CBSE Class 10 · Maths · board exam 2027
CBSE Class 10 Maths Question Paper 2024
The board’s 2024 question paper, set 30/1/1 — all 38 questions as printed; 20 objective answers from the board’s own key; full model answers for 3 written questions, how the answer begins for 11 more. Each question names its chapter, where that chapter’s most-asked questions are.
Section A · Objective questions (20)
If the sum of zeroes of the polynomial p(x) = 2x^2 - ksqrt(2)x + 1 is sqrt(2), then value of k is :
- (A) sqrt(2)
- (B) 2 ✓
- (C) 2sqrt(2)
- (D) 1/2
Answer (the board’s key): (B)
If the probability of a player winning a game is 0.79, then the probability of his losing the same game is :
- (A) 1.79
- (B) 0.31
- (C) 0.21%
- (D) 0.21 ✓
Answer (the board’s key): (D)
If the roots of equation ax^2 + bx + c = 0, a != 0 are real and equal, then which of the following relation is true ?
- (A) a = b^2/c
- (B) b^2 = ac
- (C) ac = b^2/4 ✓
- (D) c = b^2/a
Answer (the board’s key): (C)
In an A.P., if the first term a = 7, nth term a_n = 84 and the sum of first n terms s_n = 2093/2, then n is equal to :
- (A) 22
- (B) 24
- (C) 23 ✓
- (D) 26
Answer (the board’s key): (C)
If two positive integers p and q can be expressed as p = 18 a^2 b^4 and q = 20 a^3 b^2, where a and b are prime numbers, then LCM (p, q) is :
- (A) 2 a^2 b^2
- (B) 180 a^2 b^4
- (C) 12 a^2 b^2
- (D) 180 a^3 b^4 ✓
Answer (the board’s key): (D)
The vertices of triangle ABC are A(5, -6), B(6, 4) and C(0, 0) and AD is a median of the triangle. The length of AD is :
- (A) sqrt(68) units ✓
- (B) 2sqrt(15) units
- (C) sqrt(101) units
- (D) 10 units
Answer (the board’s key): (A)
If sec theta - tan theta = m, then the value of sec theta + tan theta is :
- (A) 1 - 1/m
- (B) m^2 - 1
- (C) 1/m ✓
- (D) -m
Answer (the board’s key): (C)
From the data 1, 4, 7, 9, 16, 21, 25, if all the even numbers are removed, then the probability of getting at random a prime number from the remaining is :
- (A) 2/5
- (B) 1/5 ✓
- (C) 1/7
- (D) 2/7
Answer (the board’s key): (B)
For any data x_1, x_2, ...... x_n where f_1, f_2, ..... f_n are respectively their frequencies, the value of sum_{1}^{n} f_i(x_i - x_bar) is :
- (A) n x_bar
- (B) 1
- (C) sum f_i
- (D) 0 ✓
Answer (the board’s key): (D)
The zeroes of a polynomial x^2 + px + q are twice the zeroes of the polynomial 4x^2 - 5x - 6. The value of p is :
- (A) -5/2 ✓
- (B) 5/2
- (C) -5
- (D) 10
Answer (the board’s key): (A)
If the distance between the points (3, -5) and (x, -5) is 15 units, then the values of x are :
- (A) 12, -18
- (B) -12, 18 ✓
- (C) 18, 5
- (D) -9, -12
Answer (the board’s key): (B)
If cos(alpha + beta) = 0, then value of cos((alpha + beta)/2) is equal to :
- (A) 1/sqrt(2) ✓
- (B) 1/2
- (C) 0
- (D) sqrt(2)
Answer (the board’s key): (A)
A solid sphere is cut into two hemispheres. The ratio of the surface area of this sphere to the total surface area of both the hemispheres is :
- (A) 1 : 1
- (B) 1 : 4
- (C) 2 : 3 ✓
- (D) 3 : 2
Answer (the board’s key): (C)
On arranging the data of each type in order, the observation lying exactly in the middle is :
- (A) Mode
- (B) Median ✓
- (C) Mean
- (D) Deviation
Answer (the board’s key): (B)
The volume of the largest right circular cone that can be carved out from a solid cube of edge 2 cm is :
- (A) 4pi/3 cu cm
- (B) 5pi/3 cu cm
- (C) 8pi/3 cu cm
- (D) 2pi/3 cu cm ✓
Answer (the board’s key): (D)
Two dice are rolled together. The probability of getting sum of numbers on the two dice as 2, 3 or 5, is :
- (A) 7/36 ✓
- (B) 11/36
- (C) 5/36
- (D) 4/9
Answer (the board’s key): (A)
The centre of a circle is at (2, 0). If one end of a diameter is at (6, 0), then the other end is at :
- (A) (0, 0)
- (B) (4, 0)
- (C) (-2, 0) ✓
- (D) (-6, 0)
Answer (the board’s key): (C)
In the given figure, graphs of two linear equations are shown. The pair of these linear equations is :
- (A) consistent with unique solution. ✓
- (B) consistent with infinitely many solutions.
