Write down the equation of the vertical asymptote.

Find \(f ′(x)\).

Using your graphic display calculator or otherwise, write down the coordinates of any point where the graph of \(y = f (x)\) has zero gradient.

Write down all intervals in the given domain for which \(f (x)\) is increasing.

Using your graphic display calculator or otherwise, find the value of a.

Find \(\frac{{dy}}{{dx}}\).

(i) Use your answer to part (b) to calculate the horizontal distance the ball has travelled from A when its height is a maximum.

(ii) Find the maximum vertical height reached by the football.

Draw a graph showing the path of the football from the point where it is kicked to the point where it hits the ground again. Use 1 cm to represent 5 m on the horizontal axis and 1 cm to represent 2 m on the vertical scale.

The goal posts are 35 m from the point where the ball is kicked.

At what height does the ball pass over the goal posts?

equation of asymptote is x = 0     (A1)

(Must be an equation.)

[1 mark]

\(f '(x) = 2x + 3x^{-2}\)     (or equivalent)     (A1) for each term     (A1)(A1)

[2 marks]

stationary point (–1.14, 3.93)     (G1)(G1)(ft)

(-1,4) or similar error is awarded (G0)(G1)(ft). Here and also as follow through in part (d) accept exact values \( - {\left( {\frac{3}{2}} \right)^{\frac{1}{3}}}\)for the x coordinate and \(3{\left( {\frac{3}{2}} \right)^{\frac{2}{3}}}\) for the y coordinate.

OR \(2x + \frac{3}{{{x^2}}} = 0\) or equivalent

Correct coordinates as above     (M1)

Follow through from candidate’s \(f ′(x)\).     (A1)(ft)

[2 marks]

In all alternative answers for (d), follow through from candidate’s x coordinate in part (c).

Alternative answers include:

–1.14 ≤ x < 0,     0 < x < 5     (A1)(A1)(ft)(A1)

OR [–1.14,0), (0,5)

Accept alternative bracket notation for open interval ] [. (Union of these sets is not correct, award (A2) if all else is right in this case.)

OR \( - 1.14 \leqslant x < 5,x \ne 0\)

In all versions 0 must be excluded (A1). -1.14 must be the left bound . 5 must be the right bound (A1). For \(x \geqslant - 1.14\) or \(x > - 1.14\) alone, award (A1). For \( - 1.4 \leqslant x < 0\) together with \(x > 0\) award (A2).

[3 marks]

a = 5.30 (3sf)   (Allow (5.30, 0) but 5.3 receives an (AP).)     (A1)

[1 mark]

\(\frac{{{\text{d}}y}}{{{\text{d}}x}} = - 0.042x + 1.245\)     (A1) for each term.     (A1)(A1)

[2 marks]

Unit penalty (UP) is applicable where indicated in the left hand column.

(i) Maximum value when \(f ' (x) = 0\), \( - 0.042x + 1.245 = 0\),     (M1)

(M1) is for either of the above but at least one must be seen.

(x = 29.6.)

Football has travelled 29.6 – 5.30 = 24.3 m (3sf) horizontally.     (A1)(ft)

For answer of 24.3 m with no working or for correct subtraction of 5.3 from candidate’s x-coordinate at the maximum (if not 29.6), award (A1)(d).

(UP) (ii) Maximum vertical height, f (29.6) = 12.4 m     (M1)(A1)(ft)(G2)

(M1) is for substitution into f of a value seen in part (c)(i). f(24.3) with or without evaluation is awarded (M1)(A0). For any other value without working, award (G0). If lines are seen on the graph in part (d) award (M1) and then (A1) for candidate’s value \( \pm 0.5\) (3sf not required.)

[4 marks]

(not to scale)

     (A1)(A1)(A1)(ft)(A1)(ft)

Award (A1) for labels (units not required) and scale, (A1)(ft) for max(29.6,12.4), (A1)(ft) for x-intercepts at 5.30 and 53.9, (all coordinates can be within 0.5), (A1) for well-drawn parabola ending at the x-intercepts.

[4 marks]

Unit penalty (UP) is applicable where indicated in the left hand column.

(UP) f (40.3) = 10.1 m (3sf).

Follow through from (a). If graph used, award (M1) for lines drawn and (A1) for candidate’s value \( \pm 0.5\). (3sf not required).     (M1)(A1)(ft)(G2)

[2 marks]

(i) An attempt at part (a) was seen only rarely. If there was an attempt, it was often not a meaningful equation. If an equation was seen, sometimes it was for y, not x.

The derivative seemed manageable for many, though with the expected mis-handling of the negative power quite often. Parts (c) and (d) proved problematical. Marking of (d) was lenient and it was reaffirmed that testing of the concept in (d) will be done in a more straightforward context in future, when done at all.

The derivative seemed manageable for many, though with the expected mis-handling of the negative power quite often. Parts (c) and (d) proved problematical. Marking of (d) was lenient and it was reaffirmed that testing of the concept in (d) will be done in a more straightforward context in future, when done at all.

The derivative seemed manageable for many, though with the expected mis-handling of the negative power quite often. Parts (c) and (d) proved problematical. Marking of (d) was lenient and it was reaffirmed that testing of the concept in (d) will be done in a more straightforward context in future, when done at all.

(ii) Many candidates failed to recognise that extensive use of the GDC was intended for this question. An indicator of this was the choice of awkward coefficients. It is recognised that the context confused some candidates and that the horizontal shift was a bit disturbing for some.

Nevertheless, a lot of candidates could have earned more marks here if they had persevered. Many gave up on the graph, and elementary marks for scale and labels were lost unnecessarily.

As this was the first time for the unit penalty, we were lenient about the units left off the labels but this is likely to change in the future.

(ii) Many candidates failed to recognise that extensive use of the GDC was intended for this question. An indicator of this was the choice of awkward coefficients. It is recognised that the context confused some candidates and that the horizontal shift was a bit disturbing for some.

Nevertheless, a lot of candidates could have earned more marks here if they had persevered. Many gave up on the graph, and elementary marks for scale and labels were lost unnecessarily.

As this was the first time for the unit penalty, we were lenient about the units left off the labels but this is likely to change in the future.

(ii) Many candidates failed to recognise that extensive use of the GDC was intended for this question. An indicator of this was the choice of awkward coefficients. It is recognised that the context confused some candidates and that the horizontal shift was a bit disturbing for some.

Nevertheless, a lot of candidates could have earned more marks here if they had persevered. Many gave up on the graph, and elementary marks for scale and labels were lost unnecessarily.

As this was the first time for the unit penalty, we were lenient about the units left off the labels but this is likely to change in the future.

(ii) Many candidates failed to recognise that extensive use of the GDC was intended for this question. An indicator of this was the choice of awkward coefficients. It is recognised that the context confused some candidates and that the horizontal shift was a bit disturbing for some.

Nevertheless, a lot of candidates could have earned more marks here if they had persevered. Many gave up on the graph, and elementary marks for scale and labels were lost unnecessarily.

As this was the first time for the unit penalty, we were lenient about the units left off the labels but this is likely to change in the future.

(ii) Many candidates failed to recognise that extensive use of the GDC was intended for this question. An indicator of this was the choice of awkward coefficients. It is recognised that the context confused some candidates and that the horizontal shift was a bit disturbing for some.

Nevertheless, a lot of candidates could have earned more marks here if they had persevered. Many gave up on the graph, and elementary marks for scale and labels were lost unnecessarily.

As this was the first time for the unit penalty, we were lenient about the units left off the labels but this is likely to change in the future.