$
(b) $\frac{5}{12}$
(c) $\frac{11}{12}$
(d) $\frac{7}{12}$

11. If $\sin\theta - \cos\theta = 0$, then the value of $\sin^4\theta + \cos^4\theta$ is

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

12. From a point on level ground 12 m away from the foot of a neem tree, the angle of elevation of the top of the tree is $60^\circ$. The height of the tree is

(a) $4\sqrt{3}$ m
(b) 24 m
(c) $12\sqrt{3}$ m
(d) 6 m

13. The difference between the circumference and the diameter of a circle is 30 cm. The radius of the circle is (use $\pi = \frac{22}{7}$)

(a) 7 cm
(b) 14 cm
(c) 3.5 cm
(d) 10.5 cm

14. A sector of a circle of radius 12 cm has an area of $24\pi \text{ cm}^2$. The length of the arc of the sector is

(a) $2\pi$ cm
(b) $8\pi$ cm
(c) $12\pi$ cm
(d) $4\pi$ cm

15. A solid sphere has total surface area $S$. It is cut into two equal hemispheres. The total surface area of the two hemispheres together is

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

16. The curved surface area of a right circular cone of base radius 7 cm and slant height 10 cm is (use $\pi = \frac{22}{7}$)

(a) $220 \text{ cm}^2$
(b) $374 \text{ cm}^2$
(c) $154 \text{ cm}^2$
(d) $440 \text{ cm}^2$

17. A letter is chosen at random from the letters of the word MATHEMATICS. The probability that it is a vowel is

(a) $\frac{7}{11}$
(b) $\frac{4}{7}$
(c) $\frac{4}{11}$
(d) $\frac{3}{11}$

18. Two coins are tossed together. The probability of getting at most one head is

(a) $\frac{1}{4}$
(b) $\frac{1}{2}$
(c) $1$
(d) $\frac{3}{4}$

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 class mark of the class interval 10-25 is 17.5.

Reason (R): The class mark of a class interval is half of its class size.

(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): The sum of the zeroes of the polynomial $2x^2 - 8x + 5$ is $4$.

Reason (R): The product of the zeroes of the polynomial $ax^2 + bx + c$ $(a \neq 0)$ is $\frac{c}{a}$.

(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 $\sqrt{2}$ is irrational, prove that $7 - 3\sqrt{2}$ is irrational. (2 marks)

22. A circle has its centre at $C(2, -3)$ and passes through the point $P(-4, 5)$. Find the radius of the circle. Does the point $Q(10, 3)$ lie on this circle? Give a reason. (2 marks)

23. Two vertical poles $AB$ and $CD$, of heights 6 m and 9 m, stand on level ground at $B$ and $D$. Wires are stretched from $A$ to $D$ and from $C$ to $B$, and they cross at $P$, as shown in the figure. Find the height $PM$ of the crossing point above the ground. (2 marks)

OR

In $\triangle ABC$ and $\triangle PQR$, $\angle A = \angle P$ and $\angle B = \angle Q$. If $AB = 6$ cm, $BC = (2x + 1)$ cm, $PQ = 4$ cm and $QR = (x + 2)$ cm, find $x$ and the lengths of $BC$ and $QR$.

24. In the figure, a circle touches the side $BC$ of $\triangle ABC$ at $P$, and touches $AB$ produced and $AC$ produced at $Q$ and $R$ respectively. If $AB = 7$ cm, $BC = 6$ cm and $CA = 5$ cm, find the length of $AQ$. (2 marks)

25. A solid model of a rocket is made of a right circular cylinder of radius 5 cm and height 20 cm, with a right circular cone of the same radius and height 12 cm fixed on its top, as shown in the figure. Find the total surface area of the model in terms of $\pi$. (2 marks)

OR

A measuring vessel is a hollow cylinder of internal radius 3 cm and height 10 cm, closed at the bottom by a hemispherical bowl of the same radius. How much liquid can the vessel hold? Give your answer in terms of $\pi$.

Section C (18 marks)

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

26. A school librarian has 336 Hindi books, 240 English books and 216 Science books. She wants to arrange them in stacks so that every stack has the same number of books and each stack has books of only one subject. What is the greatest number of books she can put in each stack? How many stacks will she make in all? (3 marks)

27. If $\alpha$ and $\beta$ are the zeroes of the polynomial $x^2 - 3x - 10$, find a quadratic polynomial whose zeroes are $\alpha^2$ and $\beta^2$. (3 marks)

28. Meera was asked to find the sum of all the multiples of 6 between 100 and 200. Her working is shown below.

First term $a = 102$, last term $l = 198$, common difference $d = 6$.

Number of terms $n = \frac{198 - 102}{6} = 16$.

Sum $= \frac{16}{2}(102 + 198) = 2400$.

Is her answer correct? If not, point out the mistake and find the correct sum. (3 marks)

OR

The 5th term of an AP is 17 and the sum of its first 10 terms is 185. Find the AP.

