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West Bengal Joint Entrance Examination (JEM) : Syllabus -
Mathematics
Algebra
A.P., G.P., H.P. : Definitions of A. P. and G.P.; General
term; Summation of first n-terms; A.M.and G.M.; Definitions of H.P. (only
3 terms) and H.M.; Finite arithmetico-geometric series.
Logarithms : Definition; General properties; Change
of base.
Complex Numbers : Definition and properties of complex
numbers; Complex conjugate; Triangle inequality; Square root of complex
numbers; Cube roots of unity; D'Moivre's theorem (statement only) and
its elementary applications.
Quadratic Equations : Quadratic equations with real
coefficients; Relations between roots and coefficients; Nature of roots;
Formation of a quadratic equation, sign and magnitude of the quadratic
expression ax2+bx+c (a,b,c are rational numbers and a
0).
Permutation and combination : Permutation of n different
things taken r at a time (r < n). Permutation of n things not
all different. Permutation with repetitions (circular permutation excluded).
Combinations of n different things taken r at a time (r <
n). Combination of n things not all different.
Basic properties.
Problems involving both permutations and combinations.
Principle of Mathematical Induction : Statement of the
principle. Proof by induction for the sum of squares, sum of cubes of
first n natural numbers, divisibility properties like 2 2n
- 1 is divisible by 3
(n >
1), 7 divides 3 2n+1+2 n+2 (n
1).
Binomial theorem (positive integral index) : Statement
of the theorem, general term, middle term, equidistant terms, properties
of binomial co-efficients.
Infinite series : Binomial theorem for negative and
fractional index. Infinite G.P. series, Exponential and Logarithmic series
with range of validity (statement only), simple applications.
Matrices : Concepts of m x n (m < 3, n <
3) real matrices, operations of addition, scalar multiplication and multiplication
of matrices. Transpose of a matrix. Determinant of a square matrix. Properties
of determinants (statement only). Minor, cofactor and adjoint of a matrix.
Nonsingular matrix. Inverse of a matrix. Finding area of a triangle. Solutions
of system of linear equations. (Not more than 3 variables).
Sets, Relations and Mappings : Idea of sets, subsets,
power set, complement, union, intersection and difference of sets, Venn
diagram, De Morgan's Laws, Inclusion / Exclusion formula for two or three
finite sets, Cartesian product of sets.
Relation and its properties. Equivalence relation - definition and elementary
examples, mappings, range and domain, injective, surjective and bijective
mappings, composition of mappings, inverse of a mapping.
Probability : Classical definition, addition rule,
conditional probability and Bayes' theorem, independence, multiplication
rule.
Trigonometric ratios, compound angles, multiple and submultiple angles,
general solution of trigonometric equations. Properties of triangles,
inverse trigonometric functions.
Co-ordinate geometry of two dimensions
Basic Ideas : Distance formula, section formula, area
of a triangle, condition of collinearity of three points in a plane.
Polar coordinates, transformation from Cartesian to polar coordinates
and vice versa. Parallel transformation of axes, concept of locus, elementary
locus problems.
Straight line : Slope of a line. Equation of lines in
different forms, angle between two lines. Condition of perpendicularity
and parallelism of two lines. Distance of a point from a line. Distance
between two parallel lines. Lines through the point of intersection of
two lines.
Circle : Equation of a circle with a given center and
radius. Condition that a general equation of second degree in x, y may
represent a circle. Equation of a circle in terms of endpoints of a diameter
. Parametric equation of a circle. Intersection of a line with a circle.
Equation of common chord of two intersecting circles.
Conics : Definition, Directrix, Focus and Eccentricity,
classification based on eccentricity.
Parabola : Standard equation. Reduction of the form
x = ay2;+by+c or y = ax2+bx+c to the standard form
y2 = 4ax or x2 = 4ay respectively. Elementary properties
and parametric equation of a parabola.
Ellipse and Hyperbola : Reduction to standard form of
general equation of second degree when xy term is absent. Conjugate hyperbola.
Simple properties. Parametric equations. Location of a point with respect
to a conic.
Differential calculus : Functions, composition of two
functions and inverse of a function, limit, continuity, derivative, chain
rule, derivatives of implicit functions and of functions defined parametrically.
Rolle's Theorem and Lagrange's Mean Value theorem (statement only).
Their geometric interpretation and elementary application. L'Hospital's
rule (statement only) and applications.
Second order derivative.
Integral calculus : Integration as a reverse process of differentiation,
indefinite integral of standard functions. Integration by parts. Integration
by substitution and partial fraction.
Definite integral as a limit of a sum with equal subdivisions. Fundamental
theorem of integral calculus and its applications. Properties of definite
integrals.
Differential Equations : Formulation and solution of
differential equations of the forms.
1) dy / dx = f(x).g(y)
2) dy / dx = f(y/x)
3) dy / dx = (ax+by) / (cx+dy)
4) dy / dx = (a1x+b1y+c1 ) / (a2x+b2y+c2
), (a1/a2 = b1/b2)
5) dy / dx + p(x)y = Q(x)
6) d2y / dx2 + p1 dy/dx + p2y
= 0 with p1 and p2 constants.
7) d2y/dx2 = f(x)
Application of Calculus : Tangents and normals, conditions
of tangency. Determination of monotonicity, maxima and minima. Differential
coefficient as a measure of rate.
Motion in a straight line with constant acceleration.
Geometric interpretation of definite integral as area, calculation of
area bounded by elementary curves and straight lines. Area of the region
included between two elementary curves.
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