Federal Public Service Commission (FPSC)
Competitive Examination for Recruitment to BPS-17 Posts under the Federal Government
Time Allowed: 3 Hours
PURE MATHEMATICS
MAXIMUM MARKS: 100 Marks
Attempt FIVE questions in all by selecting at least TWO Questions From SECTION-A & B and ONE Questions from SECTION-C, All questions carry EQUAL marks.
Use of Scientific Calculator is Allowed.
SECTION-A
Q. 1. (a) Let Ψ be a homomorphism of group G into group G′ with kernel K, prove that K is a normal subgroup of G.
(b) Prove that if H and K are two subgroups of a group G, then HK is a subgroup of G if and only if HK=KH.
Q. 2. (a) Find elements of the cyclic group generated by the permutation α = ( 1 2 3 4 5 6 ; 3 4 5 2 6 1 ).
(b) Verify that the polynomials 2-x², x³-x, 2-3x² and 3-x³ form a basis for the set P₃(x); the set of all polynomials of degree three. Also express the vectors 1+x² and x+x³ as a linear combination of these basis vectors.
Q. 3. (a) Let V be the real vector space of all functions from ℝ to ℝ. Show that {cos² x, sin² x, cos 2x} is linearly dependent while {cos x, sin x, cosh x, sinh x} are linearly independent.
(b) Solve the system of linear equations:
x₁ – 2x₂ – 7x₃ + 7x₄ = 5
– x₁ + 2x₂ + 8x₃ – 5x₄ = – 7
3x₁ – 4x₂ – 17x₃ + 13x₄ = 14
2x₁ – 2x₂ + 11x₃ + 8x₄ = 7
SECTION-B
Q. 4. (a) If f(x, y) = (x² – y²) tan⁻¹(x/y) – (x² – y²) tan⁻¹(y/x). Show that ∂²f / ∂y∂x = 2(x² – y²) / (x² + y²).
(b) Evaluate ∫₀⁶ f(x) dx where f(x) = { x² when x < 2
{ 3x – 2 when x > 2.
Q. 5. (a) Let I_n = ∫₀^∞ xⁿ e⁻ˣ dx where n is an integer. Prove that I_n = n I_{n-1}. Hence show that I_n = n!.
(b) i. Write r = 8 / (2 – cos θ) in rectangular coordinates.
ii. Write x⁴ + 2x²y² + y⁴ – 2x³ – 6x²y – 2xy² = 0 in polar coordinates.
Q. 6. (a) Evaluate ∬_D dy dx and ∬_D dx dy where D is the region bounded by the y-axis, the line x=2 and the curve y = eˣ.
(b) Investigate the curve y = (x³ – 3x + 1) / x² for points of inflexion.
SECTION-C
Q. 7. (a) Sum the series 1 + 1/2 cos θ + (1·3)/(2·4) cos 2θ + (1·3·5)/(2·4·6) cos 3θ + ···.
(b) Prove that cos(π/7) – cos(2π/7) + cos(3π/7) = 1/2.
Q. 8. (a) Construct the analytic function f whose real part is U = x³ – 3xy² + x² – 3y² + 1.
(b) Evaluate ∮_C 1 / (z² + 2z + 2) dz where C is a square with corners (0,0), (-2,0), (-2,-2) and (0,-2).
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