A-Math 2024 Worked Solutions

Connected rates of change

3 marks · calculus (connected rates of change)

Chain rule on a surd

4 marks: 2 + 2 · calculus (chain rule, then reverse to integrate)

Compound-angle formula for ( + )

5 marks · trigonometry (compound-angle tangent)

Intersection of a line and a circle

6 marks · coordinate geometry (line meets curve, midpoint)

Term independent of x

7 marks: 4 + 3 · algebra (binomial: independent term, then a coefficient)

Angle in a semicircle

6 marks · geometry (alternate segment theorem, tangents, semicircle)

Rationalising a surd denominator

7 marks · algebra (surds in mensuration; no calculator)

Form a model with a constraint

7 marks: 3 + 4 · calculus (form a volume model, then maximise)

Completing the square

8 marks: 2 + 2 + 4 · algebra (completing the square; quadratic conditions)

Amplitude and period

8 marks: 2 + 3 + 3 · trigonometry (graph of a cosine; solve by drawing a line)

Resolve a length into components

9 marks: 2 + 5 + 2 · trigonometry (r-formula, maximum, solve)

Partial fractions with a repeated linear factor

10 marks: 5 + 5 · algebra / calculus (partial fractions, then integrate)

Tangent and normal

10 marks: 4 + 6 · calculus (tangent and normal, area)

Exponential-decay model

4 marks · algebra (exponential-decay model)

Line-and-curve intersection

5 marks · algebra (line meets curve, discriminant)

Change of base

5 marks · algebra (logarithms, change of base)

Quotient rule

7 marks: 3 + 1 + 3 · calculus (quotient rule; integrate ^2)

Stationary point (derivative =0)

6 marks: 2 + 4 · algebra / calculus (stationary point; exponential equation)

Factor theorem (two simultaneous conditions)

9 marks: 4 + 5 · algebra (factor theorem; cubic roots in surd form)

Circle from a diameter

9 marks: 3 + 2 + 1 + 3 · coordinate geometry (circle, closest point on a line)

Interpret constants from initial/boundary conditions

9 marks: 3 + 2 + 4 · calculus (kinematics: speed model)

Prove a double-angle identity

12 marks: 4 + 6 + 2 · trigonometry (double-angle identity; solve; no-solution range)

Linear law R vs v

12 marks: 4 + 2 + 2 + 4 · coordinate geometry (linear law)

Product rule with e^-2x and x

12 marks: 2 + 5 + 5 · calculus (product rule, second derivative, verify an equation)