BIT Entrance Examination — Mathematics Notes

Total Marks: 50

A complete, unit-wise breakdown of the BIT entrance mathematics syllabus with definitions, key formulas, and exam-focused notes for every topic. Algebra and Arithmetic together carry nearly half of the total marks, making them the highest priority units. This document provides exhaustive coverage of each unit, including strategic tips and formula summaries to streamline your preparation.

Marks Distribution

Unit	Marks Weightage
Algebra	12 - 14
Calculus	6 - 8
Set Theory	6 - 8
Arithmetic	10 - 12
Vector	2 - 4
Trigonometry	2 - 4
Probability	1 - 2
Statistics	1 - 2
Linear Programming	1 - 2

1. Algebra (12 – 14 Marks)

The single highest-weightage unit. Master matrices, sequences, and complex numbers first — they carry the most repeat questions. Algebra forms the backbone of the exam, and a strong grasp here will also benefit other units like calculus and arithmetic.

1.1 Relation and Function

A relation is a fundamental concept that connects elements of two sets. Formally, a relation R from set A to set B is a subset of the Cartesian product A × B. The domain of R is the set of all first elements of the ordered pairs, and the range is the set of all second elements. A function is a special kind of relation where every element in the domain maps to exactly one element in the codomain. This means that no two ordered pairs in a function can have the same first element with different second elements.

Relation: a relation R from set A to set B is any subset of the Cartesian product A × B.
Domain / Range: the set of first elements (domain) and second elements (range) of the ordered pairs in a relation.
Function: a special relation f: A → B where every element of A maps to exactly one element of B.

Types of Relations

Relations can be classified based on specific properties. A relation is reflexive if every element is related to itself; symmetric if whenever a is related to b, b is related to a; transitive if a related to b and b related to c implies a related to c. An equivalence relation satisfies all three properties simultaneously and is crucial in partitioning sets.

Reflexive: (a, a) ∈ R for every a in A.
Symmetric: if (a, b) ∈ R then (b, a) ∈ R.
Transitive: if (a, b) ∈ R and (b, c) ∈ R then (a, c) ∈ R.
Equivalence Relation: a relation that is reflexive, symmetric, and transitive at the same time.

Types of Functions

Functions can be injective (one-one), surjective (onto), or bijective (both). Injective functions ensure that different inputs produce different outputs; surjective functions cover every element of the codomain; bijective functions have a unique inverse. Composite functions apply one function after another, and the inverse function reverses the mapping, existing only for bijective functions.

One-One (Injective): different elements of A map to different elements of B.
Onto (Surjective): every element of B has a pre-image in A.
Bijective: a function that is both one-one and onto; it has a well-defined inverse.
Composite Function: (f ∘ g)(x) = f(g(x)), applying g first and then f.
Inverse Function: f⁻¹ exists only when f is bijective, and satisfies f(f⁻¹(x)) = x.

$$f: A \to B, \qquad (f \circ g)(x) = f(g(x))$$

1.2 Logarithmic Function

The logarithm is the inverse of exponentiation. If aˣ = N, then x = logₐN. The base a must be positive and not equal to 1. Common logarithms use base 10, while natural logarithms use base e (approximately 2.71828). Logarithmic rules allow simplification of products, quotients, and powers, and the change of base formula enables conversion between different bases.

Definition: if aˣ = N (a > 0, a ≠ 1), then x = logₐN. Logarithm is the inverse operation of exponentiation.
Common Logarithm: base 10, written as log N.
Natural Logarithm: base e (≈ 2.71828), written as ln N.

Laws of Logarithms

These laws are essential for simplifying logarithmic expressions and solving logarithmic equations. They mirror the laws of exponents.

$$\log_a(mn) = \log_a m + \log_a n$$

$$\log_a\left(\frac{m}{n}\right) = \log_a m - \log_a n$$

$$\log_a m^n = n\log_a m$$

$$\log_a m = \frac{\log_b m}{\log_b a} \quad \text{(change of base)}$$

logₐ a = 1 and logₐ 1 = 0 for any valid base a.

