CEE - Delhi - Combined Entrance Examination, University of Delhi (Faculty of Technology) |
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1. Sets Sets and their representations, Finite and infinite sets, Empty set, Equal sets, Subsets, Power set, Universal set, Venn diagrams, Complements of a set, Operations on sets (union, intersection and difference of two sets), Applications of sets. 2. Relations and Functions Ordered pairs, Cartesian product of sets! Relations, domain, co-domain and range, Functions into and onto functions, one-one into and one-one onto functions, Constant function, Identity function, Composition of functions, Invertible functions, Binary operations. 3. Mathematical Induction The principle of mathematical induction, Simple applications. 4. Logarithms Meaning of logarithm of a number of a given base a, a>O, a = 1, Laws of logarithms including change of base! Common logarithm (Base 10), Characteristic and mantissa, Antilogarithms, Logarithmic tables, Application of logarithms to problems of compound interest, growth and decay ( depreciation) . 5. Complex Numbers Complex numbers of the form a+ ib, Real and imaginary parts of a complex number, Complex conjugate, Argand diagram, Representation of a complex number by a point in a plane, Modulus and argument of a complex number. Algebra of complex numbers, Triangle inequality: IZ, + Z21 ~ IZ,I + IZ2I and also IZ1 ,Z21 = IZ,I.IZ21, Polar representation of a complex number, Square root of a complex number, Cube roots of unity. 6. Linear Inequations Solution of a linear inequation in one variable and its graphical representation, Solution of system of linear inequations in one variable, Graphical solutions of linear inequations in two variables, Solution of system of linear inequations in two variables. 7. Quadratic Equations Solution of a quadratic equation in the complx number system by (i) Factorization! (ii) Using formula, Relation between roots and coefficients! Nature of roots, Formation of quadratic equations with given roots, Symmetric functions of roots, Equations reducible to quadratic forms. 8. Sequences and Series Sequence and examples of finite and infinite sequences, Arithmetic progression (A.P) -first term, common difference and nth term. Sum to n terms of an A.P Arithmetic mean (A.M.)! insertion of arithmetic means between any two given numbers,. Geometric progression (GP), first term, common ratio and nth term, Sum of n terms and infinite number of terms of a GP, Recurring demical numbers as geometric series, Geometric mean (GM.), insertion of Geometric means between any two given numbers. Harmonic Progression, Harmonic Mean (H.M.), relationship among A.M., G.M. and H.M., Arithmetic - geometric series, sum to n terms and sum of infinite arithmeticgeometric series, Special series: Sn, Sn2, Sn3, Sum of series using above special series. 9. Matrices and Determinants Concept of a matrix, types of matrices, Equality of matrices (only real entries may be considered). Operations of addition, scalar multiplication and multiplication of matrices. Statements of important results on operations of matrices and their verification by numerical problems only. Determinant of a square matrix, Properties of determinants. Minors and cofactors. Applications of determinants in (i) finding area of a triangle, (ii) solving a system of linear equations, Transpose, adjoint and inverse of a matrix, Consistency and inconsistency of system of linear equations, Solving system of linear equations, in two or three variables using inverse of a matrix. 10. Functions, Limits and Continuity Concept of a real function, its domain and range, Types of functions and their graphs, Limit of a function, meaning and related notations. Left and right hand limits. Fundamental theorems on limits (statement only), Proof of the standard limits: Limits at infinity and infinite limits, Continuity of a function (i) at a point, (ii) over an open/closed intervals, Sum, product and quotient of continuous functions. Continuity of special functionspolynomial, trigonometric, exponential, logarithmic, inverse trigonometric functions. 11. Trigonometry Degree measure and radian measure of positive and negative angle, relation between degree and radian, Definition of trigonometric functions with the help of a unit circle, Periodic functions, concept of periodicity of Trigonometric functions, Values of trigonometric functions of x for x=0, p/6, p/4, p/3, p/2, 3p/2, Trigonometric functions of sum of difference of humers : 12. Cartesian System of Rectangular Coordinates Recall of Cartesian system of coordinates in a plane. Distance formula, Section formula, centroid and incentre, Area of a triangle, condition for the collinearity of three points in a plane. Slope of a line, parallel and perpendicular lines, Intercepts of a line on the coordinate axes, Locus and its equation. 13. Straight Line and Family of Straight Lines Various forms of equations of a line - Parallel to axes, Slope Intercept form, one point slope form. Symmetric form, parametric equations of a line, Two point form, Intercept form, Normal form, General form, Intersection of lines, Equations of bisectors of angle between two lines. Angle between two lines, condition for concurrency of three lines. Distance of a point from a line, Equations of family of lines through the intersection of two lines, Translation of axes. 14. Circles Standard form of the equation of a circle, General form of the equation of a circle, its radius and centre, Equation of the circle in the parametric form, Equation of a circle when the end points of a diameter are given, Points of intersection of a line and a circle with centre at the origin, Condition for a line to be tangent to the given circle, Equation of a tangent to a circle and length of the tangent. 15. Conic Sections Sections of a cone, Equations of conic sections (Parabola, Ellipse and Hyperbola) in standard form, Applications. 16. Permutations and Combinations Fundamental principle of counting, The factorial notation, Permutation as an arrangement, meaning of P (n,r,) Combination, meaning of C(n,r), Applications of permutations and combinations. 17. Binomial Theorem Statement of Binomial Theorem, Proof of Binomial theorem for positive integral exponent using principle of mathematical induction and also by combinatorial method, General and middle terms in binomial expansions, Properties of Binomial coefficients, Binomial theorem for any index (without proof), Application of Binomial theorem. 18. Exponential and Logarithmic Series Concept of ‘e’ as the sum of an infinite series, proof of 2 < e < 3, Exponential function (eX) as the infinite series, and its graph Logarithmic function (log x) and its graph. The infinite series for loge (1 + x), loge (1-x). 19. Mathematical Logic Statements, Use of Venn diagrams in logic, Negation operation, Basic logical connectives and compound statements including their negations. Truth tables, Tautology, duality, Algebra of statements, Applications of logic in solving simple problems.
