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) Show that the order and the index of a subgroup divides the order of a finite group.
(b) Show that every finite integral domain is a field.
Q. 2. (a) Show that the characteristic of an integral domain R is either zero or a prime.
Answer:
(b) Determine whether or not the set {(1, 2, −1), (0, 3, 1), (1, −5, 3)} of vectors is a basis for R3.
Q. 3. (a) Show that a one-to-one linear transformation preserves basis and dimension.
(b) Solve the system of linear equations:
2x₁ + x₂ + 5x₃ = 4
3x₁ – 2x₂ + 2x₃ = 2
5x₁ – 8x₂ + 2x₃ = 1. (10) (20)
SECTION-B
Q. 4. (a) Solve ∫₀^{π/2} sin² 6x cos⁴ 3x dx. (10)
(b) Find the area enclosed by y = 6 / (2 – cos θ). (10) (20)
Q. 5. (a) Show that in any conic semi-latusrectum is the harmonic mean between the (10)
segments of focal chord.
(b) Prove that the evolute of hyperbola (10) (20)
2xy = a is (x + y)^{2/3} – (x – y)^{2/3} = 2a^{2/3}.
Q. 6. (a) Define Supremum and Infimum of a sequence. Find the supremum and infimum (10)
of the set
{ (-1)ⁿ (1 – 1/n), n = 1, 2, 3 … }.
(b) Evaluate (10) (20)
lim_{x→0} ( (1 + x)^{1/x} – e ) / x.
SECTION-C
Q. 7. (a) Show that Log(1 + cos θ + i sin θ) = ln(2 cos(θ/2)) + i θ/2. (10)
(b) Find v such that f(z) = u + iv is analytic. (10) (20)
Q. 8. (a) Prove that the series z(1 – z) + z²(1 – z) + z³(1 – z) + ··· converges (10)
for |z| < 1, and find its sum.
(b) Find the residues of f(z) = (z² – 2z) / ((z + 1)² (z² + 4)) at all its poles in the finite plane. (10) (20)
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