Given that \(f(1) = 2\), show that \(k = 4\).

Write down the values of \(q\) and \(r\) for the following table.

As \(x\) increases from \( - 1\), the graph of \(y = f(x)\) reaches a maximum value and then decreases, behaving asymptotically.

Draw the graph of \(y = f(x)\) for \( - 1 \leqslant x \leqslant 8\). Use a scale of \({\text{1 cm}}\) to represent 1 unit on both axes. The position of the maximum, \({\text{M}}\), the \(y\)-intercept and the asymptotic behaviour should be clearly shown.

Using your graphic display calculator, find the coordinates of \({\text{M}}\), the maximum point on the graph of \(y = f(x)\).

Write down the equation of the horizontal asymptote to the graph of \(y = f(x)\).

(i) Draw and label the line \( y = 1\) on your graph.

(ii) The equation \(f(x) = 1\) has two solutions. One of the solutions is \(x = 4\). Use your graph to find the other solution.

Find \(C'(x)\).

Show that the cost per person is a minimum when \(10\) people are invited to the party.

Calculate the minimum cost per person.

\(f(1) = \frac{k}{{{2^1}}}\)     (M1)

Note: (M1) for substituting \(x = 1\) into the formula.

\(\frac{k}{2} = 2\)     (M1)

Note: (M1) for equating to 2.

\(k = 4\)     (AG)

[2 marks]

\(q = 2\), \(r = 0.125\)     (A1)(A1)

[2 marks]

     (A4)

Notes: (A1) for scales and labels.
(A1) for accurate smooth curve passing through \((0, 0)\) drawn at least in the given domain.
(A1) for asymptotic behaviour (curve must not go up or cross the \(x\)-axis).
(A1) for indicating the position of the maximum point.

[4 marks]

\({\text{M}}\) (\(1.44\), \(2.12\))     (G1)(G1)

Note: Brackets required, if missing award (G1)(G0). Accept \(x = 1.44\) and \(y = 2.12\).

[2 marks]

\(y = 0\)     (A1)(A1)

Note: (A1) for ‘\(y = \)’ provided the right hand side is a constant. (A1) for 0.

[2 marks]

(i) See graph     (A1)(A1)

Note: (A1) for correct line, (A1) for label.

(ii) \(x = 0.3\) (ft) from candidate’s graph.     (A2)(ft)

Notes: Accept \( \pm 0.1\) from their x. For \(0.310\) award (G1)(G0). For other answers taken from the GDC and not given correct to 3 significant figures award (G0)(AP)(G0) or (G1)(G0) if (AP) already applied.

[4 marks]

\(C'(x) = 1 - \frac{{100}}{{{x^2}}}\)     (A1)(A1)(A1)

Note: (A1) for 1, (A1) for \( - 100\) , (A1) for \({x^2}\) as denominator or \({{x^{ - 2}}}\) as numerator. Award a maximum of (A2) if an extra term is seen.

[3 marks]

For studying signs of the derivative at either side of \(x = 10\)     (M1)

For saying there is a change of sign of the derivative     (M1)(AG)

OR

For putting \(x = 10\) into \(C'\) and getting zero     (M1)

For clear sketch of the function or for mentioning that the function changes from decreasing to increasing at \(x = 10\)     (M1)(AG)

OR

For solving \(C'(x) = 0\) and getting \(10\)     (M1)

For clear sketch of the function or for mentioning that the function changes from decreasing to increasing at \(x = 10\)     (M1)(AG)

Note: For a sketch with a clear indication of the minimum or for a table with values of \(x\) at either side of \(x = 10\) award (M1)(M0).

[2 marks]

\(C(10) = 10 + \frac{{100}}{{10}}\)     (M1)

\(C(10) = 20\)     (A1)(G2)

[2 marks]

There were many well drawn graphs using correctly scaled and labelled axes with a good curve drawn. A number of students did not label the maximum point. Although many students showed in their graph the asymptotic behaviour of the curve, they did not know how to describe the asymptote. It was noticed that some students were tracing the curve to find the coordinates of the maximum instead of finding the maximum directly. The intersection between the line \(y = 1\) and the curve was not always read from their graph but from their GDC's graph.

There were many well drawn graphs using correctly scaled and labelled axes with a good curve drawn. A number of students did not label the maximum point. Although many students showed in their graph the asymptotic behaviour of the curve, they did not know how to describe the asymptote. It was noticed that some students were tracing the curve to find the coordinates of the maximum instead of finding the maximum directly. The intersection between the line \(y = 1\) and the curve was not always read from their graph but from their GDC's graph.

There were many well drawn graphs using correctly scaled and labelled axes with a good curve drawn. A number of students did not label the maximum point. Although many students showed in their graph the asymptotic behaviour of the curve, they did not know how to describe the asymptote. It was noticed that some students were tracing the curve to find the coordinates of the maximum instead of finding the maximum directly. The intersection between the line \(y = 1\) and the curve was not always read from their graph but from their GDC's graph.

There were many well drawn graphs using correctly scaled and labelled axes with a good curve drawn. A number of students did not label the maximum point. Although many students showed in their graph the asymptotic behaviour of the curve, they did not know how to describe the asymptote. It was noticed that some students were tracing the curve to find the coordinates of the maximum instead of finding the maximum directly. The intersection between the line \(y = 1\) and the curve was not always read from their graph but from their GDC's graph.

There were many well drawn graphs using correctly scaled and labelled axes with a good curve drawn. A number of students did not label the maximum point. Although many students showed in their graph the asymptotic behaviour of the curve, they did not know how to describe the asymptote. It was noticed that some students were tracing the curve to find the coordinates of the maximum instead of finding the maximum directly. The intersection between the line \(y = 1\) and the curve was not always read from their graph but from their GDC's graph.

There were many well drawn graphs using correctly scaled and labelled axes with a good curve drawn. A number of students did not label the maximum point. Although many students showed in their graph the asymptotic behaviour of the curve, they did not know how to describe the asymptote. It was noticed that some students were tracing the curve to find the coordinates of the maximum instead of finding the maximum directly. The intersection between the line \(y = 1\) and the curve was not always read from their graph but from their GDC's graph.

Finding the derivative was done at least partially correctly by most of the candidates. However, using it to find the minimum and to justify why it is a minimum was troublesome for the majority of the candidates. Even those who used a graph in their reasoning neglected to mention the change from decreasing to increasing or to supply a sign diagram. Many candidates recovered in the last part of the question when finding the minimum cost.

Finding the derivative was done at least partially correctly by most of the candidates. However, using it to find the minimum and to justify why it is a minimum was troublesome for the majority of the candidates. Even those who used a graph in their reasoning neglected to mention the change from decreasing to increasing or to supply a sign diagram. Many candidates recovered in the last part of the question when finding the minimum cost.

Finding the derivative was done at least partially correctly by most of the candidates. However, using it to find the minimum and to justify why it is a minimum was troublesome for the majority of the candidates. Even those who used a graph in their reasoning neglected to mention the change from decreasing to increasing or to supply a sign diagram. Many candidates recovered in the last part of the question when finding the minimum cost.