CSS Pure Mathematics Past Paper 2018

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.
• Find the values of a for which the system has unique solution.
• Find the values of the pair (a, b) for which the system has more than one
• = sinx in a Fourier cosine series in the interval 0 ≤ x ≤ π . (10) (20)

Questions

1. Q. 210 marks

   (a) Show that every finite integral domain is a field. (b) Consider the following linear system, (10) (20) x + 2y + z = 3 ay + 5 z = 10 2 x + 7 y + az = b (i) Find the values of a for which the system has unique solution. (ii) Find the values of the pair (a, b) for which the system has more than one solution.

2. Q. 310 marks

   (a) Find condition on a,b,c so that vector (a,b,c) in R3 belongs to W= span {u 1 ,u 2 ,u 3 } where u 1 = (1,2,0), u 2 = (-1,1,2), u 3 = (3,0,-4). (10) (20) (b) Let W 1 and W2 be finite dimensional subspaces of a vector space V. Show that dimW 1 + dimW2 = dim ( W1  W 2 ) + dim ( W 1 + W 2 ) x 2 if x ≤ 1 (10)

3. Q. 410 marks

   (a) Let f ( x ) = { x if x > 1 1  Does the Mean Value Theorem hold for f on  ,2 . 2  lnsin3x (b) Calculate the. lim (20) x →0 lnsinx 5

4. Q. 510 marks

   (a) Evaluate ∫−1 x − 2 dx. (b) Prove that f xy (0,0) ≠ f yx (0,0) if (10) (20)  2 1 f ( x, y ) =  x y sin when x, y are not both 0 x  0 when x, y are both 0 PURE MATHEMATICS

5. Q. 610 marks

   (a) Find the area of the region bounded by the cycloid x = a (θ − sin θ ), y = a (1 − cosθ ) and its base. (b) Find the equation of a plane through (5,-1,4) and perpendicular to each of the (10) (20) planes x + y − 2 z − 3 = 0 and 2 x − 3 y + z = 0

6. Q. 710 marks

   (a) Express cos5 θ sin3 θ in a series of sines of multiples of θ . 5z − 2 (b) Use Cauchy’s Residue Theorem to evaluate the integral ∫c Z (Z − 1) dz where C (10) (20) is the circle z = 2 , described counter clock wise. z +1

7. Q. 820 marks

   (a) Find the Laurent series that represent the function f ( z ) = in the domain (10) z −1 1< z < ∞ . (b) Expand f(x) = sinx in a Fourier cosine series in the interval 0 ≤ x ≤ π . (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.