MATHEMATICS (SCIENCE) — MARCH 2026
ANSWER KEY | Maximum : 60 Scores
📘 Questions 1 – 8 3 marks each
1. Matrices 3 marks
(i) Order of PR: 2×3 (1)
(ii) Show PR = 3P: R = 3I₃ → PR = P(3I₃) = 3P (1)
(iii) 3P + Q = [[13,-5,7],[-11,6,13]] (1)
2. Relation R 3 marks
(i) Reflexive: (x,x)∈R; Symmetric: x=y ⇒ y=x; Transitive: x=y, y=z ⇒ x=z. Hence equivalence. (2)
(ii) Equivalence classes: [1]={1}, [2]={2}, [3]={3}, [4]={4} (1)
3. Definite integral 3 marks
(i) ∫₀ᵖⁱ sin x dx = [–cos x]₀ᵖⁱ = 2 (2)
(ii) Area 0 to 2π = 2 × 2 = 4 sq.units (1)
4. 3D geometry 3 marks
(i) PQ = 7j + 2k (1)
(ii) Perpendicular: î (option b) (1)
(iii) cos θ = 28/(5√53) (1)
5. Inverse trigonometry 3 marks
(i) cos⁻¹(–½) = 2π/3 (1)
(ii) tan⁻¹[(cos x)/(1–sin x)] = π/4 + x/2 (2)
6. Continuity & differentiation 3 marks
(i) Discontinuous at x=2 (f(2) not defined) (2)
(ii) dy/dx = cos(2x+1)/√[sin(2x+1)] (1)
7. f(x) on ℕ 3 marks
Not one‑one: f(2)=2, f(3)=2 (1.5)
Onto: range = ℕ (1.5)
8. Probability (coin & die) 3 marks
P(A)=1/2, P(B)=1/6, P(A∩B)=1/12 ⇒ independent (since product = 1/12) (3)
📗 Questions 9 – 16 4 marks each (any six)
9. Matrix A 4 marks
(i) C₃₁ = –1, C₃₂ = 2, C₃₃ = 8 (1)
(ii) 5·C₃₁+1·C₃₂+3·C₃₃ = 21 = |A| → option (c) (1)
(iii) ½(A+Aᵀ) = [[2,0.5,3],[0.5,–2,1],[3,1,3]] (2)
10. Integration 4 marks
∫ (3x+1)/[(x-1)(x²+1)] dx = 2ln|x-1| – ln|x²+1| + tan⁻¹x + C (4)
11. Graph & extrema 4 marks
(i)(a) decreasing (0,2) (1)
(i)(b) local max at x=0, local min at x=2 (1)
(ii) abs max = 6 (x=4), abs min = 2 (x=0) (2)
12. Rates & cylinder in cone 4 marks
(i) dP/dt = –2 cm/s (decreasing) (1)
(ii) using similar triangles → S = (2πh/R)(Rr–r²); dS/dr=0 → r=R/2 (max) (3)
13. Area function & bounded area 4 marks
(i) A'(2) = 4 → option (a) (1)
(ii) Intersection (0,0),(1,1); area = ∫₀¹(√x – x²)dx = 1/3 (3)
14. 3D & parallelogram 4 marks
(i) n=0 → angle 90° (d) (1)
(ii)(a) diagonal = 7i + j + 8k (1)
(ii)(b) area = |a×b| = 10√3 (2)
15. Shortest distance (skew lines) 4 marks
a₂–a₁ = (3,–2,–10); b₁×b₂ = (1,–1,–1); |cross| = √3; distance = |(3,–2,–10)·(1,–1,–1)|/√3 = 15/√3 = 5√3 (4)
16. Total probability 4 marks
P(C) = P(S)·P(C|S) + P(NS)·P(C|NS) = 0.65·0.32 + 0.35·0.80 = 0.208 + 0.28 = 0.488 (4)
📙 Questions 17 – 20 6 marks each
17. Solve system (matrix method) 6 marks
|A| = –10; adj(A) = [[9,1,–11],[–16,–4,23],[–1,1,–1]]; A⁻¹ = –1/10 adj(A); X = A⁻¹B → x = –4, y = 5, z = 2 (6)
18. Differentiation & integral 6 marks
(i) dy/dx = cosθ/(1–cosθ) (1)
(ii) from eʸ(x+1)=1 → dy/dx = –1/(x+1); d²y/dx² = 1/(x+1)² = (dy/dx)² (2)
(iii) I = ∫₀ᵖⁱ x/(1+sin x) dx = π (3)
19. Linear programming 6 marks
(i) feasible region polygon O(0,0)–A(8,0)–P(6,2)–B(0,5) (2)
(ii) corner points: O, A, B, P (2)
(iii) Z = 2x+3y → max = 18 at (6,2) (2)
20. Differential equations 6 marks
(i) dy/dx = xy → y = A·exp(x²/2) (3)
(ii) slope = x/y² → y³/3 = x²/2 + C ; through (1,2) → 2y³ = 3x² + 13 (3)