1.3 Matrix and Determinants

Matrices are rectangular arrays of numbers that can be added, subtracted, and multiplied under certain conditions. The determinant of a square matrix provides important information about the matrix, such as invertibility. Matrices are classified by their shape and properties: row/column matrices, square matrices, diagonal matrices, identity matrices, symmetric, and skew-symmetric matrices.

Types of Matrices

Row / Column Matrix: a matrix with only one row, or only one column.
Square Matrix: number of rows equals number of columns.
Diagonal Matrix: a square matrix where all non-diagonal elements are zero.
Identity Matrix (I): a diagonal matrix with all diagonal elements equal to 1.
Symmetric Matrix: A = Aᵀ (transpose equals the matrix itself).
Skew-Symmetric Matrix: Aᵀ = −A, with all diagonal elements equal to 0.

Operations and Determinants

Matrix Operations: addition, subtraction, and multiplication are defined only when the dimensions are compatible.
Transpose (Aᵀ): obtained by interchanging the rows and columns of A.

$$A = \begin{vmatrix} a_{11} & a_{12} \\ a_{21} & a_{22} \end{vmatrix}, \qquad |A| = a_{11}a_{22} - a_{12}a_{21}$$

Singular Matrix: a square matrix whose determinant is zero — it has no inverse.
Adjoint (adj A): the transpose of the matrix of cofactors of A.

$$A^{-1} = \frac{\operatorname{adj}(A)}{|A|}, \qquad |A| \neq 0$$

Tip: Matrix inversion is the most frequently tested part of this topic — practice 2×2 and 3×3 inverses until they're automatic.

1.4 Sequence and Series

Sequences are ordered lists of numbers, and series are the sums of sequences. Arithmetic progressions (AP) have a constant difference between consecutive terms, while geometric progressions (GP) have a constant ratio. Special sums of squares and cubes are also frequently used. Understanding the formulas for the nth term and sum of n terms is essential for solving problems on sequences and series.

Arithmetic Progression (A.P.)

A.P.: a sequence where the difference between consecutive terms (common difference, d) is constant.

$$t_n = a + (n-1)d$$

$$S_n = \frac{n}{2}\big[2a + (n-1)d\big] = \frac{n}{2}(a + l)$$

Geometric Progression (G.P.)

G.P.: a sequence where the ratio between consecutive terms (common ratio, r) is constant.

$$t_n = ar^{\,n-1}$$

$$S_n = \frac{a(r^n - 1)}{r-1} \;\; (r \neq 1), \qquad S_\infty = \frac{a}{1-r} \;\; (|r| < 1)$$

Special Series

$$\sum n = \frac{n(n+1)}{2}, \qquad \sum n^2 = \frac{n(n+1)(2n+1)}{6}, \qquad \sum n^3 = \left(\frac{n(n+1)}{2}\right)^2$$

1.5 Complex Numbers

A complex number is of the form z = a + ib, where i² = −1. The real part is a and the imaginary part is b. The conjugate z̄ = a − ib is obtained by changing the sign of the imaginary part. The modulus |z| = √(a²+b²) gives the distance from the origin in the complex plane. Complex numbers can be expressed in polar form as z = r(cosθ + i sinθ), and De Moivre's theorem allows easy computation of powers and roots. The cube roots of unity (1, ω, ω²) satisfy 1+ω+ω²=0 and are important in solving cubic equations.

