André wants to get from point A located in the sea to point Y located on a straight stretch of beach. P is the point on the beach nearest to A such that AP = 2 km and PY = 2 km. He does this by swimming in a straight line to a point Q located on the beach and then running to Y.

When André swims he covers 1 km in $5\sqrt 5$ minutes. When he runs he covers 1 km in 5 minutes.

(a)     If PQ = x km, $0 \leqslant x \leqslant 2$ , find an expression for the time T minutes taken by André to reach point Y.

(b)     Show that $\frac{{{\text{d}}T}}{{{\text{d}}x}} = \frac{{5\sqrt 5 x}}{{\sqrt {{x^2} + 4} }} - 5$.

(c)     (i)     Solve $\frac{{{\text{d}}T}}{{{\text{d}}x}} = 0$.

(ii)     Use the value of x found in part (c) (i) to determine the time, T minutes, taken for André to reach point Y.

(iii)     Show that $\frac{{{{\text{d}}^2}T}}{{{\text{d}}{x^2}}} = \frac{{20\sqrt 5 }}{{{{({x^2} + 4)}^{\frac{3}{2}}}}}$ and hence show that the time found in part (c) (ii) is a minimum.

(a)     ${\text{AQ}} = \sqrt {{x^2} + 4} {\text{ (km)}}$     (A1)

${\text{QY}} = (2 - x){\text{ (km)}}$     (A1)

$T = 5\sqrt 5 {\text{AQ}} + 5{\text{QY}}$     (M1)

${\text{ = 5}}\sqrt 5 \sqrt {({x^2} + 4)} + 5(2 - x){\text{ (mins)}}$     A1

[4 marks]

(b)     Attempting to use the chain rule on ${\text{5}}\sqrt 5 \sqrt {({x^2} + 4)}$     (M1)

$\frac{{\text{d}}}{{{\text{d}}x}}\left( {{\text{5}}\sqrt 5 \sqrt {({x^2} + 4)} } \right) = 5\sqrt 5 \times \frac{1}{2}{({x^2} + 4)^{ - \frac{1}{2}}} \times 2x$     A1

$\left( { = \frac{{5\sqrt 5 x}}{{\sqrt {{x^2} + 4} }}} \right)$

$\frac{{\text{d}}}{{{\text{d}}x}}\left( {5(2 - x)} \right) = - 5$     A1

$\frac{{{\text{d}}T}}{{{\text{d}}x}} = \frac{{5\sqrt 5 x}}{{\sqrt {{x^2} + 4} }} - 5$     AG     N0

[3 marks]

(c)     (i)     $\sqrt 5 x = \sqrt {{x^2} + 4}$     A1

Squaring both sides and rearranging to obtain $5{x^2} = {x^2} + 4$     M1

x = 1     A1     N1

Note: Do not award the final A1 for stating a negative solution in final answer.

(ii)     $T = 5\sqrt 5 \sqrt {1 + 4}  + 5(2 - 1)$     M1

= 30 (mins)     A1     N1

Note: Allow FT on incorrect x value.

(iii)     METHOD 1

Attempting to use the quotient rule     M1

$u = x{\text{ , }}v = \sqrt {{x^2} + 4} {\text{, }}\frac{{{\text{d}}u}}{{{\text{d}}x}} = 1{\text{ and }}\frac{{{\text{d}}v}}{{{\text{d}}x}} = x{({x^2} + 4)^{ - 1/2}}$     (A1)

$\frac{{{{\text{d}}^2}T}}{{{\text{d}}{x^2}}} = 5\sqrt 5 \left[ {\frac{{\sqrt {{x^2} + 4} - \frac{1}{2}{{({x^2} + 4)}^{ - 1/2}} \times 2{x^2}}}{{({x^2} + 4)}}} \right]$     A1

Attempt to simplify     (M1)

$= \frac{{5\sqrt 5 }}{{{{({x^2} + 4)}^{3/2}}}}[{x^2} + 4 - {x^2}]\,\,\,\,\,$or equivalent     A1

$= \frac{{20\sqrt 5 }}{{{{({x^2} + 4)}^{3/2}}}}$     AG

When $x = 1{\text{ , }}\frac{{20\sqrt 5 }}{{{{({x^2} + 4)}^{3/2}}}} > 0$ and hence T = 30 is a minimum     R1     N0

Note: Allow FT on incorrect x value, $0 \leqslant x \leqslant 2$.

METHOD 2

Attempting to use the product rule     M1

$u = x{\text{ , }}v = \sqrt {{x^2} + 4} {\text{, }}\frac{{{\text{d}}u}}{{{\text{d}}x}} = 1{\text{ and }}\frac{{{\text{d}}v}}{{{\text{d}}x}} = x{({x^2} + 4)^{ - 1/2}}$     (A1)

$\frac{{{{\text{d}}^2}T}}{{{\text{d}}{x^2}}} = 5\sqrt 5 {({x^2} + 4)^{ - 1/2}} - \frac{{5\sqrt 5 x}}{2}{({x^2} + 4)^{ - 3/2}} \times 2x$     A1

$\left( { = \frac{{5\sqrt 5 }}{{{{({x^2} + 4)}^{1/2}}}} - \frac{{5\sqrt 5 {x^2}}}{{{{({x^2} + 4)}^{3/2}}}}} \right)$

Attempt to simplify     (M1)

$= \frac{{5\sqrt 5 ({x^2} + 4) - 5\sqrt 5 {x^2}}}{{{{({x^2} + 4)}^{3/2}}}}\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\left( { = \frac{{5\sqrt 5 ({x^2} + 4 - {x^2})}}{{{{({x^2} + 4)}^{3/2}}}}} \right)$     A1

$= \frac{{20\sqrt 5 }}{{{{({x^2} + 4)}^{3/2}}}}$     AG

When $x = 1{\text{ , }}\frac{{20\sqrt 5 }}{{{{({x^2} + 4)}^{3/2}}}} > 0$ and hence T = 30 is a minimum     R1     N0

Note: Allow FT on incorrect x value, $0 \leqslant x \leqslant 2$.

[11 marks]

Total [18 marks]

Most candidates scored well on this question. The question tested their competence at algebraic manipulation and differentiation. A few candidates failed to extract from the context the correct relationship between velocity, distance and time.