Class 10 Maths · 3 hours · 80 marks

CBSE Class 10 Maths Sample Paper 1

Set in the board's pattern for 2024 to 2026. Write it in 3 hours, then check with the marking scheme and solutions.

All questions are compulsory. Internal choice is given in some questions; attempt one of the alternatives.
Section A has 20 objective questions of 1 mark each: multiple-choice and assertion-reason.
Section B has very short answer questions of 2 marks; Section C short answer questions of 3 marks; Section D long answer questions of 5 marks; Section E case-based questions of 4 marks.
Wherever necessary, draw neat, labelled diagrams. Calculators are not allowed.
For the assertion-reason questions, choose: (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.

Section A (20 × 1 mark)

Question 1 · 1 mark

The line drawn from the eye of an observer to the point of the object being viewed is called the

(a) horizontal(b) vertical(c) line of sight(d) angle of elevation

Question 2 · 1 mark

Assertion (A): 5x² − 3 = 0 is a quadratic equation. Reason (R): ax² + bx + c = 0 is a quadratic equation whenever a ≠ 0, even if b = 0.

(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.

Question 3 · 1 mark

To draw the line 3x − 2y = 0, a suitable pair of points is

(a) (2, 0) and (0, 3)(b) (0, 0) and (3, 2)(c) (0, 0) and (0, 0)(d) (0, 0) and (2, 3)

Question 4 · 1 mark

Two angles of a triangle are 48° and 67°. A triangle similar to it has angles

(a) 67°, 67°, 46°(b) 67°, 56°, 57°(c) 48°, 66°, 66°(d) 65°, 48°, 67°

Question 5 · 1 mark

In a right triangle one acute angle is 45° and the hypotenuse is 10√2 cm. The length of each of the other two sides is

(a) 5√2 cm(b) 10√2 cm(c) 20 cm(d) 10 cm

Question 6 · 1 mark

Which term of the AP 8, 14, 20, . . . is 176?

(a) 30th(b) 28th(c) 29th(d) 31st

Question 7 · 1 mark

For a grouped distribution, Σfᵢ = 40 and Σfᵢxᵢ = 1260. The mean is

(a) 31.5(b) 32(c) 3.15(d) 30

Question 8 · 1 mark

From a point on level ground 18 m from the foot of a water tower, the angle of elevation of its top is 45°. The height of the water tower is

(a) 18 m(b) 18√3 m(c) 6√3 m(d) 9 m

Question 9 · 1 mark

The probability that the morning bus from Bhopal reaches Indore late is 0.37. The probability that it is not late is

(a) 0.37(b) 0.63(c) 1.37(d) 0.73

Question 10 · 1 mark

If 2^a × 3^b = 432, where a and b are natural numbers, then (a, b) is

(a) (3, 4)(b) (4, 3)(c) (2, 4)(d) (4, 2)

Question 11 · 1 mark

ΔABC and ΔDEF are equiangular, with A, B, C matching D, E, F. AB = 4 cm, BC = 6 cm and DE = 10 cm. Then EF is

(a) 2.4 cm(b) 12 cm(c) 15 cm(d) 4 cm

Question 12 · 1 mark

In the triangle with vertices A(2, 3), B(6, 6) and C(−1, 7), the right angle is at

(a) A(b) B(c) C(d) no vertex: the triangle has no right angle

Question 13 · 1 mark

A circular mirror with centre O and radius 28 cm has a strip cut off along a straight line AB, where ∠AOB = 60°. The area of the strip cut off is. Take π = 22/7.

(a) (12323 + 196√3) cm²(b) (12323 − 196√3) cm²(c) (24643 − 196√3) cm²(d) (12323 − 392√3) cm²

Question 14 · 1 mark

Which of the following is a zero of x² + 2x − 15?

