Table of Contents
Khyber Pakhtunkhwa, Public Service Commission
Competitive Examination for the Posts of Provincial Management Services (BPS-17), 2012
Pure Mathematics, Paper-I
Note: Attempt five questions in all, selecting at least two questions from each Section. Extra attempt of a question or part will not be considered. All questions carry equal marks.
Section-A
Q.1
(a) Define a cyclic group and show that every subgroup of a cyclic group is cyclic.
(10)
(10)
(b) Let ℤ be the group of integers under addition. Let
G = {
G = {
1n
01
, n ∈ ℤ }.
Show that G is a group under matrix multiplication and that ℤ ≅ G.
(10)
Q.2
(a) Let H be a subgroup of a finite group G. Then show that order of H divides the order of G.
(10)
(10)
(b) Let α: G → G1 be a group homomorphism. Let K = Ker(α). Then show that α(G) is isomorphic to G/Ker(α).
(10)
(10)
Q.3
(a) Let A be an ideal of a ring R. Then show that A is maximal in R if and only if R/A is a simple ring.
(10)
(10)
(b) Let W be a subspace of a finite-dimensional vector space V, then show that W is finite-dimensional, dim W ≤ dim V and dim(V/W) = dim V – dim W.
(10)
(10)
Q.4
(a) Solve the following system of equations:
x + y + z = 1
2x + 3y + 4z = 1
x – y – z = 0
2x + 3y + 4z = 1
x – y – z = 0
(10)
(b) Let R be a commutative ring with identity, whose only ideals are {0} and R itself, then show that R… [text cut off]
Section B
Q.5
(a) Find the equations of the straight line passing through the point (6, -2, 3) and perpendicular to the yz-plane.
(10)
(10)
(b) Determine whether the following pair of lines intersect or not. If they intersect, then find their point of intersection.
(x + 3) / 2 = y / -2 = (z – 7) / 6 ; (x + 6) / 1 = (y + 5) / -3 = (z – 1) / 2
(10)
Q.6
(a) Find the equation of the plane passing through the line of intersection of the planes
2x – y + 3z = 0 , x + 2y – 2z = 3
and which is perpendicular to the xz-plane.
(10)
(b) The points A(3, 2, -4), B(-1, 1, -2), C(-2, 3, 3) and D(-3, -2, 1) are corners of a tetrahedron. Find the shortest distance between AC and BD.
(10)
(10)
Q.7
(a) Find the total arc length of the cardioid
r = 4 + 4 cos θ
(10 points)
(b) A point P has rectangular coordinates (-1, 1, 0). Find its Cylindrical coordinates as well as Spherical coordinates.
(10)
(10)
Q.8
(a) The endpoints of a diameter of a sphere are (3, 1, -2) and (5, 7, -4). Find its equation.
(10 points)
(10 points)
(b) Find the angle between the tangents to the following curves at their point of intersection.
r = √2 sin θ , r2 = cos 2θ
(10)
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