Cambridge International AS & A Level Mathematics: every topic, by paper

Give the quadratic formula.

x = (−b ± √(b² − 4ac)) ÷ 2a.

What does the discriminant b² − 4ac tell you?

> 0: two distinct real roots; = 0: one repeated root; < 0: no real roots.

Complete the square for x² + 6x + 11.

(x + 3)² + 2.

What is a function?

A mapping in which each input has exactly one output.

Define the domain and range of a function.

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

If f(x) = 2x + 1 and g(x) = x², find fg(x).

fg(x) = f(x²) = 2x² + 1.

Give the gradient of the line through (x₁, y₁) and (x₂, y₂).

m = (y₂ − y₁) ÷ (x₂ − x₁).

Give the equation of a line with gradient m through (x₁, y₁).

y − y₁ = m(x − x₁).

What is the condition for two lines to be perpendicular?

m₁ × m₂ = −1.

How many radians are in 180°?

π.

Convert 150° to radians.

150 × π ÷ 180 = 5π ÷ 6.

Give the arc length of a sector with radius r and angle θ radians.

s = rθ.

Give the identity linking sin, cos and tan.

tan θ = sin θ ÷ cos θ.

Give the Pythagorean identity.

sin²θ + cos²θ = 1.

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

√3/2, ½ and √3.

Give the nth term of an arithmetic progression.

uₙ = a + (n − 1)d.

Give the sum of the first n terms of an AP.

Sₙ = ½n(2a + (n − 1)d).

Give the nth term of a geometric progression.

uₙ = arⁿ⁻¹.

Differentiate xⁿ.

nxⁿ⁻¹.

Differentiate y = 3x⁴ − 2√x + 5/x.

dy/dx = 12x³ − x^(−½) − 5x⁻².

What does dy/dx = 0 identify?

Stationary points.

Integrate xⁿ (n ≠ −1).

xⁿ⁺¹ ÷ (n + 1) + c.

Integrate 6x² − 4x + 3.

2x³ − 2x² + 3x + c.

What does a definite integral represent geometrically?

The signed area between the curve and the x-axis.

State the factor theorem.

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

State the remainder theorem.

The remainder when f(x) is divided by (x − a) is f(a).

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

f(2) = 8 − 12 + 4 = 0.

What does logₐ b = c mean?

aᶜ = b.

State the three laws of logarithms.

log(xy) = log x + log y; log(x/y) = log x − log y; log xⁿ = n log x.

Solve 2ˣ = 20.

x = log 20 ÷ log 2 ≈ 4.32.

Define sec x, cosec x and cot x.

sec x = 1/cos x, cosec x = 1/sin x, cot x = 1/tan x.

Give the two identities from dividing sin²x + cos²x = 1.

1 + tan²x = sec²x; 1 + cot²x = cosec²x.

Give sin(A + B) and cos(A + B).

sin A cos B + cos A sin B; cos A cos B − sin A sin B.

Differentiate xⁿ.

nxⁿ⁻¹.

Differentiate y = 3x⁴ − 2√x + 5/x.

dy/dx = 12x³ − x^(−½) − 5x⁻².

What does dy/dx = 0 identify?

Stationary points.

Integrate xⁿ (n ≠ −1).

xⁿ⁺¹ ÷ (n + 1) + c.

Integrate 6x² − 4x + 3.

2x³ − 2x² + 3x + c.

What does a definite integral represent geometrically?

The signed area between the curve and the x-axis.

What does a change of sign show?

If f(a) and f(b) have opposite signs and f is continuous, a root lies between a and b.

Show a root of x³ − x − 1 = 0 lies between 1 and 2.

f(1) = −1 and f(2) = 5; a sign change and f continuous, so a root lies between.

What is an iterative formula?

xₙ₊₁ = g(xₙ), rearranged from f(x) = 0, used repeatedly to approach a root.

Find the magnitude of 3i + 4j.

√(9 + 16) = 5.

Find the unit vector in the direction of 2i − j + 2k.

Magnitude 3, so ⅓(2i − j + 2k).

Give the scalar product a · b in two forms.

|a||b| cos θ = a₁b₁ + a₂b₂ + a₃b₃.

What is a first-order differential equation?

An equation involving a first derivative, such as dy/dx = f(x, y).

What does separating the variables mean?

Rearranging to g(y) dy = f(x) dx, then integrating both sides.

Solve dy/dx = 2xy given y = 1 when x = 0.

∫dy/y = ∫2x dx: ln y = x² + c; c = 0, so y = e^(x²).

Define i.

i = √(−1), so i² = −1.

Simplify (3 + 2i)(1 − i).

3 − 3i + 2i − 2i² = 5 − i.

What is the complex conjugate of a + bi?

a − bi.

What does the gradient of a displacement–time graph give?

Velocity.

What does the area under a velocity–time graph give?

Displacement.

List the constant-acceleration equations.

v = u + at; s = ut + ½at²; s = vt − ½at²; v² = u² + 2as; s = ½(u + v)t.

State Newton's second law.

F = ma, where F is the resultant force.

What is the weight of a 5 kg mass (g = 10)?

50 N.

What is the normal reaction on a 4 kg mass on a horizontal floor (g = 10)?

40 N, if no other vertical forces act.

Define momentum.

Mass × velocity (N s or kg m s⁻¹).

State the principle of conservation of momentum.

In a collision with no external forces, total momentum before equals total momentum after.

Define impulse.

Change in momentum: I = mv − mu = Ft for a constant force.

Define work done by a constant force.

Force × distance moved in the direction of the force.

Give the formulas for kinetic and gravitational potential energy.

KE = ½mv²; GPE = mgh.

Define power.

Rate of doing work; P = Fv for a vehicle moving at speed v.

Distinguish discrete and continuous data.

Discrete data take specific values (counts); continuous data can take any value in a range.

How is frequency density calculated for a histogram?

Frequency ÷ class width.

How is the median found from grouped data?

By interpolation or from a cumulative frequency graph at n/2.

What is the probability of the complement of A?

P(A′) = 1 − P(A).

State the addition rule.

P(A ∪ B) = P(A) + P(B) − P(A ∩ B).

When are events mutually exclusive?

When they cannot happen together: P(A ∩ B) = 0.

What must the probabilities of a discrete random variable sum to?

1.

Give E(X) for a discrete random variable.

E(X) = Σ x P(X = x).

Give Var(X).

Var(X) = E(X²) − [E(X)]².

Describe the shape of a normal distribution.

Symmetrical and bell-shaped about the mean.

How do you standardise X ~ N(μ, σ²)?

Z = (X − μ) ÷ σ, which follows N(0, 1).

What percentage of data lies within 1, 2 and 3 standard deviations of the mean?

About 68%, 95% and 99.7%.

What is a null hypothesis?

The assumption of no change, e.g. H₀: p = 0.5, tested against an alternative H₁.

Distinguish one-tailed and two-tailed tests.

One-tailed: H₁ specifies a direction (p > or <). Two-tailed: H₁ is p ≠ value.

Define significance level.

The probability of rejecting H₀ when it is true (the threshold, e.g. 5%).