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) Prove that the normaliser of a subset of a group G is a Subgroup of G.(b) Let A be a normal subgroup and B a subgroup of a group G. Then prove that < A,B > = AB
Q. 2. (a) Let a be a fixed point of a group G and consider the mapping Ia : G→G defined by Ia(g)= aga-1 where g∈G. Show that Ia is an automorphism of G. Also show that for a, b ∈ G, Ia .Ib=Iab
(b) Let M2 (R) = { [ a b ; c d ] : a, b, c, d ∈ R } be the set of all 2×2 matrices with real entries. Show that ( M2(R), +, ∙ ) forms a ring with identity. Is ( M2(R), +, ∙ ) a field?
Q. 3. (a) Let T: X→Y be a linear transformation from a vector space X into a Vector space Y. Prove that Kernal of T is a subspace.
(b) Find the value of λ such that the system of equations x + λy + 3z = 0, 4x + 3y + λz = 0, 2x + y + 2 z = 0 has non-trivial solution.
SECTION-B
Q. 4. (a) Using δ-ε definition of continuity, prove that the function Sin² x is continuous for all x ∈ R.
(b) Find the asymptotes of the curve (x²-y²)(x+2y) + 5 (x²+y²) + x+y =0
Q. 5. (a) Prove that the maximum value of (1/x)ˣ is e^(1/e).
(b) Find the area enclosed between the curves y=x³ and y=x.
Q. 6. (a) A plane passes through a fixed point (a, b, c) and cuts the coordinate axes in A,B,C. Find the locus of the centre of the sphere OABC for different positions of the plane, O is the origin.
SECTION-C
Q. 7. (a) Determine P(z) where P(z) =(z-z₁)(z-z₂)(z-z₃)(z-z₄) with z₁=e^(iπ/4), z₂=z̄₁, z₃=-z₁ and z₄=-z̄₁.
(b) Find value of the integral ∫_c (z – z₀)ⁿ dz (n any integer) along the circle C with centre z₀ and radius r, described in the counter clock wise direction.
Q. 8. (a) Use Cauchy Integral Formula to evaluate ∫_c (cosh z + sin² z) / (z – iπ/2) dz along the simple closed contour C: | z |=3 described in the positive direction.
(b) State and prove Cauchy Residue Theorem.
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