CSS Pure Mathematics Past Paper 2019

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) Show that the characteristic of an integral domain is R is either zero or a prime. (b) Determine whether or not the set {(1, 2, −1), (0, 3, 1), (1, −5, 3)} of vectors (10) (20) is a basis for 𝑅3 .

2. Q. 310 marks

   (a) Show that a one-to-one linear transformation preserves basis and dimension. (b) Solve the system of linear equations: (10) (20) 2 x1 + x2 + 5 x3 = 4 3 x1 − 2 x2 + 2 x3 = 2 5 x1 − 8 x2 + 2 x3 = 1. 𝜋 2 (10)

3. Q. 410 marks

   (a) Solve ∫0 sin 6𝑥 cos4 3𝑥 𝑑𝑥. 2 6 (b) Find the area enclosed by 𝑦 = . (20) 2−cos 𝜃

4. Q. 510 marks

   (a) Show that in any conic semi-latusrectum is the harmonic mean between the segments of focal chord. (b) Prove that the evolute of hyperbola (10) (20) 2 2 2 2𝑥𝑦 = 𝑎 is (𝑥 + 𝑦) − (𝑥 − 𝑦) = 2𝑎 . 3 3 3

5. Q. 610 marks

   (a) Define Supremum and Infimum of a sequence. Find the supremum and infimum of the set 1 �(−1)𝑛 �1 − 𝑛� , 𝑛 = 1,2,3 … �. (b) Evaluate (10) (20) 1 (1+𝑥)𝑥 −𝑒 lim𝑥→0 . 𝑥 PURE MATHEMATICS 𝜃 𝜃 (10)

6. Q. 720 marks

   (a) Show that 𝐿𝑜𝑔(1 + cos 𝜃 + 𝑖 sin 𝜃) = ln(2 cos ) + 𝑖 . 2 2 (b) Find 𝑣 such that 𝑓 (𝑧) = 𝑢 + 𝑖𝑣 is analytic. (10)

7. Q. 820 marks

   (a) Prove that the series 𝑧(1 − 𝑧) + 𝑧 2 (1 − 𝑧) + 𝑧 3 (1 − 𝑧) + ⋯ converges (10) for |𝑧| < 1, and find its sum. 𝑧 2 −2𝑧 (b) Find the residues of 𝑓(𝑧) = (𝑧+1)2 at all its poles in the finite plane. (𝑧 2 +4) (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.