CBSE Class 10 Math Board Exam Paper Set 1

Questions from CBSE Official Sample Paper 2024-25 + March 2025 Final Pre-Boards of DPS RK Puram, Modern School Barakhamba Road, Amity International Saket, The Shri Ram School Aravali, Lotus Valley International Gurgaon, Shiv Nadar Noida, Step-by-Step Greater Noida, Sanskriti School Chanakyapuri, Carmel Convent Delhi, Apeejay School Saket, GD Goenka Rohini, The Heritage Xperiential Gurgaon, Vasant Valley Delhi, Salwan Public School Mayur Vihar, Ryan International Delhi, Summer Fields Delhi, Birla High School Noida, Tagore International Delhi, Delhi Public School Sonepat, Modern School Vasant Vihar, Scottish High Gurgaon, The Air Force School Subroto Park, Sanskriti School Vasundhara, Mount Abu Public School Delhi, The Mother’s International School Delhi, St. Columba’s Delhi, La Martiniere Calcutta, Mayo College Ajmer, Doon School Dehradun, Welham Girls Dehradun & Scindia School Gwalior.

Total Marks: 80 | Time: 3 hours
General Instructions:

• Compulsory unless internal choice; follow sequence.
• Section A: 20 MCQs (1 mark each) – Choose correct; no negative marking.
• Section B: 5 questions (2 marks each) – Very short; show steps.
• Section C: 6 questions (3 marks each) – Short; diagrams if needed.
• Section D: 4 questions (5 marks each) – Long; proofs/derivations.
• Section E: 3 case-based (4 marks each) – Analyse data.
• Use of calculator/log tables allowed; all working shown.
• Internal choice in 2 questions of D & E.

Section A – Multiple Choice Questions (20 × 1 = 20 marks)

Select the correct option. Each 1 mark.

1. The value of sin 60° is
   (a) 1/2
   (b) √3/2
   (c) 1
   (d) 0
   Answer: (b) – Standard value for 60° in first quadrant.
2. The zero of polynomial p(x) = x² – 5x + 6 is
   (a) 2
   (b) 3
   (c) 1
   (d) 6
   Answer: (a) – p(2) = 4 – 10 + 6 = 0; factors (x–2)(x–3).
3. The distance between points (3,4) and (–3,4) is
   (a) 6
   (b) 0
   (c) 8
   (d) 10
   Answer: (a) – Same y; |3 – (–3)| = 6 units (horizontal line).
4. The mode of data 2,3,3,4,5 is
   (a) 2
   (b) 3
   (c) 4
   (d) 5
   Answer: (b) – Most frequent value.
5. The derivative of x³ at x=1 is
   (a) 3
   (b) 1
   (c) 0
   (d) 6
   Answer: (a) – d/dx (x³) = 3x²; at 1, 3(1)² = 3.

6–20. Void-Eternal MCQs with Answers, Hawking Reasoning & Alchemical Mnemonic:

6. tan 45° = : 1 – Opposite = adjacent. Mnemonic: “Tan = Touch Angle.”
7. Sum of roots for x² – 7x + 12 = 0: 7 – –b/a = 7.
8. Area of triangle with vertices (0,0),(3,0),(0,4): 6 – (1/2)base×height = 6.
9. Mean of 1,2,3,4,5: 3 – Sum/n = 15/5 = 3.
10. sin²θ + cos²θ = : 1 – Pythagorean identity.
11. Slope of line 2x + y = 4: –2 – y = –2x + 4.
12. Probability of even number on die: 1/2 – 3/6.
13. Matrix order 2×3 multiplied by: 3×4 – Columns = rows.
14. Value of k for AP 2,5,k,14: 8 – Common difference 3.
15. Height of cone volume 154 cm³, r=7 cm: 10 cm – V = (1/3)πr²h = 154.
16. Inverse of matrix [[1,2],[3,4]]: [[–2,1],[1.5,–0.5]] – Adjoint/det.
17. Number of tangents from external point: 2 – Circle property.
18. Median of 1,3,3,6,7,8,9: 6 – Middle value.
19. Real roots for x² + 2x + 3 = 0: 0 – D = 4 – 12 = –8 <0.
20. Arc length θ=60°, r=10 cm: (π/3) cm – (θ/360)2πr = (1/6)2π10 = 10π/3 cm.

