Sample Paper

Exam: Comprehensive Examination for Sessions (2013-2015 2014-16) (So Called Lahore Campus, BZU)

Program: M. Sc. Subject: Mathematics

Maximum Marks: 100 Pass Marks: 50

Time Duration: 3 Hours

Note: Attempt all questions.

Real Analysis-I

1 / If are real number with , then there exists an irrational number such that . / 04
2 / Define the following terms with at least one example:
(i) Uniformly continuous function
(ii) Differentiable function / 04
3 / Give arguments to prove or disprove the following statements:
(i) The sequence (n) is divergent.
(ii) . / 04
4 / State and Prove the Mean-Value Theorem. / 04
5 / Define the following terms with at least one example:
(i) Bounded function
(ii) Convergent Sequence / 04

Ordinary Differential Equations

1 / Define exact differential equation and give one example. Determine a function so that the following differential equation is exact . / 04
2 / Define initial value problem and give one example. Solve the following initial value problem / 04
3 / The roots of an auxiliary equation are . What is the corresponding differential equation? / 04
4 / Solve the differential equation. / 04
5 / Define Wronskian of n functions. Check whether the given functions are linearly independent or linear dependent on the indicated interval.
(i)
(ii) / 04

Algebra III

1 / Show that in a vector space of dimension , any vectors are linearly dependent. / 04
2 / Define/Describe the following:
(i) Cauchy Schwartz Inequality
(ii) Rank Nullity Theorem / 04
3 / If S is an orthogonal set of nonzero vectors then S is linearly independent. / 04
4 / The vector space of real polynomials can not be spanned by a finite number of polynomials. / 04
5 / Let be a linear transformation. Show that image of T is a subspace of V. / 04

Functional Analysis-I

1 / Prove that a discrete metric space is complete. / 04
2 / Define isometric spaces. Prove that for two isometric spaces “X” and “Y”, completeness of “Y” implies that “X” is also complete. / 04
3 / Prove that a closed subset “M” of a compact metric space “X” is also compact. / 04
4 / Prove that space is not a Hilbert space. / 04
5 / Prove that a Bounded linear operator is always continuous. / 04

Modeling and Simulation

1 / State which of the following represents a system of differential equations. Justify your answer:
i.  Predator-Prey model
ii.  Radioactive Decay model / 04
2 / Differentiate between:
i.  Malthusian population growth model and Logistic growth model
ii.  Heat Equation and Laplace Equation. / 04
3 / Describe the free damped motion of a spring-mass system and give its mathematical formulation. / 04
4 / Derive the differential equation for Newton’s law of cooling/warming. Explain the nature of the constant of proportionality k. / 04
5 / Give the types of Boundary conditions for solving PDEs and give examples. / 04