Mathematics BS Mathematics/MS Mathematics QAU — Quaid-i-Azam University 2018

Department of Mathematics, Quaid-i-Azam University, Islamabad

M.Phil Admission Test - Spring 2018

Note: Attempt all questions.

Q.1) Calculate the arc-length of the curve

(t) = a(t - sint, 1 - cost),

as t goes from 0 to 2, here a is a constant.

Q.2) Prove that the unit circle S1 = {(x, y) € R? | x2 + y? = 1} is a one dimensional manifold.

Q.3) If A and B are any subsets of a topological space (X, r). Then show that AUB = AUB

Q.4) Solve

dy

dz

= y(xy? - 1).

Q.5) Reduce the following equation to canonical form and bence solve it

อน+ (2+9)

a?u

Ozoy

+*ay2=0.

Q.o) Define the term constraint. Also discuss the major classifications of constraints.

Q.7) Give an example(with explanation) of a non-abelian group in which every proper subgroup is normal.

Q.8) Show that (1, 1, 1)) is a linearly independent subeet of R?. Moreover expand it to form a basis of R3.

Q.9) Find residue at z = 0 of the function f(z) = Corec?z

Q.10) Prove that Q is not complete by telling a sequence of rational numbers whose limit is irrational

number.

Q.11) Consider the function f(x) = (»(a).

(a) What is the domain of f(x)?

(b) Discuas lim, -o+ f(x).

(c) Write the interval on which f is increasing?

Q.12) Show the the integral f° sin(x?) dz converges.

decimal places.

Q.13) Use Newton-Raphson Method to find the solution of the equation 5x + In(r) = 10000, correct to 4

Good Luck