(a) 3(b) 5(c) −3(d) 15

Question 15 · 1 mark

After simplifying, the equation (x + 3)² = x(x − 2) + 5 is

(a) a quadratic equation(b) an equation with no x in it(c) a cubic equation(d) a linear equation

Question 16 · 1 mark

In practice at a court in Kochi, a basketball player attempts 80 free throws and scores on 52 of them. The empirical probability that she scores on a free throw is

(a) 720(b) 1320(c) 0.52(d) 2013

Question 17 · 1 mark

To factorise 6x² + 7x − 3 by splitting the middle term, 7x is written as

(a) 9x − 2x(b) 6x + x(c) −9x + 2x(d) 3x + 4x

Question 18 · 1 mark

A solid cone has base radius 3 cm and height 7 cm. Take π = 22/7. Its volume is

(a) 198 cm³(b) 66 cm³(c) 22 cm³(d) 132 cm³

Question 19 · 1 mark

A line l is at a distance of 7 cm from the centre of a circle of radius 6.5 cm. Then l is

(a) a tangent to the circle(b) a secant of the circle(c) a line that does not meet the circle(d) a line through the centre

Question 20 · 1 mark

Assertion (A): A line drawn parallel to one side of a triangle, meeting the other two sides at distinct points, divides them in the same ratio. Reason (R): Any line meeting two sides of a triangle divides them in the same ratio.

(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.

Section B (5 × 2 marks)

Question 21 · 2 marks

A circle touches all four sides of a quadrilateral PQRS in which PQ = (2x + 3) cm, QR = 3x cm, RS = 9 cm and SP = 10 cm. Find x and the side QR.

Question 22 · 2 marks

If tan 2θ = √3, where 0° < 2θ < 90°, find θ. Hence show that tan 2θ is not equal to 2 tan θ.

Question 23 · 2 marks

A storage tank is a cylinder 10 m long with a hemisphere of radius 7 m at each end. Take π = 22/7. Find the outer surface area of the tank.

Or A boiler is a cylinder 6 m long with a hemisphere of radius 3.5 m at each end. Take π = 22/7. Find the outer surface area of the boiler.

Question 24 · 2 marks

An AP has first term −12, common difference 5 and last term 108. Find the number of terms and its middle term.

Question 25 · 2 marks

In a grouped distribution with class size 5, the modal class is 25-30 with frequency 18. The class before it has frequency 12 and the class after it has frequency 9. Find the mode.

Section C (6 × 3 marks)

Question 26 · 3 marks

A card is drawn from a well-shuffled deck of 52 cards. Find the probability that it is (i) a black card that is not a face card, (ii) a number card (2 to 10) of hearts, (iii) neither a club nor a king.

Question 27 · 3 marks

α and β are the zeroes of ax² + bx + c, a ≠ 0. Starting from ax² + bx + c = k(x − α)(x − β), show that α + β = −ba and αβ = ca.

Question 28 · 3 marks

A circular window with centre O and radius 30 cm has a straight wooden bar AB fixed across it, where ∠AOB = 120°. Find the length of the bar and the area of the smaller glass pane. (Take π = 3.14 and √3 = 1.73.)

Question 29 · 3 marks

A mobile mast CM stands on the flat roof of a building BC. From a point P on the ground 12 m from the foot B of the building, the angles of elevation of the foot C and the top M of the mast are 30° and 60°. Find the height of the building and the height of the mast. (Use √3 ≈ 1.73.)

Question 30 · 3 marks

In ΔABC, D is a point on AB. The line through D parallel to BC meets AC at E, and the line through E parallel to AB meets BC at F, as in the figure. Prove that ADDB = BFFC.

Question 31 · 3 marks

Solve graphically: x + y = 5 and 2x − y = 4.

Section D (4 × 5 marks)

Question 32 · 5 marks

The daily earnings of 50 vegetable vendors at a market in Mumbai are given below. (a) Find the mean by the step deviation method. (b) Find the mode. (c) Compare the two.

Earnings (₹)	Number of vendors
200-300	4
300-400	8
400-500	14
500-600	12
600-700	8
700-800	4

Or The lengths of 40 fish in a catch landed at Kochi harbour are given below. (a) Find the mean by the step deviation method. (b) Find the mode. (c) Compare the two.

