BITSAT Mathematics Syllabus

Mathematics syllabus — full chapter and topic list

Expand any chapter to see its topics and sub-topics. This is the whole examinable outline for Mathematics in BITSAT, not a summary of it.

1. 1

   Algebra

   11 topics

   • Complex numbers
     • Addition
     • Multiplication
     • Conjugation
     • Polar representation
     • Properties of modulus and principal argument
     • Triangle inequality
     • Roots of complex numbers
     • Geometric interpretations
   • Theory of Quadratic equations
     • Quadratic equations in real and complex number system and their solutions
   • Arithmetic and geometric progressions
     • Arithmetic, geometric and arithmetico geometric series
     • 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
   • Exponential series
   • Permutations and combinations
     • Permutations as an arrangement and combination as selection
     • Simple applications
   • Binomial theorem for a positive integral index
     • Properties of binomial coefficients
     • Pascal’s triangle
   • Matrices and determinants of order two or three
     • Properties and evaluation of determinants
     • Addition and multiplication of matrices
     • Adjoint and inverse of matrices
     • Solutions of simultaneous linear equations in two or three variables
     • Elementary row and column operations of matrices
     • Types of matrices
     • Applications of determinants in finding the area of triangles
   • Sets, Relations and Functions
     • Algebra of sets applications
     • Equivalence relations
     • Mappings
     • One to one, into and onto mappings
     • Composition of mappings
     • Binary operation
     • Inverse of function
     • Functions of real variables like polynomial, modulus, signum and greatest integer
   • Mathematical reasoning and methods of proofs
     • Mathematically acceptable statements
     • Connecting words/phrases – consolidating the understanding of “if and only if (necessary and sufficient) condition”, “implies”, “and/or”, “implied” by”, “and”, “or”, “there exists” and through variety of examples related to real life and Mathematics
     • Validating the statements involving the connecting words – difference between contradiction, converse and contra positive
     • Mathematical induction
   • Linear Inequalities
     • Solution of linear inequalities in one variable (Algebraic) and two variables (Graphical)
2. 2

   Trigonometry

   3 topics

   • Measurement of angles in radians and degrees, positive and negative angles, trigonometric ratios, functions with their graphs and identities
   • Solution of trigonometric equations
   • Inverse trigonometric functions
3. 3

   Two-dimensional Coordinate Geometry

   4 topics

   • Cartesian coordinates, distance between two points, section formulae, shift of origin
   • Straight lines and pair of straight lines
     • Equation of straight lines 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
     • Concurrent lines
   • Circles
     • Equation of circle in standard form
     • Parametric equations of a circle
   • Conic sections
     • Parabola, ellipse and hyperbola their eccentricity
     • Directrices & foci
4. 4

   Three dimensional Coordinate Geometry

   3 topics

   • Co-ordinate axes and co-ordinate planes, distance between two points, section formula, direction cosines and direction ratios, equation of a straight line in space and skew lines
   • Angle between two lines whose direction ratios are given, shortest distance between two lines
   • Equation of a plane, distance of a point from a plane, condition for coplanarity of three lines, angles between two planes, angle between a line and a plane
5. 5

   Differential calculus

   5 topics

   • Domain and range of a real valued function, Limits and Continuity of the sum, difference and product of two functions, Differentiability
   • Derivative of different types of functions
     • Polynomial
     • Rational
     • Trigonometric
     • Inverse trigonometric
     • Exponential
     • Logarithmic
     • Implicit functions
   • Geometric interpretation of derivative, Tangents and Normal
   • Increasing and decreasing functions, Maxima and minima of a function
   • Rolle’s Theorem, Mean Value Theorem and Intermediate Value Theorem
6. 6

   Integral calculus

   4 topics

   • Integration as the inverse process of differentiation, indefinite integrals of standard functions
   • Methods of integration
     • Integration by substitution
     • Integration by parts
     • Integration by partial fractions
     • Integration by trigonometric identities
   • Definite integrals and their properties, Fundamental Theorem of Integral Calculus, applications in finding areas under simple curves
   • Application of definite integrals to the determination of areas of regions bounded by simple curves
7. 7

   Ordinary Differential Equations

   3 topics

   • Order and degree of a differential equation, formulation of a differential equation whole general solution is given, variables separable method
   • Solution of homogeneous differential equations of first order and first degree
   • Linear first order differential equations
8. 8

   Probability

   4 topics

   • Various terminology in probability, axiomatic and other approaches of probability, addition and multiplication rules of probability
   • Conditional probability, total probability and Baye’s theorem
   • Independent events
   • Discrete random variables and distributions with mean and variance
9. 9

   Vectors

   3 topics

   • Direction ratio/cosines of vectors, addition of vectors, scalar multiplication, position vector of a point dividing a line segment in a given ratio
   • Dot and cross products of two vectors, projection of a vector on a line
   • Scalar triple products and their geometrical interpretations
10. 10

    Statistics

    2 topics

    • Measures of dispersion
    • Analysis of frequency distributions with equal means but different variances
11. 11

    Linear Programming

    2 topics

    • Various terminology and formulation of linear Programming
    • Solution of linear Programming using graphical method, feasible and infeasible regions, feasible and infeasible solutions, optimal feasible solutions
12. 12

    Mathematical modeling

    1 topic

    • Formulation of simple real life problem, solution using matrices, calculus and linear programming

Mathematics flashcards for BITSAT

21 of 52 cards from the Mathematics deck — real questions with worked answers.

