FEDERAL PUBLIC SERVICE COMMISSION

COMPETITIVE EXAMINATION FOR RECRUITMENT TO POSTS
IN BPS–17, UNDER THE FEDERAL GOVERNMENT, 2007

PURE MATHEMATICS, PAPER–I

TIME ALLOWED: THREE HOURS
NOTE: (i) Attempt ONLY FIVE questions in all, including QUESTION NO.8, which is COMPULSORY.
All questions carry EQUAL marks. Select TWO questions from each SECTION.
(ii) Extra attempt of any question or any part of the attempted question will not be considered.
(iii) Candidate must draw two straight lines ( ) at the end to separate each
question attempted in Answer Books.

SECTION – I

Q.1. (a) Prove that the center of a group G is a normal subgroup of G. (10)
(b) Prove that A(G), the set of all automorphisms of a group G, is also a group.(10)

Q.2. (a) Let R be a commutative ring with unit element whose only ideals are (0) and R itself. Prove that R is a field. (10)
(b) If D is integral domain of finite characteristic, prove that the characteristic of D is a prime number. (10)

Q.3. (a) If an n-dimensional vector space V has a linearly independent set of m vectors, prove that n ³ m. (10)
(b) Let R3 be the usual 3-dimensional vector space over R and let T :R3 ®R2 be the linear transformation: (10)

Q.4. (a) Define a finite extension of a field of L is a finite extension of a field K and K is a finite extension of a field F, prove that L is a finite extension of F. (10)
(b) Let A(V) be the algebra of all linear transformations from V to V, where V is a finite dimensional vector space over F. For T in A(V), r (T) denotes rank of T. If S,T are in A(V), prove: (10)

(i) r (ST) £ r (T)
(ii) r (TS) £ r (T)

SECTION – II

Q.5. (a) Find the center of curvature of the parabola x2 = 4ay at any point (x,y) on it.(10)
(b) Find the length of the cardiode r= a (1+sin q ). (10)

Q.6. (a) Find the equation to the tangent plane and equations of the normal to the surface
x2 + 2z2 = y2 at the point (1,3, – 2). (10)
(b) Find the envelope of the family of paraboloids x2 + y2 = 4a (z – a). (10)

Q.7. (a) Prove that the curvature and torsion of the circular helix (10)
r = a(Cosq ,Sinq ,q Cot B),
Where B is a constant, are
(b) If the curve of intersection of two surfaces is a line of curvature on both, prove that the surfaces cut a constant angle. (10)

COMPULSORY QUESTION

Q.8. Write only the correct answer in the Answer Book. Do not reproduce the question.
(1) The additive group of integers has:
(a) 3 quotient groups of order 2 each
(b) 5 quotient groups of order 5 each
(c) one quotient groups of order 5
(d) None of these

(2) Let Q and Z be the additive groups of rationals and integers respectively. Then the
group Q/Z:
(a) is cyclic (b) is a finite group
(c) has no element of order 6 (d) None of these

(3) Every field contains more than:
(a) one element (b) three element
(c) two element (d) None of these

(4) Suppose A,B are matrices such that AB exists and is zero matrix. Then:
(a) a must be zero matrix (b) B must be zero matrix
(c) Neither A nor B needs be zero matrix
(d) None of these

(5) The unit matrix of order n has rank:
(a) zero (b) n
(c) 1 (d) None of these

(6) Let V be the real vector space of all functions on R to R, and let A = {x2, Sin x}.
Then:
(a) A spans V (b) A is linearly independent
(c) A is linearly dependent (d) None of these

(7) If the matrix equation AX = 0, where A is an n´ n matrix, has a non-trivial solution,
then:
(a) determinant A is zero (b) Matrix A is non-singular
(c) determinant A is non zero (d) None of these

(8) Let Jn denote the ring of integers mod n. Then:
(a) J6 is a field (b) J5 is a field
(c) J8 is an integral domain (d) None of these

(11) The rectangular coordinates of the point with spherical coordinates (6,
6P, 6P) are:
(a) (6,0,0) (b) (0,6,0)
(c) (0,0,6) (d) None of these

(12) The only space curve whose curvature and torsion are both constant is:
(a) parabola (b) a circular helix
(c) a circle (d) None of these

(13) If the torsion at all points of a curve is zero, then the curve is:
(a) a helix (b) a straight line
(c) all in one plane (d) None of these

(14) Let G be a group of order 17. Then:
(a) G is no cyclic (b) G is non abelian
(c) G is commutative (d) None of these


(16) If V is n-dimensional vector spaces, then any set of n+1 vectors in V is:
(a) linearly dependent (b) linearly independent
(c) a basis of V (d) None of these

(17) If f :V -- W is a linear map of an n-dimensional vector space V onto W, then:
(a) dim W = dim Ker f + dim V
(b) dim Ker f + dim W = dim V
(c) Dim Ker f = dim W
(d) None of these

(18) If determinant | A | = 2, then:
(a) | A4 | = 12 (b) | A5 | = 32
(c) | A6 | = 60 (d) None of these