CSS written syllabus

CSS Applied Mathematics Syllabus

Applied Mathematics is a 100-mark Group II optional subject in the CSS written exam, examined in one three-hour paper. Below is FPSC's official syllabus, word for word: every section, topics and suggested readings.

Marks: 100
Papers: 1
Time: 3 hours a paper
Group: Group II
Subject code: 18

The paper

100 marks

I.
Vector Calculus (10%) Vector algebra; scalar and vector products of vectors; gradient divergence and curl of a vector; line, surface and volume integrals; Green’s, Stokes’ and Gauss theorems.
II.
Statics (10%) Composition and resolution of forces; parallel forces and couples; equilibrium of a system of coplanar forces; centre of mass of a system of particles and rigid bodies; equilibrium of forces in three dimensions.
III.
Dynamics (10%)
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  Motion in a straight line with constant and variable acceleration; simple harmonic motion; conservative forces and principles of energy.
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  Tangential, normal, radial and transverse components of velocity and acceleration; motion under central forces; planetary orbits; Kepler laws;
IV.
Ordinary differential equations (20%)
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  Equations of first order; separable equations, exact equations; first order linear equations; orthogonal trajectories; nonlinear equations reducible to linear equations, Bernoulli and Riccati equations.
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  Equations with constant coefficients; homogeneous and inhomogeneous equations; Cauchy-Euler equations; variation of parameters.
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  Ordinary and singular points of a differential equation; solution in series; Bessel and Legendre equations; properties of the Bessel functions and Legendre polynomials.
V.
Fourier series and partial differential equations (20%)
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  Trigonometric Fourier series; sine and cosine series; Bessel inequality; summation of infinite series; convergence of the Fourier series.
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  Partial differential equations of first order; classification of partial differential equations of second order; boundary value problems; solution by the method of separation of variables; problems associated with Laplace equation, wave equation and the heat equation in Cartesian coordinates.
VI.
Numerical Methods (30%)
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  Solution of nonlinear equations by bisection, secant and Newton-Raphson methods; the fixed- point iterative method; order of convergence of a method.
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  Solution of a system of linear equations; diagonally dominant systems; the Jacobi and Gauss-Seidel methods.
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  Numerical differentiation and integration; trapezoidal rule, Simpson’s rules, Gaussian integration formulas.
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  Numerical solution of an ordinary differential equation; Euler and modified Euler methods; Runge- Kutta methods.

Past papers

FPSC's Applied Mathematics papers from 2016 to 2026, as text. Open a year, or see them all.

Applied MathematicsAll papers

The paper

Suggested readings

Past papers