Iteration, growth and decay – Year 11 GCSE maths test answers to download as PDF

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1. Answer to Question 1 (4 marks)

   1. \(3500\times1.028^3\) (2) \(=\) £3 802.31 (1) 2.8% interest gives the multiplier 1.028, used once for each of the 3 years. Round to the nearest penny.
   2. \(3802.31-3500=\) £302.31 (1) The interest is the final amount minus the £3500 invested.

   A common error is to use simple interest, \(3500\times0.028\times3=294\), giving £3794, which earns only the method mark if the working is shown. Check: after year 1 the value is £3598, after year 2 £3698.74, and after year 3 £3802.31.

2. Answer to Question 2 (4 marks)

   1. \(24\,000\times0.85^4\) (2) \(=\) £12 528.15 (1) A 15% fall each year leaves 85% of the value, so the multiplier 0.85 is used 4 times.
   2. \(\dfrac{24000-12528.15}{24000}=\) 47.8% (1) Divide the loss by the original value and multiply by 100. The van has lost 47.8% of its value.

   The fall is not \(4\times15=60\%\), because each year's 15% is taken from a smaller value. A quicker method for (b): \(1-0.85^4=0.478\).

3. Answer to Question 3 (4 marks)

   1. \(67.20\div1.12=\) £60 (1 for the multiplier, 1 for the answer) The price after the increase is 112% of the original, so we divide by 1.12. Check: \(60\times1.12=67.20\).
   2. \(67.20\times0.88=£59.14\), which is not £60 (1); the 12% is taken from a larger amount (1) The 12% reduction is worked out on £67.20, which is bigger than £60, so it takes off more than the 12% increase added on. The final price is 64p below the original.

4. Answer to Question 4 (4 marks)

   1. 1.651, 1.669, 1.671 (1 each) Start with \(x_0=1.5\) and put each answer back into the formula: \(\sqrt[3]{1.5+3}=\sqrt[3]{4.5}\), and so on. Use the full calculator value each time.
   2. \(f(x)=x^3-x-3\): \(f(1.6)\lt 0\) and \(f(1.7)>0\) (1) \(f(1.6)=-0.504\) and \(f(1.7)=0.213\). The sign changes, so a root lies between 1.6 and 1.7.

   In (a), each step uses the previous answer, so do not round between steps. The values settle near 1.672, which is the root of \(x^3=x+3\).

5. Answer to Question 5 (4 marks)

   1. \(800\times1.06^3=\) £952.81 (2) 6% compound interest gives the multiplier 1.06, used 3 times. Hana has £952.81.
   2. Brother: \(800+3\times48=\) £944 (1); difference £8.81 (1) Simple interest is 6% of £800, which is £48 each year. Compound interest earns £8.81 more because interest is also paid on interest.

   In (a), the interest is £48 in year 1, £50.88 in year 2 and £53.93 in year 3. Simple interest is always £48.

Total : 20 marks