For equation ax² + bx + c = 0:

Roots = [-b ± √(b² – 4ac)] / 2a

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

Where D = b² – 4ac is called the discriminant

nth term: aₙ = a₁ + (n-1)d

Sum of first n terms: Sₙ = n/2 [2a₁ + (n-1)d] = n/2 (a₁ + aₙ)

a_n = a_1 + (n-1)d \\ S_n = \frac{n}{2} [2a_1 + (n-1)d]

nth term: aₙ = a₁ × rⁿ⁻¹

Sum of first n terms: Sₙ = a₁(1 – rⁿ)/(1 – r) when r ≠ 1

a_n = a_1 \cdot r^{n-1} \\ S_n = \frac{a_1(1 – r^n)}{1 – r}

(a + b)ⁿ = ⁿC₀aⁿb⁰ + ⁿC₁aⁿ⁻¹b¹ + … + ⁿCₙa⁰bⁿ

(a + b)^n = \sum_{k=0}^n \binom{n}{k} a^{n-k}b^k

logₐ(mn) = logₐm + logₐn

logₐ(m/n) = logₐm – logₐn

logₐmⁿ = n logₐm

logₐb = logₖb / logₖa (Change of base)

\log_a(mn) = \log_a m + \log_a n \\ \log_a\left(\frac{m}{n}\right) = \log_a m – \log_a n

Area = ½ × base × height

Pythagoras theorem: a² + b² = c² (for right triangles)

Heron’s formula: Area = √[s(s-a)(s-b)(s-c)] where s = (a+b+c)/2

A = \frac{1}{2} \times b \times h \\ a^2 + b^2 = c^2 \\ A = \sqrt{s(s-a)(s-b)(s-c)}

Area = πr²

Circumference = 2πr

Arc length = (θ/360) × 2πr

A = \pi r^2 \\ C = 2\pi r \\ L = \frac{\theta}{360} \times 2\pi r

Distance between (x₁,y₁) and (x₂,y₂): √[(x₂-x₁)² + (y₂-y₁)²]

Slope of line: m = (y₂-y₁)/(x₂-x₁)

Equation of line: y – y₁ = m(x – x₁)

d = \sqrt{(x_2-x_1)^2 + (y_2-y_1)^2} \\ m = \frac{y_2-y_1}{x_2-x_1} \\ y – y_1 = m(x – x_1)

sin²θ + cos²θ = 1

1 + tan²θ = sec²θ

1 + cot²θ = cosec²θ

sin(A±B) = sinA cosB ± cosA sinB

cos(A±B) = cosA cosB ∓ sinA sinB

\sin^2\theta + \cos^2\theta = 1 \\ 1 + \tan^2\theta = \sec^2\theta \\ \sin(A \pm B) = \sin A \cos B \pm \cos A \sin B

Cube volume = a³

Cuboid volume = l × b × h

Sphere volume = (4/3)πr³

Cylinder volume = πr²h

V_{cube} = a^3 \\ V_{cuboid} = l \times b \times h \\ V_{sphere} = \frac{4}{3}\pi r^3

d/dx (xⁿ) = nxⁿ⁻¹

d/dx (sin x) = cos x

d/dx (cos x) = -sin x

d/dx (tan x) = sec² x

d/dx (eˣ) = eˣ

d/dx (ln x) = 1/x

\frac{d}{dx} (x^n) = nx^{n-1} \\ \frac{d}{dx} (\sin x) = \cos x \\ \frac{d}{dx} (e^x) = e^x

∫xⁿ dx = xⁿ⁺¹/(n+1) + C, n ≠ -1

∫(1/x) dx = ln|x| + C

∫eˣ dx = eˣ + C

∫sin x dx = -cos x + C

∫cos x dx = sin x + C

\int x^n dx = \frac{x^{n+1}}{n+1} + C \\ \int \frac{1}{x} dx = \ln|x| + C \\ \int \sin x dx = -\cos x + C

∫ₐᵇ f(x) dx = F(b) – F(a) where F'(x) = f(x)

∫ₐᵇ f(x) dx = ∫ₐᵇ f(a+b-x) dx

\int_a^b f(x) dx = F(b) – F(a) \\ \int_a^b f(x) dx = \int_a^b f(a+b-x) dx

Mean = (Σxᵢ)/n

Median (odd n) = value at (n+1)/2 position

Median (even n) = average of values at n/2 and (n/2)+1 positions

Standard Deviation = √[Σ(xᵢ – x̄)²/n]

\bar{x} = \frac{\sum x_i}{n} \\ \sigma = \sqrt{\frac{\sum (x_i – \bar{x})^2}{n}}

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

P(A∪B) = P(A) + P(B) – P(A∩B)

P(A’) = 1 – P(A)

Conditional Probability: P(A|B) = P(A∩B)/P(B)

P(E) = \frac{n(E)}{n(S)} \\ P(A \cup B) = P(A) + P(B) – P(A \cap B) \\ P(A|B) = \frac{P(A \cap B)}{P(B)}

Nikhilam Sutra (Base Method):

For numbers near powers of 10 (e.g., 98 × 97)

1. Write deviations from base (100): 98 (-2), 97 (-3)

2. Multiply deviations: (-2)×(-3) = 6

3. Cross add: 98 + (-3) = 95 or 97 + (-2) = 95

4. Answer: 9506

(a5)² = a×(a+1) followed by 25

Example: 35² = 3×4 followed by 25 = 1225

Example: 75² = 7×8 followed by 25 = 5625

For 2-digit multiplication (e.g., 21 × 32):

1. Multiply vertically (left digits): 2×3 = 6

2. Multiply crosswise and add: (2×2)+(1×3) = 4+3 = 7

3. Multiply vertically (right digits): 1×2 = 2

4. Combine: 6 (7) 2 → 672 (middle digit may need carry)

Chapter 1: Real Numbers

Chapter 2: Polynomials

Chapter 3: Pair of Linear Equations

Chapter 4: Quadratic Equations

Chapter 5: Arithmetic Progressions

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Chapter 1: Relations and Functions

Chapter 2: Inverse Trigonometric Functions

Chapter 3: Matrices

Chapter 4: Determinants

Chapter 5: Continuity and Differentiability

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