1. What is the value of i raised to the powers i^1, i^2, i^3, and i^4?

   i^1 = i, i^2 = -1, i^3 = -i, i^4 = 1 (and the pattern repeats with period 4).

2. For a complex number z = a + bi, what is its modulus |z| and how is the conjugate defined?

   Modulus |z| = sqrt(a^2 + b^2); conjugate z-bar = a - bi. Also z·z-bar = |z|^2.

3. State De Moivre's Theorem for a complex number in polar form.

   For z = r(cos θ + i sin θ), z^n = r^n(cos nθ + i sin nθ) for any integer n.

4. What are the n nth roots of unity and what is their sum?

   The nth roots of unity are e^(2πik/n) = cos(2πk/n) + i sin(2πk/n) for k = 0,1,...,n-1. Their sum is 0 (for n ≥ 2).

5. For the quadratic ax^2 + bx + c = 0, give the formulas for the sum and product of the roots.

   Sum of roots = -b/a; product of roots = c/a.

6. How does the discriminant D = b^2 - 4ac determine the nature of roots of a real quadratic?

   D > 0: two distinct real roots; D = 0: two equal real roots; D < 0: two complex conjugate roots.

7. For a polynomial of degree n with roots, what does Vieta's give for the sum of roots and product of all roots?

   Sum of roots = -a_(n-1)/a_n; product of all roots = (-1)^n · a_0/a_n, where a_n is the leading coefficient.

8. What is the nth term and the sum of n terms of an arithmetic progression (AP)?

   nth term a_n = a + (n-1)d; sum S_n = n/2 [2a + (n-1)d] = n/2 (first + last).

9. What is the nth term and sum of n terms of a geometric progression (GP)?

   nth term a_n = a·r^(n-1); sum S_n = a(r^n - 1)/(r - 1) for r ≠ 1.

10. What is the sum to infinity of a GP and when does it exist?

    S_∞ = a/(1 - r), valid only when |r| < 1.

11. State the AM-GM inequality for two positive numbers a and b.

    (a + b)/2 ≥ sqrt(ab), with equality if and only if a = b. (Arithmetic mean ≥ geometric mean.)

12. Give the formulas for the sum of the first n natural numbers, their squares, and their cubes.

    Σn = n(n+1)/2; Σn^2 = n(n+1)(2n+1)/6; Σn^3 = [n(n+1)/2]^2.

13. What are the formulas for nPr (permutations) and nCr (combinations)?

    nPr = n!/(n-r)!; nCr = n!/[r!(n-r)!]. Also nPr = nCr · r!.

14. State the Binomial Theorem for (a + b)^n where n is a positive integer.

    (a + b)^n = Σ_(r=0)^(n) nCr · a^(n-r) · b^r. The general (r+1)th term is T_(r+1) = nCr · a^(n-r) · b^r.

15. State two key identities for binomial coefficients: the sum of all coefficients and Pascal's rule.

    Sum: Σ nCr = 2^n. Pascal's rule: nCr + nC(r-1) = (n+1)Cr.

16. What conditions make a square matrix invertible, and what is the formula for A inverse?

    A is invertible iff det(A) ≠ 0 (non-singular). Then A^(-1) = adj(A)/det(A).

17. What are symmetric, skew-symmetric, and orthogonal matrices?

    Symmetric: A^T = A; skew-symmetric: A^T = -A (diagonal entries are 0); orthogonal: A^T·A = I (so A^(-1) = A^T).

18. State the key properties of determinants regarding row operations and scalar multiples.

    Swapping two rows changes the sign; multiplying a row by k multiplies det by k; det(AB) = det(A)·det(B); det(kA) = k^n·det(A) for n×n matrix.

19. State Cramer's Rule for solving a system of linear equations.

    For AX = B with det(A) = D ≠ 0, each variable x_i = D_i/D, where D_i is D with the i-th column replaced by B.

20. State De Morgan's laws for sets.

    (A ∪ B)' = A' ∩ B' and (A ∩ B)' = A' ∪ B'.

21. How many subsets does a set with n elements have, and how many proper subsets?

    Total subsets = 2^n; proper subsets = 2^n - 1 (excluding the set itself).

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