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 G be a group and H be a subgroup of index 2 in G. Show that H is normal in G.
(b) Let G be any group, g a fixed element in G. Define Tg : G → G by Tg(x) = g x g⁻¹, ∀ x ∈ G. Prove that Tg is an automorphism of G onto G.
Q. 2. (a) Prove that a finite integral domain is a field.
(b) Let W be the subspace of ℝ⁵ spanned by v₁ = (1, 2, -1, 3, 4), v₂ = (2, 4, -2, 6, 8), v₃ = (1, 3, 2, 2, 6), v₄ = (1, 4, 5, 1, 8), v₅ = (2, 7, 3, 3, 9). Find a subset of the vectors that form a basis of W. Also extend the basis of W to a basis of ℝ⁵.
Q. 3. (a) Let T : ℝ⁴ → ℝ³ be defined by T(x, y, z, w) = (x – y + z + w, 2x – 2y + 3z + 4w, 3x – 3y + 4z + 5w). Find the rank and nullity of T.
(b) Find all possible solutions of the following homogeneous system of equations:
x₁ + x₂ + x₃ – x₄ = 0
x₁ + 2x₂ – 2x₃ + x₄ = 0
2x₁ + 4x₂ – 3x₃ + x₄ = 0
4x₁ + 7x₂ – 4x₃ + x₄ = 0
SECTION-B
Q. 4. (a) Find lim_{x→0} ( (1 + x) / (1 – x) )^{1/x}.
Answer:
(b) Evaluate the integral ∫₀^{π/2} cos² x sin³ x dx.
Q. 5. (a) If z = f(x, y) and x = r cos θ, y = r sin θ, then show that ∂²z/∂x² + ∂²z/∂y² = ∂²z/∂r² + (1/r) ∂z/∂r + (1/r²) ∂²z/∂θ².
(b) Evaluate ∬_R x dx dy over the region R bounded by y = x² and y = x.
Q. 6. (a) Find the area of the region bounded above by y = x + 6, bounded below by y = x², and bounded on the sides by the lines x = 0 and x = 2.
Answer:
(b) Find the foci, vertices and center of the ellipse: 9x² + 16y² − 72x − 96y + 144 = 0.
SECTION-C
Q. 7. (a) Prove that the function u(x, y) = e⁻ˣ (x sin y – y cos y) is harmonic. Also find a function v(x, y) such that f(z) = u(x, y) + i v(x, y) is analytic.
(b) Evaluate ∮_C z̄ dz around the circle |z| = 1.
Q. 8. (a) Use residues to prove that ∫₀^∞ dx / (x⁴ + 1) = π / (2√2).
Answer:
(b) Find the Fourier series of the following function f(x) which is assumed to have the period 2π: f(x) = |x|, −π < x < π.
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