CSS Pure Mathematics Past Paper 2016

Instructions specific to this paper

• Attempt FIVE questions in all by selecting TWO Questions each from SECTION-A&B and
• Use of Calculator is allowed.

Questions

1. Q. 210 marks

   (a) Let a be a fixed point of a group G and consider the mapping Ia : G→G defined -1 by Ia(g)= aga where g∈G. Show that Ia is an automorphism of G. Also show that for a, b ∈ G, Ia .Ib=Iab (10) (20) a b  (b) Let M2 (R) =   c  : a , b , c , d  R  be the set of all 2×2 matrices with  d   real entries. Show that ( M2(R), +, ∙ ) forms a ring with identity. Is ( M2(R), +, ∙ ) a field? 

2. Q. 310 marks

   (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 (10) (20) x + λy + 3z = 0 4x + 3y + λz = 0 2x + y + 2 z = 0 has non-trivial solution. 2

3. Q. 410 marks

   (a) Using - ∈ definition of continuity, prove that the function Sin x is continuous for all x ∈ R. 2 2 2 2 (10) (20) (b) Find the asymptotes of the curve (x -y )(x+2y) + 5 (x +y ) + x+y =0

4. Q. 510 marks

   ( 1x ) x (a) Prove that the maximum value of is e1/e . PURE MATHEMATICS 3

5. Q. 610 marks

   (a) Find the area enclosed between the curves y=x and y=x. (b) A plane passes through a fixed point (a, b, c) and cuts the coordinate axes in (10) (20) A,B,C. Find the locus of the centre of the sphere OABC for different positions of the plane, O is the origin.

6. Q. 820 marks

   (a) Use Cauchy Integral Formula to evaluate ∫ dz along the simple z closed counter C: | z |=3 described in the positive direction. (b) State and prove Cauchy Residue Theorem. (10)

Standard FPSC instructions

Printed on this and every CSS paper.

• Part-II is to be attempted on the separate Answer Book.
• All the parts (if any) of each question must be attempted at one place instead of at different places.
• Write Q. No. in the Answer Book in accordance with Q. No. in the question paper.
• No page/space be left blank between the answers. All the blank pages of the Answer Book must be crossed.
• Extra attempt of any question or any part of the attempted question will not be considered.