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. No. 1. (a) Let N be a normal subgroup of a group G. If H is a subgroup of G, then prove that
NH = { nh : n ∈ N and h ∈ H } is a subgroup of G. (10)
(b) If ϕ is an epimorphism from a group G onto a group H then prove that G / K is
isomorphic to H, where K = Ker ϕ. (10) (20)
Q. No. 2. (a) Let R be a ring. If every x ∈ R satisfies x² = x then prove that R is a
commutative ring. (10)
(b) For which value(s) of a will the following system have no solution? Exactly
one solution? Infinitely many solutions?
x + 2y – 3z = 4
3x – y + 5z = 2
4x + y + (a² – 14)z = a + 2. (10) (20)
Q. No. 3. (a) Determine a basis for and the dimension of the solution space of the system:
x – 2z + w = 0
3x + y – 5z = 0
x + 2y – 5w = 0. (10)
(b) Let v₁ = (1, 1, 1), v₂ = (1, 1, 0) and v₃ = (1, 0, 0) be a basis for ℝ³. Find a
linear transformation T : ℝ³ → ℝ² such that T(v₁) = (1, 0), T(v₂) = (2, -1)
and T(v₃) = (4, 3). (10) (20)
SECTION-B
Q. No. 4. (a) Evaluate the limit: (10)
(i) lim_{x→π/2} (1 + cos x)^{tan x}
(ii) lim_{x→0} ( 1 / sin x – 1 / x )
(b) State and prove the Mean Value Theorem. (10) (20)
Q. No. 5. (a) If w = f(x² + y²) then show that y (∂w/∂x) – x (∂w/∂y) = 0. (10)
(b) Find all the local maxima, local minima and saddle points of the given
function 2x³ + y² – 9x² – 4y + 12x – 2. (10) (20)
Q. No. 6. (a) Evaluate the integral ∫₀^∞ x^(1/2) (1 + 2x)⁻⁵ dx and show that the result is 9π / 1024, (10)
using Beta function.
(b) Find the vertices and foci of the hyperbola (10) (20)
25x² – 16y² + 250x + 32y + 109 = 0.
SECTION-C
Q. No. 7. (a) Verify that u(x,y) = cos x cosh y is harmonic function and find a (10)
corresponding analytic function f(z) = u(x,y) + i v(x,y).
(b) Use Residue theorem to evaluate the integral ∮_C (9z² + z – 1) / ((z² – 1)(z – 3)) dz, where C is the (10) (20)
circle |z| = 4.
Q. No. 8. (a) Use the Cauchy’s integral formula to evaluate the integral (10)
∮_C (z + 4) / (z² + 2z + 5) dz, where C is the circle |z + 1 – i| = 2.
(b) Find the three cube roots of √3 + i. (10) (20)
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