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Class 10 Maths (Standard): questions from CBSE board papers, by chapter

Every question of the 4 CBSE board exam papers Shishya has solved (2026 Q.P. 30/1/1, 2025 Q.P. 30/1/1, 2024 Q.P. 30/1/1, 2023 Q.P. 30/1/1), listed under the NCERT chapter it was asked from. The question lines are in Shishya's own shorter words; each links to its step-by-step solution, and the paper page links CBSE's own question paper. Marks per chapter count every listed question, including both sides of an either/or choice.

Real Numbers

15 questions, 26 marks across 4 papers

Asked in CBSE 2026 · 30/1/1 · Q1 · 1 markFind the HCF of 960 and 432.
Asked in CBSE 2026 · 30/1/1 · Q2 · 1 markIs the natural number 2 prime, composite, both, or neither?
Asked in CBSE 2026 · 30/1/1 · Q3 · 1 markFor every natural number n, what is the last digit of 6^n?
Asked in CBSE 2026 · 30/1/1 · Q26 · 3 marksShow that √3 is irrational.
Asked in CBSE 2025 · 30/1/1 · Q6 · 1 markm = HCF(98, 28) and n = LCM(98, 28). Find n − 7m.
Asked in CBSE 2025 · 30/1/1 · Q9 · 1 mark(−1)ⁿ + (−1)⁸ = 0. What kind of number is n?
Asked in CBSE 2025 · 30/1/1 · Q14 · 1 markPick the rational number lying between √3 and √5 (choices include 1.4142387954012..., 2.32̄6, π and 1.857142).
Asked in CBSE 2025 · 30/1/1 · Q29 · 3 marksShow that 1/√5 is irrational.
Asked in CBSE 2024 · 30/1/1 · Q5 · 1 markp = 18a²b⁴ and q = 20a³b², with a, b prime. Find LCM(p, q).
Asked in CBSE 2024 · 30/1/1 · Q25(A) · 2 marksGiven that √3 is irrational, prove that 5 − 2√3 is irrational.
Asked in CBSE 2024 · 30/1/1 · Q25(B) · 2 marksShow that 5 × 11 × 17 + 3 × 11 is composite.
Asked in CBSE 2024 · 30/1/1 · Q27 · 3 marksA workshop has 48 French, 80 Hindi and 144 English teachers. Each room seats an equal number of teachers, all of one subject. Least number of rooms?
Asked in CBSE 2023 · 30/1/1 · Q20 · 1 markRight triangle ABC with AB = 2 cm, BC = 3 cm, ∠B = 90°. A: its perimeter is rational. R: a sum of squares of two rationals is rational.
Asked in CBSE 2023 · 30/1/1 · Q24 · 2 marksLargest number that leaves remainder 1 on dividing 85 and remainder 2 on dividing 72.
Asked in CBSE 2023 · 30/1/1 · Q27 · 3 marksShow that √5 is irrational.

Polynomials

12 questions, 14 marks across 4 papers

Asked in CBSE 2026 · 30/1/1 · Q4 · 1 markFrom the given graph of y = f(x), how many zeroes does f(x) have?
Asked in CBSE 2026 · 30/1/1 · Q20 · 1 markAssertion: p(y) = y² + 4y + 3 has two zeroes. Reason: a quadratic polynomial has at most two zeroes.
Asked in CBSE 2026 · 30/1/1 · Q21 · 2 marksα and β are zeroes of p(x) = x² − 3x − 1. Find 1/α + 1/β.
Asked in CBSE 2025 · 30/1/1 · Q1 · 1 markα, β are zeroes of 3x² + 6x + k and α + β + αβ = −2/3. Find k.
Asked in CBSE 2025 · 30/1/1 · Q10 · 1 markA graph shows two polynomials. How many distinct zeroes do they have together?
Asked in CBSE 2025 · 30/1/1 · Q22 · 2 marksZeroes of p(x) = x² + (4/3)x − 4/3.
Asked in CBSE 2024 · 30/1/1 · Q1 · 1 markThe zeroes of 2x² − k√2 x + 1 add up to √2. Find k.
Asked in CBSE 2024 · 30/1/1 · Q10 · 1 markZeroes of x² + px + q are double those of 4x² − 5x − 6. Find p.
Asked in CBSE 2024 · 30/1/1 · Q20 · 1 markAssertion: a polynomial whose graph touches the x-axis at just one point cannot be quadratic. Reason: a degree-n polynomial (n > 1) has at most n zeroes.
Asked in CBSE 2023 · 30/1/1 · Q1 · 1 markFrom the graph of y = p(x), how many zeroes does p(x) have?
Asked in CBSE 2023 · 30/1/1 · Q7 · 1 markα, β are zeroes of x² + x − 1. Find 1/α + 1/β.
Asked in CBSE 2023 · 30/1/1 · Q14 · 1 markWhich quadratic has zeroes −2/3 and 2/3?

