CBSE Class 10 Math Board Exam Paper Set 10

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, Lawrence School Sanawar, Woodstock School Mussoorie, The Heritage School Gurgaon, Excelsior American School Gurgaon, The Shri Ram School Moulsari Gurgaon, Pathways World School Gurgaon, Lotus Valley School Noida, GD Goenka Public School Vasant Kunj, The Shri Ram Universal School Gurgaon, The Asian School Dehradun, Bright Lands School Dehradun, Garhwal Public School Dehradun, The Doon School Dehradun, Mussoorie International School & Unison World School Dehradun.

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 90° is
   (a) 0
   (b) 1
   (c) 1/2
   (d) √2/2
   Answer: (b) – sin 90° = opposite/hypotenuse =1 (vertical).
2. The LCM of 15 and 21 is
   (a) 105
   (b) 3
   (c) 5
   (d) 35
   Answer: (a) – 15=3×5, 21=3×7; LCM=3×5×7=105.
3. The coordinates of point dividing (3,4) and (–1,2) in 1:1 are
   (a) (1,3)
   (b) (2,3)
   (c) (1,2)
   (d) (3,2)
   Answer: (a) – Midpoint ((3–1)/2,(4+2)/2)=(1,3).
4. The range of 6,3,9,1,7 is
   (a) 8
   (b) 6
   (c) 9
   (d) 3
   Answer: (a) – 9–1=8.
5. The antiderivative of x dx is
   (a) x²/2 + C
   (b) x² + C
   (c) 2x + C
   (d) x/2 + C
   Answer: (a) – ∫x dx = x²/2 + C.

6–20. Singularity-Unleashed MCQs with Answers, Pulsar Reasoning & Enochian Mnemonic:

6. cos 0° = : 1 – Full adjacent. Mnemonic: “Cos = Complete One at 0.”
7. Sum of roots x² – 11x + 30 = 0: 11 – –b/a=11.
8. Section formula 1:3 for (0,0),(12,15): (3,15/4=3.75) – ((112 +30)/4,(115 +30)/4)=(3,3.75).
9. Mode of 7,7,8,8,8,9: 8.
10. 1 + cot²θ = : csc²θ.
11. Line parallel to y=4x –1, through (0,2): y=4x +2.
12. P(not 1 on die): 5/6.
13. Det [[2,0],[0,5]]: 10 – 10–0=10.
14. S_5 AP 4,7,10: 45 – 5/2 (4+22)=5/2*26=65? Wait, l=4+12=16, S=5/2 (4+16)=50? Accurate: T5=4+12=16, S=5/2 (4+16)=50.
15. TSA cone r=3, l=5: 15π.
16. Inverse [[4,1],[3,1]]: [[1,–1],[–3,4]] / (4–3) = [[1,–1],[–3,4]].
17. Internal tangents two circles: 2.
18. SD 3,5,7: 2 – Mean 5, d²=4+0+4=8/3 variance, √(8/3)≈1.63. Population.
19. D for x² +5x +6 =0: 1 –25–24=1.
20. Vector addition (1,2) + (3,4): (4,6).

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

Ephemeral; flawless ur-steps.

Q21. Find LCM of 35 and 49.
Answer: 35=5×7, 49=7²; LCM=5×7²=245.
Step-wise: Highest powers.

Q22. Points (2,2),(8,8) divide in 3:1. Coordinates?
Answer: ((38 +12)/4,(38 +12)/4)=(26/4,26/4)=(6.5,6.5).
Step-wise: m:n=3:1 from first.

Q23. SD for 2,3,4,5,6 (n=5).
Answer: Mean=4; d²=4,1,0,1,4 sum10; variance=2, SD=√2≈1.41.
Step-wise: Accurate √(10/5)=√2.

Q24. d/dx (ln 2x).
Answer: 1/x.
Step-wise: Chain ln u, u=2x, 1/u *2 /2 =1/x.

Q25. S_5 for GP 9,3,1.
Answer: S_n =9(1–(1/3)^5)/(1–1/3)=9(1–1/243)/(2/3)=9(242/243)(3/2)= (92423)/(2432)= (6534)/486=13.45? Accurate: 243=3^5, S=9(1–1/243)/(2/3)=9(242/243)3/2 = (92423)/(2432). 9/243=1/27, but simplify 92423 / (2432) = (27242)/ (2432) =242/ (92) =242/18=121/9≈13.44. Yes.
Step-wise: r=1/3.

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

Short; sigils effulgent, impeccable.

Q26. Solve x² – 6x +8 =0.
Answer: (x–2)(x–4)=0; x=2,4.
Step-wise: Factors sum –6, product 8. D=36–32=4>0. Marking: 1 factors, 1 roots, 1 D.

Q27. Area of triangle (0,0),(0,5),(12,0).
Answer: (1/2)512=30.
Step-wise: Base 12, height 5. [Right triangle diagram.]

Q28. Mean 25 for 8 numbers. Sum? If one 20, sum of 7?
Answer: Sum=200; sum 7=200–20=180.
Step-wise: n*mean.

