Federal Public Service Commission (FPSC)
Competitive Examination for Recruitment to BPS-17 Posts under the Federal Government
Time Allowed: 3 Hours
PURE MATHEMATICS
MAXIMUM MARKS: 100 Marks
Attempt FIVE questions in all by selecting at least TWO Questions From SECTION-A & B and ONE Questions from SECTION-C, All questions carry EQUAL marks.
Use of Scientific Calculator is Allowed.
SECTION-A
Q. No. 1. (a) Let ℚ⁺ be the set of positive rational numbers and define * by a * b = ab/2. Then prove that (ℚ⁺, *) is a group.
(b) Find all the cyclic subgroups of ℤ₁₈.
Q. No. 2. (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 dependent or independent.
Q. No. 3. (a) Let V be a vector space of all 2×2 matrices. W be a subspace of V which consists of all symmetric matrices then find two different bases 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.
SECTION- B
Q. No. 4. (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) / (x² + 1) = 4/5.
Q. No. 5. (a) Use Mean Value Theorem to show that eˣ ≥ 1 + x ∀ x ∈ ℝ.
(b) Find the absolute extrema of the function f(x,y) = xy – 2x on the region R given by vertices (0, 4), (4, 0) and (0, 0).
Q. No. 6. (a) Change the order of integration in double integral ∫₀² ∫₀ˣ f(x,y) dy dx.
(b) Draw the graph of the conic r = 2 cos θ.
SECTION-C
Q. No. 7. (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 simple closed contour OABO with O = 0+0i, A = 0+i and B = 1+i.
Q. No. 8. (a) Use Cauchy Residue Theorem to evaluate the integral ∮_C (z² – 1) / [z(z – 3)] dz where C is the circle |z| = 2.
Answer:
(b) Find the Maclaurin series for the function f(z) = z.
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