Pair of Linear Equations in Two Variables

11 questions, 26 marks across 4 papers

Asked in CBSE 2026 · 30/1/1 · Q5 · 1 markTwo linear equations in two variables are shown by coincident lines. How many solutions does the pair have?
Asked in CBSE 2026 · 30/1/1 · Q32 · 5 marksUsing graphs, find the vertices of the triangle formed by 2x − 3y + 6 = 0, 2x + 3y − 18 = 0 and x = 0, and its area.
Asked in CBSE 2025 · 30/1/1 · Q2 · 1 markx = 1, y = 2 solves 2x − 3y + a = 0 as well as 2x + 3y − b = 0. Which relation between a and b holds?
Asked in CBSE 2025 · 30/1/1 · Q32 · 5 marksVijay put money in scheme A (8% p.a.) and scheme B (9% p.a.) and got ₹1,860 interest a year. Swapping the amounts would give ₹20 more. Find each amount.
Asked in CBSE 2024 · 30/1/1 · Q18 · 1 markThe figure shows the graphs of two linear equations. What kind of pair is it?
Asked in CBSE 2024 · 30/1/1 · Q21 · 2 marksSolve 7x − 2y = 5 and 8x + 7y = 15, then verify.
Asked in CBSE 2024 · 30/1/1 · Q29 · 3 marks3 years ago Rashmi's age was 3 times Nazma's; 10 years from now it will be 2 times Nazma's. Find their present ages.
Asked in CBSE 2023 · 30/1/1 · Q2 · 1 markFor which k do kx = y + 2 and 6x = 2y + 3 have infinitely many solutions?
Asked in CBSE 2023 · 30/1/1 · Q21(a) · 2 marksSolve x = 3 and y = −4 by drawing their graphs.
Asked in CBSE 2023 · 30/1/1 · Q21(b) · 2 marksGraphically decide whether x = 0, y = −7 is a consistent system.
Asked in CBSE 2023 · 30/1/1 · Q26 · 3 marksTwo numbers: half their difference is 2, and the larger plus twice the smaller is 13. Find them.

Quadratic Equations

14 questions, 34 marks across 4 papers

Asked in CBSE 2026 · 30/1/1 · Q33(A) · 5 marksOver 200 km, the faster of two trains needs 1 hour less; the slower is 10 km/h slower. Find both speeds.
Asked in CBSE 2026 · 30/1/1 · Q33(B) · 5 marksTwo squares have total area 640 m² and perimeters differing by 64 m. Find their sides.
Asked in CBSE 2025 · 30/1/1 · Q34(a) · 5 marksA right triangle has perimeter 60 cm and hypotenuse 25 cm. Find its other two sides.
Asked in CBSE 2025 · 30/1/1 · Q34(b) · 5 marksA train covers 480 km at steady speed; 8 km/h slower it would need 3 h extra. Find its speed.
Asked in CBSE 2024 · 30/1/1 · Q3 · 1 markax² + bx + c = 0 (a ≠ 0) has real and equal roots. Which relation is true?
Asked in CBSE 2024 · 30/1/1 · Q36(i) · 1 markA floor takes 200 square tiles of side x; with side x + 1 it takes 128. Form the equation in x.
Asked in CBSE 2024 · 30/1/1 · Q36(ii) · 1 markSame tiles set-up. Put the equation in standard form.
Asked in CBSE 2024 · 30/1/1 · Q36(iii)(a) · 2 marksSame tiles set-up. Find the tile side x by factorising.
Asked in CBSE 2024 · 30/1/1 · Q36(iii)(b) · 2 marksSame tiles set-up. Solve 9x² − 32x − 16 = 0 with the quadratic formula.
Asked in CBSE 2023 · 30/1/1 · Q8 · 1 markSmallest positive k for which 2x² + kx − 4 = 0 has rational roots.
Asked in CBSE 2023 · 30/1/1 · Q36(I) · 1 markAn 18 cm × 12 cm photo has both length and width increased by x so that its area doubles. Write the equation.
Asked in CBSE 2023 · 30/1/1 · Q36(II) · 1 markWrite that equation as a quadratic in standard form.
Asked in CBSE 2023 · 30/1/1 · Q36(III)(A) · 2 marksFind the dimensions of the enlarged photo.
Asked in CBSE 2023 · 30/1/1 · Q36(III)(B) · 2 marksCould a rational x make the new area exactly 220 cm²?

