SPSC 2014 B.Sc Forestry Course 2014-16 Selection Exam — Mathematics Paper

B.Sc Forestry Course 2014-16 Selection Exam — Mathematics. Wednesday 13 August 2014, 10:00 am to 01:00 pm · Maximum marks 200. Attempt any SIX questions, selecting at least THREE questions from each section. All questions carry equal marks.

1. 1Section I — (a) Let A = {a, n, t}, B = {t, a, p} and C = {s, a, p}. Then find A ∪ (B ∩ C) = (A ∪ B) ∩ (A ∪ C). (b) Solve any two of the following: (i) 2x² + x = 1/2 (ii) (t + 1/t)² = 4(t − 1/t) (iii) x + y = 5, 2x + 3y = 2xy (c) If α, β are the roots of px² + qx + r = 0, q ≠ 0, form the equation whose roots are −1/α³ and −1/β³.

2. 2Section I — (a) (i) Define Matrix, rectangular and square matrix, scalar matrix and identity matrix. (ii) Find the inverse of the matrix [4 7; 2 6] by Adjoint method. (b) Without expansion prove that |α βγ αβγ; β γα αβγ; γ αβ αβγ| = |α α² α³; β β² β³; γ γ² γ³|. (c) Use Cramer's rule to solve the following system of three linear equations: x + 2y + z = 8, 2x − y + z = 3, x + y − z = 0.

3. 3Section I — (a) Use mathematical induction to show that 1 + 5 + 9 + … + (4n − 3) = n(2n − 1). (b) Evaluate (3 + √2)⁴ − (3 − √2)⁴.

4. 4Section I — (a) (i) Find the 70th term of the following Arithmetic Progression (A.P.): 1, 5, 9, 13, 15, … (ii) What term of the following G.P is 4096? 1, 4, 16, 64, 256, 1024, … (b) Find the arithmetic mean, geometric mean and harmonic mean of 8 and 4. (c) If A, G, H are respectively the arithmetic, the geometric and the harmonic means between any two given numbers, then prove that G² = AH.

5. 5Section I — (a) Define equilateral triangle, isosceles triangle and scalene triangle. Without using table find the value of sin 60°. (b) If cos θ = 2/3 and P(θ) is in the first quadrant, find the remaining trigonometric functions. (c) Prove any two of the following identities: (i) 1 + tan²(π/4) = sec²(π/4) (ii) (sin θ − cos θ)² = 1 − 2 cos θ sin θ (iii) √(1 − sin θ) / √(1 + sin θ) = sec θ − tan θ

6. 6Section II — (a) Evaluate any three of the following limits: (i) lim x→1 [√(5x − 4) − √x] / (x − 1) (ii) lim x→∞ [x²/(x + 1) − x²/(x + 3)] (iii) lim x→0 [(cosec x − cot x)/x] (iv) lim x→∞ [x − √(x² − a²)] (b) Examine the continuity of the following functions: (i) f(x) = x sin(1/x), x ≠ 0; = 0, x = 0 (ii) f(x) = x − 1, x < 2; = 2x − 3, x ≥ 2

7. 7Section II — Find dy/dx of any three of the following functions: (i) y = √[(1 + x)/(1 − x)] (ii) y = ln(x + √(1 + x²)) (iii) x³ + y³ − 3axy = 0 (iv) x = 3at/(1 + t²), y = 3at²/(1 + t²)

8. 8Section II — Evaluate any three of the following integrals: (i) ∫ dx/√(a² − x²), x ≠ a (ii) ∫ (1 − sin x)/(1 + cos x) dx (iii) ∫ e^(ax) sin(bx + c) dx (iv) ∫ (2x − 3)/[(x² − 1)(2x + 3)] dx

9. 9Section II — (a) Determine the extreme values of the function f(x) = (x − 2)(x − 3)/x². (b) Find the area of the triangle whose vertices are A(−1, 3), B(2, 3), and C(2, 6). (c) Find the equation of the circle containing the points (−1, −2) and (6, −1) and touches x-axis.

10. 10Section II — (a) Find the equations of a straight line which is parallel and perpendicular to the straight line 2x − 3y = 9 and passing through the point (4, −1). (b) Find the equations of the tangent and normal of the circle x² + y² + 6x + 4y − 132 = 0 at (6, 6).