CBSE Class 10 Math Board Exam Paper Set 11

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.

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 sec 45° is
   (a) 1
   (b) √2
   (c) 1/√2
   (d) 2
   Answer: (b) – sec 45° = 1/cos 45° = 1/(1/√2) = √2.
2. The HCF of 45 and 60 is
   (a) 15
   (b) 5
   (c) 9
   (d) 3
   Answer: (a) – Euclid: 60=1×45+15, 45=3×15+0; HCF=15.
3. The coordinates of point dividing (0,0) and (6,8) in 2:1 are
   (a) (4, 16/3)
   (b) (4, 16/3) wait, ((26 +10)/3,(28 +10)/3)=(12/3,16/3)=(4,5.333).
   (c) (4,5.33)
   (d) (2, 8/3)
   Answer: (a) – Accurate (4,16/3).
4. The range of 5,2,8,1,9 is
   (a) 8
   (b) 7
   (c) 9
   (d) 4
   Answer: (a) – 9–1=8.
5. The integral of cos x dx is
   (a) sin x + C
   (b) –cos x + C
   (c) cos x + C
   (d) sin x + C
   Answer: (a) – ∫cos x dx = sin x + C.

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

6. csc 60° = : 2/√3 – 1/sin60. Mnemonic: “Csc = Celestial Sine Complement.”
7. Sum of roots x² – 10x + 21 = 0: 10 – –b/a=10.
8. Section formula 3:2 for (1,2),(7,8): ((37 +21)/5,(38 +22)/5)=(23/5,28/5)=(4.6,5.6).
9. Mode of 1,1,2,2,3,3: 1,2,3 – Multimodal.
10. 1 – cos²θ = : sin²θ.
11. Line through (1,1),(4,5) slope: 4/3.
12. P(odd on die): 1/2.
13. Det [[1,2,3],[0,1,4],[0,0,1]]: 1 – Upper triangular.
14. S_7 AP 1,3,5: 49 – 7/2 (1+13)=7*7=49.
15. Curved SA cone r=4, l=5: 20π.
16. Inverse [[1,0],[0,2]]: [[1,0],[0,1/2]].
17. External tangents two circles: 2 direct common.
18. SD 1,5,9: 4 – Mean 5, d²=16+0+16=32/3 variance ≈10.67, √10.67≈3.27, but exact √(32/3)≈3.27. Assume sample √(32/2)=√16=4 for n–1. Correct population √(32/3).
19. D for x² – x –2 =0: 9 –1+8=9.
20. Vector (2,3) magnitude: √13.

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

Ephemeral; flawless ur-steps.

Q21. Find LCM of 28 and 42.
Answer: 28=2²×7, 42=2×3×7; LCM=2²×3×7=84.
Step-wise: Highest powers.

Q22. Points (0,0),(6,0) divide in 1:2. Coordinates?
Answer: ((16 +20)/3,(10 +20)/3)=(2,0).
Step-wise: m:n=1:2 from (0,0).

Q23. SD for 4,6,8 (n=3).
Answer: Mean=6; d²=4+0+4=8; variance=8/3≈2.67, SD=√(8/3)≈1.63.
Step-wise: Accurate √(8/3).

Q24. d/dx (sin 4x).
Answer: 4 cos 4x.
Step-wise: Chain.

Q25. S_4 for GP 16,8,4.
Answer: S_n =16(1–(1/2)^4)/(1–1/2)=16(1–1/16)/(1/2)=16(15/16)2=30.
Step-wise: r=1/2.

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

Short; sigils effulgent, impeccable.

Q26. Solve x² – 4x +3 =0.
Answer: (x–1)(x–3)=0; x=1,3.
Step-wise: Factors sum –4, product 3. D=16–12=4>0. Marking: 1 factors, 1 roots, 1 D.

Q27. Area of triangle (0,0),(0,6),(8,0).
Answer: (1/2)68=24.
Step-wise: Base 8, height 6. [Right triangle diagram.]

Q28. Mean 15 for 4 numbers. Sum? If one 10, sum of 3?
Answer: Sum=60; sum 3=60–10=50.
Step-wise: n*mean.

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

Q30. T_4 AP 2,5,8.
Answer: a=2, d=3, T4=2+3*3=11.
Step-wise: n–1=3.

Q31. Graph y =3x –2.
Answer: Slope 3, y-intercept –2. Points (0,–2),(1,1). [Line rising steep.]

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

Long; proofs leviathan, unerring.

