Quick Reference: A Level Maths Formulas

How to Master Binomial Expansion for Any Power

At A Level, binomial expansion extends far beyond positive whole-number powers. You are expected to work confidently with rational, fractional, and negative powers using the general expansion for expressions of the form (1 + x)ⁿ. This expansion produces an infinite series and is only valid within a specific range of x values, known as the validity condition. The general form begins with 1 + nx + n(n−1)/2 x² + n(n−1)(n−2)/6 x³ and continues indefinitely. Because it is infinite, examiners require you to state the condition |x| < 1, adjusted appropriately if the expression is rewritten. For example, expanding (1 − 3x)⁻² requires stating |x| < 1/3. Missing this condition is one of the most common ways students lose easy marks. Binomial expansion is frequently used for approximations. Expressions such as √1.04 can be rewritten in the form (1 + 0.04)¹ᐟ², allowing you to approximate values using just the first few terms. These questions reward precision, structure, and correct use of validity conditions rather than lengthy calculations.

When to Use Double Angle Formulas vs Rcos(x − α)

Choosing the correct trigonometric identity is often the difference between a short solution and a complicated one. The key signal is the structure of the angles involved. If an equation mixes angles such as x and 2x, your aim should be to rewrite everything in terms of a single angle. This is where double angle formulas are essential. Using identities like sin(2x) = 2sin(x)cos(x) or cos(2x) = 1 − 2sin²(x) allows you to reduce the equation to a quadratic in sin(x) or cos(x). Once simplified, standard algebraic techniques can be applied to find solutions within a given interval. When expressions involve a cos(x) + b sin(x), the harmonic form is the correct tool. Writing the expression as Rcos(x − α) converts two trigonometric terms into one, where R represents the maximum magnitude of the function. This approach is especially powerful for solving equations and finding maximum or minimum values without calculus.

“Strong trigonometry isn’t about remembering every identity, but about recognising which form simplifies the problem fastest.”

Choosing Between Product, Quotient, and Chain Rules

Differentiation errors often occur because students apply the wrong rule. Before differentiating, the first step should always be simplification. Cancelled factors or simplified expressions can remove the need for complex rules entirely. If two functions are multiplied together, such as x² ln(x), the Product Rule must be used. When one function is divided by another, such as sin(x)/eˣ, the Quotient Rule applies. In both cases, clearly defining u and v before differentiating helps avoid mistakes. The Chain Rule is required whenever one function is embedded inside another, as in (3x + 5)⁷ or cos(x³). This rule is fundamental at A Level and underpins many advanced differentiation and integration problems. Recognising composite functions early prevents incorrect methods and unnecessary loss of marks.

The LATE Rule and Vector Shortcuts for Exam Efficiency

Integration by Parts is the inverse of the Product Rule and is used when integrating the product of two functions. The most important decision is choosing which part to label as u. The LATE rule provides a reliable priority order: Logarithmic, Algebraic, Trigonometric, Exponential. Selecting u using this order ensures the resulting integral is simpler than the original. Vectors are another area where formula recall must be paired with interpretation. To find the angle between two vectors, the scalar product formula links algebraic components with geometry. Calculating the dot product and magnitudes separately before substitution reduces arithmetic errors. A key shortcut to remember is that if the scalar product of two vectors is zero, the vectors are perpendicular. This observation appears frequently in exams and can save significant time when answering proof or reasoning-based questions.