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 and G′ be two groups and f : G → G′ be a homomorphism then prove the following:
(i) f(e) = e′ where e and e′ are the identities of G and G′ respectively.
(ii) f(a⁻¹) = [f(a)]⁻¹, ∀ a ∈ G.
(b) Prove that every homomorphic image of a group is isomorphic to some quotient group.
Q. 2. (a) A ring R is without zero divisor if and only if the cancellation law hold. (10)
(b) Prove that arbitrary intersection of subrings is a subring. (10) (20)
Q. 3. (a) Let T : ℝ³ ⟶ ℝ³ be the linear transformation defined by
T(x₁, x₂, x₃) = (x₁ – x₂, x₁ + x₃, x₂ + x₃). Find a basis and dimension of Range of T. (10)
(b) Prove that every finitely generated vector space has a basis. (10) (20)
SECTION-B
Q. 4. (a) Find the critical points of f(x) = x³ – 12x – 5 and identify the open intervals
on which f is increasing and on which f is decreasing. (10)
(b) Find the horizontal and vertical asymptotes of the graph of f(x) = – 8 / (x² – 4). (10) (20)
Q. 5. (a) Calculate ∫ (–2x + 4) / ((x² + 1)(x – 1)²) dx. (10)
(b) Find ∂w/∂x at the point (x, y, z) = (2, -1, 1) if w = x² + y² + z², z³ – xy + yz + y³ = 1
and x and y are the independent variables. (10) (20)
Q. 6. (a) Determine the focus, vertex and directrix of the parabola x² + 6x -8y + 17 = 0. (10)
(b) Find polar coordinates of the point p whose rectangular coordinates are (3√2, –3√2). (10) (20)
SECTION-C
Q. 7. (a) Show that (cos θ + i sin θ)ⁿ = cos(nθ) + i sin(nθ) for all integers n. (10)
(b) Find the n, nth roots of unity. (10) (20)
Q. 8. (a) Find the Taylor series generated by f(x) = 1/x at a = 2. Where, if anywhere,
does the series converge to 1/x? (10)
(b) Show that the p-series ∑_{n=1}^∞ 1/nᵖ, (p a real constant) converges if p > 1, and
diverges if p < 1. (10) (20)
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