CSS Pure Mathematics Past Paper 2023

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) For any group G , show that ≅ G and ≅ {e} . {e} G (b) Suppose U and W are distinct four dimensional subspaces of a vector space V of (10) (20) dimension six. Find the possible dimension of U ∩ W .  − α α −1 α +1  

2. Q. 310 marks

   (a) For what value of α is the matrix  1 2 3  is singular?   2 − α α + 3 α + 7 (b) Define T : ℜ3 → ℜ3 by T ( x1 , x2 , x3 ) = (− x3 , x1 , x1 + x3 ) . Find N (T ) . Is T one-to- (10) (20) one? sin 2 x + θ sin x

3. Q. 410 marks

   (a) Find the value of θ and the limit in order that lim be finite. x →0 x3 x (b) Show that x < sin −1 x < , 0<x<1. (10) (20) 1 − x2 1 (10)

4. Q. 520 marks

   (a) Given that U = . Verify that U xx + U yy + U zz = 0. x2 + y2 + z 2 (b) Evaluate ∫∫ ( x 2 + y 2 )dxdy , over the domain bounded by y = x 2 and x = y 2 . (10)

5. Q. 610 marks

   (a) Evaluate ∫∫ ( x 2 + y 2 )dxdy , over the region bounded by xy=1, y=0, y=x and x=2. x2 y2 (b) Find an equation of a normal to the hyperbola − = 1 in the form (10) (20) a 2 b2 ax cosθ + by cot θ = a 2 + b 2 . Prove that the normal is external bisector of the angle between the focal distances of its foot. \ PURE MATHEMATICS \

6. Q. 710 marks

   (a) Determine k such that U = e 2 x cos ky is harmonic and find a conjugate harmonic. 1 (b) Evaluate ∫ ( 5 + z 3 )dz from 1 to -1 along the upper arc of the unit circle. (10) (20) C z 4 1

7. Q. 810 marks

   (a) Find the Laurent Series of in the region 0< z − 1 <2. 1− z2 − Z 2 − 22 z + 8 (b) Find the residues at the singular points of which lie inside the (10) (20) Z 3 − 5z 2 + 4 z circle z =2.

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.