Find \(f'(x)\) .

Find \(g(4)\) .

(i)     Write down the value of \(h\) .

(ii)     Find the value of \(a\) .

Find the value of \(x\) for which the tangent to the graph of \(f\) is parallel to the tangent to the graph of \(g\) .

\(f'(x) = \cos x + x - 2\)     A1A1A1     N3

Note: Award A1 for each term.

[3 marks]

recognizing \(g(0) = 5\) gives the point (\(0\), \(5\))     (R1)

recognize symmetry     (M1)

eg vertex, sketch

\(g(4) = 5\)     A1     N3

[3 marks]

(i)     \(h = 2\)     A1 N1

(ii)     substituting into \(g(x) = a{(x - 2)^2} + 3\) (not the vertex)     (M1)

eg   \(5 = a{(0 - 2)^2} + 3\) , \(5 = a{(4 - 2)^2} + 3\)

working towards solution     (A1)

eg   \(5 = 4a + 3\) , \(4a = 2\)

\(a = \frac{1}{2}\)     A1     N2

[4 marks]

\(g(x) = \frac{1}{2}{(x - 2)^2} + 3 = \frac{1}{2}{x^2} - 2x + 5\)

correct derivative of \(g\)     A1A1

eg   \(2 \times \frac{1}{2}(x - 2)\) , \(x - 2\)

evidence of equating both derivatives     (M1)

eg   \(f' = g'\)

correct equation     (A1)

eg   \(\cos x + x - 2 = x - 2\)

working towards a solution     (A1)

eg   \(\cos x = 0\) , combining like terms

\(x = \frac{\pi }{2}\)    A1     N0

Note: Do not award final A1 if additional values are given.

[6 marks]

In part (a), most candidates were able to correctly find the derivative of the function.

In part (b), many candidates did not understand the significance of the axis of symmetry and the known point (\(0\), \(5\)), and so were unable to find \(g(4)\) using symmetry. A few used more complicated manipulations of the function, but many algebraic errors were seen.

In part (c), a large number of candidates were able to simply write down the correct value of \(h\), as intended by the command term in this question. A few candidates wrote down the incorrect negative value. Most candidates attempted to substitute the \(x\) and \(y\) values of the known point correctly into the function, but again many arithmetic and algebraic errors kept them from finding the correct value for \(a\).

Part (d) required the candidates to find the derivative of \(g\), and to equate that to their answer from part (a). Although many candidates were able to simplify their equation to \(\cos x = 0\), many did not know how to solve for \(x\) at this point. Candidates who had made errors in parts (a) and/or (c) were still able to earn follow-through marks in part (d).