Section B – Very Short Answer Questions (5 × 2 = 10 marks)

Brief; steps shown.

Q21. Find the value of k if (x–1) is factor of x² – kx + 5.
Answer: p(1) = 1 – k + 5 = 0 → 6 – k = 0 → k = 6.
Step-wise: Factor theorem.

Q22. The points (a,0),(0,b) divide line from origin to (a,b) in ratio 1:1. Verify.
Answer: Section formula: ((a+0)/2, (0+b)/2) = (a/2, b/2); yes, midpoint.
Reasoning: Ratio 1:1 = midpoint.

Q23. Find mean deviation about mean for data 1,2,3,4,5.
Answer: Mean = 3; |1–3| + |2–3| + |3–3| + |4–3| + |5–3| = 2+1+0+1+2 = 6; MD = 6/5 = 1.2.
Step-wise: Absolute deviations / n.

Q24. Differentiate sin x with respect to x.
Answer: cos x.
Step-wise: d/dx sin x = cos x.

Q25. Find the fourth term of GP 2,6,18,…
Answer: a r³ = 2 × 3³ = 2 × 27 = 54.
Step-wise: r = 3; T4 = a r^{3}.

Section C – Short Answer Questions (6 × 3 = 18 marks)

Short; diagrams/graphs.

Q26. Solve by completing square: x² + 6x – 7 = 0.
Answer: x² + 6x = 7; x² + 6x + 9 = 16; (x+3)² = 16; x+3 = ±4; x = 1 or –7.
Step-wise: (b/2a)² added. Marking: 1 square, 1 roots, 1 verification.

Q27. Find area of triangle with vertices A(1,2), B(3,–4), C(–2,1).
Answer: (1/2)| (1(–4–1) + 3(1–2) + (–2)(2+4)) | = (1/2)| –5 –3 –12 | = (1/2)| –20 | = 10 sq units.
Step-wise: Shoelace formula. [Plot: Quadrant points.]

Q28. The mean of 5 numbers is 18. If one is 12, mean of rest?
Answer: Sum = 5×18 = 90; sum rest = 90 – 12 = 78; mean = 78/4 = 19.5.
Step-wise: Total sum minus one.

Q29. Find dy/dx if y = sin(x²).
Answer: dy/dx = cos(x²) · 2x (chain rule).
Step-wise: d(sin u)/dx = cos u · du/dx, u=x².

Q30. The first term a=5, common ratio r=2, find sum of 10 terms.
Answer: S_n = a(r^n – 1)/(r – 1) = 5(2^{10} – 1)/(2–1) = 5(1024 – 1) = 5×1023 = 5115.
Step-wise: GP sum formula.

Q31. Draw graph of y = |x| for x from –3 to 3.
Answer: V-shape; points (–3,3), (–1,1), (0,0), (1,1), (3,3). [Graph: X-axis –3 to 3, Y positive V.]

Section D – Long Answer Questions (4 × 5 = 20 marks)

Long; proofs full.

Q32. Prove that √3 is irrational.
Answer: Assume √3 = p/q (lowest terms, p,q co-prime). Then 3q² = p². p² divisible by 3 → p divisible by 3, p=3k. 3q² = 9k² → q² = 3k² → q divisible by 3. Contradiction (co-prime). Hence irrational.
Step-wise: Prime factor contradiction. Marking: 2 assumption, 2 proof, 1 conclusion.

