bit entrance notes
BitMathematics•Updated: 5/7/2026
📘 MATHEMATICS NOTES
1. RELATION AND FUNCTION
Definition
A relation is a set of ordered pairs, while a function is a special relation where each input has exactly one output.
Types of Relations
• Reflexive • Symmetric • Transitive
Types of Functions
• One-one Function • Many-one Function • Onto Function • Into Function
Domain and Range
Domain → Set of input values
Range → Set of output values
Example
If $$f(x) = x^2$$ then $$f(2) = 4$$
2. LOGARITHMIC FUNCTION
Definition
A logarithmic function is the inverse of an exponential function.
Form
$$\log_a(x) = y \iff a^y = x$$
Rules
Product Rule
$$\log(ab) = \log a + \log b$$
Quotient Rule
$$\log\left(\frac{a}{b}\right) = \log a - \log b$$
Power Rule
$$\log(a^n) = n \log a$$
Examples
$$\log_{10}(100) = 2$$
$$\log_{2}(8) = 3$$
3. MATRIX AND DETERMINANTS
Definition
A matrix is a rectangular arrangement of numbers; a determinant is a scalar value of a square matrix.
Types of Matrices
• Row Matrix • Column Matrix • Square Matrix • Identity Matrix
Operations
• Addition • Subtraction • Multiplication
Determinant (2×2)
$$ \begin{vmatrix} a & b \ c & d \end{vmatrix} = ad - bc $$
Example
Matrix: [ \begin{bmatrix} 1 & 2 \ 3 & 4 \end{bmatrix} ]
Determinant: $$1×4 - 2×3 = -2$$
4. SEQUENCE AND SERIES
Definition
A sequence is an ordered list; a series is the sum of terms.
Types
• Arithmetic Progression (AP) • Geometric Progression (GP)
AP Formula
$$a_n = a + (n-1)d$$
GP Formula
$$a_n = ar^{n-1}$$
Example
AP: 2, 4, 6, 8
GP: 2, 4, 8, 16
5. COMPLEX NUMBERS
Definition
Numbers of the form: $$z = a + bi$$ where (i^2 = -1)
Parts
• Real part = a • Imaginary part = b
Modulus
$$|z| = \sqrt{a^2 + b^2}$$
Example
3 + 4i → Modulus = 5
6. POLYNOMIAL EQUATIONS
Definition
An equation involving variables with powers and coefficients.
Degree
Highest power of variable.
Types
• Linear • Quadratic • Cubic
Quadratic Formula
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Example
x² − 5x + 6 = 0 Roots: 2 and 3
7. SYSTEM OF LINEAR EQUATIONS
Definition
A set of linear equations solved together.
Methods
• Substitution • Elimination • Matrix Method
Example
x + y = 5 x − y = 1
Solution: x = 3, y = 2
8. BINOMIAL THEOREM
Definition
Expands expressions of the form (a + b)ⁿ.
Formula
$$ (a+b)^n = \sum_{k=0}^{n} \binom{n}{k} a^{n-k} b^k $$
Example
(a + b)² = a² + 2ab + b²
9. CALCULUS
LIMITS AND CONTINUITY
Definition
Limit is the value a function approaches; continuity means no breaks.
$$\lim_{x \to a} f(x)$$
Example: $$\lim_{x \to 2} x^2 = 4$$
DERIVATIVES
Definition
Rate of change of a function.
$$\frac{d}{dx}(x^n) = nx^{n-1}$$
Example: d/dx (x²) = 2x
APPLICATIONS
• Slope • Maxima/Minima • Increasing/Decreasing functions
INTEGRATION
Definition
Reverse of differentiation.
$$\int x^n dx = \frac{x^{n+1}}{n+1} + C$$
Example: ∫x dx = x²/2 + C
APPLICATIONS
• Area under curve • Volume
10. SET THEORY
Definition
A set is a collection of elements.
Types
• Finite • Infinite • Empty
Operations
• Union (A ∪ B) • Intersection (A ∩ B) • Complement
Example: A = {1,2}, B = {2,3} A ∪ B = {1,2,3}
11. VECTOR
Definition
A quantity with magnitude and direction.
Types
• Zero Vector • Unit Vector • Position Vector
Magnitude
$$|\vec{a}| = \sqrt{x^2 + y^2 + z^2}$$
Example: (3,4) → 5
12. TRIGONOMETRY
Definition
Study of angles and ratios.
Ratios
$$\sin\theta = \frac{opposite}{hypotenuse}, \quad \cos\theta = \frac{adjacent}{hypotenuse}$$
Example: sin 30° = 1/2
Equations
Example: sin x = 0 → x = nπ
13. ARITHMETIC
Basic operations: addition, subtraction, multiplication, division.
Topics: • Percentage • Ratio • Profit & Loss
Example: 20% of 100 = 20
14. PROBABILITY
Definition
Chance of an event.
$$P(A) = \frac{\text{favorable outcomes}}{\text{total outcomes}}$$
Example: Probability of head = 1/2
15. STATISTICS
Definition
Study of data collection and analysis.
Measures
• Mean • Median • Mode
Mean Formula
$$\bar{x} = \frac{\sum x}{n}$$
Example: 2, 4, 6 → Mean = 4
16. LINEAR PROGRAMMING
Definition
Method to find maximum or minimum value under constraints.
Components
• Objective Function • Constraints
Example: Maximize profit with limited resources