A-Math 2021 Worked Solutions

Rationalising a denominator

3 marks · algebra (surds: rationalising a quotient)

Simultaneous equations (one non-linear)

3 marks · algebra (curve and line intersection)

Completing the square

4 marks · algebra (completing the square; turning point)

Integrate x^-2

4 marks · calculus (integration of x^-2 and 1/ax+b)

Partial fractions

6 marks · algebra (partial fractions with a repeated factor)

Remainder theorem

6 marks: 2 + 4 · algebra (remainder & factor theorem)

Amplitude

6 marks: 2 + 2 + 2 · trigonometry (amplitude, period, midline of a sine curve)

Area as rectangle minus corner triangles

7 marks: 3 + 4 · calculus (area model; stationary value)

Axis of symmetry of a kite

9 marks: 3 + 4 + 2 · coordinate geometry (kite)

Prove a trig identity

8 marks: 4 + 4 · trigonometry (prove an identity, then apply it)

Alternate segment theorem

7 marks: 4 + 3 · geometry (tangents, alternate segment theorem)

Index laws

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

Related rates with dV/dt

9 marks: (2 + 3) + 4 · calculus (connected & related rates of change)

Chain rule for (4+3x)^1/2

10 marks: 5 + 5 · calculus (normal to a curve; area by integration)

Substitution u=e^x

5 marks · algebra (exponential equation; quadratic in e^x)

Factor theorem

5 marks · algebra (factor theorem; cubic)

Product/chain rule

8 marks: 4 + 4 · calculus (differentiation then integration)

Resolve vertical components

9 marks: 2 + 4 + 3 · trigonometry (r-formula; maximum height)

Quadratic inequality

9 marks: 4 + 3 + 2 · algebra (quadratic inequality; tangency by discriminant)

Binomial general term

9 marks: 4 + 2 + 3 · algebra (binomial; coefficients in a product)

Linearise d=av^2+bv via d/v

11 marks: 4 + (4 + 2 + 1) · algebra (linear law; exponential model)

Solve exponential equation

10 marks: 3 + 3 + 4 · calculus (kinematics; exponential velocity)

Equation of a circle

12 marks: 3 + 2 + 1 + 6 · coordinate geometry (circle; tangents)

Stationary point of a trig curve

12 marks · calculus (area between a curve and a chord)