CBSE Class 10 Maths (Standard) 2026 — Paper 30(B)

Question 1

Q1. At which of the following points, the quadratic polynomial p(x) = -3x + 18x² - 1 intersects the positive x -axis? (a) (1/6, 0) (b) (-1/3, 0) (c) (-1/6, 0) (d) (1/3, 0)

Accepted Answer

(d) (1/3, 0)

Question 2

Q3. If pair of linear equations given by a₁x + b₁y + c₁ = 0 and a₂x + b₂y + c₂ = 0 has infinitely many solutions, then which of the following is definitely true? (a) a₁/a₂ = c₂/c₁ (b) a₁ a₂ ≠ b₁ b₂ (c) a₁/a₂ ≠ b₁/b₂ (d) a₁ b₂ = a₂ b₁

Accepted Answer

(d) a₁ b₂ = a₂ b₁

Question 3

Q4. The 31st term of the A.P.: -5/6, -3/4, -2/3, -7/12, … is (a) -5/3 (b) 5/3 (c) 12/20 (d) -12/20

Accepted Answer

(b) 5/3

Question 4

Q5. Four tangents drawn to a circle are extended from both the sides to form a quadrilateral. Which of these quadrilateral is not possible? (a) Trapezium (b) Square (c) Rectangle (d) Rhombus

Accepted Answer

(c) Rectangle

Question 5

Q6. ABCD is a trapezium in which AB ∥ DC and E, F are points on AD and BC respectively such that EF ∥ DC. If ED = 36 cm, BF = 70 cm and FC = 30 cm, then the length of AD is: (a) 124 cm (b) 120 cm (c) 110 cm (d) 114 cm

Accepted Answer

(b) 120 cm

Question 6

Q7. If x · tan 45° · sin 30°=cos 30° · cot 60°, then x is equal to (a) √3 (b) 1/√3 (c) 1 (d) 1/2

Accepted Answer

(c) 1

Question 7

Q8. If △DEF ~ △PQR such that 3 DE = PQ and EF = 6 cm, then the length of QR is: (a) 12 cm (b) 3 cm (c) 2 cm (d) 18 cm

Accepted Answer

(d) 18 cm

Question 8

Q9. If the mean of first n natural numbers is 6n/11, then n is: (a) 11 (b) 6 (c) 12 (d) 22

Accepted Answer

(a) 11

Question 9

Q10. If a, b, c and d are consecutive terms of an A.P., then c - b is equal to: (a) d - a (b) d - b (c) d - c (d) c - a

Accepted Answer

(c) d - c

Question 10

Q11. If in a △ABC, AB = 6 cm and DE ∥ BC such that AE = (1/3)AC, then the length of BD is (a) 2 cm (b) 3 cm (c) 4 cm (d) 5 cm

Accepted Answer

(c) 4 cm

Question 11

Q12. From a point P, tangents PQ and PR are drawn to a circle with centre O and radius 6 cm. If OP = 10 cm, then area of quadrilateral PQOR is: (a) 48 cm² (b) 24 cm² (c) 96 cm² (d) 72 cm²

Accepted Answer

(a) 48 cm²

Question 12

Q13. The perimeter of sector of a circle of radius 21 cm and central angle 60°, is (a) 22 cm (b) 44 cm (c) 64 cm (d) 273 cm

Accepted Answer

(c) 64 cm

Question 13

Q14. The number of conical bottles of radius 2 cm and height 1.2 cm that can be filled from a cylindrical bottle of radius 6 cm and height 8 cm, full of liquid, is (a) 60 (b) 130 (c) 180 (d) 18

Accepted Answer

(c) 180

Question 14

Q15. One ticket is drawn at random from a bag containing 50 tickets numbered 1 to 50. The probability that the drawn ticket has a number which is a multiple of 7 is: (a) 1/5 (b) 7/50 (c) 3/25 (d) 4/25

Accepted Answer

(b) 7/50

Question 15

Q16. Two dice are thrown simultaneously. The probability of getting a sum of 7 is: (a) 2/9 (b) 1/9 (c) 5/36 (d) 1/6

Accepted Answer

(d) 1/6

Question 16

Q17. If HCF of 66 and 99 is expressible in the form of 55m - 132, then the value of m is: (a) 4 (b) 2 (c) 1 (d) 3

Accepted Answer

(d) 3

Question 17

Q18. sin 2θ = 2sinθ is true, when θ is equal to (a) 90° (b) 60° (c) 45° (d) 0°

Accepted Answer

(d) 0°

Question 18

Q19. In the following question, a statement of Assertion (A) is followed by a statement of Reason (R). Choose the correct option. Assertion (A): In a circle of radius 21 cm, an arc of length 22 cm subtends an angle of 60° at the centre. Reason (R): The length of arc of a sector of a circle of radius r and central angle θ is 2πrθ/360. (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.

