Table of Contents
KHYBER PAKHTUNKHWA PUBLIC SERVICE COMMISSION
COMPETITIVE EXAMINATION FOR THE POSTS OF PMS OFFICERS (BPS-17)
PURE MATHEMATICS (PAPER-I)
TIME ALLOWED: THREE HOURS
Max. Marks: 100
Instructions: Attempt three questions from Section A and two questions from Section B.
SECTION A
Q1 (a) Let G be a group of non-zero complex numbers under multiplication and let N = {a + ib ∈ G : a2 + b2 = 1} be a subset of G. (10)
- Show that N is a subgroup of G.
- Show that G/N ≅ R+, where R+ is the group of positive real numbers under multiplication.
(b) Let C12 = {1, b, b2, …, b11 ; b12 = 1} be a cyclic group of order 12 generated by b. Then, find all the generators of C12 other than b. (10)
Q2 (a) Let S = {a + ib : a, b ∈ Z, b is even}. Show that S is a subring of Z[i] = {r + is : r, s ∈ Z} but not an ideal of Z[i]. (10)
(b) Find all k ∈ Z3 such that Z3[x] / (x3 + kx2 + 1) is a field. (10)
Q3 (a) Let V be a vector space with dim(V) = 6. Suppose U and W are two different vector subspaces of V with dim(U) = 4 = dim(W). Find the possible dimensions of U ∩ W. (10)
(b) Find the eigenvalues and eigen vectors of the following matrix and diagonalize it if possible: (10)
| 1 | 2 | 3 |
| 0 | -1 | 2 |
| 0 | 0 | 2 |
Q4 (a) For what values of λ the following homogeneous system of linear equations has non-trivial solution: (10)
(λ + 2)x1 – 2x2 + x3 = 0
-2x1 + (λ – 1)x2 + 6x3 = 0
(b) Find the rank and nullity of T : R3 → R3 defined by T(x1, x2, x3) = (x1 + 2x2 + 3x3, 2x1 + 3x2 + 4x3, 3x1 + 5x2 + 7x3). (10)
SECTION B
Q5. (a) Find the center, vertices, foci, and asymptotes of the hyperbola 4x2 – y2 – 8x – 4y – 4 = 0. (10)
(b) Find a Cartesian equation for the hyperbola centered at the origin that has a focus at (3, 0) and the line x = 1 as the corresponding directrix. (10)
Q6 (a) Find the parametric equations for the lines in which the planes 3x – 6y – 2z = 15 and 2x + y – 2z = 5 intersect and find the angle between these two planes. (10)
(b) Write the cartesian equation of r2 = 4r sin(θ) and then graph it. (10)
Q7 (a) Define principal tangent vector T(t), principal normal vector N(t), binormal vector B(t) and curvature κ(t). (8)
(b) Let r(t) = 2 cos(t)i + 2 sin(t)j + 3tk be a vector function. Find T(t), N(t), B(t) and κ(t). (6)
(c) Using vector function given in part (b), find an equation of the osculating plane, the normal plane and the rectifying plane at the point corresponding to t = π/2. (6)