Complex Number: z = a + ib, where i² = −1; a is the real part and b is the imaginary part.
Conjugate: z̄ = a − ib, obtained by reversing the sign of the imaginary part.
Modulus: |z|, the distance of z from the origin in the complex plane.

$$|z| = \sqrt{a^2+b^2}, \qquad \bar z = a - ib$$

Polar Form and De Moivre's Theorem

$$z = r(\cos\theta + i\sin\theta), \qquad r = |z|,\;\; \theta = \arg(z)$$

$$z^n = r^n(\cos n\theta + i\sin n\theta) \quad \text{(De Moivre's Theorem)}$$

Cube Roots of Unity: the three solutions of x³ = 1, namely 1, ω, ω², where 1 + ω + ω² = 0.

1.6 Polynomial Equations

Polynomial equations are expressions of the form p(x) = 0. The Remainder Theorem states that when a polynomial p(x) is divided by (x − a), the remainder is p(a). The Factor Theorem is a special case: (x − a) is a factor if and only if p(a) = 0. For quadratic equations, the discriminant D = b²−4ac determines the nature of roots. For cubic equations, the relationships between coefficients and sums/products of roots are useful.

Remainder Theorem: if a polynomial p(x) is divided by (x − a), the remainder equals p(a).
Factor Theorem: (x − a) is a factor of p(x) if and only if p(a) = 0.

$$x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$$

Nature of Roots (Discriminant D = b² − 4ac)

D > 0: two distinct real roots.
D = 0: two equal real roots.
D < 0: roots are complex conjugates.

Cubic Equations: for ax³ + bx² + cx + d = 0, sum of roots = −b/a, sum of products of roots taken two at a time = c/a, product of roots = −d/a.

1.7 System of Linear Equations

Systems of linear equations can be solved using algebraic methods (substitution or elimination), matrix inversion, or Cramer's rule. For matrix inversion, the system is written as AX = B, and if A is non-singular, X = A⁻¹B. Cramer's rule expresses each variable as a ratio of determinants. The consistency of the system is determined by the determinant of the coefficient matrix: if D ≠ 0, the system has a unique solution; if D = 0 and all Dₓ, Dᵧ, D₂ are zero, there are infinitely many solutions; otherwise, no solution exists.

Substitution / Elimination: algebraic methods best suited for two-variable systems.
Matrix Inversion Method: solve AX = B as X = A⁻¹B, valid only when A is non-singular.
Cramer's Rule: expresses each unknown as a ratio of determinants.

$$x = \frac{D_x}{D}, \qquad y = \frac{D_y}{D}, \qquad z = \frac{D_z}{D}$$

Consistency of Equations

D ≠ 0: unique solution (consistent, independent).
D = 0, Dₓ = Dy = Dz = 0: infinitely many solutions (consistent, dependent).
D = 0, any of Dₓ, Dy, Dz ≠ 0: no solution (inconsistent).

1.8 Binomial Theorem

The binomial theorem provides the expansion of (x + y)ⁿ as a sum of terms involving binomial coefficients. The general term T(r+1) = ⁿCᵣ xⁿ⁻ʳ yʳ allows finding specific terms without full expansion. The middle term depends on whether n is even or odd. To find the term independent of x, set the exponent of x in the general term to zero. Binomial coefficients are symmetric and follow Pascal's triangle.

$$(x+y)^n = \sum_{r=0}^{n} \binom{n}{r} x^{\,n-r} y^r$$

General Term: T(r+1) = ⁿCᵣ xⁿ⁻ʳ yʳ, used to find any specific term of the expansion.
Middle Term: if n is even, there is one middle term, the (n/2 + 1)ᵗʰ term; if n is odd, there are two middle terms.
Term Independent of x: found by setting the power of x in the general term equal to zero.

Note: Binomial coefficients ⁿC₀, ⁿC₁, …, ⁿCₙ follow Pascal's Triangle and satisfy ⁿCᵣ = ⁿCₙ₋ᵣ.

2. Calculus (6 – 8 Marks)

Fewer marks than Algebra, but the concepts here (derivatives especially) are foundational and quick to score once the formula list is memorized. Focus on limits, standard derivatives, and integration rules to secure these marks efficiently.

