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
Q.1. (a) Let W be a subspace of a finite dimensional vector space V, then W is finite dimensional and
dim (W) ≤ dim (V). Also if dim (W) = dim (V), then V = W. (10)
(b) Let V & W be vector spaces and let T : V → W be a linear map. If V is finite dimensional, then
nullity (T) + rank (T) = dim V. (10)
Q.2. (a) Show that there exist a homomorphism from Sₙ onto the multiplication group {–1,1} of
2 elements (n ≥ 1). (7)
(b) If H is the only subgroup of a given finite order in a group G. Prove that H is normal in G. (7)
(c) Show that a field K has only two ideals (namely K & (0)). (6)
Q.3. (a) Find all possible Jordan canonical forms for 3×3 matrix whose eigenvalues are -2, 3, 3. (10)
(b) Show that matrix
⎡ 1 3 0 ⎤
⎢ 0 2 1 ⎥
⎣ 0 1 1 ⎦
is diagonalizable with minimum calculation. (10)
Q.4. (a) Every group is isomorphic to a permutation group. (7)
(b) Show that for n ≥ 3, Z(Sₙ) = {I}. (6)
(c) Let A, B be two ideals of a ring, then (A + B) / A ≅ B / (A ∩ B). (7)
Q.5. (a) Verify Cayley – Hamilton theorem for the matrix
A = ⎡ 0 1 2 ⎤
⎢ 2 -3 0 ⎥
⎣ 1 1 1 ⎦. (7)
(b) Prove that ring A = ℤ, the set of all integers, is a principal ideal ring. (7)
(c) Under what condition on the scalar ξ, do the vectors (1, 1, 1), (1, ξ, ξ²), (1, -ξ, ξ²)
form a basis of ℂ³? (6)
SECTION-B
Q.6. (a) Show that T · N = 0 for the helix
R(t) = (a cos ωt) î + (a sin ωt) ĵ + (bt) k̂. (10)
(b) The vector equation of ellipse : r(t) = (2 cos t) î + (3 sin t) ĵ ; (0 ≤ t ≤ 2π)
Find the curvature of ellipse at the end points of major & minor axes. (10)
Q.7. (a) Discuss & sketch the surface
x² + 4y² = 4x – 4z². (12)
(b) Show that an equation to the right circular cone with vertex at 0, axis oz & semi –
vertical angle α is x² + y² = z² tan² α. (8)
Q.8. (a) Show that hyperboloids of one sheet and hyperbolic paraboloids are ruled surfaces. (6+6)
(b) Find an equation of the plane which passes through the point (3, 4, 5), has an x – intercept
equal to -5 and is perpendicular to the plane 2x + 3y – z = 8. (8)
PURE MATHEMATICS, PAPER-II
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
Q.1. (a) If f is continuous on [a,b] and if α is of bounded variation on [a,b], then f ∈ R(α) on [a, b] i.e. f
is Riemann–integrable with respect to α on [a,b]. (10)
(b) Let ∑ aₙ be an absolutely convergent series having sum S, then every rearrangement of ∑ aₙ
also converges absolutely & has sum S. (10)
Q.2. (a) For what +ve value of P, ∫₀¹ dx / (1 – x)ᴾ is convergent? (10)
(b) Evaluate ∫₁⁵ dx / ∛(x – 2). (10)
Q.3. (a) Find the vertical and horizontal asymptotes of the graph of function:
f(x) = (2x + 3) / √(x² – 2x + 3) (10)
(b) Let (i) y = f(x) = ((x + 2)(x – 1)) / (x – 3)²
(ii) y = f(x) = (x – 1) / ((x + 3)(x – 2)) (10)
Examine what happens to y when x → –∞ & x → +∞.
Q.4. (a) Find a power series about 0 that represent x / (1 – x³). (6)
(b) Let ∑ Sₙ be any series, Justify: (5+5+4)
(i) if Lim_{n→∞} |S_{n+1} / Sₙ| = r < 1, then ∑ Sₙ is absolutely convergent.
(ii) if Lim_{n→∞} |S_{n+1} / Sₙ| = r and (r > 1 or r = ∞), then Sₙ diverges.
(iii) if Lim_{n→∞} |S_{n+1} / Sₙ| = 1, then we can draw no conclusion about the convergence or
divergence.
Q.5. (a) Show that ∫₀^{π/2} sin²ᵐ⁻¹(θ) cos²ⁿ⁻¹(θ) dθ = (Γ(m)Γ(n)) / (2Γ(m + n)) ; m, n > 0. (10)
(b) Prove that β(m, n) = (Γ(m)Γ(n)) / Γ(m + n) ; m, n > 0. (10)
Q.6. (a) Let A be a sequentially compact subset of a metric space X. Prove that A is totally
bounded. (10)
(b) Let A be compact subset of a metric space (X, d) and let B be a closed subset of X such
that A ∩ B = ∅, show that d(A, B) > 0. (10)
SECTION-B
Q.7. (a) Show that if tan Z is expanded into Laurent series about Z = π/2, then (10)
(i) Principal part is –1 / (Z – π/2)
(ii) Series converges for 0 < |Z – π/2| < π/2
(b) Evaluate 1 / (2πi) ∮_C e^{zt} / (z²(z² + 2z + 2)) dz around the circle with equation |z| = 3. (10)
Q.8. (a) Expand f(x) = x² ; 0 < x < 2π in a Fourier series if period is 2π. (10)
(b) Show that ∫₀^∞ cos(kx) / (x² + 1) dx = π / 2 * e⁻ᵏ ; k ≥ 0. (10)
Q.9. (a) Let f(z) be analytic inside and on the simple closed curve C except at a pole of
order m inside C. Prove that the residue of f(z) at a is given
by a₋₁ = Lim_{z→a} 1 / (m – 1)! * dᵐ⁻¹/dzᵐ⁻¹ {(z – a)ᵐ f(z)}. (10)
(b) If f(z) is analytic inside a circle C with center at a, then for all Z inside C.
f(z) = f(a) + f′(a)(z – a) + f′′(a)/2! * (z – a)² + f′′′(a)/3! * (z – a)³ + …. (10)
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