Question 33 · 5 marks

(a) Prove that the lengths of the two tangents drawn from an external point to a circle are equal. (b) Show also that the line joining the external point to the centre bisects the angle between the tangents. (c) For a circle of radius 33 cm whose centre O is 65 cm from the external point P, find the length of each tangent and the distance from P to the nearest point of the circle.

Or (a) PA and PB are tangents from P to a circle with centre O. Prove that the centre lies on the bisector of ∠APB. (b) Deduce that PA = PB. (c) If the radius is 11 cm and PA = 60 cm, find OP and the distance from P to the nearest point of the circle.

Question 34 · 5 marks

A science museum in Kolkata sells adult and student tickets. On Saturday, 40 adult and 60 student tickets brought in ₹5,200. On Sunday, 50 adult and 90 student tickets brought in ₹7,100. (a) Form the pair of equations. (b) Find the price of each ticket. (c) Find what a group of 12 adults and 30 students pays.

Or At a plant nursery in Pune, 5 rose plants and 8 tulsi plants cost ₹490, while 3 rose plants and 10 tulsi plants cost ₹450. (a) Form the pair of equations. (b) Find the price of each plant. (c) Find the cost of 4 rose and 6 tulsi plants.

Question 35 · 5 marks

In ΔPQR, right-angled at Q, PQ = 12 cm and sin R = 35 (see the figure). (a) Find PR and QR. (b) Find cos P and tan P. (c) Verify that 1 + tan²R = sec²R.

Or In ΔABC, right-angled at C, AC = 20 cm and cos A = 45 (see the figure). (a) Find AB and BC. (b) Find sin B and tan B. (c) Verify that cot²A + 1 = cosec²A.

Section E (3 × 4 marks)

Question 36 · 4 marks

A coastguard drone D hovers 90 m above the sea, directly above the point F on the water. Its camera picks up a fishing boat A on one side at an angle of depression of 60°, and a second boat B on the other side, in the same straight line with F, at an angle of depression of 45°, as in the figure. The crew on the boats use these readings to fix their positions. (Use √3 ≈ 1.73.)

(a) The angle of elevation of the drone, as seen from boat A, is [1]

(a) 30°(b) 45°(c) 60°(d) 90°

(b) The distance FA is [1]

(a) 90√3 m(b) 30√3 m(c) 45 m(d) 90 m

Or A third boat C, on the same side of F as A and in line with it, is seen at an angle of depression of 30°. Find the distance AC.

Question 37 · 4 marks

A gram panchayat near Hassan is fencing a rectangular plot for a community vegetable garden. The surveyor notes that the length of the plot is 6 m more than its breadth and that its area is 160 m². The panchayat needs the dimensions to order the fencing wire and to plan the beds. Let the breadth of the plot be x metres.

(a) The quadratic equation satisfied by x is [1]

(a) x² + 6x − 160 = 0(b) x² − 6x − 160 = 0(c) x² + 6x + 160 = 0(d) 2x + 6 = 160

(b) The discriminant of this equation is [1]

(a) 676(b) 604(c) 26(d) −604

Or Find the length of wire needed to fence the plot once around.

Question 38 · 4 marks

A printing unit in Surat makes a triangular logo ABC for a cooperative dairy and a large banner PQR of exactly the same shape. The banner's longest side RP had not been measured when the order sheet below was written. A third piece, a sticker XYZ, was cut by a new worker from his own measurements.

Item	Sides
Logo ABC	AB = 4 cm, BC = 6 cm, CA = 7 cm
Banner PQR	QR = 12 cm, PQ = 18 cm, RP = ?
Sticker XYZ	XY = 8 cm, YZ = 12 cm, ZX = 13 cm

(a) The scale factor of the banner to the logo is [1]

(a) 3(b) 13(c) 2(d) 32

(b) The length of RP is [1]

(a) 21 cm(b) 28 cm(c) 18 cm(d) 14 cm

(c) Write the similarity between the logo and the banner with the vertices in the correct order, and name the criterion. [2]

Or Is the sticker XYZ similar to the logo ABC? Justify.