G3 Additional Mathematics revision reference

This selected lookup belongs to 2027 G3 Additional Mathematics (K341). It is a revision aid; examination instructions determine which formulas are supplied and which methods or answer forms are required.

Quadratics, surds and polynomials

For a ≠ 0, completing the square gives:

The vertex has x = -b/(2a). A quadratic is strictly positive for every real x when a > 0 and b²-4ac < 0; strictly negative when a < 0 and b²-4ac < 0. A zero discriminant allows a zero value, so it does not establish strict positivity or negativity.

To solve a quadratic inequality, find the roots and use the sign on each interval. Do not simply write the roots as the answer.

For nonnegative real a,b, square root of ab = square root of a square root of b. Rationalise a denominator such as a + square root of b by multiplying numerator and denominator by a- square root of b, when that conjugate is nonzero. Check an equation in its original form after squaring it.

Remainder theorem: division of a polynomial f(x) by x-a leaves remainder f(a). Factor theorem: x-a is a factor exactly when f(a) = 0.

For partial fractions, include every power of a repeated linear factor. For an irreducible quadratic factor, use a linear numerator. Preserve the original denominator exclusions.

Coordinate geometry

The midpoint of (x₁,y₁) and (x₂,y₂) is:

Nonvertical parallel lines have equal gradients. For perpendicular lines with finite gradients, m₁m₂ = -1; handle a horizontal/vertical pair separately.

A circle with centre (a,b) and radius r > 0 has equation (x-a)² + (y-b)² = r². In x² + y² + 2gx + 2fy + c = 0, the centre is (-g,-f) and r² = g² + f²-c; check that the latter is positive for a nondegenerate real circle.

For a straight-line transformation, label the new variables explicitly. If y = axⁿ for positive x,y,a, then ln y = ln a + n ln x: plotting ln y against ln x gives gradient n and intercept ln a.

Read more: Algebra and Coordinates.

Binomial expansion and logarithms

For a positive integer n:

Here 0! = 1. The term with power b^r is the (r + 1)th term. Include signs in b before raising it to a power.

For positive x,y and base a > 0, a ≠ 1:

• log ₐ(xy) = log ₐ x + log ₐ y.
• log ₐ(x/y) = log ₐ x- log ₐ y.
• log ₐ(x^k) = k log ₐ x.
• log ₐ x = (ln x)/(ln a); ln x = log ₑ x.

Logarithms turn products into sums, not sums into products: log ₐ(x + y) has no matching sum rule. Check positivity of every original logarithm argument after solving.

Read more: Binomial expansions and Exponential and logarithmic functions.

Trigonometry

Radians: π radians = 180°. Reciprocals: sec x = 1/cos x, cosecx = 1/sin x and cot x = 1/tan x, wherever their denominators are nonzero.

Use tangent formulas only where every expression involved is defined.

Exact values: at 30°,45°,60°, sine is 1/2,1/(square root of 2),(square root of 3)/2; cosine is (square root of 3)/2,1/(square root of 2),1/2; tangent is 1/(square root of 3),1, square root of 3.

Inverse notation: sin⁻¹ x means the inverse-sine function, not a reciprocal. Its principal range is [-π/2,π/2]; cosine’s is [0,π] and tangent’s is (-π/2,π/2). Check all solutions in the given interval, including endpoints where permitted.

For y = a sin(bx) + c or y = a cos(bx) + c, amplitude is |a| and period is 2π/b for b > 0 with x in radians. For a sin(x/b) + c, the period is 2π b. A nonconstant tangent graph has period π/b for a tan(bx).

For nonzero (a,b), write a cos x + b sin x = R cos(x-α) using R = square root of (a² + b²), R cos α = a and R sin α = b. Choose the quadrant from both signs.

The midpoint theorem joins the midpoints of two sides of a triangle with a segment parallel to the third side and half its length. The tangent–chord theorem relates the angle between a tangent and a chord to the angle in the alternate segment; identify the correct segment before equating angles.

Read more: Trigonometry and Geometry and reasoning.

Differentiation and integration

Derivative f'(x) or dy/(dx) gives a tangent gradient and a rate of change. Second derivative f''(x) describes the rate of change of that gradient.

Work on an interval where the real-valued power is defined and differentiable. A constant differentiates to zero.

For differentiable u(x),v(x):

Chain rule: differentiate the outer function and multiply by the derivative of the inner function.

At (a,f(a)), the tangent is y-f(a) = f'(a)(x-a). A normal has gradient -1/f'(a) when f'(a) ≠ 0; if the tangent is horizontal, the normal is vertical.

A stationary point satisfies f'(a) = 0. Then f''(a) > 0 gives a local minimum and f''(a) < 0 a local maximum. If f''(a) = 0, the test is inconclusive; examine the gradient on either side. Check interval endpoints in an optimisation problem.

C is a constant of integration. Differentiate an antiderivative to check it. A definite integral ∫ₐ^b f(x) dx = F(b)-F(a) when F' = f on the relevant interval. Above the horizontal axis it gives area; check the sign and the course’s permitted region before using it geometrically.

Trigonometric, exponential and logarithmic calculus

Use radians for trigonometric derivatives.

• (sin x)' = cos x; (cos x)' = - sin x; (tan x)' = sec² x.
• (e^x)' = e^x; (ln x)' = 1/x for x > 0.
• ∫ sin x dx = - cos x + C; ∫ cos x dx = sin x + C.
• ∫ sec² x dx = tan x + C; ∫ e^x dx = e^x + C.
• For a nonzero constant a, integrating sin(ax + b), cos(ax + b) or e^(ax + b) introduces a factor 1/a.

For area below the axis, split the interval at roots and take the positive area of each part. This edition’s region problems do not require area between two curves.

In straight-line motion, v = ds/(dt) and a = dv/(dt). Displacement is signed; total distance adds the magnitudes of displacements after splitting at direction changes.

Read more: Calculus.