2.1 Limits and Continuity

The limit of a function describes its behavior as x approaches a particular value. Left-hand and right-hand limits must be equal for the limit to exist. Indeterminate forms (0/0, ∞/∞) require algebraic manipulation or L'Hôpital's rule. A function is continuous at a point if the limit exists and equals the function's value at that point.

$$\lim_{x \to a} f(x) = L$$

Left-Hand Limit: the value f(x) approaches as x → a from values less than a.
Right-Hand Limit: the value f(x) approaches as x → a from values greater than a.
Indeterminate Forms: expressions like 0/0 or ∞/∞ that require simplification (factoring, rationalizing, or L'Hôpital's rule) before evaluating.
Continuity: f is continuous at x = a if lim(x→a) f(x) exists and equals f(a).

2.2 Derivatives (Differentiation)

The derivative represents the instantaneous rate of change of a function. It is defined as the limit of the difference quotient. Standard derivatives of common functions (power, trigonometric, exponential, logarithmic) should be memorized. Differentiation rules—sum, product, quotient, and chain rule—are essential for handling composite functions.

$$f'(x) = \lim_{h \to 0} \frac{f(x+h) - f(x)}{h}$$

Standard Derivatives

Function	Derivative
xⁿ	n·xⁿ⁻¹
sin x	cos x
cos x	−sin x
tan x	sec²x
eˣ	eˣ
ln x	1/x

Rules: sum/difference rule, product rule, quotient rule, and chain rule for composite functions.

2.3 Applications of Derivatives

Derivatives are used to find the slope of a curve (tangent) and the equation of the normal (perpendicular to the tangent). They also help determine where a function is increasing or decreasing. Critical points (where f'(x)=0) are potential maxima or minima; the second derivative test confirms their nature.

Tangent & Normal: the derivative f'(x₀) gives the slope of the tangent at x = x₀; the normal is perpendicular to it.
Increasing / Decreasing: f is increasing where f'(x) > 0, and decreasing where f'(x) < 0.
Maxima & Minima: critical points occur where f'(x) = 0; the second derivative test (f''(x) < 0 → maximum, f''(x) > 0 → minimum) confirms the nature.

2.4 Anti-Derivatives (Integration)

Integration is the reverse process of differentiation. The indefinite integral (anti-derivative) of a function is a family of functions differing by a constant. Standard integrals of basic functions should be memorized. Techniques such as substitution, integration by parts, and partial fractions are used for more complex integrands.

$$\int x^n\,dx = \frac{x^{\,n+1}}{n+1} + C \quad (n \neq -1)$$

Function	Integral
xⁿ	xⁿ⁺¹/(n+1) + C
cos x	sin x + C
sin x	−cos x + C
sec²x	tan x + C
1/x	ln\|x\| + C
eˣ	eˣ + C

Methods: substitution, integration by parts (∫u dv = uv − ∫v du), and partial fractions for rational functions.

2.5 Applications of Integration

The definite integral is used to compute the area under a curve. The area between two curves is found by integrating the difference of the upper and lower functions over the appropriate interval. Applications also include finding volumes of revolution and solving problems involving rates of change.

$$\text{Area} = \int_a^b f(x)\,dx$$

Area Between Curves: computed as the definite integral of the difference of the two functions over the interval where one lies above the other.

3. Set Theory (6 – 8 Marks)

Set Theory is the study of collections of objects. It includes operations like union, intersection, difference, and complement. De Morgan's laws relate the complement of unions and intersections. The power set is the set of all subsets. The real number system is a hierarchy of sets, and intervals represent continuous ranges. Mathematical logic deals with statements and connectives, using truth tables to evaluate compound statements.