Arithmetic Progressions

16 questions, 36 marks across 4 papers

Asked in CBSE 2026 · 30/1/1 · Q6 · 1 markCommon difference of the AP √2, 2√2, 3√2, 4√2, ...
Asked in CBSE 2026 · 30/1/1 · Q36(i) · 1 markPotato race: bucket 5 m from the first potato, 10 potatoes 3 m apart in a line; each potato is fetched back to the bucket one at a time. Distance run for the first potato?
Asked in CBSE 2026 · 30/1/1 · Q36(ii) · 1 markSame potato race (bucket 5 m from first potato, gaps 3 m): distance run for the second potato?
Asked in CBSE 2026 · 30/1/1 · Q36(iii)(a) · 2 marksSame potato race with 10 potatoes: total distance the competitor runs.
Asked in CBSE 2026 · 30/1/1 · Q36(iii)(b) · 2 marksSame potato race: at an average speed of 5 m/s, how long to collect all 10 potatoes?
Asked in CBSE 2025 · 30/1/1 · Q11 · 1 markAn AP has Sₘ = 2m² + 3m (sum of its first m terms). Find a₂.
Asked in CBSE 2025 · 30/1/1 · Q36(i) · 1 markCharity run: round 1 is 300 m and each next round is 50 m longer (10 rounds). Write rounds 4, 5 and 6.
Asked in CBSE 2025 · 30/1/1 · Q36(ii) · 1 markSame run (300 m, then +50 m per round). Length of the 8th round?
Asked in CBSE 2025 · 30/1/1 · Q36(iii)(a) · 2 marksSame run (300 m, then +50 m per round). Total distance for all 10 rounds?
Asked in CBSE 2025 · 30/1/1 · Q36(iii)(b) · 2 marksSame run (300 m, then +50 m per round). Total distance if a runner stops after 6 rounds?
Asked in CBSE 2024 · 30/1/1 · Q4 · 1 markAn AP has a = 7, last (nth) term 84 and sum of n terms 2093/2. Find n.
Asked in CBSE 2024 · 30/1/1 · Q33(A) · 5 marksIn an AP, first term + eighth term = 32 and their product = 60. Find a and d, and the sum of the first 20 terms.
Asked in CBSE 2024 · 30/1/1 · Q33(B) · 5 marksAn AP has 40 terms. Its first 9 terms add to 153 and its last 6 terms add to 687. Find a, d and the sum of all terms.
Asked in CBSE 2023 · 30/1/1 · Q3 · 1 markp − 1, p + 1, 2p + 3 are consecutive AP terms. Find p.
Asked in CBSE 2023 · 30/1/1 · Q35(a) · 5 marksIn an AP, a₁₁ : a₁₇ = 3 : 4. Find a₅ : a₂₁ and S₅ : S₂₁.
Asked in CBSE 2023 · 30/1/1 · Q35(b) · 5 marks250 logs sit in rows: 22 at the base, then 21, then 20, each row one fewer. How many rows, and how many logs in the top row?

Triangles

17 questions, 53 marks across 4 papers

Asked in CBSE 2026 · 30/1/1 · Q7 · 1 markΔABC ~ ΔDEF with DE = 2AB and BC = 8 cm. Find EF.
Asked in CBSE 2026 · 30/1/1 · Q22(A) · 2 marksIn ΔABC, DE ∥ BC with AD = x, DB = x − 2, AE = x + 2, EC = x − 1. Find x.
Asked in CBSE 2026 · 30/1/1 · Q22(B) · 2 marksΔABC ~ ΔXYZ with AB = 4 cm, BC = 6 cm, AC = y; XY = x, YZ = 7.2 cm, XZ = 6 cm. Find x and y.
Asked in CBSE 2026 · 30/1/1 · Q34(A) · 5 marksState the Basic Proportionality (Thales) Theorem and prove it.
Asked in CBSE 2026 · 30/1/1 · Q34(B) · 5 marksΔABC ~ ΔPQR and CM, RN are their medians. Prove (i) ΔAMC ~ ΔPNR and (ii) ΔCMB ~ ΔRNQ.
Asked in CBSE 2025 · 30/1/1 · Q8 · 1 markTriangles ABC, DEF have ∠B = ∠E and ∠C = ∠F, with AB = 3 DE. How are they related?
Asked in CBSE 2025 · 30/1/1 · Q24(a) · 2 marksΔABC ~ ΔPQR with AC = 8 cm, AB = 6 cm, BC = 4 cm and PR = 6 cm. Find PQ + QR.
Asked in CBSE 2025 · 30/1/1 · Q24(b) · 2 marksIn the figure, QR/QS = QT/PR and ∠1 = ∠2. Prove ΔPQS ~ ΔTQR.
Asked in CBSE 2025 · 30/1/1 · Q33(a) · 5 marksABCD is a parallelogram, E lies on BC, and AE meets diagonal BD at F. Prove DF × EF = FB × FA.
Asked in CBSE 2025 · 30/1/1 · Q33(b) · 5 marksIn ΔABC, AD ⊥ BC and AD² = BD × DC. Prove ∠BAC = 90°.
Asked in CBSE 2024 · 30/1/1 · Q24 · 2 marksIn quadrilateral ABCD, diagonal BD bisects both ∠B and ∠D. Show (i) △ABD ~ △CBD and (ii) AB = BC.
Asked in CBSE 2024 · 30/1/1 · Q34(A) · 5 marksProve: a line drawn parallel to one side of a triangle, cutting the other two sides at distinct points, divides those sides proportionally.
Asked in CBSE 2024 · 30/1/1 · Q34(B) · 5 marksPA, QB, RC are perpendicular to AC with AP = x, BQ = y, CR = z (Q on both PC and AR). Prove 1/x + 1/z = 1/y.
Asked in CBSE 2023 · 30/1/1 · Q22 · 2 marksIn triangle ABC, X on AB and Z on AC with XZ ∥ BC; AM (M on BC) cuts XZ at Y. Given ZC = 2, AZ = 3, MC = 5, BM = 3 (all cm), find XY.
Asked in CBSE 2023 · 30/1/1 · Q32 · 5 marksProve: a line parallel to one side of a triangle, cutting the other two sides at distinct points, divides them in the same ratio.
Asked in CBSE 2023 · 30/1/1 · Q38(c)(A) · 2 marksSame set-up, with Q where OB meets the circle and PQ ∥ OA (P on AB). Find AP.
Asked in CBSE 2023 · 30/1/1 · Q38(c)(B) · 2 marksSame set-up: find PQ.