- (C) inconsistent.
- (D) inconsistent but can be made consistent by extending these lines.
Answer (the board’s key): (A)
Assertion (A) : The tangents drawn at the end points of a diameter of a circle, are parallel. Reason (R) : Diameter of a circle is the longest chord.
- (A) Both, Assertion (A) and Reason (R) are true and Reason (R) is correct explanation of Assertion (A).
- (B) Both, Assertion (A) and Reason (R) are true but Reason (R) is not correct explanation for 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): (B)
Assertion (A) : If the graph of a polynomial touches x-axis at only one point, then the polynomial cannot be a quadratic polynomial. Reason (R) : A polynomial of degree n(n > 1) can have at most n zeroes.
- (A) Both, Assertion (A) and Reason (R) are true and Reason (R) is correct explanation of Assertion (A).
- (B) Both, Assertion (A) and Reason (R) are true but Reason (R) is not correct explanation for 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): (D)
Section B · Written questions (18)
Solve the following system of linear equations 7x - 2y = 5 and 8x + 7y = 15 and verify your answer.
In a pack of 52 playing cards one card is lost. From the remaining cards, a card is drawn at random. Find the probability that the drawn card is queen of heart, if the lost card is a black card.
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Model answer · Playing cards, lost card
- 1.Given: a deck of 52 playing cards, and one black card is lost from it. Total number of remaining cards = 52 – 1 = 51 So the number of possible outcomes = 51. These outcomes are equally likely because the card is drawn at random.1 mark
- 2.The lost card is black, and hearts are red, so the queen of hearts is still in the deck. Number of outcomes favourable to getting the queen of hearts = 1 P(E) = Number of outcomes favourable to E / Number of all possible outcomes P (getting queen of heart) = 1/511 mark
- Write "Total number of remaining cards = 52 – 1 = 51" before you work out the probability.
- Say that hearts are red, so a lost black card cannot be the queen of hearts. That is why the number of favourable outcomes is 1.
Marks split as in the board’s published marking scheme
(A) Evaluate : 2sqrt(2) cos 45° sin 30° + 2sqrt(3) cos 30° OR (B) If A = 60° and B = 30°, verify that : sin (A + B) = sin A cos B + cos A sin B
In the given figure, ABCD is a quadrilateral. Diagonal BD bisects ∠B and ∠D both. Prove that : (i) ΔABD ~ ΔCBD (ii) AB = BC
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Model answer · AA similarity criterion proof
- 1.(i) Let ∠1 = ∠ABD, ∠2 = ∠CBD, ∠3 = ∠ADB and ∠4 = ∠CDB. In ΔABD and ΔCBD: ∠1 = ∠2 (BD bisects ∠B) ∠3 = ∠4 (BD bisects ∠D) Two angles of one triangle are respectively equal to two angles of the other, so by the AA similarity criterion, ∴ ΔABD ~ ΔCBD1 mark
- 2.(ii) In ΔABD and ΔCBD: ∠1 = ∠2 (proved above) BD = BD (common side) ∠3 = ∠4 (proved above) ∴ ΔABD ≅ ΔCBD (ASA congruence rule) ∴ AB = BC (corresponding parts of congruent triangles)1 mark
Diagram: Draw quadrilateral ABCD with diagonal BD. Mark ∠1 (∠ABD) and ∠2 (∠CBD) at B, and ∠3 (∠ADB) and ∠4 (∠CDB) at D.
- Give the reason beside each equal pair of angles: "BD bisects ∠B" and "BD bisects ∠D", then name the AA similarity criterion.
- In part (ii), write "BD = BD (common side)" so you can prove ΔABD ≅ ΔCBD before you conclude AB = BC.
