Table of Contents
Khyber Pakhtoon Khwa, Public Service Commission, Peshawar
Competitive Examination for Provincial Management Service, 2013
Pure Mathematics, Paper-II
Instructions:
- Attempt FIVE questions in all, selecting at least TWO questions from each section.
- Do not use any list of formulae.
- All questions carry equal marks.
SECTION – A
Q1(a): Evaluate the following:
(i) limx → 0 1x2 − 1sin2 x
(06)
(06)
(ii) limx → 1 x2 − x1 − x + ln x
(06)
(06)
(b): Use Beta integral to evaluate
∫π/20sin4t cos5t dt.
(08)
∫π/20sin4t cos5t dt.
(08)
Q2(a): Determine the area bounded by the curve whose equation is y = 1√8−x, the x-axis, y-axis and the asymptote x = 8.
(10)
(10)
(b): Find the extreme values of f(x,y) = x3 + y3 − 63(x + y) + 12xy.
(10)
(10)
Q3(a): Determine the convergence or divergence of the integral
∫∞0e−x x5 dx.
(10)
∫∞0e−x x5 dx.
(10)
(b): Derive Reduction Formula for ∫ sinnx dx and use it to evaluate
∫π/20sin4x dx.
(10)
∫π/20sin4x dx.
(10)
Q4(a): Prove that
π396 <
∫π/20x27 + sin2x dx <
π384
(10)
π396 <
∫π/20x27 + sin2x dx <
π384
(10)
(b): Calculate by double integration, the volume generated by the revolution of the cardioids r = a(1 − cos θ) about its axis.
(10)
(10)
SECTION – B
Q5(a): Derive Cauchy-Riemann equation in the polar form.
(10)
(10)
(b): If ω = φ + iψ is an analytic function then determine φ where ψ = x2 − y2 + xx2+y2
(10)
(10)
Q6(a): Find the Fourier series of the function f(x) = x − x2 where −π < x < π and deduce 1 − 122 + 132 − 142 + ………………. = π212.
(10)
(10)
(b): Expand f(x) =
{
14 − x 0 < x < 12
x − 34 12 < x < 1
as the Fourier series of sine terms.
(10)
Q7: Evaluate the following by Residue theorem:
(i) ∫2π0 cos 3θ5 − 4 cos θ dθ
(10)
(10)
(ii) ∫∞0 x−a1 + x dx
(10)
(10)
Q8(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(x1, x2) = |x1 − x2| then show that (ℝ, d) is a metric space.
(06)
(06)
(b): Prove that an open sphere in a metric space ‘X’ is an open set.
(06)
(06)
(c): Define Cauchy sequence and Bounded sequence. Also show that the sequence {Sn} defined by S1 = 1, Sn+1 = 4 + 3Sn3 + Sn , n ∈ ℕ is convergent and find its limit.
(08)
(08)
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