NWFP, PUBLIC SERVICE COMMISSION, PESHAWAR
COMPETITIVE EXAMINATION FOR PROVINCIAL MANAGEMENT SERVICE, 2008
PURE MATHEMATICS, PAPER-II
- (i) Attempt FIVE questions in all, selecting at least TWO questions from each section.
- (ii) Do not use any list of formulae.
- (iii) All questions carry equal marks.
Q.1 (a) Evaluate the following:
04
limx → 0
| 1 |
| x2 |
– cot2 x
06
(b) State & prove Cauchy’s Mean value theorem and also show that the equation
6x5 + 5x4 + 4x3 + 3x2 = 0 has at least one real root in (0, 1). 10
Q.2 (a) Find an approximate formula for the change in total surface area of a right circular cylinder if small changes are made in the base radius ‘r‘ and the height ‘h‘. 06
(b) Find the extreme values of f(x, y) = 2(x – y)2 – x4 – y4. 08
(c) Evaluate ∫∫R (x2 + y2) dx dy where R is the interior of one half of the four-leaved rose r = a cos 2θ. 06
Q.3 (a) If p > 1 then show that
∑n=1∞
| 1 |
| np |
converges.
08
(b) Determine whether the function ‘f‘ defined below:
f(x) = 1, if x is irrational
= 0, if x is rational
is Riemann integrable on. 08
(c) Show that
∫0∞
| cos mx |
| 1 + x2 |
dx =
| π e–m |
| 2 |
.
04
Q.4 (a) Define a metric space. Let X = ℝ be the set of all real numbers and let d : ℝ × ℝ → ℝ be defined by d(x, y) = |x – y|, then show that (ℝ, d) is a metric space. 06
(b) Prove that an open sphere in a metric space ‘X’ is an open set. 06
(c) Define Cauchy sequence and Bounded sequence. Also show that the sequence {Sn} defined by S1 = 1,
Sn+1 =
| 4 + 3Sn |
| 3 + Sn |
,
n ∈ ℕ is convergent and find its limit.
08
Q.5 (a) Define analytic function and prove that the real and imaginary parts of an analytic function of a complex variable when expressed in polar form satisfy the equation
| ∂2Ψ |
| ∂r2 |
+
| 1 |
| r |
| ∂Ψ |
| ∂r |
+
| 1 |
| r2 |
| ∂2Ψ |
| ∂θ2 |
= 0
05
(b) If U = (x – 1)3 – 3xy2 + 3y2 then determine ‘V‘ so that U + iV is an analytic function of z = x + iy. 05
(c) State & prove De Moivre’s theorem for integral exponent. 10
Q.6 (a) Find the Fourier series of the function
| 1 |
| 1 + a cos x |
, -π ≤ x ≤ π
12
(b) Expand f(x) = Sin x in a Fourier Cosine series in the interval 0 ≤ x ≤ π. 08
Q.7 Evaluate the following by Residue theorem:
(a)
| x2 dx |
| (x2 + 1)2(x2 + 2x + 2) |
10
(b)
| cos 3x |
| 5 – 4 cos x |
dx
10