Coordinate Geometry

15 questions, 27 marks across 4 papers

Asked in CBSE 2026 · 30/1/1 · Q8 · 1 markWhere does the mid-point of the segment from (5, −4) to (6, 4) lie?
Asked in CBSE 2026 · 30/1/1 · Q23 · 2 marksA circle has centre (x − 7, 2x) and radius 5√2 and passes through (−9, 11). Find x.
Asked in CBSE 2026 · 30/1/1 · Q27 · 3 marksIn what ratio does the x-axis divide the segment from (−6, 5) to (−4, −1)? Find the point where they meet.
Asked in CBSE 2025 · 30/1/1 · Q3 · 1 markWhere does the mid-point of Q(4, 6) and P(−4, 5) lie?
Asked in CBSE 2025 · 30/1/1 · Q23 · 2 marksA circle has centre (2a, a − 7), passes through (11, −9) and has diameter 10√2. Find a.
Asked in CBSE 2025 · 30/1/1 · Q28 · 3 marksIn what ratio does the y-axis cut the segment from (5, −6) to (−1, −4)? Give the meeting point too.
Asked in CBSE 2024 · 30/1/1 · Q6 · 1 markTriangle ABC has A(5, −6), B(6, 4), C(0, 0). Find the length of median AD.
Asked in CBSE 2024 · 30/1/1 · Q11 · 1 markPoints (3, −5) and (x, −5) are 15 units apart. Find the values of x.
Asked in CBSE 2024 · 30/1/1 · Q17 · 1 markA circle has centre (2, 0); one end of a diameter is (6, 0). Find the other end.
Asked in CBSE 2024 · 30/1/1 · Q26(A) · 3 marksIn what ratio does (8/5, y) divide the join of (1, 2) and (2, 3)? Also find y.
Asked in CBSE 2024 · 30/1/1 · Q26(B) · 3 marksRectangle ABCD: A(−1, −1), B(−1, 6), C(3, 6), D(3, −1). P, Q, R, S are midpoints of AB, BC, CD, DA. Show the diagonals of PQRS bisect each other.
Asked in CBSE 2023 · 30/1/1 · Q4 · 1 markIn what ratio does the x-axis cut the segment from A(3, 6) to B(−12, −3)?
Asked in CBSE 2023 · 30/1/1 · Q13 · 1 markDistance between (0, 2√5) and (−2√5, 0).
Asked in CBSE 2023 · 30/1/1 · Q19 · 1 markA: P(0, 2) is where the line 3x + 2y = 4 meets the y-axis. R: P(0, 2) is 2 units from the x-axis.
Asked in CBSE 2023 · 30/1/1 · Q28 · 3 marks(−5, 3) and (5, 3) are vertices of an equilateral triangle that contains the origin. Find the third vertex (√3 = 1.7).

