CSS Pure Mathematics Past Paper 2020

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.
• f ( e ) = e′ where e and e′ are the identities of G and G ′ respectively
• f a −1 = [ f (a) ] , ∀ a ∈G
• = at a = 2. Where, if anywhere, (10)

Questions

1. Q. 220 marks

   (a) A ring R is without zero divisor if and only if the cancellation law hold. (10) (b) Prove that arbitrary intersection of subrings is a subring. (10)

2. Q. 310 marks

   (a) Let T : R3 R3 be the linear transformation defined by T ( x1 , x2 , x3 ) =( x1 − x2 , x1 + x3 , x2 + x3 ) . Find a basis and dimension of Range of T. (b) Prove that every finitely generated vector space has a basis. (10) (20)

3. Q. 410 marks

   (a) Find the critical points of f ( x ) =x3 − 12 x − 5 and identify the open intervals on which f is increasing and on which f is decreasing. 8 (b) Find the horizontal and vertical asymptotes of the graph of f ( x ) = − (10) (20) x −4 2 −2 x + 4 (10)

4. Q. 510 marks

   (a) Calculate ∫ (x 2 + 1)( x − 1) 2 dx . ∂w (b) Find at the point (x, y, z) = (2, -1, 1) if w = x 2 + y 2 + z 2 , z 3 − xy + yz + y 3 = 1 (20) ∂x and x and y are the independent variables.

5. Q. 610 marks

   (a) Determine the focus, vertex and directrix of the parabola x2 + 6x -8y +17=0 (b) Find polar coordinates of the point p whose rectangular coordinates are (10) (20) ( 3 2, − 3 2 ) PURE MATHEMATICS Show that ( cos θ + i sin θ ) = cos(n θ ) + isin(n θ ) for all integers n. n

6. Q. 710 marks

   (a) (b) Find the n, nth roots of unity. (10) (20) 1

7. Q. 810 marks

   (a) Find the Taylor series generated by f (x) = at a = 2. Where, if anywhere, x 1 does the series converge to ? x ∞ 1 (b) Show that the p-series ∑n n =1 p , (p a real constant) converges if p> 1, and (10) (20) diverges if P < 1

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.