3.1 Set, Real Number System, and Logic

Sets

Set Operations: union (∪), intersection (∩), difference (−), and complement ( ' ).
Power Set: the set of all subsets of a set A, containing 2ⁿ elements if A has n elements.

$$(A \cup B)' = A' \cap B', \qquad (A \cap B)' = A' \cup B' \quad \text{(De Morgan's Laws)}$$

Real Number System

N ⊂ W ⊂ Z ⊂ Q ⊂ R: natural numbers, whole numbers, integers, rationals, and reals form a nested hierarchy; irrational numbers (Q′) complete the real line together with Q.
Intervals: open (a, b), closed [a, b], and half-open [a, b) or (a, b] represent continuous ranges of real numbers.

Mathematical Logic

Statement: a sentence that is either true or false, but not both.
Connectives: negation (¬), conjunction (∧, 'and'), disjunction (∨, 'or'), conditional (→, 'if...then'), biconditional (↔, 'if and only if').
Tautology: a compound statement that is always true, regardless of the truth values of its components.
Truth Table: a table listing the truth value of a compound statement for every combination of its component truth values.

4. Vector (2 – 4 Marks)

Vectors represent quantities with both magnitude and direction. They are expressed in terms of unit vectors i, j, k. The dot product (scalar product) yields a scalar and is used to find angles and test perpendicularity. The cross product (vector product) yields a vector perpendicular to both inputs and is used to compute areas. The section formula gives the position vector of a point dividing a line segment in a given ratio.

Vector: a quantity with both magnitude and direction, typically written as a⃗ = xi + yj + zk.
Unit Vector: a vector of magnitude 1, obtained by dividing a vector by its own magnitude.
Position Vector: the vector from the origin to a given point.

Dot Product

$$\vec a \cdot \vec b = |\vec a|\,|\vec b|\cos\theta$$

Use: finding the angle between two vectors, and testing perpendicularity (a⃗ · b⃗ = 0).

Cross Product

$$\vec a \times \vec b = |\vec a|\,|\vec b|\sin\theta\;\hat n$$

Use: finding a vector perpendicular to two given vectors, and computing area of a parallelogram or triangle.
Section Formula: a point dividing a line segment in ratio m : n has position vector (m·b⃗ + n·a⃗)/(m+n).

5. Trigonometry (2 – 4 Marks)

Trigonometry deals with the relationships between angles and sides of triangles. The standard angle table (0°, 30°, 45°, 60°, 90°) with sine, cosine, and tangent values is fundamental. Pythagorean identities and compound angle formulas are frequently used. General solutions of trigonometric equations are written using periodic formulas, taking into account domain restrictions.

5.1 Basic Trigonometry and Circular Functions

θ	0°	30°	45°	60°	90°
sin θ	0	1/2	1/√2	√3/2	1
cos θ	1	√3/2	1/√2	1/2	0
tan θ	0	1/√3	1	√3	undefined

$$\sin^2\theta + \cos^2\theta = 1, \qquad 1 + \tan^2\theta = \sec^2\theta, \qquad 1+\cot^2\theta = \csc^2\theta$$

Compound and Multiple Angle Formulas

$$\sin(A \pm B) = \sin A \cos B \pm \cos A \sin B$$

$$\cos(A \pm B) = \cos A \cos B \mp \sin A \sin B$$

$$\sin 2\theta = 2\sin\theta\cos\theta, \qquad \cos 2\theta = 1 - 2\sin^2\theta = 2\cos^2\theta - 1$$

5.2 Trigonometric Equations and General Solutions

$$\sin\theta = \sin\alpha \;\Rightarrow\; \theta = n\pi + (-1)^n\alpha, \;\; n \in \mathbb{Z}$$

$$\cos\theta = \cos\alpha \;\Rightarrow\; \theta = 2n\pi \pm \alpha$$

$$\tan\theta = \tan\alpha \;\Rightarrow\; \theta = n\pi + \alpha$$

Note: Always check the domain restrictions of the original equation before writing the final general solution.