Introduction to Trigonometry

22 questions, 42 marks across 4 papers

Asked in CBSE 2026 · 30/1/1 · Q9 · 1 markIf sin θ = a/b, express cos θ in terms of a and b.
Asked in CBSE 2026 · 30/1/1 · Q10 · 1 markGiven cos A = 1/2, evaluate sin²A + 2cos²A.
Asked in CBSE 2026 · 30/1/1 · Q24(A) · 2 marksGiven tan θ = 24/7, find sin θ + cos θ.
Asked in CBSE 2026 · 30/1/1 · Q24(B) · 2 marksGiven cot θ = 7/8, evaluate (1 + sin θ)(1 − sin θ) ÷ [(1 + cos θ)(1 − cos θ)].
Asked in CBSE 2026 · 30/1/1 · Q28(A) · 3 marksGiven x = h + a cos θ and y = k + b sin θ, show that ((x − h)/a)² + ((y − k)/b)² = 1.
Asked in CBSE 2026 · 30/1/1 · Q28(B) · 3 marksShow that tan A/(1 + sec A) − tan A/(1 − sec A) = 2 cosec A.
Asked in CBSE 2025 · 30/1/1 · Q4 · 1 markθ is acute and 7 + 4 sin θ = 9. Find θ.
Asked in CBSE 2025 · 30/1/1 · Q5 · 1 markValue of tan²θ − (1/cos θ × sec θ).
Asked in CBSE 2025 · 30/1/1 · Q21(a) · 2 marksGiven y cos 0° + x cos 60° + sin 30° − cot 45° = 5, find x + 2y.
Asked in CBSE 2025 · 30/1/1 · Q21(b) · 2 marksFind tan²60° ÷ (sin²60° + cos²30°).
Asked in CBSE 2025 · 30/1/1 · Q27(a) · 3 marksProve tan θ/(1 − cot θ) + cot θ/(1 − tan θ) = 1 + sec θ cosec θ.
Asked in CBSE 2025 · 30/1/1 · Q27(b) · 3 marksLet s = sin A, c = cos A. Prove (s + c)/(s − c) + (s − c)/(s + c) = 2/(2s² − 1).
Asked in CBSE 2024 · 30/1/1 · Q7 · 1 markGiven sec θ − tan θ = m, find sec θ + tan θ.
Asked in CBSE 2024 · 30/1/1 · Q12 · 1 markIf cos(α + β) = 0, find cos((α + β)/2).
Asked in CBSE 2024 · 30/1/1 · Q23(A) · 2 marksFind the value of 2√2 cos 45° sin 30° + 2√3 cos 30°.
Asked in CBSE 2024 · 30/1/1 · Q23(B) · 2 marksWith A = 60° and B = 30°, check that sin(A + B) = sin A cos B + cos A sin B.
Asked in CBSE 2024 · 30/1/1 · Q28 · 3 marksProve tan θ/(1 − cot θ) + cot θ/(1 − tan θ) = 1 + sec θ cosec θ.
Asked in CBSE 2023 · 30/1/1 · Q6 · 1 markGiven 2 tan A = 3, evaluate (4 sin A + 3 cos A)/(4 sin A − 3 cos A).
Asked in CBSE 2023 · 30/1/1 · Q9 · 1 markEvaluate (3/4)tan²30° − sec²45° + sin²60°.
Asked in CBSE 2023 · 30/1/1 · Q23(a) · 2 marksGiven sin θ + cos θ = √3, find sin θ · cos θ.
Asked in CBSE 2023 · 30/1/1 · Q23(b) · 2 markssin α = 1/√2 and cot β = √3. Find cosec α + cosec β.
Asked in CBSE 2023 · 30/1/1 · Q30 · 3 marksProve (tan θ + sec θ − 1)/(tan θ − sec θ + 1) = (1 + sin θ)/cos θ.

Some Applications of Trigonometry

13 questions, 29 marks across 4 papers

Asked in CBSE 2026 · 30/1/1 · Q11 · 1 markA tower is 30 m high. Find the angle of elevation of its top from a car 10√3 m from its base.
Asked in CBSE 2026 · 30/1/1 · Q37(i) · 1 markTower PA stands at P with B on it below A; point O is 6 m from P on level ground. Elevation of B from O is 30°, of A is 60°. Length of wire OB?
Asked in CBSE 2026 · 30/1/1 · Q37(ii) · 1 markSame tower (OP = 6 m, elevation of A from O is 60°): length of wire OA?
Asked in CBSE 2026 · 30/1/1 · Q37(iii)(a) · 2 marksSame tower (OP = 6 m, elevations 30° to B and 60° to A): find AB.
Asked in CBSE 2026 · 30/1/1 · Q37(iii)(b) · 2 marksSame tower (OP = 6 m, elevation of B is 30°): find the area of ΔOPB.
Asked in CBSE 2025 · 30/1/1 · Q17 · 1 markA kite is 150 m above the ground; its string makes 30° with the horizontal. Find the string length.
Asked in CBSE 2025 · 30/1/1 · Q38(i) · 1 markLighthouse CE; Amrita at A (foot level) sees top C at 60°, then from B, 40 m above A, sees C at 45°. With CD = h, find BD in terms of h.
Asked in CBSE 2025 · 30/1/1 · Q38(ii) · 1 markSame lighthouse set-up (CD = h, angle at B is 45°). Find BC in terms of h.
Asked in CBSE 2025 · 30/1/1 · Q38(iii)(a) · 2 marksSame set-up (B is 40 m above A; elevations 60° from A, 45° from B). Find the lighthouse height CE, taking √3 = 1.73.
Asked in CBSE 2025 · 30/1/1 · Q38(iii)(b) · 2 marksSame set-up (angle CAE = 60°). If AC = 100 m, find AE.
Asked in CBSE 2024 · 30/1/1 · Q35 · 5 marksA 6 m pole stands on a tower. From P on the ground the pole's top has elevation 60°; from the tower's top, P has depression 45°. Find the tower's height and its distance from P (√3 = 1.73).
Asked in CBSE 2023 · 30/1/1 · Q33(a) · 5 marksA 24 m tower's top is seen at 60° elevation from the foot of a second tower; the second tower's top is seen at 30° from the foot of the first. Find the distance between them, the second tower's height, and the length of a wire joining the two tops.
Asked in CBSE 2023 · 30/1/1 · Q33(b) · 5 marksA spherical balloon of radius r subtends 60° at an observer's eye, and its centre is at 45° elevation. Show the centre's height is √2 r.