Marks split as in the board’s published marking scheme
(A) Prove that 5 - 2sqrt(3) is an irrational number. It is given that sqrt(3) is an irrational number. OR (B) Show that the number 5 x 11 x 17 + 3 x 11 is a composite number.
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Model answer · proving irrationality of numbers
- 1.Let us assume, to the contrary, that 5 − 2√3 is a rational number. Then we can find coprime integers a and b (b ≠ 0) such that 5 − 2√3 = a/b0.5 marks
- 2.Rearranging this equation: 2√3 = 5 − a/b = (5b − a)/b ⇒ √3 = (5b − a)/2b0.5 marks
- 3.Since a and b are integers, the RHS, (5b − a)/2b, is rational, so √3 would be rational. But the LHS, √3, is irrational. This contradicts the fact that √3 is irrational. Therefore our assumption that 5 − 2√3 is rational is wrong.0.5 marks
- 4.Hence, 5 − 2√3 is an irrational number.0.5 marks
- Begin with "Let us assume, to the contrary, that 5 − 2√3 is a rational number" and state that a and b are integers with b ≠ 0.
- Rearrange until √3 stands alone: √3 = (5b − a)/2b.
- Say that the RHS is rational but the LHS, √3, is irrational, then end with "Hence, 5 − 2√3 is an irrational number."
Marks split as in the board’s published marking scheme
(A) Find the ratio in which the point (8/5, y) divides the line segment joining the points (1, 2) and (2, 3). Also, find the value of y. OR (B) ABCD is a rectangle formed by the points A (-1, -1), B (-1, 6), C (3, 6) and D (3, -1). P, Q, R and S are mid-points of sides AB, BC, CD and DA respectively. Show that diagonals of the quadrilateral PQRS bisect each other.
How the model answer begins
1. Given: the point P(8/5, y) divides the line segment joining A(1, 2) and B(2, 3). Let P divide AB internally in the ratio AP : PB = k : 1. By the section formula, the coordinates of P are ((2k + 1)/(k + 1), (3k + 2)/(k +…
The full answer from Medha Ma’am →In a teachers' workshop, the number of teachers teaching French, Hindi and English are 48, 80 and 144 respectively. Find the minimum number of rooms required if in each room the same number of teachers are seated and all of them are of the same subject.
How the model answer begins
1. Given: there are 48 French teachers, 80 Hindi teachers and 144 English teachers. Every room has the same number of teachers, and all of them teach the same subject. Minimum number of rooms required means there should be…
The full answer from Medha Ma’am →Prove that : tan theta / (1 - cot theta) + cot theta / (1 - tan theta) = 1 + sec theta cosec theta
How the model answer begins
1. Write the LHS in terms of sin θ and cos θ, using tan θ = sin θ/cos θ and cot θ = cos θ/sin θ: LHS = (sinθ/cosθ) / (1 − cosθ/sinθ) + (cosθ/sinθ) / (1 − sinθ/cosθ)
The full answer from Medha Ma’am →Three years ago, the age of Rashmi was three times the age of Najma. Ten years hence, the age of Rashmi will be twice the age of Najma. Find their present ages.
How the model answer begins
1. At present, Rashmi is 42 years old and Najma is 16 years old. This is found as follows. Let the present age of Rashmi be x years and the present age of Najma be y years. Three years ago, Rashmi was (x − 3) years old and…
The full answer from Medha Ma’am →(A) In the given figure, AB is a diameter of the circle with centre O. AQ, BP and PQ are tangents to the circle. Prove that angle POQ = 90 degree. OR (B) A circle with centre O and radius 8 cm is inscribed in a quadrilateral ABCD in which P, Q, R, S are the points of contact as shown. If AD is perpendicular to DC, BC = 30 cm and BS = 24 cm, then find the length DC.
How the model answer begins
1. Given: AB is a diameter of the circle with centre O. AQ is the tangent at A, BP is the tangent at B, and PQ is a tangent that touches the circle at R. To prove: ∠POQ = 90°. Construction: Join OR, OQ and OP.
The full answer from Medha Ma’am →The difference between the outer and inner radii of a hollow right circular cylinder of length 14 cm is 1 cm. If the volume of the metal used in making the cylinder is 176 cm^3, find the outer and inner radii of the cylinder.
