# Syllabus for General Aptitude (GA)

*(COMMON TO ALL)***Verbal Ability:**English grammar, sentence completion, verbal analogies, word groups, instructions, critical reasoning and verbal deduction.

**Numerical Ability:**Numerical computation, numerical estimation, numerical reasoning and data interpretation.

**Sample Questions****Verbal Ability**

**Q.1.**Choose the appropriate answer to complete the following sentence:

After several ……. attempts to send the missile into space, the spacecraft was finally launched successfully.

(A) abortive (B) difficult (C) experimental (D) preliminary

**Ans. (A)**

**Q.2.**Choose the appropriate answer to complete the following sentence:

Medicine is to illness as law is to _________

(A) discipline (B) anarchy (C) treason (D) etiquette

**Ans. (B)**

**Q.3**. Read the following paragraph :

“The ordinary form of mercury thermometer is used for temperature ranging from –40

^{o}F to 500^{o}F. For measuring temperature below –40^{o}F, thermometers filled with alcohol are used. These are, however, not satisfactory for use in high temperatures. When a mercury thermometer is used for temperature above 500^{o}F, the space above the mercury is filled with some inert gas, usually nitrogen or carbon dioxide, placed in the thermometer under pressure. As the mercury rises, the gas pressures is increased, so that it is possible to use these thermometers for temperatures as high as 1000^{o}F.”
With what, besides mercury, would a thermometer be filled if it wasdesigned to be used for measuring temperature of about 500

^{o}F?
(A) Pyrometer (B) Inert gas (C) Iron and brass (D) Gas

**Ans. (B)**

**Q.4.**The cost of manufacturing tractors inKoreais twenty percent less than the cost of manufacturing tractors inGermany. Even after transportation fees and import taxes are added, it is still cheaper to import tractors fromKoreatoGermanythan to produce tractors inGermany.

Which of the following assertions is best supported by the above information?

(A) Labour costs inKoreaare twenty percent below those inGermany.

(B) Importing tractors intoGermanywill eliminate twenty percent of the manufacturing jobs inGermany.

(C) The costs of transporting a tractor fromKoreatoGermanyis more than twenty percent ofthe cost of manufacturing the tractor inKorea.

(D) The import taxes on a tractor imported fromKoreatoGermanyis less than twenty percentof the cost of manufacturing the tractor inGermany.

**Ans. (D)**

**Numerical Ability**

**Q.5.**In a survey, 3/16 of the people surveyed told that they preferred to use publictransport whilecommuting daily to office. 5/8 of the people surveyed told that theypreferred to use their own vehicles. Theremaining 75respondents said thatthey had no clear preference. How many people preferred to usepublictransport?

(A) 75 (B) 100 (C) 125 (D) 133

**Ans. (A)**

# Syllabus for Mathematics (MA)

**Linear Algebra:**Finite dimensional vector spaces; Linear transformations and their matrix representations, rank; systems of linear equations, eigen values and eigen vectors, minimal polynomial, Cayley-Hamilton Theroem, diagonalisation, Hermitian, Skew-Hermitian and unitary matrices; Finite dimensional inner product spaces, Gram-Schmidt orthonormalization process, self-adjoint operators.

**Complex Analysis:**Analytic functions, conformal mappings, bilinear transformations; complex integration: Cauchy’s integral theorem and formula; Liouville’s theorem, maximum modulus principle; Taylor and Laurent’s series; residue theorem and applications for evaluating real integrals.

**Real Analysis:**Sequences and series of functions, uniform convergence, power series, Fourier series, functions of several variables, maxima, minima; Riemann integration, multiple integrals, line, surface and volume integrals, theorems of Green, Stokes and Gauss; metric spaces, completeness, Weierstrass approximation theorem, compactness; Lebesgue measure, measurable functions; Lebesgue integral, Fatou’s lemma, dominated convergence theorem.

**Ordinary Differential Equations:**First order ordinary differential equations, existence and uniqueness theorems, systems of linear first order ordinary differential equations, linear ordinary differential equations of higher order with constant coefficients; linear second order ordinary differential equations with variable coefficients; method of Laplace transforms for solving ordinary differential equations, series solutions; Legendre and Bessel functions and their orthogonality.

**Algebra:**Normal subgroups and homomorphism theorems, automorphisms; Group actions, Sylow’s theorems and their applications; Euclidean domains, Principle ideal domains and unique factorization domains. Prime ideals and maximal ideals in commutative rings; Fields, finite fields.

**Functional Analysis:**Banach spaces, Hahn-Banach extension theorem, open mapping and closed graph theorems, principle of uniform boundedness; Hilbert spaces, orthonormal bases, Riesz representation theorem, bounded linear operators.

**Numerical Analysis:**Numerical solution of algebraic and transcendental equations: bisection, secant method, Newton-Raphson method, fixed point iteration; interpolation: error of polynomial interpolation, Lagrange, Newton interpolations; numerical differentiation; numerical integration: Trapezoidal and Simpson rules, Gauss Legendrequadrature, method of undetermined parameters; least square polynomial approximation; numerical solution of systems of linear equations: direct methods (Gauss elimination, LU decomposition); iterative methods (Jacobi and Gauss-Seidel); matrix eigenvalue problems: power method, numerical solution of ordinary differential equations: initial value problems: Taylor series methods, Euler’s method, Runge-Kutta methods.

**Partial Differential Equations:**Linear and quasilinear first order partial differential equations, method of characteristics; second order linear equations in two variables and their classification; Cauchy, Dirichlet and Neumann problems; solutions of Laplace, wave and diffusion equations in two variables; Fourier series and Fourier transform and Laplace transform methods of solutions for the above equations.

**Mechanics:**Virtual work, Lagrange’s equations for holonomic systems, Hamiltonian equations.

**Topology:**Basic concepts of topology, product topology, connectedness, compactness, countability and separation axioms, Urysohn’s Lemma.

**Probability and Statistics:**Probability space, conditional probability, Bayes theorem, independence, Random variables, joint and conditional distributions, standard probability distributions and their properties, expectation, conditional expectation, moments; Weak and strong law of large numbers, central limit theorem; Sampling distributions, UMVU estimators, maximum likelihood estimators, Testing of hypotheses, standard parametric tests based on normal

*, X*– distributions; Linear regression; Interval estimation.

^{2 }, t, F**Linear programming:**Linear programming problem and its formulation, convex sets and their properties, graphical method, basic feasible solution, simplex method, big-M and two phase methods; infeasible and unbounded LPP’s, alternate optima; Dual problem and duality theorems, dual simplex method and its application in post optimality analysis; Balanced and unbalanced transportation problems, u -u method for solving transportation problems; Hungarian method for solving assignment problems.

**Calculus of Variation and Integral Equations:**Variation problems with fixed boundaries; sufficient conditions for extremum, linear integral equations of Fredholm and Volterra type, their iterative solutions.

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