Circles

22 questions, 42 marks across 4 papers

Asked in CBSE 2026 · 30/1/1 · Q12 · 1 markTP and TQ are tangents from T to a circle with centre O, and ∠POQ = 120°. Find ∠PTQ.
Asked in CBSE 2026 · 30/1/1 · Q13 · 1 markPA is a tangent at A from external point P to a circle with centre O; AOB is a diameter and ∠POB = 125°. Find ∠APO.
Asked in CBSE 2026 · 30/1/1 · Q25 · 2 marksConcentric circles have radii 5 cm and 4 cm. Find the length of a chord of the bigger circle that is tangent to the smaller one.
Asked in CBSE 2026 · 30/1/1 · Q29(A) · 3 marksΔABC is right-angled at B with AB = 4 cm and BC = 3 cm. Find the radius of its incircle.
Asked in CBSE 2026 · 30/1/1 · Q29(B) · 3 marksA circle touches side QR of ΔPQR at S and sides PQ, PR produced at M, N. Show PM = ½(PQ + QR + PR).
Asked in CBSE 2025 · 30/1/1 · Q7 · 1 markHow are the tangents at the two ends of a diameter of a circle related?
Asked in CBSE 2025 · 30/1/1 · Q16 · 1 markPA touches a circle (centre O) at A from outside point P; AOB is a diameter and ∠POB = 115°. Find ∠APO.
Asked in CBSE 2025 · 30/1/1 · Q25 · 2 marksP is 26 m from the centre O of a circular ground; tangents PA and PB are each 10 m. Find the radius.
Asked in CBSE 2025 · 30/1/1 · Q26(a) · 3 marksBCD touches a circle with centre O at C, and A is a point on the circle with B, O, A in one line. Prove ∠BAC + ∠ACD = 90°.
Asked in CBSE 2025 · 30/1/1 · Q26(b) · 3 marksA quadrilateral is drawn around a circle. Prove each pair of opposite sides makes angles at the centre adding to 180°.
Asked in CBSE 2024 · 30/1/1 · Q19 · 1 markAssertion: tangents at the two ends of a diameter are parallel. Reason: a diameter is the longest chord.
Asked in CBSE 2024 · 30/1/1 · Q30(A) · 3 marksAB is a diameter of a circle with centre O; AQ, BP (tangents at A, B) and PQ (touching the circle at R) are tangents. Prove ∠POQ = 90°.
Asked in CBSE 2024 · 30/1/1 · Q30(B) · 3 marksA circle (centre O, radius 8 cm) is inscribed in quadrilateral ABCD, touching AD, AB, BC, DC at P, S, R, Q. AD ⊥ DC, BC = 30 cm, BS = 24 cm. Find DC.
Asked in CBSE 2024 · 30/1/1 · Q38(i) · 1 markRight triangle ABC (right angle at B), AB = 7 m, BC = 15 m, with incircle touching AC, BC, AB at P, Q, R; AP = x m. Find AR.
Asked in CBSE 2024 · 30/1/1 · Q38(ii) · 1 markSame incircle set-up (O is the centre). What kind of quadrilateral is BQOR?
Asked in CBSE 2024 · 30/1/1 · Q38(iii)(a) · 2 marksSame incircle set-up. Express PC in terms of x and then find x.
Asked in CBSE 2024 · 30/1/1 · Q38(iii)(b) · 2 marksSame incircle set-up. Find x and then the radius r.
Asked in CBSE 2023 · 30/1/1 · Q5 · 1 markPQ touches the circle (centre O) at B; chord AB has ∠AOB = 95°. Find ∠ABQ.
Asked in CBSE 2023 · 30/1/1 · Q29(a) · 3 marksTP and TQ are tangents from external point T to a circle with centre O. Show ∠PTQ = 2∠OPQ.
Asked in CBSE 2023 · 30/1/1 · Q29(b) · 3 marksA circle (centre O, radius r) is inscribed in quadrilateral ABCD with ∠B = 90°. AD = 17 cm, AB = 20 cm, DS = 3 cm. Find r.
Asked in CBSE 2023 · 30/1/1 · Q38(a) · 1 markAB touches a circle (centre O, radius 75 cm) at A, and ∠ABO = 30°. Find AB.
Asked in CBSE 2023 · 30/1/1 · Q38(b) · 1 markSame set-up: find OB.

