📘 Mathematics Semester 1 – Full Notes

1. Algebra (बीजगणित)

1.1 Set Theory (समुच्चय सिद्धांत)

Set – कोई भी well-defined collection of objects.

Representation:

Roster form {1,2,3}

Set builder form {x | x ∈ N, x < 5}

Types:

Finite, Infinite, Empty set (∅), Singleton, Universal set (U).

Operations:

Union (A ∪ B), Intersection (A ∩ B), Difference (A – B), Complement (A′).

Laws of sets:

Idempotent, Commutative, Associative, Distributive, De Morgan’s laws.

Example:

If A = {1,2,3}, B = {2,3,4}, then

A ∪ B = {1,2,3,4},

A ∩ B = {2,3},

A – B = {1}.

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1.2 Functions (फलन)

A function f: X → Y is a relation such that every element of X has a unique image in Y.

Types:

One-one (Injective), Onto (Surjective), Bijective.

Composition: (f ∘ g)(x) = f(g(x))

Inverse function: f⁻¹(y) = x such that f(x) = y.

Important Functions:

Polynomial, Rational, Exponential, Logarithmic, Trigonometric.

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1.3 Matrices (आव्यूह)

Definition: Rectangular array of numbers.

Types: Square, Diagonal, Identity, Null, Row, Column, Symmetric, Skew-symmetric.

Operations: Addition, Subtraction, Scalar multiplication, Matrix multiplication.

Properties: (AB)C = A(BC), A(B+C)=AB+AC.

Determinant: |A| used to check invertibility.

Inverse: A⁻¹ = (1/|A|)Adj(A), if |A| ≠ 0.

Applications:

Solving system of linear equations (Cramer’s Rule, Matrix Inverse method).

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2. Calculus (कलन)

2.1 Limits (सीमा)

Definition: lim (x→a) f(x) = L

Important limits:

lim (x→0) (sin x)/x = 1

lim (x→0) (1 – cos x)/x² = 1/2

lim (x→∞) (1 + 1/x)^x = e

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2.2 Continuity & Differentiability

A function f(x) is continuous at x=a if:

lim (x→a⁻) f(x) = f(a) = lim (x→a⁺) f(x).

Differentiability ⇒ Continuity but not vice versa.

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2.3 Derivatives (अवकलज)

Definition: f′(x) = lim (h→0) (f(x+h) – f(x))/h

Rules:

(xⁿ)′ = n xⁿ⁻¹

(sin x)′ = cos x, (cos x)′ = –sin x

(tan x)′ = sec²x

(log x)′ = 1/x, (e^x)′ = e^x

Product Rule: (uv)′ = u′v + uv′

Quotient Rule: (u/v)′ = (u′v – uv′)/v²

Chain Rule: dy/dx = (dy/du)·(du/dx)

Applications:

Slope of tangent, rate of change, maxima & minima.

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2.4 Integration (समाकलन)

Indefinite Integral: ∫f(x) dx = F(x) + C

Formulas:

∫xⁿ dx = (xⁿ⁺¹)/(n+1) + C (n ≠ –1)

∫1/x dx = log|x| + C

∫e^x dx = e^x + C

∫sin x dx = –cos x + C

∫cos x dx = sin x + C

Definite Integral: ∫[a,b] f(x) dx = F(b) – F(a)

Properties:

∫[a,a] f(x) dx = 0

∫[a,b] f(x) dx = –∫[b,a] f(x) dx

Applications:

Area under curves, volume of revolution.

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3. Trigonometry (त्रिकोणमिति)

3.1 Basic Identities

sin²x + cos²x = 1

1 + tan²x = sec²x

1 + cot²x = csc²x

3.2 Multiple & Submultiple Angles

sin(2x) = 2sinx cosx

cos(2x) = cos²x – sin²x

tan(2x) = 2tanx / (1–tan²x)

3.3 Inverse Trigonometric Functions

sin⁻¹(sin x) = x (if x ∈ [–π/2, π/2])

cos⁻¹(cos x) = x (if x ∈ [0, π])

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4. Coordinate Geometry (निर्देशांक ज्यामिति)

4.1 Straight Line

General Equation: ax + by + c = 0

Slope (m) = –a/b

Distance between two points (x₁,y₁), (x₂,y₂):

√((x₂–x₁)² + (y₂–y₁)²)

Midpoint formula: ((x₁+x₂)/2, (y₁+y₂)/2)

4.2 Circle

Standard Equation: (x–h)² + (y–k)² = r²

General form: x² + y² + 2gx + 2fy + c = 0

Center (–g, –f), Radius √(g²+f²–c).

4.3 Conic Sections

Parabola: y² = 4ax (axis along x-axis)

Ellipse: (x²/a²) + (y²/b²) = 1

Hyperbola: (x²/a²) – (y²/b²) = 1

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5. Differential Equations (अवकल समीकरण)

5.1 Definition

Equation involving derivatives.

Order: Highest order derivative present.

Degree: Power of highest order derivative.

5.2 Types

1. First Order Linear DE: dy/dx + Py = Q

Solution: y·e^(∫P dx) = ∫Q·e^(∫P dx) dx + C

2. Variable Separable: dy/dx = f(x)g(y)

Solution: ∫ dy/g(y) = ∫ f(x) dx

3. Homogeneous DE: dy/dx = F(y/x).

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6. Probability & Statistics (संभाव्यता एवं सांख्यिकी)

6.1 Probability

If S = sample space, n(S)= total outcomes, E = event, n(E) = favorable outcomes,

P(E) = n(E)/n(S).

Laws:

0 ≤ P(E) ≤ 1

P(S) = 1

P(A∪B) = P(A)+P(B)–P(A∩B)

6.2 Statistics

Mean (x̄) = (Σx)/n

Median = middle value of ordered data.

Mode = most repeated value.

Standard Deviation = √(Σ(x–x̄)²/n).

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7. Vector Algebra (सदिश बीजगणित)

7.1 Basics

A vector has magnitude & direction.

Unit vector: a/|a|

Position vector of P(x,y,z) is OP = xi + yj + zk

7.2 Operations

Dot Product: a·b = |a||b|cosθ

Cross Product: a×b = |a||b|sinθ n̂

Scalar triple product: a·(b×c) = Volume of parallelepiped.

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8. Applications in Real Life

Matrices → Economics, Computer Graphics.

Calculus → Physics (velocity, acceleration), Biology (growth models).

Probability → Risk analysis, Games, Weather forecasting.

Trigonometry → Architecture, Navigation.

Differential Equations → Population growth, Radioactive decay.

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