20. Boolean Algebra Boolean algebra as an algebraic structure, Principle of duality, Boolean function, Conditional and bioconditional statements, Valid arguments, Switching circuits, Application of Boolean algebra to switching circuits. 21. Statistics Mean deviation for undergrouped data, Variation for grouped and ungrouped data, Standard deviatiol1. Introduction, basic concepts and basic laws of mechanics, force, resultant of forces acting a point, parallelogram law of forces, resolved parts of a force, Equilibrium of a particle under three concurrent forces, triangle law of forces and its converse, Lami’s theorem and its converse. Two parallel forces, like and unlike parallel forces, couple and its moment. 22. Probability Random experiments and sample space, events as subsets of sample space, occurrence of an event, sure and impossible events, exhaustive events, algebra of events, meaning of equally likely outcomes, Probability of an event, theorem on probability, addition rule, multiplication rule, independent experiments and independent events [finding P (A or B), P (A and B)), Random variables, Probability distribution of a random varible. Conditional probability, Baye’s theorem and its applications, Recall of concept of random variables and its probability distribution, Mean and variance of random variables, Binomial and Poisson’s distributions, their mean, variance and applications, Applications of these distributions in commerce and industry. 23. Differentiation Derivative of a function, its geometrical and physical significance, Relationship between continuity and differentiability. Derivative of some simple functions from first principle, Derivative of sum, difference, product and quotient of functions, Derivative of polynomial, trigonometric, exponential, logarithmic, inverse trigonometric and implicit functions. Logarithmic differentiation. Derivative of functions expressed in parametric form, chain rule and differentiation by substitution. Derivatives of second order. 24. Application of Derivatives Rates of change of quantities, Tangents and normals, increasing and decreasing functions and sign of the derivatives. Maxima and minima, greatest and least values, Rolle’s theorem and Mean Value theorem (without proof). Approximation by differentials. Curve tracing of simple curves. 25. Indefinite Integrals Integration as inverse of differentiation. Properties of integrals. Integration by substitution. Partial fractions and their use in integrating rational functions, Integral of the type: 26. Definite Integrats Definite integral as limit of a sum, Fundamental theorems of integral calculus (without proof), Evaluation of definite integrals by : (i) substitution, (ii) Using properties of definite integrals : 27. Differential Equations Definition, order and degree, General and particular solution of a differential equation. Formation of differential equations whose general solultion is given, Solution of differential equations by method of separation of variables, Homogeneous differential equations of first order and their solutions. Solution of linear differential equation of the type : where P(x), and Q(x) are functions of x, Solution of second order differential equations : 28. Three Dimensional Geometry Coordinate axes and coordinate planes in three dimensional space, Coordinate of a point in space. Distance between two points, Section formula, Direction cosines and direction ratios of a line joining two points, Projection of the join of two points on a given line, Angle between two lines whose direction ratios are given. Cartesian and vector equation of a line through (i) a point and parallel to a given vector, (ii) through two points. Collinearity of three points, Coplanar and skew lines, shortest distance between two lines, condition for the intersection of two lines. Cartesian and vector equation of a plane (i) when the normal vector and the distance of the plane from the origin is given. (ii) passing through a point and perpendicular to a given vector, (Hi) passing through a point and parallel to two given lines or through the intersection of two other planes, (iv) containing two lines, (v) passing through three points. Angle between (i) two lines, (H) two planes, (Hi) a line and a plane, Condition of coplanarity of two lines in vector and Cartesian form, length of perpendicular from a point on a plane by both vector and Cartesian methods, vector and Cartesian equations of a sphere, its centre and radius, diameter form of the equation of a sphere. 29. Vectors Vectors and scalars, Magnitude and direction of a vector, Types of vectors - equal vectors, unit vector, zero vector, position vector of a point, localized and free vectors, parallel and collinear vectors, negative of a vector, Components of a vector, Addition of vectors, Multiplication of a vector by a scalar, Position vector of a point dividing a tine segment in a given ratio, Application of vectors in geometry. Scalar (or dot) product of vectors, Projection of a vector on a line, Vector (or cross) product of two vectors. Application of dot and cross products in (i) finding areas. of triangle and parallelogram, (ii) problems of plane geometry and trigonometry, (iii) finding work done by a force, (iv) vector moment of a vector about a point, Scalar triple product and its applications, Moment of a vector about a line. Coplanarity of three vectors or four points using scalar triple product. Vector triple product. 30. Elementary Dynamics Basic concepts displacement, speed and velocity, average speed, instantaneous speed, acceleration and retardation, resultant of two velocities, Motion of a particle along a line when moving with constant acceleration, motion of a particle under gravity, Projectile motion - the path of a projectile, its horizontal range, velocity at any instant, greatest height and time of flight 31. Linear Programming Introduction, definition of related terminology such as constraints, objective function, optimization, isoprofit, isocostlines. Advantages of linear programming, Limitations of linear programming. Application areas of linear programming, Different types of linear programming (L.P.) Problems, Mathematical formulation of L.P. problems, Graphicai method of solution for problems in two variables, feasible and infeasible regions, feasible and infeasible solutions, optimum feasible solution.
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