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. 1. (a) Let H, K be subgroups of a group G. Prove that HK is a subgroup of G if and only if HK=KH.
(b) If N, M are normal subgroups of a group G, prove that NM/M ≅ N/N∩M.
Q. 2. (a) If R is a commutative ring with unit element and M is an ideal of R then show that M is a maximal ideal of R if and only if R/M is a field.
(b) If F is a finite field and α ≠ 0, β ≠ 0 are two elements of F then show that we can find elements a and b in F such that 1 + αa² + βb² = 0.
Q. 3. (a) Let V be a finite-dimensional vector space over a field F and W be a subspace of V. Then show that W is finite-dimensional, dim W ≤ dim V and dim V/W = dim V – dim W.
(b) Suppose V is a finite-dimensional vector space over a field F. Prove that a linear transformation T ∈ A(V) is invertible if and only if the constant term of the minimal polynomial for T is not 0.
SECTION-B
Q. 4. (a) Use the Mean-Value Theorem to show that if f is differentiable on an interval I, and if |f′(x)| ≤ M for all values of x in I, then |f(x) − f(y)| ≤ M|x − y| for all values of x and y in I. Use this result to show further that |sin x − sin y| ≤ |x − y|.
(b) Prove that if x = x(t) and y = y(t) are differentiable at t, and if z = f(x, y) is differentiable at the point (x, y) = (x(t), y(t)), then z = f(x(t), y(t)) is differentiable at t and dz/dt = (∂z/∂x)(dx/dt) + (∂z/∂y)(dy/dt) where the ordinary derivatives are evaluated at t and the partial derivatives are evaluated at (x, y).
Q. 5. (a) Evaluate the double integral ∬_R (3x − 2y) dx dy where R is a region enclosed by the circle x² + y² = 1.
Answer:
(b) Find the area of the region enclosed by the curves y = sin x, y = cos x, x = 0, x = 2π.
Q. 6. (a) Find an equation of the ellipse traced by a point that moves so that the sum of its distance to (4,1) and (4,5) is 12.
Answer:
(b) Show that if a, b and c are nonzero, then the plane whose intercepts with the coordinate axes are x = a, y = b, and z = c is given by the equation x/a + y/b + z/c = 1.
SECTION-C
Q. 7. (a) Prove that a necessary and sufficient condition that w = f(z) = u(x, y) + iv(x, y) be analytic in a region R is that the Cauchy-Riemann equations ∂u/∂x = ∂v/∂y and ∂u/∂y = −∂v/∂x are satisfied in R where it is supposed that these partial derivatives are continuous in R.
(b) Show that the function f(z) = z̅ is not analytic anywhere in the complex plane Z.
Q. 8. (a) Let f(z) be analytic inside and on the boundary C of a simply-connected region R. Prove that f′(a) = (1 / 2πi) ∮_C f(z) / (z−a)² dz.
Answer:
(b) Show that ∫₀² virulence dθ / (5−3 sin θ)² = 5π / 32.
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