Federal Public Service Commission (FPSC)
Competitive Examination for Recruitment to BPS-17 Posts under the Federal Government
Time Allowed: 3 Hours
PURE MATHEMATICS, PAPER-I
MAXIMUM MARKS: 100 Marks
Attempt FIVE questions in all by selecting at least THREE Questions From SECTION-A and TWO Questions from SECTION-B, All questions carry EQUAL marks.
Use of Scientific Calculator is Allowed.
SECTION-A
(a) For any integer n let αₙ : ℤ → ℤ by such that αₙ(m) = m+n, m ∈ ℤ. (10)
Let A = {αₙ ; n ∈ ℤ}. Show that A is the group under the usual composition of
mappings.
(b) Show that the group of all inner automorphisms of a group G is isomorphic to (10)
the factor group of G by its center.
(a) Let A and B be cyclic groups of order n. Show that the set Hom(A,B) of all (10)
homomorphisms of A to B is a cyclic group.
(b) Prove that group G is abelian iff G/Z(G) is cyclic, where Z(G) is Centre of the (10)
group.
(a) Define the dimension of a vector space V, prove that all bases of a finite (10)
dimension vector space contain same number of elements.
(b) Show that the vectors (3, 0, -3), (-1, 1, 2), (4, 2, -2) and (2, 1, 1) are linearly (10)
dependent.
(a) The set {v₁, v₂, …, vₙ} of vectors is a vector space V is linearly dependent if (10)
and only if some vᵢ is the linear combination of the other vectors.
(b) Let A, B be two ideals of the ring R. Then show that (A + B) / A ≅ B / (A ∩ B). (10)
(a) If A is n x n matrix then (10)
(i) Determinant of (A – λI) where λ is a scalar is a polynomial P(λ).
(ii) The eigenvalues of A are the solutions of P(λ) = 0.
(b) If A is an ideal of the ring R with unity such that 1 ∈ A, then A = R. (10)
SECTION-B
(a) Find an equation of the straight line joining two points on the ellipse (10)
x²/a² + y²/b² = 1 whose eccentric angles are given. Hence find equations of the
tangent and normal at any point ‘θ’ on the ellipse.
(b) Prove that an equation of the normal to the astroid x^(2/3) + y^(2/3) = a^(2/3) (10)
is x sin t – y cos t + a cos 2t = 0, t being parameter.
(a) Show that the pedal equation of the curve (10)
x = 2a cos θ – a cos 2θ, y = 2a sin θ – a sin 2θ is 9(r² – a²) = 8p²
(b) Find the length of the arc of the curve x = e^θ sin θ, y = e^θ cos θ (10)
from θ = 0 to θ = π/2.
(a) Find the shortest distance between the straight line joining the points A(3, 2, -4) (10)
and B(1, 6, -6) and the straight line joining the points C(-1, 1, -2) and
D(-3, 1, -6). Also find equation of the line of shortest distance and coordinates
of the feet of the common perpendicular.
(b) Find an equation of the sphere for which the circle (10)
x² + y² + z² + 7y – 2z + 2 = 0, 2x + 3y – 4z – 8 = 0 is a great circle.
PURE MATHEMATICS, PAPER-II
Attempt FIVE questions in all by selecting at least THREE Questions From SECTION-A and TWO Questions from SECTION-B, All questions carry EQUAL marks.
Use of Scientific Calculator is Allowed.
SECTION-A
(a) Let ℓᵖ (p ≥ 1) be the set of all sequences (ξⱼ) of complex numbers such
that the series ∑_{j=1}^∞ |ξⱼ|ᵖ converges. Let the real valued function
d : ℓᵖ × ℓᵖ → ℝ be defined by
d(x, y) = ( ∑_{j=1}^∞ |ξⱼ – ηⱼ|ᵖ )^(1/p)
where x = (ξⱼ) and y = (ηⱼ). Show that d is a metric on ℓᵖ. (8)
(b) If d is the usual metric on ℝⁿ (the set of all ordered n-tuples of real numbers) then
prove that (ℝⁿ, d) is a complete metric space. (7)
(c) Prove that the function f : (X, dₓ) → (Y, d_y) is continuous ⇔ f⁻¹(G) is closed in X
whenever G is closed in Y. (5)
(a) Prove that there exists no rational number x such that x² = 2. (5)
(b) Examine the continuity of f at x = 0 when
f(x) = { x sin(1/x) if x ≠ 0
{ 0 if x = 0 (5)
(c) Find the nth derivative of the function eˣ ln x. (5)
(d) Show that f(x) = ln(x + 1) / x decreases on ]0, ∞[. (5)
(a) If f(x) = x(x – 1)(x – 2), a = 0, b = 1/2 ; find c of the Mean Value
Theorem. (4)
(b) Examine the series ∑_{n=1}^∞ n! / nⁿ for convergence or divergence. (5)
(c) Determine whether the series ∑_{n=1}^∞ 2ⁿ / n⁵ converges or diverges. (5)
(d) Let f(x) = |x|. Check the differentiability of f at x = 0. (4)
If u = sin⁻¹( (x² + y²) / (x + y) ) then show that x ∂u/∂x + y ∂u/∂y = tan u.
Find the percentage error in calculating the area of a rectangle when there is error of 1
percent in measuring its sides.
An open rectangular box is to be made from a sheet of cardboard 8dm by
5dm, by cutting equal squares from each corner and turning up the sides.
Find the edge of the square which makes the volume maximum.
Find the asymptotes of the curve, y = (x³ + x – 2) / (x – x²)
Evaluate the double integral of F(x, y) = x² + xy, over the triangle with
vertices (0, 0), (0, 1) and (1, 1).
Let f be Riemann integrable on [a, b]. Prove that |f| is also Riemann integrable on
[a, b] and
| ∫_a^b f(x) dx | ≤ ∫_a^b |f(x)| dx
Examine the convergence of the improper integral ∫₀² dx / (2x – x²)
SECTION-B
Solve the equation, z² + (2i – 3)z + 5 – i = 0
Prove that
cos⁻¹(cos θ + i sin θ) = sin⁻¹(√sin θ) + i ln( √(1 + sin θ) – √sin θ )
If w = f(z) is differentiable then prove that f(z) is continuous.
Prove that the essential characteristic for a function f(z) to be analytic is
that ∂f/∂z̄ = 0.
If u(x, y) is a harmonic function then prove that it satisfies the differential equation
∂²u / (∂z ∂z̄) = 0.
Show that the function f(z) = cos(z + 1/z) can be expanded as a Laurent’s series,
f(z) = a₀ + ∑_{n=1}^∞ aₙ (zⁿ + 1/zⁿ),
where aₙ = 1/(2π) ∫₀^{2π} cos(2 cos θ) cos nθ dθ
Prove that ∫_{-∞}^∞ (a cos x + x sin x) / (x² + a²) dx = 2π / eᵃ , a > 0
Q.8 (a) Prove that ∫_{-∞}^∞ (a cos x + x sin x) / (x² + a²) dx = 2π / eᵃ , a > 0 (8)
(b) Prove that ∫₀^∞ (sin x) / x dx = π / 2 (6)
(c) Let f(z) be analytic on a closed contour C : |z – a| = r. If |f(z)| ≤ M then
prove that |f⁽ⁿ⁾(a)| ≤ n! / rⁿ * M. (6)
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