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

For ax² + bx + c = 0 with roots α, β, give α + β and αβ.

α + β = −b/a; αβ = c/a.

For ax³ + bx² + cx + d = 0, give Σα, Σαβ and αβγ.

Σα = −b/a; Σαβ = c/a; αβγ = −d/a.

Express α² + β² in terms of α + β and αβ.

(α + β)² − 2αβ.

What is a vertical asymptote of y = f(x) ÷ g(x)?

A line x = a where g(a) = 0 (and f(a) ≠ 0).

Find the asymptotes of y = (2x + 1) ÷ (x − 3).

Vertical x = 3; horizontal y = 2.

When does a rational function have an oblique asymptote?

When the numerator's degree is one more than the denominator's.

Give Σr from r = 1 to n.

½n(n + 1).

Give Σr² from r = 1 to n.

⅙n(n + 1)(2n + 1).

Give Σr³ from r = 1 to n.

¼n²(n + 1)².

When can matrices A and B be multiplied as AB?

When the number of columns of A equals the number of rows of B.

Find the determinant of (3 2 ; 1 4).

3 × 4 − 2 × 1 = 10.

Give the inverse of a 2 × 2 matrix (a b ; c d).

(1 ÷ (ad − bc)) (d −b ; −c a).

How are polar coordinates (r, θ) converted to Cartesian?

x = r cos θ, y = r sin θ.

Convert Cartesian (x, y) to polar.

r = √(x² + y²), tan θ = y/x (choosing θ by quadrant).

Give the Cartesian equation of r = 4.

x² + y² = 16, a circle of radius 4.

What does the vector product a × b give?

A vector perpendicular to both a and b, with magnitude |a||b| sin θ.

Compute (1, 0, 0) × (0, 1, 0).

(0, 0, 1), i.e. i × j = k.

Give the vector equation of a plane in scalar-product form.

r · n = a · n, where n is a normal vector and a a point on the plane.

Define sinh x and cosh x.

sinh x = ½(eˣ − e⁻ˣ); cosh x = ½(eˣ + e⁻ˣ).

Define tanh x.

sinh x ÷ cosh x = (eˣ − e⁻ˣ) ÷ (eˣ + e⁻ˣ).

State the identity linking cosh and sinh.

cosh²x − sinh²x = 1.

When can matrices A and B be multiplied as AB?

When the number of columns of A equals the number of rows of B.

Find the determinant of (3 2 ; 1 4).

3 × 4 − 2 × 1 = 10.

Give the inverse of a 2 × 2 matrix (a b ; c d).

(1 ÷ (ad − bc)) (d −b ; −c a).

Differentiate sin⁻¹x.

1/√(1 − x²).

Differentiate tan⁻¹x.

1/(1 + x²).

Integrate 1/(a² + x²).

(1/a) tan⁻¹(x/a) + c.

Give the general Maclaurin series.

f(x) = f(0) + f′(0)x + f″(0)x²/2! + f‴(0)x³/3! + …

Give the Maclaurin series for eˣ.

1 + x + x²/2! + x³/3! + …, valid for all x.

Give the Maclaurin series for sin x.

x − x³/3! + x⁵/5! − …

Write z = 1 + i√3 in modulus–argument form.

2(cos π/3 + i sin π/3).

State de Moivre's theorem.

(r(cos θ + i sin θ))ⁿ = rⁿ(cos nθ + i sin nθ).

Compute (1 + i)⁸.

(√2)⁸ (cos 2π + i sin 2π) = 16.

What is an integrating factor for dy/dx + P(x)y = Q(x)?

e^(∫P dx).

Solve dy/dx + 2y = 0.

y = Ae^(−2x).

Solve dy/dx + y/x = x (x > 0).

IF = x: d(xy)/dx = x², so xy = x³/3 + c and y = x²/3 + c/x.

Give the horizontal and vertical displacement of a projectile launched at speed u, angle θ.

x = ut cos θ; y = ut sin θ − ½gt².

Give the time of flight of a projectile on level ground.

T = 2u sin θ ÷ g.

Give the range on level ground.

R = u² sin 2θ ÷ g, maximum at θ = 45°.

What is a probability density function?

A function f(x) ≥ 0 for a continuous random variable, with total area under it equal to 1.

Give E(X) for a continuous random variable.

∫ x f(x) dx over the range.

Give Var(X) for a continuous random variable.

∫ x² f(x) dx − μ².