Federal Public Service Commission (FPSC)
Competitive Examination for Recruitment to BPS-17 Posts under the Federal Government
Time Allowed: 3 Hours
PURE MATHEMATICS
MAXIMUM MARKS: 100 Marks
Attempt FIVE questions in all by selecting at least TWO Questions From SECTION-A & B and ONE Questions from SECTION-C, All questions carry EQUAL marks.
Use of Scientific Calculator is Allowed.
SECTION-A
Q. 1. (a) Let H and K be normal subgroups of a group G. Show that HK is a normal subgroup of G.
Answer:
(b) Let H and K be normal subgroups of a group G such that H ⊆ K. Then show that (G / H) / (K / H) ≅ G / K.
Q. 2. (a) Show that every finite integral domain is a field.
Answer:
(b) Consider the following linear system:
x + 2y + z = 3
a y + 5z = 10
2x + 7y + a z = b
(i) Find the values of a for which the system has unique solution.
(ii) Find the values of the pair (a, b) for which the system has more than one solution.
Q. 3. (a) Find condition on a,b,c so that vector (a,b,c) in ℝ³ belongs to W = span {u₁, u₂, u₃} where u₁ = (1, 2, 0), u₂ = (-1, 1, 2), u₃ = (3, 0, -4).
Answer:
(b) Let W₁ and W₂ be finite dimensional subspaces of a vector space V. Show that dim W₁ + dim W₂ = dim(W₁ ∩ W₂) + dim(W₁ + W₂).
SECTION-B
Q. 4. (a) Let f(x) = { x² if x ≤ 1
{ x if x > 1
Does the Mean Value Theorem hold for f on [1/2, 2]? (10)
(b) Calculate the lim_{x→0} ( ln sin 3x ) / ( ln sin x ). (10)
Q. 5. (a) Evaluate ∫_{-1}^5 |x – 2| dx. (10)
(b) Prove that f_{xy}(0,0) ≠ f_{yx}(0,0) if
f(x,y) = { x² y sin(1/x) when x, y are not both 0
{ 0 when x, y are both 0 (10)
Q. 6. (a) Find the area of the region bounded by the cycloid
x = a(θ – sin θ), y = a(1 – cos θ) and its base. (10)
(b) Find the equation of a plane through (5, -1, 4) and perpendicular to each of the
planes
x + y – 2z – 3 = 0 and 2x – 3y + z = 0. (10)
SECTION-C
Q. 7. (a) Express cos⁵ θ sin³ θ in a series of sines of multiples of θ. (10)
(b) Use Cauchy’s Residue Theorem to evaluate the integral ∮_C (5z – 2) / (z(z – 1)) dz where C (10)
is the circle |z| = 2, described counter clock wise.
Q. 8. (a) Find the Laurent series that represent the function f(z) = (z + 1) / (z – 1) in the domain (10)
1 < |z| < ∞.
(b) Expand f(x) = sin x in a Fourier cosine series in the interval 0 ≤ x ≤ π. (10)
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