Bsc.CSIT Entrance Notes
Bsc-csitMathematics•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
ALGEBRA
1. SET, REAL NUMBER SYSTEM AND LOGIC
Definition of Set
A set is a well-defined collection of objects.
Types of Sets
Empty Set
A set with no elements.
$$ A = {} $$
Finite Set
A set having limited elements.
$$ A = {1,2,3,4} $$
Infinite Set
A set having unlimited elements.
$$ A = {1,2,3,4,\dots} $$
Operations on Sets
Union
$$ A \cup B = {x : x \in A \text{ or } x \in B} $$
Intersection
$$ A \cap B = {x : x \in A \text{ and } x \in B} $$
Difference
$$ A - B = {x : x \in A, x \notin B} $$
Complement
$$ A' = U - A $$
De Morgan's Laws
$$ (A \cup B)' = A' \cap B' $$
$$ (A \cap B)' = A' \cup B' $$
Real Number System
Natural Numbers
$$ N = {1,2,3,4,\dots} $$
Whole Numbers
$$ W = {0,1,2,3,4,\dots} $$
Integers
$$ Z = {\dots,-2,-1,0,1,2,\dots} $$
Rational Numbers
$$ Q = \left{\frac{p}{q} : q \neq 0\right} $$
Irrational Numbers
Numbers that cannot be written as fractions.
Example:
$$ \sqrt{2}, \pi $$
Logic
Statement
A sentence that is either true or false.
Logical Connectives
Symbol | Meaning |
$\land$ | AND |
$\lor$ | OR |
$\neg$ | NOT |
$\Rightarrow$ | Implies |
$\Leftrightarrow$ | If and only if |
Truth Table for AND
P | Q | $P \land Q$ |
T | T | T |
T | F | F |
F | T | F |
F | F | F |
2. RELATION AND FUNCTION
Relation
A relation from set $A$ to set $B$ is a subset of:
$$ A \times B $$
Function
A function assigns exactly one output to every input.
$$ f : A \to B $$
Example
If
$$ f(x) = x^2 $$
then
$$ f(2) = 4 $$
Types of Functions
One-One Function
Different inputs have different outputs.
Onto Function
Every element in codomain has a preimage.
Bijective Function
A function that is both one-one and onto.
Domain and Range
Domain
Set of all possible inputs.
Range
Set of all outputs.
Composite Function
$$ (f \circ g)(x) = f(g(x)) $$
Inverse Function
$$ f^{-1}(x) $$
If
$$ f(x)=2x+3 $$
then
$$ f^{-1}(x)=\frac{x-3}{2} $$
3. LOGARITHMIC FUNCTION
Definition
A logarithmic function is the inverse of an exponential function.
Form
$$ \log_a(x)=y \iff a^y=x $$
Common Logarithm
$$ \log_{10}(x) $$
Natural Logarithm
$$ \ln(x)=\log_e(x) $$
Rules of Logarithm
Product Rule
$$ \log(ab)=\log a+\log b $$
Quotient Rule
$$ \log\left(\frac{a}{b}\right)=\log a-\log b $$
Power Rule
$$ \log(a^n)=n\log a $$
Change of Base Formula
$$ \log_a b = \frac{\log_c b}{\log_c a} $$
Example
$$ \log_2 8 = 3 $$
because
$$ 2^3 = 8 $$
4. MATRIX AND DETERMINANTS
Matrix
A rectangular arrangement of numbers.
$$ A = \begin{bmatrix} 1 & 2 \ 3 & 4 \end{bmatrix} $$
Order of Matrix
If matrix has $m$ rows and $n$ columns, then order is:
$$ m \times n $$
Addition of Matrices
$$ A+B=[a_{ij}+b_{ij}] $$
Multiplication of Matrices
$$ AB \neq BA $$
Determinant of $2 \times 2$ Matrix
$$ \begin{vmatrix} a & b \ c & d \end{vmatrix} =ad-bc $$
Inverse of Matrix
For matrix
$$ A=\begin{bmatrix} a & b \ c & d \end{bmatrix} $$
inverse is:
$$ A^{-1}=\frac{1}{ad-bc} \begin{bmatrix} d & -b \ -c & a \end{bmatrix} $$
5. SEQUENCE AND SERIES
Sequence
An ordered list of numbers.