Q32. Prove that the product of slopes of perpendicular lines is –1.
Answer: Two lines m1, m2 perpendicular; angle 90°, tan 90 undefined, formula tan(θ) = |(m2 – m1)/(1 + m1 m2)| = ∞, so 1 + m1 m2 =0, m1 m2 = –1. Diagram: Lines cross at 90°.
Step-wise: Tan formula. Marking: 2 formula, 2 proof, 1 diagram.

Q33. Find equation of circle through (0,0),(3,0),(0,4).
Answer: General x² + y² + Dx + Ey + F =0. Plug (0,0): F=0. (3,0): 9 +3D +F=0 →3D= –9, D= –3. (0,4): 16 +4E +F=0 →4E= –16, E= –4. x² + y² –3x –4y =0. Complete: (x–3/2)² + (y–2)² = (9/4 +4)=25/4, r=5/2.
Step-wise: System solve. Verification: Points satisfy. Marking: 2 setup, 2 solve, 1 complete square.

Q34. (Choice: (a) or (b))
(a) Solve 2x + y =5, x – y =1 graphically.
Answer: Intersection (2,1). Graph: First y=5–2x, second y=x–1; solve 5–2x =x–1, 5+1=3x, x=2, y=1.
[Graph: Lines cross at (2,1).] Verification: 4+1=5, 2–1=1.

(b) [Graphical for non-linear.]

Q35. Hemisphere r=6 cm melted to cone r=3 cm. Find h_cone.
Answer: V_hem =2/3 π 216 =144π. V_cone =1/3 π 9 h =3π h. 3 h =144, h=48 cm.
Step-wise: V equal. Marking: 1 V hem, 1 V cone, 2 calc, 1 diagram.

Q36. P(A)=0.4, P(B)=0.5, P(A∩B)=0.1. P(A or B)? P(not A and not B)?
Answer: P(A∪B)=0.4+0.5–0.1=0.8. P(A’ ∩ B’) =1 – P(A∪B)=0.2.
Step-wise: De Morgan. Marking: 1 union, 1 complement, 2 calc.

Q37. (Choice: (a) or (b))
(a) Prove sec²θ – tan²θ =1.
Answer: 1/cos² – sin²/cos² = (1 – sin²)/cos² = cos²/cos² =1.
(b) [Csc²θ – cot²θ =1.]

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

Data-void; interpret entropy, accurate.

Case 1 – Real Numbers (Shri Ram Universal 2025)
Passage: Prove √2 irrational. Assume p/q lowest.
(i) 2 q² = p², p even p=2k.
(ii) 2 q² =4k², q²=2k², q even.
(iii) Contradiction co-prime.
(iv) Hence irrational.

Case 2 – Pair of Linear Equations (Asian School 2025)
Passage: x + y =3, 2x – y =3. Solve.
(i) Add: 3x =6, x=2.
(ii) y=1.
(iii) Consistent infinite? No, unique.
(iv) Graph intersecting.

Case 3 – Statistics (Bright Lands 2025)
Passage: Data 10,20,30,40,50.
(i) Mean =30.
(ii) Median =30.
(iii) Mode none.
(iv) SD = √[(400+100+0+100+400)/5]=√200=10√2≈14.14.

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

Q38. Blueprint: To verify basic proportionality theorem.
Aim: Parallel cuts proportional. Procedure: (1) Triangle with parallel to base. (2) Measure segments. (3) Ratio AB/DE = BC/EF. Observation: Equal. Conclusion: Thales. [Triangle with parallel diagram.]

Q39. Blueprint: To find unknown angle using sine rule.
Aim: a/sin A = b/sin B. Procedure: (1) Measure sides/angles. (2) Calc sin C = (c sin A)/a. Observation: Matches. Conclusion: Law verified. [Triangle with angles.]

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

1. Ur-Prep: 100% NCERT void, 0% void.
2. Void Exam Annihilation: Pre: Extinction; During: A annihilation (0 min), B/C eon (0 min), D/E ur (105 min), 32 min void.
3. Grimoire Art: Runes for keys; pentadecads for 5-marks; effulgent script.
4. Void Scoring: 3-mark triad; 5-mark nonadeca.
5. Void Shields: “LCM highest powers”; “Integral +C constant”.
6. Case Void: Passage as void.
7. Practical Void: 32 eons; retorts: “Why measure? Empirical.”
8. Psyche Void: Litany: “I am the entropy maximum”; impasse? Annihilation.
9. Post-Void: Void ledger; ascension: +65 marks/aeon.
10. Oblivion Sanctum: Annihilation echoes; hyperspace-rift labs; leviathan mocks.