(a)Let G and G' be two groups and f : G → G' be a homomorphism then prove the following: f (e) = e' where e and e' are the identities of G and G' respectively f(a⁻¹) = [f(a)]⁻¹, ∀a∈G [10]
(b)Prove that every homomorphic image of a group is isomorphic to some quotient group. [10]
(10)
Q. 2
(a)A ring R is without zero divisor if and only if the cancellation law hold. [10]
(b)Prove that arbitrary intersection of subrings is a subring. [10]
(10)
Q. 3
(a)Let T: R³ → R³ be the linear transformation defined by T(x₁, x₂, x₃) = (x₁ - x₂, x₁ + x₃, x₂ + x₃). Find a basis and dimension of Range of T. [10]
(b)Prove that every finitely generated vector space has a basis. [10]
(10)
Section B
Q. 4
(a)Find the critical points of f(x) = x³ - 12x - 5 and identify the open intervals on which f is increasing and on which f is decreasing. [10]
(b)Find the horizontal and vertical asymptotes of the graph of f(x) = x / (x² - 4). [10]
(10)
Q. 5
(a)Calculate ∫ (-2x+4) / ((x²+1)(x-1)²) dx. [10]
(b)Find ∂w/∂x at the point (x, y, z) = (2, -1, 1) if w = x² + y² + z², z³ – xy + yz + y³ = 1 and x and y are the independent variables. [10]
(10)
Q. 6
(a)Determine the focus, vertex and directrix of the parabola x² + 6x -8y +17=0 [10]
(b)Find polar coordinates of the point p whose rectangular coordinates are (3√2, -3√2). [10]
(10)
Section C
Q. 7
(a)Show that (cos θ + i sin θ)ⁿ = cos(nθ) + i sin(nθ) for all integers n. [10]
(b)Find the n, nth roots of unity. [10]
(10)
Q. 8
(a)Find the Taylor series generated by f(x) = 1/x at a = 2. Where, if anywhere, does the series converge to 1/x ? [10]
(b)Show that the p-series ∑ (1/nᵖ) (p a real constant) converges if p > 1, and diverges if p < 1. [10]
The 2020 CSS Pure Mathematics paper set by the FPSC. Question wording only; questions marked “Not yet checked” have not been compared with the official paper yet.
Disclaimer Prep Right is independent and not affiliated with FPSC.