Accepted Answer

(a) Both Assertion (A) and Reason (R) are true and Reason (R) is the correct explanation of Assertion (A).

Question 19

Q20. In the following question, a statement of Assertion (A) is followed by a statement of Reason (R). Choose the correct option. Assertion (A): If the Mode and Mean of a data are 12k and 15k, then Median of the data is 14k. Reason (R): The relation between the Mean, Mode and Median of a data is: Mean = 3 Median - 2 Mode. (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.

Accepted Answer

(c) Assertion (A) is true, but Reason (R) is false.

Question 20

Q21. Two numbers are in the ratio 3: 5 and their LCM is 180. Find the HCF of these two numbers.

Accepted Answer

The HCF of the two numbers is 12.

Question 21

Q22. Find the quadratic polynomial the sum of whose zeroes is 1 and their product is -12. Hence find the zeroes of the polynomial.

Accepted Answer

The quadratic polynomial is x² - x - 12, and its zeroes are 4 and -3.

Question 22

Q23. A circle is inscribed in a right triangle ABC, right angled at B. If the lengths of the two sides containing the right angle are 8 cm and 15 cm, find the radius of the incircle.

Accepted Answer

The radius of the incircle is 3 cm.

Question 23

Q24(a). If 7sin² A + 3cos² A = 4, then find the value of tan A.

Accepted Answer

tan A = 1/√3 (equivalently √3/3), which corresponds to A = 30°.

Question 24

Q24(b). If 4tan A = 3, then find the value of (cosec² A + 1)/(cosec² A - 1).

Accepted Answer

(cosec² A + 1)/(cosec² A - 1) = 17/8.

Question 25

Q25(a). A chord AB of a circle of radius 14 cm makes a right angle at the centre of the circle. Find the area of the minor segment.

Accepted Answer

The area of the minor segment is 56 cm².

Question 26

Q25(b). An arc of length 22 cm subtends an angle of 60° at the centre of the circle. Find the area of the sector of the circle made by the arc.

Accepted Answer

The area of the sector is 231 cm².

Question 27

Q26. Prove that (2+3√5)/7 is an irrational number, given that √5 is an irrational number.

Accepted Answer

Assuming (2+3√5)/7 is rational forces √5 = (7p-2q)/(3q) to be rational, which contradicts the given fact that √5 is irrational. Therefore (2+3√5)/7 is irrational. Hence proved.

Question 28

Q27(a). Find the coordinates of the points of trisection of the line segment joining the points P(5, -4) and Q(-4, 2).

Accepted Answer

The points of trisection are (2, -2) and (-1, 0).

Question 29

Q27(b). Find the ratio in which the point P (6, k) divides the line segment joining the points M (4, 2) and N (8, -4). Hence find the value of k.

Accepted Answer

P divides MN in the ratio 1:1 (P is the midpoint), and k=-1.

Question 30

Q28. Find a relation between x and y such that the point P(x, y) is equidistant from the points A(5, 3) and B(1, 7).

Accepted Answer

The required relation is x - y + 2 = 0 (equivalently y = x + 2), which is the perpendicular bisector of AB.

Question 31

Q29(b). AB is a chord of length 24 cm of a circle of radius 15 cm. The tangents at A and B intersect at a point P. Find the length PA.

Accepted Answer

PA = 20 cm.

Question 32

Q30. If cos A + sin A = √2 cos A, prove that cos A - sin A = √2 sin A.

Accepted Answer

Squaring the given relation gives 2sin Acos A = 2cos²A - 1; substituting into (cos A-sin A)² yields 2sin²A, so cos A - sin A = √2 sin A. Hence proved.

Question 33

Q31. Find the mean of the following distribution: Table: Class | Frequency; 30 - 40 | 6; 40 - 50 | 13; 50 - 60 | 8; 60 - 70 | 12; 70 - 80 | 11.

Accepted Answer

The mean of the distribution is 56.8.