Example:
$$ 1,2,3,4,\dots $$
Arithmetic Progression (AP)
nth Term
$$ a_n=a+(n-1)d $$
Sum of n Terms
$$ S_n=\frac{n}{2}[2a+(n-1)d] $$
Geometric Progression (GP)
nth Term
$$ a_n=ar^{n-1} $$
Sum of n Terms
$$ S_n=a\frac{r^n-1}{r-1} $$
Infinite GP
$$ S_\infty=\frac{a}{1-r} $$
where
$$ |r|<1 $$
6. COMPLEX NUMBER
Definition
A number of the form:
$$ z=a+ib $$
where
$$ i=\sqrt{-1} $$
Real Part
$$ Re(z)=a $$
Imaginary Part
$$ Im(z)=b $$
Conjugate
$$ \bar z=a-ib $$
Modulus
$$ |z|=\sqrt{a^2+b^2} $$
Polar Form
$$ z=r(\cos\theta+i\sin\theta) $$
De Moivre's Theorem
$$ (\cos\theta+i\sin\theta)^n =\cos(n\theta)+i\sin(n\theta) $$
7. POLYNOMIAL EQUATION
Polynomial
An expression of the form:
$$ a_nx^n+a_{n-1}x^{n-1}+\dots+a_1x+a_0 $$
Quadratic Equation
$$ ax^2+bx+c=0 $$
Quadratic Formula
$$ x=\frac{-b\pm\sqrt{b^2-4ac}}{2a} $$
Nature of Roots
If
$$ b^2-4ac>0 $$
roots are real and distinct.
If
$$ b^2-4ac=0 $$
roots are equal.
If
$$ b^2-4ac<0 $$
roots are imaginary.
8. SYSTEM OF LINEAR EQUATIONS
General Form
$$ a_1x+b_1y=c_1 $$
$$ a_2x+b_2y=c_2 $$
Cramer's Rule
$$ x=\frac{D_x}{D} $$
$$ y=\frac{D_y}{D} $$
where
$$ D=\begin{vmatrix} a_1 & b_1 \ a_2 & b_2 \end{vmatrix} $$
9. BINOMIAL EQUATIONS
Binomial Theorem
$$ (a+b)^n=\sum_{r=0}^{n}\binom{n}{r}a^{n-r}b^r $$
General Term
$$ T_{r+1}=\binom{n}{r}a^{n-r}b^r $$
Middle Term
If $n$ is even:
$$ \left(\frac{n}{2}+1\right)^{th} text{ term} $$
10. PERMUTATION AND COMBINATION
Permutation
Arrangement of objects.
$$ {}^nP_r=\frac{n!}{(n-r)!} $$
Combination
Selection of objects.
$$ {}^nC_r=\frac{n!}{r!(n-r)!} $$
Relation
$$ {}^nP_r={}^nC_r\times r! $$
TRIGONOMETRY
11. BASIC TRIGONOMETRY AND CIRCULAR FUNCTION
Trigonometric Ratios
$$ \sin\theta=\frac{\text{Perpendicular}}{\text{Hypotenuse}} $$
$$ \cos\theta=\frac{\text{Base}}{\text{Hypotenuse}} $$
$$ \tan\theta=\frac{\text{Perpendicular}}{\text{Base}} $$
Reciprocal Ratios
$$ \csc\theta=\frac{1}{\sin\theta} $$
$$ \sec\theta=\frac{1}{\cos\theta} $$
$$ \cot\theta=\frac{1}{\tan\theta} $$
Fundamental Identities
$$ \sin^2\theta+\cos^2\theta=1 $$
$$ 1+\tan^2\theta=\sec^2\theta $$
$$ 1+\cot^2\theta=\csc^2\theta $$
Degree and Radian
$$ 180^\circ=\pi \text{ radians} $$
12. TRIGONOMETRIC EQUATION AND GENERAL SOLUTION
General Solution
For
$$ \sin x=\sin a $$
$$ x=n\pi+(-1)^na $$
For
$$ \cos x=\cos a $$
$$ x=2n\pi\pm a $$
For
$$ \tan x=\tan a $$
$$ x=n\pi+a $$
where
$$ n\in Z $$
13. PROPERTIES OF TRIANGLES
Sine Rule
$$ \frac{a}{\sin A}=\frac{b}{\sin B}=\frac{c}{\sin C} $$
Cosine Rule
$$ c^2=a^2+b^2-2ab\cos C $$
Area of Triangle
$$ \Delta=\frac{1}{2}ab\sin C $$
Heron's Formula
$$ \Delta=\sqrt{s(s-a)(s-b)(s-c)} $$
where
$$ s=\frac{a+b+c}{2} $$
COORDINATE GEOMETRY
14. STRAIGHT LINES
Slope Formula
$$ m=\frac{y_2-y_1}{x_2-x_1} $$
Equation of Line
Slope-Intercept Form
$$ y=mx+c $$
Point-Slope Form
$$ y-y_1=m(x-x_1) $$
Two Point Form
$$ \frac{y-y_1}{y_2-y_1}=\frac{x-x_1}{x_2-x_1} $$
Distance Between Two Points
$$ D=\sqrt{(x_2-x_1)^2+(y_2-y_1)^2} $$
15. PAIR OF STRAIGHT LINES
Homogeneous Equation
$$ ax^2+2hxy+by^2=0 $$
represents pair of straight lines through origin.
