Pure Mathematics for CSS in 2017 Past Papers

Pure Mathematics for CSS in 2017 Past Papers

Pure Mathematics for CSS in 2017 Past Papers


Q. 1.

(a) Let H, K be subgroups of a group G. Prove that HK is a subgroup of G if and only if HK=KH. (10)
(b) If N, M are normal subgroups of a group G, prove that NM/M ≅N/N∩M. (10)

Q. 2.

(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. (10)

(b) If F is a finite field and α ≠ 0, β ≠ 0 are two elements of F then show that we
can find elements a and b in F such that 1 + αa^2 + βb^2 = 0. (10)

Q. 3.

(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. (10)

(b) Suppose V is a finite-dimensional vector space over a field F. Prove that a linear transformation T ∈ A(V) is invertible if and only if the constant term of the minimal polynomial for T is not 0. (10)


Q. 4.

(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|. (10)

(b) Prove that if x = x(t) and y = y(t) are differentiable at t, and if 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/dt = (∂z/∂x) x (dx/dt) + (∂z/∂y) x (dy/dt) where the ordinary derivatives are evaluated at t and the partial derivatives are evaluated at (x, y). (10)


Q. 5.

(a) Evaluate the double integral ∫∫ (3x − 2y) dx dy R (10)

(b) Where R is a region enclosed by the circle x^2 + y^2 = 1.
Find the area of the region enclosed by the curves y = sin x, y = cos x, x = 0, x = 2π. (10)

Q. 6.

(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. (10)
(b) Show that if a, b and c are nonzero, then the plane whose intercepts with the
coordinate axes are x = a, y = b, and z = c is given by the equation.

x/a + y/b + z/c = 1. (10)


Q. 7.

(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/∂x = ∂v/∂y and ∂u/∂y = − ∂v/∂x are satisfied in R where it is supposed that these partial derivatives are continuous in R. (10)

(b) Show that the function f(z) = z̅is not analytic anywhere in the complex plane Z. (10)

Q. 8.

(a) Let f(z) be analytic inside and on the boundary C of a simply-connected region R. Prove that f′(a) = (1/2πi )∮C (f(z)/ (z−a) ^2) dz. (20)


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