OCR MEI C4 (Core Mathematics 4) 2012 June

1 Solve the equation \(\frac { 4 x } { x + 1 } - \frac { 3 } { 2 x + 1 } = 1\).

2 Find the first four terms in the binomial expansion of \(\sqrt { 1 + 2 x }\). State the set of values of \(x\) for which the expansion is valid.

3 The total value of the sales made by a new company in the first \(t\) years of its existence is denoted by \(\pounds V\). A model is proposed in which the rate of increase of \(V\) is proportional to the square root of \(V\). The constant of proportionality is \(k\).

1. Express the model as a differential equation. Verify by differentiation that \(V = \left( \frac { 1 } { 2 } k t + c \right) ^ { 2 }\), where \(c\) is an arbitrary constant, satisfies this differential equation.
2. The value of the company’s sales in its first year is \(\pounds 10000\), and the total value of the sales in the first two years is \(\pounds 40000\). Find \(V\) in terms of \(t\).

5 Given the equation \(\sin \left( x + 45 ^ { \circ } \right) = 2 \cos x\), show that \(\sin x + \cos x = 2 \sqrt { 2 } \cos x\).
Hence solve, correct to 2 decimal places, the equation for \(0 ^ { \circ } \leqslant x \leqslant 360 ^ { \circ }\).

6 Solve the differential equation \(\frac { \mathrm { d } y } { \mathrm {~d} x } = \frac { y } { x ( x + 1 ) }\), given that when \(x = 1 , y = 1\). Your answer should express \(y\) explicitly in terms of \(x\).

7 Fig. 7a shows the curve with the parametric equations $$x = 2 \cos \theta , \quad y = \sin 2 \theta , \quad - \frac { \pi } { 2 } \leqslant \theta \leqslant \frac { \pi } { 2 } .$$ The curve meets the \(x\)-axis at O and P . Q and R are turning points on the curve. The scales on the axes are the same. \begin{figure}[h] \end{figure}

1. State, with their coordinates, the points on the curve for which \(\theta = - \frac { \pi } { 2 } , \theta = 0\) and \(\theta = \frac { \pi } { 2 }\).
2. Find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) in terms of \(\theta\). Hence find the gradient of the curve when \(\theta = \frac { \pi } { 2 }\), and verify that the two tangents to the curve at the origin meet at right angles.
3. Find the exact coordinates of the turning point Q . When the curve is rotated about the \(x\)-axis, it forms a paperweight shape, as shown in Fig. 7b. \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{9001b0d0-8d06-43f4-8831-23c0d6aef59d-3_321_385_1758_831} \captionsetup{labelformat=empty} \caption{Fig. 7b}
   \end{figure}
4. Express \(\sin ^ { 2 } \theta\) in terms of \(x\). Hence show that the cartesian equation of the curve is \(y ^ { 2 } = x ^ { 2 } \left( 1 - \frac { 1 } { 4 } x ^ { 2 } \right)\).
5. Find the volume of the paperweight shape.

8 With respect to cartesian coordinates Oxyz, a laser beam ABC is fired from the point A(1, 2, 4), and is reflected at point B off the plane with equation \(x + 2 y - 3 z = 0\), as shown in Fig. 8. \(\mathrm { A } ^ { \prime }\) is the point (2, 4, 1), and M is the midpoint of \(\mathrm { AA } ^ { \prime }\). \begin{figure}[h] \end{figure}

1. Show that \(\mathrm { AA } ^ { \prime }\) is perpendicular to the plane \(x + 2 y - 3 z = 0\), and that M lies in the plane. The vector equation of the line AB is \(\mathbf { r } = \left( \begin{array} { l } 1
   2
   4 \end{array} \right) + \lambda \left( \begin{array} { r } 1
   - 1
   2 \end{array} \right)\).
2. Find the coordinates of B , and a vector equation of the line \(\mathrm { A } ^ { \prime } \mathrm { B }\).
3. Given that \(\mathrm { A } ^ { \prime } \mathrm { BC }\) is a straight line, find the angle \(\theta\).
4. Find the coordinates of the point where BC crosses the Oxz plane (the plane containing the \(x\) - and \(z\)-axes).