Condition of Perpendicular Lines
$$ a+b=0 $$
16. CIRCLE, PARABOLA, ELLIPSE, HYPERBOLA
Circle
Standard Equation
$$ (x-h)^2+(y-k)^2=r^2 $$
Parabola
Standard Equation
$$ y^2=4ax $$
Ellipse
$$ \frac{x^2}{a^2}+\frac{y^2}{b^2}=1 $$
Hyperbola
$$ \frac{x^2}{a^2}-\frac{y^2}{b^2}=1 $$
17. COORDINATES IN SPACE/PLANE
Distance Formula in Space
$$ D=\sqrt{(x_2-x_1)^2+(y_2-y_1)^2+(z_2-z_1)^2} $$
Midpoint Formula
$$ \left(\frac{x_1+x_2}{2},\frac{y_1+y_2}{2},\frac{z_1+z_2}{2}\right) $$
CALCULUS
18. LIMIT AND CONTINUITY
Limit
$$ \lim_{x\to a}f(x) $$
Important Limits
$$ \lim_{x\to0}\frac{\sin x}{x}=1 $$
$$ \lim_{x\to0}\frac{1-\cos x}{x}=0 $$
$$ \lim_{x\to\infty}\left(1+\frac{1}{x}\right)^x=e $$
Continuity
A function is continuous if:
$$ \lim_{x\to a}f(x)=f(a) $$
19. DERIVATIVES AND APPLICATIONS
Definition
$$ \frac{dy}{dx}=\lim_{h\to0}\frac{f(x+h)-f(x)}{h} $$
Basic Derivatives
$$ \frac{d}{dx}(x^n)=nx^{n-1} $$
$$ \frac{d}{dx}(\sin x)=\cos x $$
$$ \frac{d}{dx}(\cos x)=-\sin x $$
$$ \frac{d}{dx}(e^x)=e^x $$
$$ \frac{d}{dx}(\ln x)=\frac{1}{x} $$
Product Rule
$$ \frac{d}{dx}(uv)=u\frac{dv}{dx}+v\frac{du}{dx} $$
Quotient Rule
$$ \frac{d}{dx}\left(\frac{u}{v}\right) =\frac{v\frac{du}{dx}-u\frac{dv}{dx}}{v^2} $$
Chain Rule
$$ \frac{dy}{dx}=\frac{dy}{du}\cdot\frac{du}{dx} $$
Maxima and Minima
Critical points occur when:
$$ \frac{dy}{dx}=0 $$
20. INTEGRATION AND APPLICATIONS
Integration
$$ \int f(x)dx $$
Basic Integrals
$$ \int x^n dx=\frac{x^{n+1}}{n+1}+C $$
$$ \int \frac{1}{x}dx=\ln|x|+C $$
$$ \int e^x dx=e^x+C $$
$$ \int \sin x,dx=-\cos x+C $$
$$ \int \cos x,dx=\sin x+C $$
Integration by Parts
$$ \int u,dv=uv-\int v,du $$
Definite Integral
$$ \int_a^b f(x)dx $$
Area Under Curve
$$ A=\int_a^b y,dx $$
IMPORTANT FORMULAS SUMMARY
Quadratic Formula
$$ x=\frac{-b\pm\sqrt{b^2-4ac}}{2a} $$
Binomial Theorem
$$ (a+b)^n=\sum_{r=0}^{n}\binom{n}{r}a^{n-r}b^r $$
Euler Formula
$$ e^{ix}=\cos x+i\sin x $$
Fundamental Trigonometric Identity
$$ \sin^2\theta+\cos^2\theta=1 $$
Derivative of Logarithm
$$ \frac{d}{dx}(\ln x)=\frac{1}{x} $$
Integral of Exponential
$$ \int e^x dx=e^x+C $$
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