CSS Pure Mathematics Past Paper 2022

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) Prove that a finite integral domain is a field. (b) Let W be the subspace of R spanned by = 1,2, −1,3,4 , = 2,4, −2,6,8 , (10) (20) = 1,3,2,2,6 , = 1,4,5,1,8 , = 2,7,3,3,9 . Find a subset of the vectors that form a basis of W. Also extend the basis of W to a basis of R . 

2. Q. 310 marks

   (a) Let : R → R be defined by , , !, " = − + ! + ", 2 − 2 + 3! + 4", 3 − 3 + 4! + 5" Find the rank and nullity of T. (b) Find all possible solutions of the following homogeneous system of equations. (10) (20) + + − =0 +2 − 2 + = 0 2 +4 −3 + =0 4 +7 −4 + =0 ⁄ *+ (

3. Q. 420 marks

   (a) Find lim 1 + 2 . (10) (→) ( (b) Evaluate the integral - . cos 2 3 . (10)

4. Q. 520 marks

   (a) If =4 , and = 5 cos 6 , = 5 sin 6, then show that (10) 9: 9: 9: 9: 89( ; + 89<; = 8 9= ; + = > 89?; (b) Evaluate ∬A 3 3 over the region bounded by = and = . (10)

5. Q. 610 marks

   (a) Find the area of the region bounded above by = + 6, bounded below by = , and bounded on the sides by the lines = 0 and = 2. (b) Find the foci, vertices and center of the ellipse: (10) (20) 9 + 16 − 72 − 96 + 144 = 0 \ PURE MATHEMATICS \ (

6. Q. 720 marks

   (a) Prove that the function , =. sin − cos is harmonic. Also find (10) a function B , such that 4 ! = , + C B , is analytic. (b) Evaluate ∮F !̅ 3! around the circle |!| = 1. (10)

7. Q. 810 marks

   (a) Use residues to prove that ) H( K -M ( I J = √ (b) Find the Fourier series of the following function 4 which is assumed to have (10) (20) the period 2N. 4 = | |, −N < < N

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.