6. Arithmetic (10 – 12 Marks)

High weightage and purely formula/practice-based — this is one of the fastest units to master with consistent problem drilling. Topics include ratio and proportion, percentages, profit and loss, simple and compound interest, time and work, time-speed-distance, and mixtures/alligation. These problems are direct applications of formulas, so memorization and practice are key.

Ratio & Proportion: comparison of two quantities of the same kind; a : b = c : d means ad = bc.
Percentage: a fraction expressed with denominator 100, used to compare relative quantities.

$$\text{Profit\% } = \frac{\text{Profit}}{\text{Cost Price}} \times 100$$

$$A = P\left(1 + \frac{r}{100}\right)^n \quad \text{(Compound Interest)}$$

$$SI = \frac{P \times r \times t}{100} \quad \text{(Simple Interest)}$$

Time and Work: if A can complete a job in x days, A's one-day work = 1/x; combined rates are added.
Time, Speed and Distance: Distance = Speed × Time; average speed for equal distances at speeds a and b is 2ab/(a+b).
Mixtures and Alligation: used to find the ratio in which two ingredients of different values must be mixed to get a mixture of a desired value.

7. Probability (1 – 2 Marks)

Probability quantifies the likelihood of an event occurring. The sample space is the set of all possible outcomes, and an event is a subset of it. The probability of an event is the ratio of favorable outcomes to total outcomes. The addition theorem handles union of events, with special cases for mutually exclusive events (P(A∩B)=0) and independent events (P(A∩B)=P(A)P(B)).

Sample Space (S): the set of all possible outcomes of a random experiment.
Event (E): any subset of the sample space.

$$P(E) = \frac{n(E)}{n(S)}$$

Key Theorems

$$P(A \cup B) = P(A) + P(B) - P(A \cap B) \quad \text{(Addition Theorem)}$$

Mutually Exclusive Events: cannot occur simultaneously, so P(A ∩ B) = 0.
Independent Events: P(A ∩ B) = P(A) · P(B), since the occurrence of one does not affect the other.

8. Statistics (1 – 2 Marks)

Statistics involves collecting, organizing, and interpreting data. Measures of central tendency (mean, median, mode) summarize the data. Measures of dispersion (range, variance, standard deviation) describe the spread. The standard deviation σ is the square root of the variance, and it indicates how much the data deviate from the mean.

Measures of Central Tendency

Mean: the sum of all observations divided by the number of observations.
Median: the middle value when data is arranged in order.
Mode: the value that occurs most frequently in the data set.

Measures of Dispersion

$$\sigma = \sqrt{\frac{\sum (x_i - \bar x)^2}{n}} \quad \text{(Standard Deviation)}$$

Variance: the square of the standard deviation, σ².
Range: the difference between the maximum and minimum values in the data set.

9. Linear Programming (1 – 2 Marks)

Linear programming deals with optimizing (maximizing or minimizing) a linear objective function subject to linear constraints. The feasible region is the intersection of all constraints, usually a convex polygon. The optimum value occurs at a corner point (vertex) of this region. The corner-point method evaluates the objective function at each vertex to find the best solution.

Objective Function: the linear function to be maximized or minimized, typically Z = ax + by.
Constraints: linear inequalities that define the feasible region within which the solution must lie.
Feasible Region: the common region satisfying all constraints simultaneously, usually a convex polygon.
Corner-Point Method: the optimum value of Z always occurs at one of the vertices (corner points) of the feasible region.

$$Z = ax + by$$

Quick Revision Tips

Priority: Algebra + Arithmetic together carry ~22–26 marks — prioritize these two units first.
Formula Recall: keep a single-page formula sheet for log rules, binomial expansion, AP/GP sums, and derivative/integral tables.
Practice: solve previous years' BIT entrance questions unit-wise to spot recurring question patterns.
Low-Hanging Fruit: Vector, Probability, Statistics, and Linear Programming are low-weightage but quick to master — don't skip them.

bit entrance notes