# Applied Mathematics (Paper A & B)

This paper consista of two papers of 100 marks each. One paper is called “Paper A” and other is called “Paper B”.

## Paper A

- NOTE: attempt two questions from each section.

### SECTION-I (4/12: 17,17,17,17)

Basic concepts of differential equations. Classification and formation of DEs. Various methods of solutions of first order ODE (linear and non-linear). The Bernoulli’s, Ricatti and Clairaut’s equations. Singular solutions. Orthogonal trajectories. Application of first order ODE in problems of decay and growth, population dynamics, logistic equations. Linear DE of higher order (homogeneous and non-homogeneous). Solution by: D-operator and undetermined co-officients Methods. Reduction of order and variation of parameters methods for 2nd order linear DE. Cauchy-Euler equation. Power series solution about an arbitrary point.

### SECTION-II (4/12: 16,16,16,16)

Laplace Transformation, solution of ODEs. Error analysis. Solution of non-linear (algebraic and transcendental) equation in one variable using bisection method, false position method, Newton - Raphson method and fixed point method. Convergence and order of convergence of Newton- Raphson method. Difference operators. Interpolation (Newton’s and Lagrange’s methods). Numerical differentiation (at a point of the data). Numerical integration (rectangular, trapezoidal and 1/3 Simpson’s rules).

### SECTION-III (4/12: 17,17,17,17)

Linear programming. Mathematical model. Graphical solution of two dimensional LP models. Simplex method. Event and Sample Space. Axioms of probability. Conditional probability. Independence. Random variable of discrete and continuous type. Construction of probability distribution. Expectation. Variance and standard deviation. Binomial distribution.

**Advice to paper setter:** - The paper setter is advised; please not to set both parts of a question from the same topic/ chapter.

## Paper B

- NOTE: attempt two questions from each section.

### SECTION-I (4/12: 17,17,17,17)

Vectors in three-dimensions. Scalar and vector products with applications. Scalar and vector triple products. Differentiation and integration of vector functions. Gradient, divergence and curl. Tensor analysis. Differential operators. Application to vector analysis. Composition and resolution of co-planar forces. Moments, couples and conditions of equilibrium under the action of co-planar forces.

### SECTION-II (4/12: 16,16,16,16)

Frictional forces. Laws of friction. Equilibrium of bodies on rough surfaces. Principle of virtual work and related problems. Center of gravity. Center of mass of various bodies. Kinematics of a particle in cartesian and polar co-ordinates. Laws of mechanics. Linear and angular velocity. Relative velocity. Rectilinear motion with uniform and variable acceleration. Simple harmonic motion.

### SECTION-III (4/12: 17,17,17,17)

Projectile motion. Motion along horizontal and vertical circles. Orbital motion. Planetary motion and Keplar’s laws. Conservative forces. Damped and forced vibrations. Co-ordinates in three dimension. Rectangular, cylindrical and spherical co-ordinates. Equations of plane, straight line, sphere, cylinder, cone, ellipsoid, hyperboloid and paraboloid. Longitude and latitudes. Spherical triangle and direction of Qibla.

**Advice to paper setter:** - The paper setter is advised; please not to set both parts of a question from the same topic/ chapter.

## Recommended Books.

- Theory of Differential Equations of Dennis G.Zill. Books Thomson Learning Academic Resource Center. USA.
- Mathematical Techniques by K.H. Dar, Irfan-ul-Haq and M.A. Jajja. The Carvan Books House. Kachehry Road, Lahore.
- Mathematics Methods by S.M. Yousaf. Illmi Kitab Khana. Urdu Bazar, Lahore.
- Numerical Analysis by R.L. Burden and J.D. Faires. PES-Kent Publishing Company. Bostan. USA
- Operations Research by H.A. Taha. Prentice-hall Inc. Englewood. Cliffs USA.(1996)
- Mathematical Statistics. By Dr. J.E.Freund. Prentice-hall Inc. Englewood. Cliffs USA.
- Vector and Tensor Methods, by Chorlton, Ellis Horwood Publishers.
- Elementary Vector Analysis. By Dr. Munawar Hussain. S.M. Hafeez. M.A. Saeed and Ch. Bashir Ahmed. The Caravan Book House, Kachhry Road , Lahore.
- A Text Book by Dynamics by Chorlton, Van Nostrand Company Ltd. London.
- Mechanics by O.K. Ghori. West Pakistan Publishing Company, Lahore.