A-Math 2022 Worked Solutions

Completing the square

6 marks: 2 + 4 · algebra (completing the square; line meets curve)

Linear law (linearise m = Ae^-kt)

6 marks: 2 + 2 + 2 · algebra (linear law; exponential model)

Division to a constant quotient

6 marks: 1 + 5 · algebra (polynomial division; partial fractions)

Evaluate a function at parametrised points

5 marks: 3 + 2 · calculus (gradient comparison on a parabola)

Binomial general term

8 marks: 4 + 2 + 2 · algebra (binomial expansion in a product)

Integrate e^-x and e^2x

7 marks · calculus (integrate a second derivative twice)

Equal tangents from an external point

6 marks: 3 + 3 · geometry (tangents, alternate segment, angle in semicircle)

Substitution u = root(x)

8 marks: 4 + 4 · algebra (curve and line; tangency)

Always-increasing condition (discriminant < 0)

7 marks: 3 + 4 · algebra & calculus (always increasing; factor theorem)

Amplitude and period of a x and a bx

6 marks: (1 + 2) + 3 · trigonometry (amplitude, period, sketch)

Area of a triangle from a horizontal base

7 marks: 3 + 4 · coordinate geometry (area and length conditions)

Amplitude and period of a sinusoidal model

8 marks: 3 + 3 + 2 · calculus (kinematics with a sinusoidal model)

Quotient rule

10 marks: 3 + 5 + 2 · calculus (no stationary point; normal; range of a rational function)

Exponential decay model

6 marks: 4 + 2 · algebra (exponential-decay model)

Factor theorem

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

Double-angle identities

8 marks: 4 + 4 · trigonometry (prove identity, then solve)

Express SA via conservation of volume

10 marks: 4 + 4 + 2 · calculus (surface-area optimisation)

Compound-angle for 75°

8 marks: 4 + 2 + 2 · trigonometry (exact values; no calculator)

Substitution u = 2^x

9 marks: 5 + (2 + 2) · algebra (exponential equation; curve intercepts)

Resolve perpendiculars

9 marks: 3 + 4 + 2 · trigonometry (r-formula in context)

Product & chain rule

10 marks: 4 + 2 + 4 · calculus (product/chain rule; connected rate; integral via a "show that")

Equation of a circle (complete the square)

11 marks: 4 + 5 + 2 · coordinate geometry (circle)

Tangent line via chain rule

12 marks: 6 + 6 · calculus (tangent, intercepts; area between curve and tangent)