## PURE MATHEMATICS paper (II) 2015

FEDERAL PUBLIC SERVICE COMMISSION
COMPETITIVE EXAMINATION FOR RECRUITMENT TO POSTS
IN BS-17, UNDER THE FEDERAL GOVERNMENT, 2015
PURE MATHEMATICS, PAPER - II
TIME ALLOWED: THREE HOURS (PART-I MCQs) Maximum Marks: 20
PART I (MCQs) Maximum 30 Minutes (PART-II) Maximum Marks: 80

Note: (i) Attempt ONLY FIVE questions in all, by selecting THREE questions from SECTION-I and TWO questions from SECTION-II. ALL questions carry EQUAL marks.
(iii) All the parts (if any) of each Question must be attempted at One Place instead of at different places.
(iv) Candidate must write Q. No. in the Answer Book in accordance with Q. No. in the Q. Paper.
(v) No Page /Space be left blank between the answers. All the blank pages of Answer Book must be crossed.
(vi) Extra attempt of any question or any part of the attempted question will not be considered.
(vii) Use of Calculator is allowed.

SECTION-I
Q. 2. (a) Use the Mean Value Theorem to show that

for any real number x and y.
(b) Use Taylors Theorem to prove that

Q.2. (a) Evaluate
(b) Find the equation of the asympototes of
(c) Evaluate the integral
Q.3. (a) Verify that for the following function.

(b) Find the points of relative extrema for f(x) = sinxcos2x.
(c) Evaluate the
Q.4.(a) Let be a metric space. Then defined by is also a metric.
(b) Show that an open ball in metric space X is an open set.
(c) Show that convergent sequence in metric space is Cauchy sequence.
Q.5. (a) Let (X,d) be a metric space, a subset A of X is dense if and only if A has non-empty intersection with any open subset of X.
(b) Determine whether the given series converges of diverges:
.
(c) Determine wheather the given series converges absolutely, converges conditionally or diverges:
.
SECTION-II
Q.6. (a) Use De Moivres Theorem to evaluate .
(b) Evaluate , where C is the circle
Q.7. (a) Find the Laurent series that represents the function:

(b) Evaluate the sum of the infinite series:
.
Q.8. (a) Find the Fourier transform of:
(ii)
(b) Find the residue at z = 0 of the function:
(i) (ii)