Hint

Divide the quadratic-coefficient equation by the square of the linear-coefficient equation to eliminate a.

Worked solution

(a)

The linear coefficient is na. The quadratic coefficient is \(\displaystyle \binom{n}{2}a^{2}= \frac{n(n- 1)a^{2}}{2}\). Equating these with the supplied values gives the two equations.

Answer: \(na=12\) and \(\displaystyle \frac{n(n- 1)a^{2}}{2}= 60\).

(b)

Since \(\displaystyle na=12\ne 0\), division is valid. It gives \(\displaystyle \frac{n- 1}{2n}= \frac{60}{144}= \frac{5}{12}\). Thus \(\displaystyle 6(n-1)=5n\), so \(\displaystyle n=6\) and \(\displaystyle a= \frac{12}{6}= 2\). The integer condition holds and the coefficients check as 12 and \(\displaystyle 15(4)=60\).

Answer: \(n=6,\ a=2\).

(c)

The coefficients are \(\binom64 2^4=240\) and \(\binom63 2^3=160\). Their ratio in the requested order is \(\displaystyle \frac{240}{160}= \frac{3}{2}\).

Answer: \(\displaystyle \frac{3}{2}\).

Maths4U indicative marking guidance

Maths4U indicative marking guidance, not an official examination-board mark scheme. Award stated criteria once each, up to the part maximum. Accept valid equivalent methods. Follow-through applies only where explicitly stated; do not award an accuracy criterion for an incorrect final result.

(a) · 2 marks

• 1 mark: Give \(\displaystyle na=12\).
• 1 mark: Give \(\displaystyle \frac{n(n- 1)a^{2}}{2}= 60\).

Alternative methods: Equivalent equations using binomial notation are accepted.

Common error: Using \(\displaystyle n^{2}a^{2}\) for the quadratic coefficient.

(b) · 3 marks

• 1 mark: Eliminate a using a valid division or substitution.
• 1 mark: Obtain the admissible integer \(\displaystyle n=6\).
• 1 mark: Obtain \(\displaystyle a=2\) and check consistency.

Alternative methods: Substitute \(\displaystyle a= \frac{12}{n}\) into the second equation; \(\displaystyle n\ge 2\) ensures division by n is valid.

Common error: Accepting a value of n without checking the integer and lower-bound conditions.

(c) · 2 marks

• 1 mark: Obtain the two correct coefficients, or their correct consecutive-coefficient ratio.
• 1 mark: Give \(\displaystyle \frac{3}{2}\) in the requested order.

Alternative methods: Use \(\displaystyle \frac{C(n, 4)}{C}(n, 3)= \frac{n- 3}{4}\) and multiply by a, giving \(\displaystyle \frac{3\cdot 2}{4}\).

Common error: Reversing the requested ratio.

Total: 7 marks.