Areas Related to Circles

14 questions, 26 marks across 4 papers

Asked in CBSE 2026 · 30/1/1 · Q14 · 1 markArc length of a sector with radius 21 cm and central angle 60°.
Asked in CBSE 2026 · 30/1/1 · Q15 · 1 markA clock's 7 cm hour hand: what angle does it turn through from 7:00 a.m. to 8:10 a.m.?
Asked in CBSE 2026 · 30/1/1 · Q38(i) · 1 markA silver-wire brooch is a circle of diameter 35 cm with 5 diameters making 10 equal sectors. Radius of the circle?
Asked in CBSE 2026 · 30/1/1 · Q38(ii) · 1 markSame brooch (diameter 35 cm): its circumference?
Asked in CBSE 2026 · 30/1/1 · Q38(iii)(a) · 2 marksSame brooch (circle of diameter 35 cm plus 5 diameters): total wire needed?
Asked in CBSE 2026 · 30/1/1 · Q38(iii)(b) · 2 marksSame brooch (diameter 35 cm, 10 equal sectors): area of one sector?
Asked in CBSE 2025 · 30/1/1 · Q15 · 1 markA sector with central angle 72° has area 40π sq. units. Find the radius.
Asked in CBSE 2025 · 30/1/1 · Q18 · 1 markA 20 cm wire is bent into an arc of radius 60/π cm. Find the angle at the centre.
Asked in CBSE 2025 · 30/1/1 · Q37(i) · 1 markA brooch is a silver-wire circle of diameter 35 mm with 5 diameters making 10 equal sectors. Central angle of each sector?
Asked in CBSE 2025 · 30/1/1 · Q37(ii) · 1 markSame brooch (diameter 35 mm, 10 equal sectors). Length of arc ACB, the arc of one sector?
Asked in CBSE 2025 · 30/1/1 · Q37(iii)(a) · 2 marksSame brooch (diameter 35 mm, 10 equal sectors). Area of one sector?
Asked in CBSE 2025 · 30/1/1 · Q37(iii)(b) · 2 marksSame brooch (diameter 35 mm, 5 diameters). Total silver wire used?
Asked in CBSE 2024 · 30/1/1 · Q32 · 5 marksIn a circle of radius 21 cm an arc subtends 60° at the centre. Find (i) the arc length and (ii) the area of the minor segment cut off by its chord.
Asked in CBSE 2023 · 30/1/1 · Q34 · 5 marksIn a circle of radius 14 cm a chord subtends 60° at the centre. Find the areas of the minor and the major segments.

Surface Areas and Volumes

11 questions, 21 marks across 4 papers

Asked in CBSE 2026 · 30/1/1 · Q16 · 1 markTotal surface area of a solid hemisphere whose diameter is 2d.
Asked in CBSE 2026 · 30/1/1 · Q30 · 3 marksA solid cylinder with a hemisphere at each end is 20 cm tall overall; the cylinder's diameter is 7 cm. Find its volume (π = 22/7).
Asked in CBSE 2025 · 30/1/1 · Q20 · 1 markAssertion: two equal hemispheres joined base to base form a sphere. Reason: a sphere of radius r has total surface area 3πr².
Asked in CBSE 2025 · 30/1/1 · Q30 · 3 marksA room is a cylinder topped by a hemispherical dome; dome radius = half the cylinder height. Air volume 1408/21 m³. Find the cylinder height (π = 22/7).
Asked in CBSE 2024 · 30/1/1 · Q13 · 1 markA solid sphere is cut into two hemispheres. Ratio of the sphere's surface area to the total surface area of both hemispheres?
Asked in CBSE 2024 · 30/1/1 · Q15 · 1 markVolume of the biggest right circular cone cut from a solid cube of edge 2 cm?
Asked in CBSE 2024 · 30/1/1 · Q31 · 3 marksA hollow cylinder of length 14 cm has outer radius 1 cm more than inner radius. Metal volume is 176 cm³. Find both radii.
Asked in CBSE 2023 · 30/1/1 · Q10 · 1 markA cylinder 5 cm tall has curved surface area 94.2 cm². Find its radius (π = 3.14).
Asked in CBSE 2023 · 30/1/1 · Q12 · 1 markCurved surface area of a cone with h = 24 cm and r = 7 cm.
Asked in CBSE 2023 · 30/1/1 · Q31(a) · 3 marksA room is a cylinder topped by a hemispherical dome; the dome radius is half the cylinder height. The room holds 1408/21 m³ of air. Find its total height (π = 22/7).
Asked in CBSE 2023 · 30/1/1 · Q31(b) · 3 marksA cone (r = 3 cm, h = 12 cm) is filled with ice-cream except its bottom 1/6 of volume, with a hemisphere of ice-cream on top. Find the ice-cream volume (π = 3.14).

