CSS 2017 Pure Mathematics Past Paper

Paper details

Examination

CSS

Year

2017

Subject

Pure Mathematics

Paper

Single Paper

Subject type

Optional

Mode

Subjective

Questions recorded from this paper

These 8 question wordings are transcribed from CSS Vista’s structured Pure Mathematics past-paper record for CSS 2017.

1. Q1 · Normal subgroups(a) Let H, K be subgroups of a group G. Prove that HK is a subgroup of G if and (10) only if HK=KH. (b) If N, M are normal subgroups of a group G, prove that (10) (20) NM/M ≅N/N∩M.
2. Q2 · Ring(a) If R is a commutative ring with unit element and M is an ideal of R then show (10) 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. 
3. Q3 · Field extension and finite fields(a) Let V be a finite-dimensional vector space over a field F and W be a subspace (10) 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.
4. Q5 · Calculus & Analytic Geometry(a) Evaluate the double integral (10) ∫∫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.
5. Q8 · Calculus & Analytic Geometry(a) Let f(z) be analytic inside and on the boundary C of a simply-connected (10) 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 .
6. Q4 · Partial derivatives(a) Use the Mean-Value Theorem to show that if f is differentiable on an (10) 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).
7. Q7 · Partial derivatives(a) Prove that a necessary and sufficient condition that (10) 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.
8. Q6 · The plane(a) Find an equation of the ellipse traced by a point that moves so that the sum (10) 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

Topic-wise Pure Mathematics past-paper analysis.