How the model answer begins
1. Given: a hollow right circular cylinder has length (height) h = 14 cm. The volume of the metal used is 176 cm^3. Let the outer radius be r2 cm and the inner radius be r1 cm. Take π = 22/7. The difference of the radii is…
The full answer from Medha Ma’am →An arc of a circle of radius 21 cm subtends an angle of 60 degree at the centre. Find : (i) the length of the arc. (ii) the area of the minor segment of the circle made by the corresponding chord.
How the model answer begins
1. Given: a circle with centre O and radius r = 21 cm. The arc AB subtends ∠AOB = θ = 60° at the centre, and AB is the corresponding chord. Let L be any point on the minor arc AB, so OALB is the sector bounded by the radii…
The full answer from Medha Ma’am →(A) The sum of the first and eighth terms of an Arithmetic Progression (A.P.) is 32 and their product is 60. Find the first term and common difference of this A.P. Also find the sum of the first 20 terms of this A.P. OR (B) The sum of the first 9 terms of an A.P. of 40 terms is 153 and the sum of its last 6 terms is 687. Find the first term and common difference of this A.P. Also find the sum of all the terms of this A.P.
How the model answer begins
1. Answering option (A) of the OR choice (only one option is to be answered): the first term and the common difference of the AP are 2 and 4, or 30 and –4, and the sum of the first 20 terms is 800 or –160 respectively. The…
The full answer from Medha Ma’am →(A) If a line is drawn parallel to one side of a triangle to intersect the other two sides in distinct points, then prove that the other two sides are divided in the same ratio. OR (B) In the given figure PA, QB and RC are each perpendicular to AC. If AP = x, BQ = y and CR = z, then prove that 1/x + 1/z = 1/y
How the model answer begins
1. Attempting (A) – Basic Proportionality Theorem. Given: A triangle ABC in which a line parallel to side BC intersects the other two sides AB and AC at D and E respectively, i.e., DE || BC. To prove: AD/DB = AE/EC.…
The full answer from Medha Ma’am →A pole 6m high is fixed on the top of a tower. The angle of elevation of the top of the pole observed from a point P on the ground is 60 degree and the angle of depression of the point P from the top of the tower is 45 degree. Find the height of the tower and the distance of point P from the foot of the tower. (Use sqrt(3) = 1.73)
A rectangular floor area can be completely tiled with 200 square tiles. If the side length of each tile is increased by 1 unit, it would take only 128 tiles to cover the floor. (i) Assuming the original length of each side of a tile be x units, make a quadratic equation from the above information. (ii) Write the corresponding quadratic equation in standard form. (iii) (a) Find the value of x, the length of side of a tile by factorisation. OR (b) Solve the quadratic equation for x, using quadratic formula.
BINGO is game of chance. The host has 75 balls numbered 1 through 75. Each player has a BINGO card with some numbers written on it. The participant cancels the number on the card when called out a number written on the ball selected at random. Whosoever cancels all the numbers on his/her card, says BINGO and wins the game. The table given below, shows the data of one such game where 48 balls were used before Tara said 'BINGO'. Numbers announced | Number of times 0-15 | 8 15-30 | 9 30-45 | 10 45-60 | 12 60-75 | 9 Based on the above information, answer the following : (i) Write the median class. (ii) When first ball was picked up, what was the probability of calling out an even number ? (iii) (a) Find median of the given data. OR (b) Find mode of the given data.
How the model answer begins
1. (i) The median class is 30 - 45. The total frequency is n = 8 + 9 + 10 + 12 + 9 = 48, so n/2 = 24. The cumulative frequency of each class is: 0 - 15: 8; 15 - 30: 17; 30 - 45: 27; 45 - 60: 39; 60 - 75: 48. The first…
The full answer from Medha Ma’am →A backyard is in the shape of a triangle ABC with right angle at B. AB = 7 m and BC = 15 m. A circular pit was dug inside it such that it touches the walls AC, BC and AB at P, Q and R respectively such that AP = x m. Based on the above information, answer the following questions : (i) Find the length of AR in terms of x. (ii) Write the type of quadrilateral BQOR. (iii) (a) Find the length PC in terms of x and hence find the value of x. OR (b) Find x and hence find the radius r of circle.
How the model answer begins
1. (i) AR = x m. A is an external point to the circle, and AP and AR are the two tangents from A, with P and R as the points of contact. The lengths of the two tangents from an external point to a circle are equal, so AR =…
The full answer from Medha Ma’am →
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