CSS Pure Mathematics Past Paper 2017

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) If R is a commutative ring with unit element and M is an ideal of R then show that M is a maximal ideal of R if and only if R/M is a field. (b) If F is a finite field and 𝛼 ≠ 0, 𝛽 ≠ 0 are two elements of F then show that we (10) (20) can find elements a and b in F such that 1 + 𝛼𝑎2 + 𝛽𝑏 2 = 0. 

2. Q. 310 marks

   (a) Let V be a finite-dimensional vector space over a field F and W be a subspace of V. Then show that W is finite-dimensional, dimW ≤ dim V and dim V/W = dim V – dim W. (b) Suppose V is a finite-dimensional vector space over a field F. Prove that a (10) (20) linear transformation 𝑇 ∈ 𝐴(𝑉) is invertible if and only if the constant term of the minimal polynomial for T is not 0.

3. Q. 410 marks

   (a) Use the Mean-Value Theorem to show that if f is differentiable on an interval I, and if |𝑓′(𝑥)| ≤ 𝑀 for all values of 𝑥 in I, then |𝑓(𝑥) − 𝑓(𝑦)| ≤ 𝑀|𝑥 − 𝑦| for all values of 𝑥 and 𝑦 in I. Use this result to show further that |sin 𝑥 − sin 𝑦| ≤ |𝑥 − 𝑦|. (b) Prove that if 𝑥 = 𝑥(𝑡) and 𝑦 = 𝑦(𝑡) are differentiable at t, and if (10) (20) 𝑧 = 𝑓(𝑥, 𝑦) is differentiable at the point (𝑥, 𝑦) = (𝑥(𝑡), 𝑦(𝑡)), then 𝑧 = 𝑓(𝑥(𝑡), 𝑦(𝑡))is differentiable at t and 𝑑𝑧 𝜕𝑧 𝑑𝑥 𝜕𝑧 𝑑𝑦 = 𝜕𝑥 𝑑𝑡 + 𝜕𝑦 𝑑𝑡 𝑑𝑡 where the ordinary derivatives are evaluated at t and the partial derivatives are evaluated at (𝑥, 𝑦).

4. Q. 510 marks

   (a) Evaluate the double integral ∬𝑅 (3𝑥 − 2𝑦) 𝑑𝑥 𝑑𝑦 (b) Where R is a region enclosed by the circle 𝑥 2 + 𝑦 2 = 1. (10) (20) Find the area of the region enclosed by the curves 𝑦 = sin 𝑥, 𝑦 = cos 𝑥, 𝑥 = 0, 𝑥 = 2𝜋. PURE MATHEMATICS

5. Q. 610 marks

   (a) Find an equation of the ellipse traced by a point that moves so that the sum of its distance to (4,1) and (4,5) is 12. (b) Show that if a, b and c are nonzero, then the plane whose intercepts with the (10) (20) coordinate axes are 𝑥 = 𝑎, 𝑦 = 𝑏, and 𝑧 = 𝑐 is given by the equation. 𝑥 𝑦 𝑧 + 𝑏 + 𝑐 = 1. 𝑎

6. Q. 710 marks

   (a) Prove that a necessary and sufficient condition that 𝑤 = 𝑓(𝑧) = 𝑢(𝑥, 𝑦) + 𝑖𝑣(𝑥, 𝑦) be analytic in a region R is that the Cauchy-Riemann equations 𝜕𝑢 𝜕𝑣 𝜕𝑢 𝜕𝑣 = 𝜕𝑦and𝜕𝑦 = − 𝜕𝑥 𝜕𝑥 are satisfied in R where it is supposed that these partial derivatives are continuous in R. (b) Show that the function 𝑓(𝑧) = 𝑧̅ is not analytic anywhere in the complex (10) (20) plane Z.

7. Q. 810 marks

   (a) Let 𝑓(𝑧) be analytic inside and on the boundary C of a simply-connected region R. Prove that 1 𝑓(𝑧) 𝑓′(𝑎) = ∮ 𝑑𝑧. 2𝜋𝑖 𝐶 (𝑧−𝑎)2 (b) Show that 2𝜋 𝑑𝜃 5𝜋 (10) (20) ∫0 (5−3 sin 𝜃)2 = . 32

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.