Statistics

15 questions, 27 marks across 4 papers

Asked in CBSE 2026 · 30/1/1 · Q17 · 1 markData has mean 12 and mode 21. Find its median.
Asked in CBSE 2026 · 30/1/1 · Q35 · 5 marksClasses 0-10, 10-20, ..., 60-70 have frequencies 1, x, 5, 7, y, 3, 1. Total frequency is 25 and mean is 35. Find x and y.
Asked in CBSE 2025 · 30/1/1 · Q12 · 1 markA data set has mode 15x and mean 18x. Find its median.
Asked in CBSE 2025 · 30/1/1 · Q35 · 5 marksDaily allowance classes 11–13, 13–15, ..., 23–25 have frequencies 7, 6, 9, 13, f, 5, 4. Mean is 18. Find f and the mode.
Asked in CBSE 2024 · 30/1/1 · Q9 · 1 markFor values x₁ … xₙ with frequencies f₁ … fₙ and mean x̄, find Σ fᵢ(xᵢ − x̄).
Asked in CBSE 2024 · 30/1/1 · Q14 · 1 markWhat is the name of the middle value of data arranged in order?
Asked in CBSE 2024 · 30/1/1 · Q37(i) · 1 markBINGO data (48 balls): 0-15: 8, 15-30: 9, 30-45: 10, 45-60: 12, 60-75: 9. Give the median class.
Asked in CBSE 2024 · 30/1/1 · Q37(iii)(a) · 2 marksSame BINGO table. Find the median.
Asked in CBSE 2024 · 30/1/1 · Q37(iii)(b) · 2 marksSame BINGO table. Find the mode.
Asked in CBSE 2023 · 30/1/1 · Q11 · 1 markMarks of 80 students as a 'less than' table: <10: 3, <20: 12, <30: 27, <40: 57, <50: 75, <60: 80. Find the modal class.
Asked in CBSE 2023 · 30/1/1 · Q15 · 1 markEvery observation is increased by 3. What happens to the mean?
Asked in CBSE 2023 · 30/1/1 · Q37(I) · 1 mark2018 monsoon rainfall classes of width 200 mm from 200 to 1800 mm have sub-division counts 2, 4, 7, 4, 2, 3, 1, 1 (in order). Give the modal class.
Asked in CBSE 2023 · 30/1/1 · Q37(II)(A) · 2 marksFind the median rainfall from the same table.
Asked in CBSE 2023 · 30/1/1 · Q37(II)(B) · 2 marksFind the mean rainfall from the same table.
Asked in CBSE 2023 · 30/1/1 · Q37(III) · 1 markSub-divisions with 1000 mm or more count as good rainfall. How many are there?

Probability

15 questions, 21 marks across 4 papers

Asked in CBSE 2026 · 30/1/1 · Q18 · 1 markOne die is thrown. Probability that the number shown is not 3.
Asked in CBSE 2026 · 30/1/1 · Q19 · 1 markAssertion: P(a leap year has 53 Mondays) = 2/7. Reason: P(a non-leap year has 53 Mondays) = 5/7.
Asked in CBSE 2026 · 30/1/1 · Q31 · 3 marksTwo different-coloured dice are rolled together. List the outcomes and find P(same number on both) and P(different numbers).
Asked in CBSE 2025 · 30/1/1 · Q13 · 1 markOne card is drawn from a pack of 52. Probability that it is a red face card?
Asked in CBSE 2025 · 30/1/1 · Q19 · 1 markAssertion: choosing a number at random from 1 to 20 has probability 1. Reason: an event with probability 1 is a sure event.
Asked in CBSE 2025 · 30/1/1 · Q31 · 3 marksTwo dice are rolled together. Probability that their numbers differ by 2?
Asked in CBSE 2024 · 30/1/1 · Q2 · 1 markA player wins a game with probability 0.79. Probability that he loses it?
Asked in CBSE 2024 · 30/1/1 · Q8 · 1 markData: 1, 4, 7, 9, 16, 21, 25. Drop the even values; probability that a value picked at random from the rest is prime?
Asked in CBSE 2024 · 30/1/1 · Q16 · 1 markTwo dice are thrown. Probability that the total is 2, 3 or 5?
Asked in CBSE 2024 · 30/1/1 · Q22 · 2 marksA black card is missing from a 52-card pack. One card is drawn from the rest. Find P(queen of hearts).
Asked in CBSE 2024 · 30/1/1 · Q37(ii) · 1 markBalls are numbered 1 to 75. For the first ball drawn, find P(even number).
Asked in CBSE 2023 · 30/1/1 · Q16 · 1 markp = P(event happens), q = P(it does not happen). How are p and q related?
Asked in CBSE 2023 · 30/1/1 · Q17 · 1 mark6000 lottery tickets are sold; a girl's chance of first prize is 0.08. How many tickets did she buy?
Asked in CBSE 2023 · 30/1/1 · Q18 · 1 markOf 20 people, 5 cannot swim. Probability that a randomly chosen person can swim.
Asked in CBSE 2023 · 30/1/1 · Q25 · 2 marksA bag has 4 red, 3 blue, 2 yellow balls; one is drawn at random. Find P(red) and P(yellow).

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