1. Algebra
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- Algebra of complex numbers, addition, multiplication, conjugation, polar representation, properties of modulus and principal argument, triangle inequality, cube roots of unity, geometric interpretations.
- Quadratic equations with real coefficients, relations between roots and coefficients, formation of quadratic equations with given roots, symmetric functions of roots.
- Arithmetic, geometric and harmonic progressions, arithmetic, geometric and harmonic means, sums of finite arithmetic and geometric progressions, infinite geometric series, sums of squares and cubes of the first n natural numbers.
- Logarithms and their properties.
- Permutations and combinations, binomial theorem for a positive integral index, properties of binomial coefficients.
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2. Matrices
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- Matrices as a rectangular array of real numbers.
- Equality of matrices, addition, multiplication by a scalar and product of matrices.
- Transpose of a matrix.
- Determinant of a square matrix of order up to three.
- Inverse of a square matrix of order up to three.
- Properties of these matrix operations, diagonal, symmetric and skew-symmetric matrices and their properties.
- Solutions of simultaneous linear equations in two or three variables.
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3. Probability
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- Addition and multiplication rules of probability, conditional probability, Bayes Theorem, independence of events, computation of probability of events using permutations and combinations.
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4. Trigonometry
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- Trigonometric functions, their periodicity and graphs, addition and subtraction formulae, formulae involving multiple and sub-multiple angles, general solution of trigonometric equations.
- Relations between sides and angles of a triangle, sine rule, cosine rule, half-angle formula and the area of a triangle, inverse trigonometric functions (principal value only).
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5. Analytical geometry
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- Two dimensions: Cartesian coordinates, distance between two points, section formulae, shift of origin.
- Equation of a straight line in various forms, angle between two lines, distance of a point from a line; Lines through the point of intersection of two given lines, equation of the bisector of the angle between two lines, concurrency of lines; Centroid, orthocentre, incentre and circumcentre of a triangle.
- Equation of a circle in various forms, equations of tangent, normal and chord. Parametric equations of a circle, intersection of a circle with a straight line or a circle, equation of a circle through the points of intersection of two circles and those of a circle and a straight line.
- Equations of a parabola, ellipse and hyperbola in standard form, their foci, directrices and eccentricity, parametric equations, equations of tangent and normal.
- Locus problems.
- Three dimensions: Direction cosines and direction ratios, equation of a straight line in space, equation of a plane, distance of a point from a plane.
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6. Differential Calculus
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- Real valued functions of a real variable, into, onto and one-to-one functions, sum, difference, product and quotient of two functions, composite functions, absolute value, polynomial, rational, trigonometric, exponential and logarithmic functions.
- Limit and continuity of a function, limit and continuity of the sum, difference, product and quotient of two functions, L’Hospital rule of evaluation of limits of functions.
- Even and odd functions, inverse of a function, continuity of composite functions, intermediate value property of continuous functions.
- Derivative of a function, derivative of the sum, difference, product and quotient of two functions, chain rule, derivatives of polynomial, rational, trigonometric, inverse trigonometric, exponential and logarithmic functions.
- Derivatives of implicit functions, derivatives up to order two, geometrical interpretation of the derivative, tangents and normals, increasing and decreasing functions, maximum and minimum values of a function, Rolle’s theorem and
- Lagrange’s mean value theorem.
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7. Integral calculus
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- Integration as the inverse process of differentiation, indefinite integrals of standard functions, definite integrals and their properties, fundamental theorem of integral calculus.
- Integration by parts, integration by the methods of substitution and partial fractions, application of definite integrals to the determination of areas involving simple curves.
- Formation of ordinary differential equations, solution of homogeneous differential equations, separation of variables method, linear first order differential equations.
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8. Vectors
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- Addition of vectors, scalar multiplication, dot and cross products, scalar triple products and their geometrical interpretations.
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