Edexcel International GCSE Additional Mathematics: every topic, by paper

What is a function?

A rule that maps each input (from the domain) to exactly one output (in the range).

What are the domain and range of a function?

Domain: the set of allowed inputs. Range: the set of possible outputs.

Find the range of f(x) = x² + 3 for all real x.

f(x) ≥ 3.

Write the quadratic formula.

x = (−b ± √(b² − 4ac)) ÷ 2a.

Complete the square for x² + 8x + 3.

(x + 4)² − 13.

Complete the square for 2x² − 12x + 5.

2(x − 3)² − 13.

What is a polynomial of degree 3 called?

A cubic.

State the factor theorem.

If f(a) = 0, then (x − a) is a factor of f(x).

Show that (x − 2) is a factor of x³ − 3x² + 4.

f(2) = 8 − 12 + 4 = 0, so (x − 2) is a factor.

Simplify a³ × a⁵ and (a²)⁴.

a⁸ and a⁸.

Evaluate 27^(2/3).

(∛27)² = 9.

Evaluate 16^(−1/2).

1/√16 = 1/4.

Give the gradient formula between (x₁, y₁) and (x₂, y₂).

m = (y₂ − y₁)/(x₂ − x₁).

Find the midpoint and length of the line from (1, 2) to (7, 10).

Midpoint (4, 6); length √(36 + 64) = 10.

Find the equation of the line through (2, −1) with gradient 3.

y + 1 = 3(x − 2), so y = 3x − 7.

How many radians are in 180°?

π radians.

Convert 60° and 270° to radians.

π/3 and 3π/2.

Convert 5π/6 radians to degrees.

150°.

State the identity linking tan θ, sin θ and cos θ.

tan θ = sin θ ÷ cos θ.

State the Pythagorean identity.

sin²θ + cos²θ = 1.

Give the exact values of sin 45°, cos 30° and tan 60°.

√2/2, √3/2 and √3.

What is a vector?

A quantity with both magnitude and direction.

Find the magnitude of 3i + 4j.

√(9 + 16) = 5.

Find a unit vector in the direction of 6i − 8j.

Magnitude 10, so 0.6i − 0.8j.

What does nCr (or (n r)) represent in the binomial expansion?

The binomial coefficient: n! ÷ (r!(n − r)!).

Expand (1 + x)⁴.

1 + 4x + 6x² + 4x³ + x⁴.

Expand (2 + x)³.

8 + 12x + 6x² + x³.

Differentiate y = axⁿ.

dy/dx = naxⁿ⁻¹.

Differentiate y = 3x⁴ − 2x + 7.

12x³ − 2.

Differentiate y = 4/x².

y = 4x⁻², so dy/dx = −8x⁻³ = −8/x³.

What is integration?

The reverse of differentiation; it also finds areas under curves.

Integrate xⁿ (n ≠ −1).

xⁿ⁺¹/(n + 1) + c.

Integrate 6x² − 4x + 3.

2x³ − 2x² + 3x + c.

How are displacement, velocity and acceleration related by calculus?

v = ds/dt and a = dv/dt; integrating goes the other way.

s = t³ − 3t² + 2t. Find the velocity at t = 2.

v = 3t² − 6t + 2 = 12 − 12 + 2 = 2 m/s.

v = 4t − t². Find the acceleration at t = 1.

a = 4 − 2t = 2 m/s².