CSS Pure Mathematics Past Paper 2014 PDF

Federal Public Service Commission (FPSC)
Competitive Examination for Recruitment to BPS-17 Posts under the Federal Government

 Time Allowed: 3 Hours 

 PURE MATHEMATICS, PAPER-I 

 MAXIMUM MARKS: 100 Marks 

Attempt FIVE questions in all by selecting at least THREE Questions From SECTION-A and TWO Questions from SECTION-B, All questions carry EQUAL marks. 

 Use of Scientific Calculator is Allowed. 

 SECTION-A 

Q. No. 1. (a) If G is a group in which (a · b)ⁱ = aⁱ · bⁱ for three consecutive integers i for all
a, b ∈ G, show that G is abelian. (10)
(b) The center Z of a group G is defined by Z = {z ∈ G | zx = xz all x ∈ G}. Prove
that Z is a subgroup of G. (10)

Q. No. 2. (a) If f : G → G′ be a homomorphism. Prove that Ker f is a normal subgroup of G. (10)
(b) Prove that any group of order 15 is cyclic. (10)

Q. No. 3. (a) If in a ring R with unity, (xy)² = x²y² for all x, y ∈ R, then show that R is commutative. (10)
(b) Prove that the set ℤ₇ = {0, 1, 2, 3, 4, 5, 6} forms a commutative ring with unit
element under addition and multiplication modulo 7. (10)

Q. No. 4. (a) Prove that a non empty subset W of a vector space V(F) is a subspace of V if
and only if αx + βy ∈ W for α, β ∈ F, x, y ∈ W. (10)
(b) Show that the vectors
v₁ = (1, -1, -4, 0), v₂ = (1, 1, 2, 4), v₃ = (2, -1, -5, 2), v₄ = (2, 1, 1, 6) are linearly
dependent in ℝ⁴(ℝ). (10)

Q. No. 5. (a) A company produces three products, each of which must be processed (10)
through three different departments. Given table summarizes the hours
required per unit of each product in each department. In addition, the
weekly capacities are stated for each department in terms of work-hours
available. What is desired is to determine whether there are any
combinations of the three products which would exhaust the weekly
capacities of the three departments.

Department	Product 1	Product 2	Product 3	Hours Available per Week
A	2	3.5	3	1,200
B	3	2.5	2	1,150
C	4	3	2	1,400

(b) Show that
| 2bc – a² c² b² |
| c² 2ca – b² a² | = (a³ + b³ + c³ – 3abc)² (10)
| b² a² 2ab – c² |

 SECTION-B 

Q. No. 6. (a) Find the equation of the straight line joining two points on the ellipse (10)
x²/a² + y²/b² = 1 whose eccentric angles are given. Hence find equations of the
tangent and normal at any point θ on the ellipse.
(b) Find the angle of intersection of the cardioids r = a(1 + cos θ) and r = b(1 – cos θ). (10)

Q. No. 7. (a) Find the equation of the line L through the point (5, 7/2, 5) and intersecting at (10)
right angles the line M with parametric equations
x = 4 + 3t, y = 1 + t, z = -3t.
(10)

Q. No. 8. (a) Find the volume of the solid obtained by revolving the area enclosed by one arc of the cycloid x = a(θ + sin θ), y = a(1 + cos θ) about x-axis. (10)
(b) Discuss the surface and make a sketch, x² – y² + z² = 1. (10)

 PURE MATHEMATICS, PAPER-II 

Attempt FIVE questions in all by selecting at least THREE Questions From SECTION-A and TWO Questions from SECTION-B, All questions carry EQUAL marks. 

 Use of Scientific Calculator is Allowed. 

 SECTION-A 

Q. No. 1. (a) Prove that if n is a positive integer which is not a perfect square, then √n is an
irrational number. (10)
(b) Show that every non-empty set of real numbers which has a lower bound has
the infimum. (10)

Q. No. 2. (a) For what value of a, m, and b does the function
f(x) = { 3 x = 0
{ -x² + 3x + a 0 < x < 1
{ mx + b 1 ≤ x ≤ 2
satisfy the hypotheses of the Mean Value Theorem on the interval [0, 2]? (10)
(b) For what value of a is
f(x) = { x² – 1 x < 3
{ 2a x x ≥ 3
continuous at every x? (10)

Q. No. 3. (a) Find the area of the surface generated by revolving r = 2a sin θ about the polar
axis. (6)
(b) Find the area enclosed by the graph of the cardioid r = a(1 – sin θ). (7)
(c) Evaluate the integral ∫₁¹⁰ dx / (x – 2)^(2/3). (7)

Q. No. 4. (a) Find the sum of the series ∑_{n=1}^∞ 1 / (n(n + 1)). (6)
(b) For what value of x does the series converges absolutely, converges
conditionally and diverges?
∑_{n=0}^∞ ( (-1)ⁿ xⁿ ) / √(n² + 3) (7)

(c) Let f(x,y) = { x³ / (x² + y²) if (x, y) ≠ (0, 0)
{ 0 if (x, y) = (0, 0) (7)
Show that f is not continuous at the origin.

Q. No. 5. (a) Let X be a non-empty set and define (10)
d : X × X → ℝ by
d(a, b) = { 1 if a ≠ b
{ 0 if a = b
Show that d is a metric on X.
Also describe open and closed balls in this metric space.
(b) Prove that a function f from a metric space (X, d) into a metric space (Y, d′) is (10)
continuous if and only if f⁻¹(A) is a closed subset of X for every closed subset
A of Y.

 SECTION-B 

Q. No. 6. (a) Using De Moivre’s Theorem evaluate (10)
( (1 + i) / (√3 + i) )⁶.
(b) Find real constants a, b, c and d so that the given function is analytic (10)
f(z) = x² + a x y + b y² + i(c x² + d x y + y²).

Q. No. 7. (a) Evaluate ∮_C dz / (z² + 1) , where C is the circle |z| = 4. (10)
(b) Expand f(z) = 1 / (z(z – 1)) in a Laurent series valid for 1 < |z – 2| < 2. (10)

Q. No. 8. (a) Find the Fourier transform of f(x) = e⁻|ˣ|. (10)
(b) Evaluate ∮_C 1 / ( (z – 1)² (z – 3) ) dz , where the contour C is the rectangle (10)
defined by x = 0, x = 4, y = -1, y = 1.

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