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-I and TWO Questions from SECTION-II, All questions carry EQUAL marks.
Use of Scientific Calculator is Allowed.
SECTION-I
Q.No.1. (a) Let H be a subgroup of a group G. Prove that the
normalizer of H in G (i.e. N(H)) is a subgroup of G. (10)
(b) Prove that a group of prime order is cyclic. (10)
Q.No.2. (a) Write three non-isomorphic groups of order 12. (10)
(b) Prove that a group G is isomorphic to a subgroup of the
group of automorphisms of G. (10)
Q.No.3. (a) Construct Cayley’s table for Multiplication Modulo 7
of ℤ₇ – {0}={1,2,3,4,4,5,6}
Show that ℤ₇ is an integral domain. (You may use
Cayley’s table.) Is ℤ₇ a field? Justify your answer. (8) (4+3+1)
(b) Give an example of zero divisor in
ℤ₆={0,1,2,3,4,4,5}
Is ℤ₆ an integral domain?
Justify your answer. (5) (2+1+2)
(c) What is FIELD EXTENSION? Verify that the field Q [√5]={x+y: x,y Q} is an
extension of Q. (7)
Q.No.4. (a) Show that W = {A ∈ M₂ₓ₂(ℝ) | Aᵀ = A} is a subspace of
the vector space M₂ₓ₂(ℝ) consisting of all 2 × 2
matrices over ℝ. (10)
(b) Prove that if a subset S = {v₁, v₂, …, v_n} of a vector space V
is linearly dependent then one vector among v_i
is a linear combination of the remaining vectors. (4)
(c) (1) What is dimension of ℝ³. (2) Write a basis of ℝ³.
(3) Is {(1,1,0),(1,1,2),(1,0,1),(0,1,2)} ⊆ ℝ³ linearly
dependent or independent? Justify your answer.
(4) Is {(0,0,0),(1,1,2),(1,0,1)} ⊆ ℝ³ linearly dependent or
independent? Justify your answer. (6) (1+1+2+2)
Q.No.5. (a) Define eigen value of a square matrix.
Find eigen values and eigen vectors of A = [ 0 1 ]
[ 1 0 ]. (10)
(b) Find reduced echelon form of the matrix
A = [ 4 3 7 ]
[ 1 1 5 ]
[ 4 5 7 ]. (10)
(c) Let λ₁, λ₂, …, λ_n be eigen values of a square matrix
A = [a_ij]. What are |A| and trace(A) in terms of λ_i’s? (5)
SECTION-II
Q.No.6. (a) Find equations of tangent plane and normal line at a
point (x₁, y₁, z₁) of ellipsoid
x²/4 + y²/9 + z²/4 = 1. (10)
(b) Find equation of the ellipse centered at the origin, a
focus at (3, 0) and vertex at (5, 0). (5)
(c) Find the polar equation of a parabola x = 8y². (5)
Q.No.7. (a) Find the equation of elliptic paraboloid
x = y² + z²
in spherical coordinates. (10)
(b) Convert the following equation of quadratic surface
to standard form. What is this surface?
4x² + y² + 4z² – 16x – 2y + 17 = 4. (10)
Q.No.8. (a) Find curvature of the space curve
r(t) = 2t i + t² j + 1/3 t³ k. (10)
(b) (1) Find first fundamental form of the surface
r(u, v) = (cos u, sin u, v).
(2) Write formulae for normal and Gaussian curvature of
a surface
r = r(u, v). (10) (6+4)
PURE MATHEMATICS, PAPER-II
Attempt FIVE questions in all by selecting at least THREE Questions From SECTION-I and TWO Questions from SECTION-II, All questions carry EQUAL marks.
Use of Scientific Calculator is Allowed.
SECTION-I
Q. No. 1. (a) Use the Mean Value Theorem to show that (10)
|sin x – sin y| ≤ |x – y|
for any real number x and y.
(b) Use Taylor’s Theorem to prove that (10)
ln sin(x + h) = ln sin x + h cot x – 1/2 h² csc² x + 1/3 h³ cot x csc² x + ···.
Q. No. 2. (a) Evaluate lim_{x→0} ( sin x – ln(eˣ cos x) ) / (x sin x). (8)
(b) Find the equation of the asymptotes of 2xy = x² + 3. (6)
(c) Evaluate the integral ∫₀³ x³ √(2x + 3) dx. (6)
Q. No. 3. (a) Verify that f_{xy} = f_{yx} for the following function: (8)
f(x,y) = e^{a x y} cos(b x + c).
(b) Find the points of relative extrema for f(x) = sin x cos 2x. (6)
(c) Evaluate the limit lim_{x→0} (1 – cos x) / x². (6)
Q. No. 4. (a) Let d : X × X → ℝ be a metric space. Then d′ : X × X → ℝ defined by (10)
d′(x,y) = d(x,y) / (1 + d(x,y))
is also a metric.
(b) Show that an open ball in metric space X is an open set. (5)
(c) Show that convergent sequence in a metric space is Cauchy sequence. (5)
Q. No. 5. (a) Let (X,d) be a metric space, a subset A of X is dense if and only if A (8)
has non-empty intersection with any open subset of X.
(b) Determine whether the given series converges or diverges: (6)
∑_{n=1}^∞ (2n)! / 4ⁿ
(c) Determine whether the given series converges absolutely, converges (6)
conditionally or diverges:
∑_{n=1}^∞ (-1)ⁿ n! / (2n)!
SECTION-II
Q. No. 6. (a) Use De Moivre’s Theorem to evaluate ( (√3 – i) / (√3 + i) )⁶. (10)
(b) Evaluate ∮_C (z + 2) / z dz , where C is the circle z = 2 e^{iθ} (0 ≤ θ ≤ 2π). (10)
Q. No. 7. (a) Find the Laurent series that represents the function: (10)
f(z) = z² sin(1 / z²).
(b) Evaluate the sum of the infinite series: (10)
cos θ – 1/2 cos 2θ + 1/3 cos 3θ – 1/4 cos 4θ + ···.
Q. No. 8. (a) Find the Fourier transform of : (10)
(i) f(t) = e^{-|t|} (ii) f(t) = sin a t²
(b) Find the residue at z = 0 of the functions: (10)
(i) f(z) = 1 / (z + z²) (ii) f(z) = z cos(1 / z)
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