Table of Contents
KHYBER PAKHTUNKHWA PUBLIC SERVICE COMMISSION
COMPETITIVE EXAMINATION FOR THE POSTS OF PMS OFFICERS (BPS-17)
PURE MATHEMATICS PAPER-II
Time Allowed: 03 Hours
Maximum Marks: 100
Instructions: Attempt three questions from Section A and two questions from Section B.
SECTION A
Q1. (a) Find (10)
(i) limx→0+ xxx (ii) limx→π/2 (tan x)cos x
(b) Is there a value of k that will make f(x) = { x + k, x < 0 ; cos x, x ≥ 0 } continuous at x = 0? Differentiable at x = 0? Justify your answer. (10)
Q2. (a) Find the extreme values of the function f(x,y) = x3y2(1 – x – y). (10)
(b) Evaluate the ∫ dx ⁄ (x2 – 2x + 1)√(x2 – 1). (10)
Q3. (a) Test the convergence of (10)
(i) ∑n=1∞ n ln(n) ⁄ 2n (ii) ∑n=1∞ (-1)n sin(n) ⁄ n2
(b) Find the value of ∫01 (∑n=1∞ xn ⁄ n(n+2)) dx. (10)
Q4. (a) Show that in a metric space (X, d), every finite set is closed. (10)
(b) Let (X, d) be a metric space, A ⊆ X and a ∈ X. Show that a ∈ A if and only if there exists a sequence {xn} in A such that xn → a, where A stands for closure of set A. (10)
SECTION B
Q5. (a) Find the solution of cos(z) = 1/2. (10)
(b) Show that tanh-1z = 1⁄2 log ( 1+z ⁄ 1-z ). (10)
Q6. (a) Verify that U(x, y) = tan-1(y⁄x) is harmonic in C and find its conjugate harmonic function. (10)
(b) Find the Laurent series of the function f(z) = 1⁄(z-1)(z-2) in |z| > 2. (10)
Q7. (a) Using Residue theorem, find ∫|z|=2 ez ⁄ z(z-1)3 dz. (10)
(b) Let γ be the upper half of a unit circle, oriented counterclockwise. Show that |∫γ ez ⁄ z dz| ≤ πe. (10)