CSS Pure Mathematics Past Paper 2021

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.
• ; the set of all polynomials of degree three. Also express the vectors 1+x2

Questions

1. Q. 210 marks

   (a) Find elements of the cyclic group generated by the permutation. 1 2 3 4 5 6    3 4 5 2 6 1 (b) (10) (20) 2 3 2 3 Verify that the polynomials 2-x , x -x, 2-3x and 3-x form a basis for the set P3 (x); the set of all polynomials of degree three. Also express the vectors 1+x2 and x+x3 as a linear combination of these basis vectors. 

2. Q. 310 marks

   (a) Let V be the real vector space of all function from R to R. Show that {cos 2 x, sin2 x, cos 2x}is linearly dependent while { cosx, sinx, coshx, sinhx} are linearly independent. (b) Solve the system of linear equations: (10) (20) x1 – 2x2 – 7x3 + 7x4 = 5 – x1 + 2x2 + 8x3 – 5x4 = – 7 3x1 – 4x2 – 17x3 + 13x4 = 14 2x1 – 2x2 + 11x3 + 8x4 = 7  y 1  x  If f  x, y   x tan  1

3. Q. 410 marks

   (a) 2   y 2 tan  . x  y 2 f  x 2  y2  Show that    2 2 x, y  yx  x +y  6  x 2 whenx  2 (b) Evaluate  f  x dx 0 where f ( x)   3x  2whenx  2 (10) (20) PURE MATHEMATICS 

4. Q. 510 marks

   (a)  Let I n  x n e  x dx where n is an integer. Prove that 0 𝐼𝑛 = 𝑛 𝐼𝑛−1 Hence show that In = n! 8 (b) i. Write r  in rectangular coordinates. (10) (20) 2  cos Write x  2 x y  y  6 x y  2 y  0 4 2 2 4 2 3 ii. in polar coordinates.

5. Q. 610 marks

   (a) Evaluate  dydx D and  dxdy where D is the region bounded by the y-axis, the D x lines x=2 and the curve e . x3  x (b) Investigate the curve y  2 for points of inflexion. (10) (20) 3x  1 \ 1 1.3 1.3.5

6. Q. 710 marks

   (a) Sum the series1  cos  cos 2  cos3  ... 2 2.4 2.4.6  2 3 1 (b) Prove that co s  co s  co s  (10) (20) 7 7 7 2 Construct the analytic function f whose real part is U  x  3xy  3x  1 3 2

7. Q. 810 marks

   (a) dz (b) Evaluate z C 2  2z  2 Where C is a square with corners (10) (20) (0,0),(-2,0),(-2,-2) and (0,-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.