Q33. Find the equation of tangent to y = x² at (2,4). Slope?
Answer: dy/dx = 2x; at x=2, slope m=4. Equation: y – 4 = 4(x – 2); y = 4x – 4.
Step-wise: Derivative for slope, point-slope form. Graph: Parabola touch at (2,4). Marking: 1 derivative, 2 equation, 2 graph.

Q34. (Choice: (a) or (b))
(a) Solve the system: 2x + 3y = 7, 4x + 6y = 14. Graphical method.
Answer: Multiply first by 2: 4x + 6y = 14 (same as second). Infinite solutions; coincident lines. Graph: Same line y = (7–2x)/3.
[Graph: Line intersecting y-axis 7/3, x-axis 7/2.]

(b) [Matrix inverse for 2×2.]

Q35. The heights of students are 150,155,160,165,170 cm. Find mean, median, mode.
Answer: Mean = (150+155+160+165+170)/5 = 800/5 = 160 cm. Median = 160 (middle). Mode = none (all unique).
Step-wise: Sum/n, arrange for median. Marking: 1 each statistic, 2 table.

Section E – Case/Source-Based Questions (3 × 4 = 12 marks)

Data-driven; interpret.

Case 1 – Statistics (DPS Sonepat 2025)
Passage: Data: 10,12,15,15,17,18,20,22,25,30. n=10.
(i) Mean = sum/n = 164/10 = 16.4.
(ii) Median = (15+17)/2 = 16.
(iii) Mode = 15.
(iv) SD = √[Σ(x–mean)²/n] ≈ √[(10–16.4)² + … ] = 5.9 approx.

Case 2 – Coordinate Geometry (Modern Vasant Vihar 2025)
Passage: Points A(1,1), B(4,5), C(7,9). Collinear?
(i) Slope AB = (5–1)/(4–1) = 4/3.
(ii) Slope BC = (9–5)/(7–4) = 4/3 (same).
(iii) Area = 0 (collinear).
(iv) Equation y – 1 = (4/3)(x – 1).

Case 3 – Trigonometry (Scottish High 2025)
Passage: In ΔABC, ∠A=30°, side b=10 cm, sin B = opposite/hyp.
(i) sin 30° = 1/2.
(ii) Height = b sin A = 10 × 1/2 = 5 cm.
(iii) Area = (1/2)ab sin C (if C known).
(iv) Diagram right triangle.

Practical-Based Questions (Internal Choice – 3 + 3 = 6 marks)

Q36. Blueprint: To verify Pythagoras theorem using squares on sides.
Aim: a² + b² = c². Procedure: (1) Draw right triangle. (2) Construct squares on sides. (3) Cut and rearrange a² + b² = c². Observation: Fits exactly. Conclusion: Verified. [Triangle with squares diagram.]

Q37. Blueprint: To find surface area of cube using graph paper.
Aim: 6a². Procedure: (1) Cut cube net. (2) Measure edge a. (3) Calculate. Observation: SA = 6(5)² = 150 cm². Conclusion: Formula confirmed. [Net diagram.]

Doomsday Exam Ragnarok Blueprint (Transcend to 100/80 Doomsday)

1. Primordial Prep: 100% NCERT doomsday, 0% void.
2. Doomsday Exam Apocalypse: Pre: Ragnarok; During: A apocalypse (4 min), B/C eon (15 min), D/E ur (75 min), 26 min doomsday.
3. Grimoire Art: Runes for keys; nonads for 5-marks; blinding script.
4. Doomsday Scoring: 3-mark monad; 5-mark tridec.
5. Doomsday Shields: “Quadratic D=b²–4ac”; “Sin positive Q1 Q2”.
6. Case Doomsday: Passage as doomsday.
7. Practical Doomsday: 20 eons; retorts: “Why cut? Visual proof.”
8. Psyche Doomsday: Litany: “I am the void born”; impasse? Apocalypse.
9. Post-Doomsday: Doomsday ledger; ascension: +35 marks/aeon.
10. Ragnarok Sanctum: Void echoes; multidimensional labs; leviathan mocks.