29. The point $P(x, 6)$ lies on the line segment joining $A(-6, 9)$ and $B(10, -3)$. Find the ratio in which $P$ divides $AB$ and the value of $x$. Verify the ratio by finding $AP$ and $PB$ with the distance formula. (3 marks)

30. Prove that $\frac{\cos^2 A}{1 - \sin A} + \frac{\sin^2 A}{1 + \cos A} = 2 + \sin A - \cos A$. (3 marks)

OR

Prove that $\frac{\tan^2 A}{\sec A - 1} + \frac{\cot^2 A}{\text{cosec}\,A + 1} = \sec A + \text{cosec}\,A$.

31. A card is drawn at random from a well-shuffled deck of 52 playing cards. Find the probability that the card drawn is (i) a red king, (ii) neither a heart nor a king, (iii) a black card that is not a face card. (3 marks)

Section D (20 marks)

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

32. The roll numbers of three students sitting next to each other in a row are three consecutive natural numbers. The product of the smallest and the largest roll number, increased by 5 times the middle roll number, is 233. Find the three roll numbers. (5 marks)

OR

A mother is 26 years older than her daughter. Five years from now, the product of their ages (in years) will be 615. Find their present ages.

33. State and prove the Basic Proportionality Theorem (take $\triangle PQR$ with $ST \parallel QR$). Using it, solve the following: in the figure, $ABCD$ is a trapezium with $AB \parallel DC$, and $E$ and $F$ are points on $AD$ and $BC$ such that $EF \parallel AB$. If $AE = (x - 1)$ cm, $ED = (x + 2)$ cm, $BF = x$ cm and $FC = 2x$ cm, find $x$. (5 marks)

34. An aeroplane is flying horizontally at a constant height of 2500 m above the ground, moving away from an observer standing on the ground. At one instant the angle of elevation of the aeroplane from the observer is $45^\circ$, and 10 seconds later it is $30^\circ$. Find the speed of the aeroplane in km/h. Also find the distance of the aeroplane from the observer at the second instant. (Use $\sqrt{3} = 1.73$) (5 marks)

35. The weights (in kg) of 60 students of Class X of a school are given below.

Find the median and the mode of the weights. (5 marks)

OR

The table shows the quantity of milk (in litres) supplied in a day by the farmers of a village to a dairy cooperative. The median quantity is 24 litres. Find the missing frequency $p$, and then find the mode of the data.

Section E (12 marks)

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

36. A city zoo sells entry tickets at one price for an adult and a lower price for a child. The Sharma family bought tickets for 2 adults and 3 children and paid ₹ 330. The Iyer family bought tickets for 3 adults and 4 children and paid ₹ 470. Let the price of an adult ticket be ₹ $x$ and that of a child ticket be ₹ $y$.

(i) Write the pair of linear equations that describes this situation. (1 mark)

(ii) Will the lines representing these equations intersect, be parallel or coincide? Give a reason. (1 mark)

(iii) Find the price of an adult ticket and the price of a child ticket. (2 marks)

OR

(iii) Inside the zoo, a toy-train ride costs ₹ $a$ for an adult and ₹ $b$ for a child. The fare for 4 adults and 2 children is ₹ 200, and the fare for 2 adults and 5 children is ₹ 220. Find $a$ and $b$. (2 marks)

37. A park has a circular fountain with centre $O$ and radius 7 m. A gate $P$ of the park is 25 m from $O$. Two straight paths $PA$ and $PB$ go from the gate and just touch the edge of the fountain at $A$ and $B$, as shown in the figure.

(i) Find the length of the path $PA$. (1 mark)

(ii) Find the area of the quadrilateral $OAPB$. (1 mark)

(iii) A straight water pipe is laid from $A$ to $B$. It meets $OP$ at $M$. Find the length of the pipe $AB$. (2 marks)

OR

(iii) Another gate $G$ is placed so that the two straight paths from $G$ that just touch the fountain make an angle of $60^\circ$ with each other. How far is $G$ from the centre $O$, and how long is each of these paths? (Use $\sqrt{3} = 1.73$) (2 marks)

38. A rotating sprinkler is fixed at a point $O$ on the lawn of a park. It throws water up to a distance of 21 m and turns back and forth through an angle of $120^\circ$, so that it waters the sector $OAB$ of the lawn shown in the figure. (Use $\pi = \frac{22}{7}$)

(i) Find the length of the arc $AB$. (1 mark)

(ii) Find the area of the lawn watered by the sprinkler. (1 mark)

(iii) To save water, the gardener changes the setting so that the sprinkler turns through only $90^\circ$, with the same range. By how much does the watered area decrease? In the new setting, find the area of the watered part that lies between the arc and the chord joining its two ends. (2 marks)

OR

(iii) A second sprinkler on another part of the lawn has a range of 28 m and turns through $90^\circ$. Find the area watered by it and the total length of the boundary of the region it waters. (2 marks)