UPSC Exam: SCRA Exam 2013 Notification issued- Part 11
Paper-III
MATHEMATICS
1. Algebra:
Concept of a set,
Union and Intersection of sets, Complement of a set, Null set, Universal set
and Power set, Venn diagrams and simple applications. Cartesian product of two
sets, relation and mapping — examples, Binary operation on a set
— examples.
Representation of real numbers on a line. Complex numbers: Modulus, Argument,
Algebraic operations on complex numbers. Cube roots of unity. Binary system of
numbers, Conversion of a decimal number to a binary number and vice-versa.
Arithmetic, Geometric and Harmonic progressions. Summation of series involving
A.P., G.P., and H.P.. Quadratic equations with real co-efficients. Quadratic
expressions: extreme values. Permutation and Combination, Binomial theorem and
its applications.
Matrices and
Determinants: Types of matrices, equality, matrix addition and scalar
multiplication - properties. Matrix multiplication — non-commutative and distributive
property over addition. Transpose of a matrix, Determinant of a matrix. Minors
and Cofactors. Properties of determinants. Singular and non-singular matrices.
Adjoint and Inverse of a square-matrix, Solution of a system of linear
equations in two and three variables-elimination method, Cramers rule and
Matrix inversion method (Matrices with m rows and n columns where m, n < to
3 are to be considered).
Idea of a Group,
Order of a Group, Abelian Group. Identitiy and inverse elements- Illustration
by simple examples.
2. Trigonometry:
Addition and
subtraction formulae, multiple and sub-multiple angles. Product and factoring formulae.
Inverse trigonometric functions — Domains, Ranges and Graphs. DeMoivre's
theorem, expansion of Sin n0 and Cos n0 in a
series of multiples of Sines and Cosines. Solution of simple trigonometric
equations. Applications: Heights and Distance.
3. Analytic Geometry (two dimensions):
Rectangular
Cartesian. Coordinate system, distance between two points, equation of a straight
line in various forms, angle between two lines, distance of a point from a
line. Transformation of axes. Pair of straight lines, general equation of
second degree in x and y — condition to represent a pair of straight lines,
point of intersection, angle between two lines. Equation of a circle in
standard and in general form, equations of tangent and normal at a point,
orthogonality of two cricles. Standard equations of parabola, ellipse and
hyperbola — parametric equations, equations of tangent and normal at a point in
both cartesian and parametric forms.
4. Differential
Calculus:
Concept of a real
valued function — domain, range and graph. Composite functions, one to one,
onto and inverse functions, algebra of real functions, examples of polynomial, rational,
trigonometric, exponential and logarithmic functions. Notion of limit, Standard
limits - examples. Continuity of functions - examples, algebraic operations on continuous
functions. Derivative of a function at a point, geometrical and physical
interpretation of a derivative - applications. Derivative of sum, product and
quotient of functions, derivative of a function with respect to another
function, derivative of a composite function, chain rule. Second order
derivatives. Rolle's theorem (statement only), increasing and decreasing
functions. Application of derivatives in problems of maxima, minima, greatest
and least values of a function.
5. Integral Calculus and Differential equations:
Integral Calculus: Integration as inverse of differential, integration by substitution
and by parts, standard integrals
involving algebraic expression, trigonometric, exponential and hyperbolic
functions. Evaluation of definite integrals-determination of areas of plane
regions bounded by curves - applications.
Differential
equations: Definition of order and
degree of a differential equation, formation of a differential equation by
examples. General and particular solution of a differential equation, solution
of first order and first degree differential equation of various types - examples. Solution of second order
homogeneous differential equation with constant co-efficients.
6. Vectors and its applications:
Magnitude and
direction of a vector, equal vectors, unit vector, zero vector, vectors in two
and three dimensions, position vector. Multiplication of a vector by a scalar,
sum and difference of two vectors, Parallelogram law and triangle law of
addition. Multiplication of vectors — scalar product or dot product of two
vectors, perpendicularity, commutative and distributive properties. Vector
product or cross product of two vectors. Scalar and vector triple products.
Equations of a line, plane and sphere in vector form - simple problems. Area of
a triangle, parallelogram and problems of plane geometry and trigonometry using
vector methods. Work done by a force and moment of a force.
7. Statistics and probability :
Statistics :
Frequency distribution, cumulative frequency distribution - examples. Graphical
representation - Histogram, frequency polygon - examples. Measure of central tendency
- mean, median and mode. Variance and standard deviation – determination and
comparison. Correlation and regression.
Probability : Random experiment, outcomes and associated sample space, events, mutually
exclusive and exhaustive events, impossible and certain events. Union and Intersection of events. Complementary,
elementary and composite events. Definition of probability : classical and
statistical - examples. Elementary theorems on probability - simple problems.
Conditional probability, Bayes' theorem - simple problems. Random variable as
function on a sample space. Binomial distribution, examples of random experiments
giving rise to Binomial distribution.
Personality Test
Each candidate will
be interviewed by a Board who will have before them a record of his career both
academic and extramural. They will be asked questions on matters of general
interest. Special attention will be paid to assessing their potential qualities
of leadership, initiative and intellectual curiosity, tact and other social
qualities, mental and physical energy, power of practical application and
integrity of character.
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