Write down the coordinates of the minimum point on the graph of f .

The points \({\text{P}}(p,{\text{ }}3)\) and \({\text{Q}}(q,{\text{ }}3){\text{, }}q > p\), lie on the graph of \(y = f(x)\) .

Find p and q .

Find the coordinates of the point, on \(y = f(x)\) , where the gradient of the graph is 3.

Find the coordinates of the point of intersection of the normals to the graph at the points P and Q.

\((3.79, - 5)\)     A1 

[1 mark] 

\(p = 1.57{\text{ or }}\frac{\pi }{2},{\text{ }}q = 6.00\)     A1A1

[2 marks]

\(f'(x) = 3\cos x - 4\sin x\)     (M1)(A1)

\(3\cos x - 4\sin x = 3 \Rightarrow x = 4.43...\)     (A1)

\((y = -4)\)     A1

Coordinates are \((4.43, -4)\)

[4 marks]

\({m_{{\text{normal}}}} = \frac{1}{{{m_{{\text{tangent}}}}}}\)     (M1)

gradient at P is \( - 4\) so gradient of normal at P is \(\frac{1}{4}\)     (A1)

gradient at Q is 4 so gradient of normal at Q is \( - \frac{1}{4}\)     (A1)

equation of normal at P is \(y - 3 = \frac{1}{4}(x - 1.570...){\text{ }}({\text{or }}y = 0.25x + 2.60...)\)     (M1)

equation of normal at Q is \(y - 3 = \frac{1}{4}(x - 5.999...){\text{ }}({\text{or }}y = -0.25x + \underbrace {4.499...}_{})\)     (M1) 

Note: Award the previous two M1 even if the gradients are incorrect in \(y - b = m(x - a)\) where \((a,b)\) are coordinates of P and Q (or in \(y = mx + c\) with c determined using coordinates of P and Q.

intersect at \((3.79,{\text{ }}3.55)\)     A1A1 

Note: Award N2 for 3.79 without other working. 

[7 marks]

Candidates answered parts (a) and (b) of this question well and, although many were also successful in part (c), just a few candidates gave answers to the required level of accuracy. Part d) was rather challenging for many candidates. The most common errors among the candidates who attempted this question were the confusion between tangents and normals and incorrect final answers due to premature rounding.

Candidates answered parts (a) and (b) of this question well and, although many were also successful in part (c), just a few candidates gave answers to the required level of accuracy. Part d) was rather challenging for many candidates. The most common errors among the candidates who attempted this question were the confusion between tangents and normals and incorrect final answers due to premature rounding.

Candidates answered parts (a) and (b) of this question well and, although many were also successful in part (c), just a few candidates gave answers to the required level of accuracy. Part d) was rather challenging for many candidates. The most common errors among the candidates who attempted this question were the confusion between tangents and normals and incorrect final answers due to premature rounding.

Candidates answered parts (a) and (b) of this question well and, although many were also successful in part (c), just a few candidates gave answers to the required level of accuracy. Part d) was rather challenging for many candidates. The most common errors among the candidates who attempted this question were the confusion between tangents and normals and incorrect final answers due to premature rounding.