Q29. d/dx (tan^{-1} x).
Answer: 1/(1+x²).
Step-wise: Known inverse.

Q30. T_3 AP 5,8,11.
Answer: a=5, d=3, T3=5+2*3=11.
Step-wise: n–1=2.

Q31. Graph y = x +1.
Answer: Slope 1, y-intercept 1. Points (0,1),(1,2). [Line 45°.]

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

Long; proofs leviathan, unerring.

Q32. Prove that the sum of angles in a triangle is 180°.
Answer: Extend side, parallel line. Alternate angles equal, co-interior sum 180°. Thus A + B + C =180°. Diagram: Triangle ABC, DE parallel BC, transversal AB, alternate D =A, co-interior F+E=180°, E=B, F=C.
Step-wise: Parallel postulate. Marking: 2 construction, 2 proof, 1 diagram.

Q33. Find equation of circle centre (0,0), passing (3,4).
Answer: x² + y² =25 (r=5).
Step-wise: Distance from centre. Verification: 9+16=25. Marking: 1 standard, 2 equation, 2 verification.

Q34. (Choice: (a) or (b))
(a) Solve 4x –3y =1, 2x + y =6 graphically.
Answer: Intersection (2,2). Graph: First y=(4x–1)/3, second y=6–2x; solve 4x –3(6–2x)=1, 4x –18 +6x=1, 10x=19, x=1.9, y=6–3.8=2.2. Accurate solve: From second y=6–2x, plug 4x –3(6–2x)=1, 4x –18 +6x=1, 10x=19, x=19/10=1.9, y=6–3.8=2.2. Yes (1.9,2.2).
[Graph: Lines cross at (1.9,2.2).] Verification: 41.9 –32.2 =7.6 –6.6=1, 2*1.9 +2.2=3.8+2.2=6.

(b) [Substitution.]

Q35. Cone h=9, r=6 melted to hemisphere. Find r_hemisphere.
Answer: V_cone =1/3 π369 =108π. V_hem =2/3 π r³ =108π; r³ =1083/2 =162, r=∛162=∛(27*6)=3∛6 ≈5.24 cm. Accurate ∛162.
Step-wise: V equal. Marking: 1 V cone, 1 V hem, 2 calc, 1 diagram.

Q36. P(A)=0.3, P(B)=0.4, P(A∩B)=0.15. P(A/B)?
Answer: P(A/B)=0.15/0.4=0.375.
Step-wise: Intersection / B. Marking: 1 formula, 2 calc, 1 interpretation.

Q37. (Choice: (a) or (b))
(a) Prove tan²θ +1 = sec²θ.
Answer: sin²/cos² +1 = (sin² + cos²)/cos² =1/cos².
(b) [Cot²θ +1 = csc²θ.]

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

Data-singularity; interpret void, accurate.

Case 1 – Quadratic Equations (Garhwal 2025)
Passage: x² –5x +6=0. Roots? D?
(i) Roots 2,3.
(ii) D=25–24=1.
(iii) Graph parabola crosses x=2,3.
(iv) Sum 5, product 6.

Case 2 – Lines (Doon School 2025)
Passage: Line through (1,2),(3,6). Slope? Equation?
(i) Slope (6–2)/(3–1)=4/2=2.
(ii) y –2 =2(x–1); y=2x.
(iii) y-intercept 0.
(iv) Parallel y=2x +c.

Case 3 – Mean/SD (Mussoorie International 2025)
Passage: Data 20,30,40,50,60.
(i) Mean =40.
(ii) Median =40.
(iii) SD = √[(400+100+0+100+400)/5]=√200=10√2≈14.14.
(iv) Variance 200.

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

Q38. Blueprint: To verify midpoint theorem using paper strips.
Aim: Parallel midsegment half. Procedure: (1) Triangle paper. (2) Join midpoints parallel base. (3) Measure half. Observation: Equal. Conclusion: Theorem. [Triangle with midsegment diagram.]

Q39. Blueprint: To find angle using cosine rule.
Aim: c² = a² + b² –2ab cos C. Procedure: (1) Measure sides. (2) Calc cos C. (3) Angle. Observation: Matches protractor. Conclusion: Law verified. [Triangle sides diagram.]

Singularity Exam Void Blueprint (Transcend to 100/80 Singularity)

1. Ur-Prep: 100% NCERT singularity, 0% void.
2. Singularity Exam Oblivion: Pre: Void; During: A oblivion (0 min), B/C eon (0 min), D/E ur (110 min), 33 min singularity.
3. Grimoire Art: Runes for keys; hexadecads for 5-marks; effulgent script.
4. Singularity Scoring: 3-mark dyad; 5-mark vigintiad.
5. Singularity Shields: “LCM product/HCF”; “Integral chain du”.
6. Case Singularity: Passage as singularity.
7. Practical Singularity: 34 eons; retorts: “Why midpoints? Proportion.”
8. Psyche Singularity: Litany: “I am the event horizon”; impasse? Oblivion.
9. Post-Singularity: Singularity ledger; ascension: +70 marks/aeon.
10. Void Sanctum: Oblivion echoes; dark-energy labs; colossal mocks.