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Binomial Expansion Formula: Theorem, Examples & Solved Problems

The binomial expansion formula gives the expanded form of (x + y)ⁿ as a sum of n + 1 terms involving binomial coefficients. The general formula is (x + y)ⁿ = Σ ⁿCₖ xⁿ⁻ᵏ yᵏ, where ⁿCₖ = n! / [k!(n − k)!]. It is used to expand any binomial raised to a positive integer, fractional, or negative power without multiplying the binomial by itself n times.

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Mean, Median, Mode Formula - Differences, Use, and Examples

The mean is the average of all values (Σx / n), the median is the middle value when data is ordered, and the mode is the value that appears most often. Use the mean for symmetric data, the median for skewed data with outliers, and the mode for categorical data. For moderately skewed distributions, the three are linked by Karl Pearson's empirical relation: Mode = 3 Median − 2 Mean.

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Perimeter Formulas: All Shapes with Variables and Worked Examples

The perimeter of a 2D shape is the total length of its boundary — found by adding the lengths of all its sides. Common shortcuts include P = 4a (square), P = 2(l + w) (rectangle), and C = 2πr (circle).

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Triangle Formula: Area, Perimeter, and All Types Explained

The most common triangle formulas are A = ½ × base × height for area and P = a + b + c for perimeter. Other formulas - Heron's formula, the trigonometric area formula, and per-type formulas for equilateral, isosceles, and right triangles - apply depending on what information you have. The decision table further down shows which formula to use in which case.

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Geometric Sequence Formulas: nth Term, Sum, and Examples

The geometric sequence formulas cover three things: the nth term aₙ = a · r^(n−1), the sum of the first n terms Sₙ = a(1 − rⁿ)/(1 − r) for r ≠ 1, and the infinite sum S∞ = a/(1 − r) when |r| < 1. The first formula finds any single term; the other two add up multiple terms.

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Arithmetic Sequence Formula: nth Term, Sum & Common Difference

The arithmetic sequence formula has two parts: aₙ = a₁ + (n−1)d for the nth term and Sₙ = (n/2) × [2a₁ + (n−1)d] for the sum of n terms. Here a₁ is the first term, d is the common difference, and n is the position. Use (n−1) - not n - when counting the jumps between terms.

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Double Angle Formulas: Sin, Cos, Tan Identities

The double angle formulas are sin 2θ = 2 sin θ cos θ, cos 2θ = cos²θ − sin²θ (also 1 − 2 sin²θ and 2 cos²θ − 1), and tan 2θ = 2 tan θ / (1 − tan²θ). All three are derived from the sum formulas by setting α = β = θ. The cosine has three equivalent forms because the Pythagorean identity converts between them.

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Volume Formulas for All 3D Shapes

Volume measures the space a 3D shape occupies, expressed in cubic units (cm³, m³, in³). Each shape has its own formula - cube uses V = a³, cuboid uses l × b × h, cylinder uses πr²h, sphere uses ⁴⁄₃ πr³, and cones and pyramids hold exactly one-third of their corresponding cylinder or prism.

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Slope Formula: How to Find the Slope of a Line

The slope formula is m = (y₂ − y₁) / (x₂ − x₁) - it calculates the steepness and direction of a line from any two points on it. Slope can be positive (line rises), negative (line falls), zero (horizontal), or undefined (vertical). The formula is also written as rise over run, where rise = Δy and run = Δx.

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Distance Formula in Maths: Definition, Derivation, and Examples

The distance formula, d = √[(x₂ − x₁)² + (y₂ − y₁)²], gives the straight-line distance between two points on a coordinate plane. It is derived from the Pythagorean theorem and extends to 3D by adding a z-coordinate term.

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Geometry Formulas: Area, Perimeter & Volume for 2D and 3D Shapes

Geometry formulas calculate the perimeter and area of 2D shapes (square, rectangle, triangle, circle) and the surface area and volume of 3D shapes (cube, cuboid, cylinder, cone, sphere). Perimeter uses linear units (cm), area and surface area use square units (cm²), and volume uses cubic units (cm³).

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Integration Formulas: Complete List with Examples

Integration formulas are the standard antiderivative results used to solve integrals across Class 11, 12, and JEE - covering basic, trigonometric, inverse trigonometric, exponential, logarithmic, hyperbolic, and special integrals. The general form is ∫f'(x) dx = f(x) + C.

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