CBSE Class 10 Maths Sample Paper 2024

If two positive integers a and b are written as a = x^3y^2 and b = xy^3, where x, y are prime numbers, then the result obtained by dividing the product of the positive integers by the LCM (a, b) is

• (A) xy
• (B) xy^2 ✓
• (C) x^3y^3
• (D) x^2y^2

Answer (the board’s key): (B)

The given linear polynomial y = f(x) has

• (A) 2 zeros
• (B) 1 zero and the zero is ‘3’ ✓
• (C) 1 zero and the zero is ‘4’
• (D) No zero

Answer (the board’s key): (B)

The lines representing the given pair of linear equations are non-intersecting. Which of the following statements is true?

• (A) a1/a2 = b1/b2 = c1/c2
• (B) a1/a2 = b1/b2 != c1/c2 ✓
• (C) a1/a2 != b1/b2 = c1/c2
• (D) a1/a2 != b1/b2 != c1/c2

Answer (the board’s key): (B)

The nature of roots of the quadratic equation 9x^2 - 6x - 2 = 0 is:

• (A) No real roots
• (B) 2 equal real roots
• (C) 2 distinct real roots ✓
• (D) More than 2 real roots

Answer (the board’s key): (C)

Two APs have the same common difference. The first term of one of these is -1 and that of the other is - 8. The difference between their 4th terms is

• (A) 1
• (B) -7
• (C) 7 ✓
• (D) 9

Answer (the board’s key): (C)

What is the ratio in which the line segment joining (2,-3) and (5, 6) is divided by x-axis?

• (A) 1:2 ✓
• (B) 2:1
• (C) 2:5
• (D) 5:2

Answer (the board’s key): (A)

A point (x,y) is at a distance of 5 units from the origin. How many such points lie in the third quadrant?

• (A) 0
• (B) 1
• (C) 2
• (D) infinitely many ✓

Answer (the board’s key): (D)

In Δ ABC, DE || AB. If AB = a, DE = x, BE = b and EC = c. Then x expressed in terms of a, b and c is:

• (A) ac/b
• (B) ac/(b+c) ✓
• (C) ab/c
• (D) ab/(b+c)

Answer (the board’s key): (B)

If O is centre of a circle and Chord PQ makes an angle 50° with the tangent PR at the point of contact P, then the angle subtended by the chord at the centre is

• (A) 130°
• (B) 100° ✓
• (C) 50°
• (D) 30°

Answer (the board’s key): (B)

A quadrilateral PQRS is drawn to circumscribe a circle. If PQ = 12 cm, QR = 15 cm and RS = 14 cm, then find the length of SP is

• (A) 15 cm
• (B) 14 cm
• (C) 12 cm
• (D) 11 cm ✓

Answer (the board’s key): (D)

Given that sin θ = a/b, then cos θ is.

• (A) b/sqrt(b^2-a^2)
• (B) b/a
• (C) sqrt(b^2-a^2)/b ✓
• (D) a/sqrt(b^2-a^2)

Answer (the board’s key): (C)

(sec A + tan A) (1 - sin A) equals:

• (A) sec A
• (B) sin A
• (C) cosec A
• (D) cos A ✓

Answer (the board’s key): (D)

If a pole 6 m high casts a shadow 2 sqrt(3)m long on the ground, then the Sun’s elevation is

• (A) 60° ✓
• (B) 45°
• (C) 30°
• (D) 90°

Answer (the board’s key): (A)

If the perimeter and the area of a circle are numerically equal, then the radius of the circle is

• (A) 2 units ✓
• (B) π units
• (C) 4 units
• (D) 7 units

Answer (the board’s key): (A)

It is proposed to build a new circular park equal in area to the sum of areas of two circular parks of diameters 16 m and 12 m in a locality. The radius of the new park is

• (A) 10m ✓
• (B) 15m
• (C) 20m
• (D) 24m

Answer (the board’s key): (A)

There is a square board of side ‘2a’ units circumscribing a red circle. Jayadev is asked to keep a dot on the above said board. The probability that he keeps the dot on the shaded region is.

• (A) π/4
• (B) (4-π)/4 ✓
• (C) (π-4)/4
• (D) 4/π

Answer (the board’s key): (B)

2 cards of hearts and 4 cards of spades are missing from a pack of 52 cards. A card is drawn at random from the remaining pack. What is the probability of getting a black card?

• (A) 22/52
• (B) 22/46 ✓
• (C) 24/52
• (D) 24/46

Answer (the board’s key): (B)

The upper limit of the modal class of the given distribution is:

• (A) 165
• (B) 160
• (C) 155
• (D) 150 ✓

Answer (the board’s key): (D)

DIRECTION: In the question number 19 and 20, a statement of assertion (A) is followed by a statement of Reason (R). Choose the correct option Statement A (Assertion): Total Surface area of the top is the sum of the curved surface area of the hemisphere and the curved surface area of the cone. Statement R( Reason) : Top is obtained by joining the plane surfaces of the hemisphere and cone together.

• (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 and 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.

Answer (the board’s key): (A)

Statement A (Assertion): -5, -5/2, 0, 5/2, .... is in Arithmetic Progression. Statement R (Reason) : The terms of an Arithmetic Progression cannot have both positive and negative rational 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 and 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.

