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) Prove that both the order and index of a subgroup of a finite group divide the order of the
group. (10)
(b) Define cyclic group. Also prove that every cyclic group is abelian. (05)
(c) Define order of a permutation in Sₙ. Find the order of α = ⎛ 1 2 3 ⎞
⎣ 2 3 1 ⎦. (05)
Q.2. (a) Let ϕ be a homomorphism of a group G onto another group H with Kernel K. Prove that
G/K is isomorphic to H. (10)
(b) Show that the vectors (3, 0, -3), (-1, 1, 2), (4, 2, -2) and (2, 1, 1) are linearly dependent over ℝ. (10)
Q.3. (a) Define the dimension of a vector space V over a field F. Also prove that all basis of a finite
dimensional vector space contain the same number of elements. (10)
(b) A linear transformation T : U → V is one-to-one iff N(T) = {0}. (10)
Q.4. (a) Examine the following system for a non-trivial solution: (10)
x₁ – x₂ + 2x₃ + x₄ = 0
3x₁ + 2x₂ + x₄ = 0
4x₁ + x₂ + 2x₃ + 2x₄ = 0
(b) Show that ℤ₃ = {0̄, 1̄, 2̄} form finite field with addition and multiplication of residue classes
modulo P. (10)
Q.5. (a) Let V be a vector space of n – square matrices over a field ℝ. Let U and W be the subspaces of
symmetric and anti symmetric matrices respectively. Then show that V = U ⊕ W. (10)
(b) Let A and B be matrices of order 6 such that det(AB²) = 72 and det(A²B²) = 144. Find
det(A) and det(AB⁶). (10)
SECTION-B
Q.6. (a) Sketch the curve r² = a² cos 2θ, a > 0. (10)
(b) Find the tangent and the normal to the circle x = a cos θ, y = a sin θ at the point P(a cos α, a sin α). (10)
Q.7. (a) Find the Pedal equation of the parabola y² = 4a(x + a). (10)
(b) Find the equations for a straight line passing through the points P₁(x₁, y₁, z₁), P₂(x₂, y₂, z₂).
Find the co-ordinates of the point where this line cuts the yz-plane. (10)
Q.8. (a) Determine the curvature of the cycloid x = a(t – sin t), y = a(1 – cos t) at the point (x, y). (10)
(b) Find the 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. (10)
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
Q.1. (a) Prove that every non-empty set of real numbers that has an upper bound also has a supremum
in ℝ. (10)
(b) If x ∈ ℝ, set of real numbers, then there exists n ∈ ℕ such that x < n. (10)
Q.2. (a) Define continuity of a function at a point and also prove that if f and g be functions on A
to ℝ, where A ⊆ ℝ then f + g and f g are continuous at C. (10)
(b) If f : I → ℝ is differentiable at C ∈ I, then f is continuous at C. (10)
Q.3. (a) Evaluate ∫₁⁵ dx / ∛(x – 2). (08)
(b) (i) Define Complete metric space. (04)
(ii) Prove that a sequence of real numbers is convergent iff it is a Cauchy sequence. This
theorem is not in metric space, for justification give one example. (08)
Q.4. (a) Let (x, d) be a metric space and A a subset of X. Then prove that
(i) Interior A° of A is an open subset of X. (05)
(ii) A° is the largest subset of X contained in A. (05)
(b) State and prove Mean value theorem. (10)
Q.5. (a) If ∑ aₙ converges absolutely then ∑ aₙ converges. (10)
(b) Find the area enclosed by the parabola y² + 16x – 71 = 0 and the line 4x + y + 7 = 0. (10)
SECTION-B
Q.6. (a) Let Z = (cos θ + i sin θ). Then prove that Zⁿ = cos nθ + i sin nθ for all n. (10)
(b) Using De Moivre’s Theorem evaluate ( (√3 – i) / (√3 + i) )⁶. (10)
Q.7. (a) Expand f(x) = x², 0 < x < 2π in a Fourier series if period is 2π. (10)
(b) If f(z) is analytic inside a circle C with centre at a, then for all Z inside C
f(z) = f(a) + f′(a)(z – a) + f′′(a)/2! * (z – a)² + … (10)
Q.8. (a) Evaluate the integral by using Cauchy integral Formula
∮_C (4 – 3z) / (z(z – 1)(z – 2)) dz where C is a circle |z| = 3/2. (10)
(b) Prove that
∫₀^{2π} dθ / (1 – 2p cos θ – p²) = 2π / (1 – p²)
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