G2 Additional Mathematics revision reference

This selected lookup is for 2027 G2 Additional Mathematics (K232). Its calculus reminders stay with the course’s algebraic power functions. Use the lesson links for full explanations and applications.

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.

Read more: Quadratics and algebra and Coordinate geometry.

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.

Read more: Trigonometry.

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.

For this edition, the integration methods here exclude the power n = -1 and areas between two curves. Areas below the x-axis, motion in a straight line, and trigonometric, exponential and logarithmic calculus are not in the G2 syllabus; they belong to G3 Additional Mathematics.

Read more: Calculus.