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 the set Sₙ of all permutations on a set X of n elements is a group under the operation
‘o’ of composition of permutations. Will (Sₙ, o) be an abelian group? How do we call this
group? (10)
(b) If G is a group, N a normal subgroup of G, then show that the set G/N of right cosets of N in G
is also a group. How we call this group? Also, if G is finite then show that
o(G/N) = o(G) / o(N). (10)
Q.2. (a) Let cp be a homomorphism of a group G onto another group H with kernel K. Prove that G/K is
isomorphic to H, that is G/K ≈ H. (10)
(b) Let Zₙ be the set of the congruence classes modulo n, that is,
Zₙ = {[0], [1], [2], ……….. [n–1]}
Define the two binary operations on Zₙ under which it is a ring. Prove that the ring Zₙ is an
integral domain ⇔ n is a prime number. (10)
Q.3. (a) Let T : ℝ³ → ℝ³ be the linear mapping defined by:
T(x,y,z) = (x+2y – z, y + z, x+2y – z)
Verify that
Rank (T) + Nullity (T) = dim D(T)
Also find a basis for each Rank (T) and Nullity (T) (10)
(b) If U and W are finite – dimensional subspaces of a vector space V over a field F then prove that
dim(U+W) + dim(U∩W) = dim U + dim W (10)
Q.4. (a) Let H and K be two subgroups of a group G. Prove that HK is a subgroup of G ⇔ HK = KH.
(10)
(b) Let v₁, v₂, …….., vₙ be non-zero eigen vectors of an operator T:V→V belonging to distinct
eigen values λ₁, λ₂, …….., λₙ. Show that the vectors v₁, v₂, ………, vₙ are linearly independent.
Q.5. (a) Let V be the vector space of n-square matrices over the field IR. Let U and W be the subspaces
of symmetric and antisymmetric matrices, respectively. Show that V = U ⊕ W. (10)
(b) Diagonalize the following matrix:
M =
⎡ 8 4 4 ⎤
⎢ 4 6 4 ⎥
⎣10 4 6 ⎦
SECTION-B
Q.6. (a) Find the lengths of the following curves: (10)
(i) 9y² = 4x³ from x = 3 to x = 8
(ii) r = a sin²(θ/2) from θ = 0 to θ = π
(b) Find the radius of curvature of the given curve at the designated point. (10)
y = a ln( (a + √(a² – x²)) / x ) – √(a² – x²) ; at (x, y)
Q.7. (a) Show that the two lines
L₁ : x = 4 – t, y = –2 +2t, z = 7 – 3t
L₂ : x = 3 + 2s, y = – 7 – 3s, z = 6 + 4s
are skew. Also find the points on the lines such that the segment joining these points is
perpendicular to both lines and hence find the shortest distance between the given lines. (10)
(b) Find the equation of the sphere through the circle x² + y² + z² = 1, 2x + 4y + 5z = 6 and touching
the plane z = 0. (10)
Q.8. (a) At a point on a curve r = r(t) at which k ≠ 0, show that
τ = [r′, r′′, r′′′] / |r′ × r′′|²
where r′ = dr/dt (10)
(b) Find the First Fundamental Form and fundamental magnitudes of first order for the sphere
r = (a cos u cos v, a cos u sin v, a sin u)
Also prove that parametric curves are orthogonal.
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) Let the function f : [–2, 2] → ℝ be defined by f(x) = |x|. Show that f is continuous at x = 0 but it
is not differentiable at x = 0. Will there exist a point c in ]–1, 1[ such that
f′(c) = 0 or f(1) − f(−1) = 2f′(c) ? (10)
(b) Evaluate Limₓ→₀ ( (1 + x)^(1/x) – e ) / x. (10)
Q.2. (a) Find the asymptotes of the curve defined by the equation
(x – y)²(x² + y²) – 10(x – y)x² + 12y² + 2x + y = 0. (10)
(b) Test the convergence of the series
∑_{n=1}^∞ 1 / nᵏ , k > 0
How do we call this series? (10)
Q.3. (a) Find the area enclosed by the parabola y² + 16x + 6y – 71 = 0 and the line 4x + y + 7 = 0. (10)
(b) Find the volume of the solid generated by revolving about the y-axis the area of the triangle with
vertices at (2,1), (6,1) and (4,5). (10)
Q.4. (a) If u = arcSin( (x² + y²) / (x + y) ), show that x ∂u/∂x + y ∂u/∂y = tan u. (10)
(b) Integrate F(x, y) = 1 / (y⁵ + 1) over the region R : 0 ≤ x ≤ 8, ∛x ≤ y ≤ 2. (10)
Q.5. (a) Let X be the set of all (bounded or unbounded) sequences of complex numbers. If d : X × X → ℝ
is defined as
d(x, y) = ∑_{j=1}^∞ (1 / 2ʲ) * ( |ξⱼ – ηⱼ| / (1 + |ξⱼ – ηⱼ|) )
where x = (ξⱼ) and y = (ηⱼ), then show that d is a metric on X. (10)
(b) Prove that the mapping:
T : (X, dₓ) → (Y, d_y)
is continuous at a point x₀ ∈ X ⇔ xₙ → x₀ implies Txₙ → Tx₀. (10)
SECTION-B
Q.6. (a) If Z = ((1 + i) + (3 + 2i)t) / (1 + it) , then show that the locus of Z is a circle. Also calculate the minimum
and maximum distance of Z from the origin. (10)
(b) Find the complex number Z satisfying the equation
Z² + (2i – 3) Z + (5 – i) = 0. (10)
Q.7. (a) Show that the function
u(x, y) = 4xy – 3x + 2
is harmonic. Construct the corresponding analytic function
f(z) = u(x, y) + i v(x, y). (10)
(b) Find the Fourier Series of the function
f(x) = { x, 0 < x ≤ π
{ 2π – x, π < x < 2π
period 2π. (10)
Q.8. (a) Evaluate the following integral by using Cauchy Integral Formula:
∮_C (4 – 3z) / (z(z – 1)(z – 2)) dz
where C is the circle |z| = 3/2. (10)
(b) Prove that
∫₀^{2π} dθ / (1 – 2p cos θ – p²) = 2π / (1 – p²)
where 0 < p < 1. (10)
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