A-Math 2021 Specimen Worked Solutions

Axis intercepts

5 marks · coordinate geometry (intercepts; gradient and length)

Range of a+b kx

5 marks: 2 + 3 · trigonometry (range and graph of a transformed sine)

Binomial general term

6 marks: 3 + 3 · algebra (binomial expansion; term independent of x)

Quotient rule

6 marks: 4 + 2 · calculus (increasing function; related rates)

Fit e^kt to a data point

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

Integrate x

6 marks · calculus (integrate a second derivative twice)

Completing the square

7 marks: 4 + 3 · algebra (completing the square; ranges and non-intersection)

Compound-angle identity

7 marks: 2 + 5 · trigonometry (exact values; no calculator)

Similar triangles to relate x and y

7 marks: 3 + 4 · calculus (inscribed rectangle; maximise area)

Factor theorem

8 marks: 4 + 4 · algebra (factor and remainder theorems; cubic)

Isosceles base angles

8 marks: 3 + 5 · geometry (circle, tangent-chord, isosceles and parallel chords)

Prove an identity via /

9 marks: 4 + 3 + 2 · trigonometry (prove identity; solve; show no solution)

Substitute initial/final conditions for constants

10 marks: 3 + 4 + 3 · calculus (kinematics: velocity, acceleration, distance)

Improper rational function

6 marks · algebra (partial fractions; improper rational function)

Log-linearising yx^n=k

10 marks: 3 + (2 + 2 + 3) · algebra (linear law; log-linearising and physical model)

Tangent at a point

10 marks: 5 + 5 · calculus (tangent to a curve; area between curve and line)

Line-and-curve non-intersection via discriminant

10 marks: 5 + (3 + 2) · algebra (non-intersection / discriminant; always-negative quadratic)

Algebraic division of a linear fraction

10 marks: 4 + 3 + 3 · calculus (algebraic division; product rule; integration by reversing)

Resolve a perpendicular distance into a - b

9 marks: 3 + 4 + 2 · trigonometry (modelling a perpendicular distance; r-formula)

Chain rule on e^f(x)

11 marks: 5 + 4 + 2 · calculus (chain rule on e^f(x); stationary point and nature)

Midpoint

12 marks: 5 + 5 + 2 · coordinate geometry (perpendicular bisector; equation of a circle; diameter)

Substitution u=2^x to a quadratic

12 marks: 4 + 4 + 4 · algebra (exponential equation by substitution; logarithm laws; no-solution argument)