Answer (the board’s key): (C)

Prove that sqrt(2) is an irrational number.

ABCD is a parallelogram. Point P divides AB in the ratio 2:3 and point Q divides DC in the ratio 4:1. Prove that OC is half of OA.

From an external point P, two tangents, PA and PB are drawn to a circle with centre O. At a point E on the circle, a tangent is drawn to intersect PA and PB at C and D, respectively. If PA = 10 cm, find the perimeter of triangle PCD.

If tan (A + B) = sqrt(3) and tan (A - B) = 1/sqrt(3); 0° < A + B < 90°; A > B, find A and B. [or] Find the value of x if 2 cosec^2(30) + x sin^2(60) - 3/4 tan^2(30) = 10

With vertices A, B and C of triangle ABC as centres, arcs are drawn with radii 14 cm and the three portions of the triangle so obtained are removed. Find the total area removed from the triangle. [or] Find the area of the unshaded region shown in the given figure.

National Art convention got registrations from students from all parts of the country, of which 60 are interested in music, 84 are interested in dance and 108 students are interested in handicrafts. For optimum cultural exchange, organisers wish to keep them in minimum number of groups such that each group consists of students interested in the same artform and the number of students in each group is the same. Find the number of students in each group. Find the number of groups in each art form. How many rooms are required if each group will be allotted a room?

If alpha, beta are zeroes of quadratic polynomial 5x^2 + 5x + 1, find the value of 1. alpha^2 + beta^2 2. alpha^-1 + beta^-1

The sum of a two digit number and the number obtained by reversing the digits is 66. If the digits of the number differ by 2, find the number. How many such numbers are there? [or] Solve : - 2/sqrt(x) + 3/sqrt(y) = 2 ; 4/sqrt(x) - 9/sqrt(y) = -1, x, y>0

PA and PB are tangents drawn to a circle of centre O from an external point P. Chord AB makes an angle of 30° with the radius at the point of contact. If length of the chord is 6 cm, find the length of the tangent PA and the length of the radius OA. [or] Two tangents TP and TQ are drawn to a circle with centre O from an external point T. Prove that angle PTQ = 2 angle OPQ.

If 1 + sin^2(theta) = 3 sin(theta) cos(theta), then prove that tan(theta) = 1 or 1/2

The length of 40 leaves of a plant are measured correct to nearest millimetre, and the data obtained is represented in the following table. Length [in mm] | Number of leaves 118 - 126 | 3 127 - 135 | 5 136 - 144 | 9 [table continues on next page] Find the mean length of the leaves.

A motor boat whose speed is 18 km/h in still water takes 1 hour more to go 24 km upstream than to return downstream to the same spot. Find the speed of stream. [or] Two water taps together can fill a tank in 9 3/8 hours. The tap of larger diameter takes 10 hours less than the smaller one to fill the tank separately. Find the time in which each tap can separately fill the tank.

(a) State and prove Basic Proportionality theorem. (b) In the given figure angle CEF = angle CFE. F is the midpoint of DC. Prove that AB/BD = AE/FD

Water is flowing at the rate of 15 km/h through a pipe of diameter 14 cm into a cuboidal pond which is 50 m long and 44 m wide. In what time will the level of water in pond rise by 21 cm? What should be the speed of water if the rise in water level is to be attained in 1 hour? [or] A tent is in the shape of a cylinder surmounted by a conical top. If the height and radius of the cylindrical part are 3 m and 14 m respectively, and the total height of the tent is 13.5 m, find the area of the canvas required for making the tent, keeping a provision of 26 m^2 of canvas for stitching and wastage. Also, find the cost of the canvas to be purchased at the rate of ₹ 500 per m^2.

The median of the following data is 50. Find the values of 'p' and 'q', if the sum of all frequencies is 90. Also find the mode of the data. Marks obtained | Number of students 20 – 30 | p 30 – 40 | 15 40 – 50 | 25 50 – 60 | 20 60 – 70 | q 70 – 80 | 8 80 - 90 | 10

[passage] (i) How many throws Sanjitha practiced on 11th day of the camp? [1] (ii) What would be Sanjitha's throw distance at the end of 6 weeks? [2] (or) When will she be able to achieve a throw of 11.16 m? (iii) How many throws did she do during the entire camp of 15 days ? [1]

[passage] (i) At an instance, the midfielders and forward formed a parallelogram. Find the position of the central midfielder (D) if the position of other players who formed the parallelogram are :- A(1,2), B(4,3) and C(6,6) [1] (ii) Check if the Goal keeper G(-3,5), Sweeper H(3,1) and Wing-back K(0,3) fall on a same straight line. [2] [or] Check if the Full-back J(5,-3) and centre-back I(-4,6) are equidistant from forward C(0,1) and if C is the mid-point of IJ. (iii) If Defensive midfielder A(1,4), Attacking midfielder B(2,-3) and Striker E(a,b) lie on the same straight line and B is equidistant from A and E, find the position of E. [1]

[passage] (i) At what distance from the foot of the tree was he observing the bird sitting on the tree? [1] (ii) How far did the bird fly in the mentioned time? [2] (or) After hitting the tree, how far did the ball travel in the sky when Kaushik saw the ball? (iii) What is the speed of the bird in m/min if it had flown 20(sqrt(3) + 1) m? [1]