A-Math 2020 Worked Solutions

Sum & product of roots

5 · algebra (sum & product of roots → new equation)

Laws of indices

5 marks: 3 + 2 · algebra (indices in prime factors; exponential growth)

Partial fractions (improper

5 · algebra (partial fractions, improper)

Quotient rule

4 · calculus (quotient rule; increasing function)

Modulus graph

4 · algebra (modulus graph; line intersection condition)

Stationary points

6 · calculus (stationary points; second-derivative test)

Reciprocal ratios (sec

7 · geometry & trigonometry (trig equation via identities)

Remainder theorem

7 marks: 3 + 4 · algebra (remainder & factor theorems)

Equation of a circle (general → centre/radius form)

7 marks: 4 + 3 · geometry & trigonometry (equation of a circle; tangent)

Solve cos 2x = k in range

9 marks: 3 + 6 · calculus + trigonometry (solve trig equation; exact area)

Find a constant via ln

10 marks: 1 + 6 + 3 · calculus (kinematics from a speed function)

Compound-angle expansion

11 marks: 2 + 1 + 2 + 1 + 2 + 3 · geometry & trigonometry (compound angles; no calculator)

R-formula

8 marks: 4 + 4 · geometry & trigonometry (r-formula)

Simultaneous (one quadratic)

9 marks: 3 + 3 + 3 · algebra (quadratics: simultaneous, sign condition, tangency)

General term ⁿCr

6 marks: 3 + 3 · algebra (binomial theorem)

Tangent-chord (alternate segment) theorem

5 · geometry & trigonometry (alternate segment theorem; cyclic quadrilateral)

Quadratic inequality on a number line

8 marks: 4 + 2 + 2 · algebra (quadratic inequality; power-function graphs & indices)

Linearise y = Aeᵏˣ as ln P = kx + ln A

9 marks: 2 + 4 + 3 · algebra (linear law; exponential model)

Product rule + chain rule

11 marks: 5 + 4 + 2 · calculus (product + chain rule; reverse to integrate; signed area)

Quadratic in eˣ

10 marks: 3 + 4 + 3 · algebra (exponential & logarithmic equations)

Distance formula

11 marks: 7 + 3 + 1 · geometry & trigonometry (coordinate geometry; parallelogram)

Integrate f″ twice

12 marks: 5 + 4 + 3 · calculus (integrate from the 2nd derivative; tangent)

Volume of similar cones

11 marks: 3 + 4 + 4 · calculus (connected rates of change)