CSS Pure Mathematics Past Paper 2025

Instructions specific to this paper

• Attempt FIVE questions in all by selecting from TWO questions each from SELECTION A&B
• Use of calculator is allowed.

Questions

1. Q. 210 marks

   (a) A homomorphism φ: Z6 → Z6 is a one to one mapping then calculate Ker(φ). (b) Check whether the vectors a = (1,2,3), b = (2,5,7)and c = (1,3,5) are linearly (10) dependent or independent.

2. Q. 310 marks

   (a) Let V be a vector space of all 2x 2 matrices. W be a subspace of V which consists of all symmetric matrices then find two different basis of W. (b) Consider the transformation T: R³ → R² given by T(x, y, z) = (x, y+z). Check whether T is linear or not. (10)

3. Q. 410 marks

   (a) Find all the real numbers x ∈ R such that x2 + x > 2. (b) Use the definition of limit to establish that lim x→2 X3-4 = 4 (10) X2+1 5

4. Q. 510 marks

   (a) Use Mean Value Theorem to show that e x ≥ 1 + x ∀ x∈R. (b) Find the absolute extrema of the function f(x,y) = xy - 2x on the region R given (10) by vertices (0, 4), (4, 0) and (0, 0). Q. No.6. (a) Change the order of integration in double integral 0∫2 0∫x f(x,y)dydx. (10) (b) Draw the graph of the conic r = 2cosθ. (10)

5. Q. 710 marks

   (a) Check that the Cauchy-Riemann Equations are satisfied in polar coordinates for f(z) = 1/z. (b) Evaluate the contour integral c∫ f(Z)dZ where f(Z) = y - x - 3ix2 and C is (10) simple closed contour OABO with O = 0+0i, A = 0+i and B = 1+i.

6. Q. 810 marks

   (a) Use Cauchy Residue Theorem to evaluate the integral c (z2-1)/[z(z-3)] dz where C is the circle |Z| = 2. (b) Find the Maclaurin series for the function f(Z) = Z2e3Z. (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.