Question 34

Q32(b). CD and GH are respectively the bisectors of ∠ACB and ∠EGF such that D and H lie on sides AB and FE of △ABC and △EFG respectively. If △ABC ~ △FEG, then show that (i) CD/(GH) = AC/(FG) and (ii) △DCB ~ △HGE.

Accepted Answer

Both results are proved. (i) △ACD ~ △FGH (AA: ∠A=∠F, ∠ACD=∠FGH) gives CD/(GH) = AC/(FG). (ii) △DCB ~ △HGE (AA: ∠DBC=∠HEG, ∠DCB=∠HGE).

Question 35

Q33(a). The sum of numerator and denominator of a fraction is 4 less than twice the denominator. If each of the numerator and denominator is decreased by 1, the fraction becomes 1/3. Find the fraction.

Accepted Answer

The required fraction is 3/7.

Question 36

Q33(b). Three consecutive positive integers are such that sum of square of the first and the product of the other two is 67, find the integers.

Accepted Answer

The three consecutive positive integers are 5, 6 and 7.

Question 37

Q34. Two poles of equal heights are standing opposite to each other on either side of the road which is 90 m wide. From a point between them on the road, the angles of elevation of the top of the poles are 30° and 60° respectively. Find the height of the poles and the distances of the point from the poles. [Use √3 = 1.732 ]

Accepted Answer

The height of each pole is 22.5√3 ≈ 38.97 m. The point is 22.5 m from the pole whose top is seen at 60° and 67.5 m from the pole whose top is seen at 30°.

Question 38

Q35. From a solid cylinder whose height is 2.8 cm and radius 2.1 cm, a conical cavity of the same height and same radius is hollowed out. Find the volume and the total surface area of the remaining solid.

Accepted Answer

Volume of the remaining solid = 25.872 cm³ and its total surface area = 73.92 cm².

Question 39

Q36. While playing badminton, Ravi has set the barrier chain hung between two posts at the edge of the walkway of a street. It is hung in the shape of a parabola. Based on the above information, answer the following questions: (a) Which type of the polynomial (linear, quadratic, cubic etc.) is graphically represented by a parabola? (b) If the polynomial represented by a parabola intersects the x -axis at -2 and 3 and the y -axis at -3, then write the zeroes of the parabola. (c) Find the expression for the above polynomial. OR If the zeroes of the polynomial are -5 and 3, find its expression.

Accepted Answer

(a) A quadratic polynomial (degree 2). (b) The zeroes are -2 and 3. (c) p(x)=(1/2)(x+2)(x-3)=(1/2)x²-(1/2)x-3. OR, for zeroes -5 and 3, p(x)=x²+2x-15.

Question 40

Q37. A friend of yours wants to buy an electric car, for which he plans to take a loan from a bank and repay the total loan and interest, ₹ 5,90,000, by paying every month. He starts with a first instalment of ₹ 5,000 and increases the instalment by ₹ 500 every month. (a) What are the first three instalments paid by him? (b) Find the amount to be paid by him in the 11th instalment. (c) Find the number of instalments in which he would clear his total loan. OR After paying the 31st instalment, find how much money he still has to pay.

Accepted Answer

(a) The first three instalments are ₹ 5000, ₹ 5500 and ₹ 6000. (b) The 11th instalment is ₹ 10,000. (c) He clears the loan in 40 instalments; OR, after the 31st instalment he still has to pay ₹ 2,02,500.

Question 41

Q38. Raghav has a collection of balls of different colours. He has a total of 35 balls in his basket, out of which seven are black and eight are yellow. Out of the remaining balls, some are white and the rest are red. (a) If the probability of drawing a red ball at random from the basket is three times that of a white ball, then find the number of red balls in the basket. (b) Find the probability of drawing a ball at random from the basket which is either a black or a white ball.

Accepted Answer

(a) The number of red balls is 15 (with 5 white balls). (b) P(black or white) = 12/35.

CBSE Class 10 Maths (Standard) 2026 — Paper 30(B)

Questions 1–20 · 1 mark each

Questions 21–25 · 2 marks each

Questions 26–31 · 3 marks each

Questions 32–35 · 5 marks each

Questions 36–38 · 4 marks each

Other 2026 papers

Frequency distribution
Class	Frequency
30 - 40	6
40 - 50	13
50 - 60	8
60 - 70	12
70 - 80	11