OCR MEI C4 (Core Mathematics 4) 2016 June

1 Express \(\cos \theta - 3 \sin \theta\) in the form \(R \cos ( \theta + \alpha )\), where \(R > 0\) and \(0 < \alpha < \frac { \pi } { 2 }\).
Hence show that the equation \(\cos \theta - 3 \sin \theta = 4\) has no solution.

2 Given that \(\left( 1 + \frac { x } { p } \right) ^ { q } = 1 - x + \frac { 3 } { 4 } x ^ { 2 } + \ldots\), find \(p\) and \(q\), and state the set of values of \(x\) for which the expansion is valid.

3 Fig. 3 shows the curve \(y = x ^ { 4 }\) and the line \(y = 4\). \begin{figure}[h] \end{figure} The finite region enclosed by the curve and the line is rotated through \(180 ^ { \circ }\) about the \(y\)-axis. Find the exact volume of revolution generated.

4 Solve the equation \(2 \sin 2 \theta = 1 + \cos 2 \theta\) for \(0 ^ { \circ } \leqslant \theta \leqslant 180 ^ { \circ }\).

5 In Fig. 5, triangles \(\mathrm { ABC } , \mathrm { ACD }\) and ADE are all right-angled, and angles \(\mathrm { BAC } , \mathrm { CAD }\) and DAE are all \(\theta\). \(\mathrm { AB } = x\) and \(\mathrm { AE } = 2 x\). \begin{figure}[h] \end{figure}

1. Show that \(\sec ^ { 3 } \theta = 2\).
2. Hence show the ratio of lengths ED to CB is \(2 ^ { \frac { 2 } { 3 } } : 1\).

6 P is a general point on the curve with parametric equations \(x = 2 t , y = \frac { 2 } { t }\). This is shown in Fig. 6. The tangent at P intersects the \(x\) - and \(y\)-axes at the points Q and R respectively. \begin{figure}[h] \end{figure} Show that the area of the triangle OQR , where O is the origin, is independent of \(t\).