OCR MEI C4 (Core Mathematics 4) 2011 June

1 Express \(\frac { 1 } { ( 2 x + 1 ) \left( x ^ { 2 } + 1 \right) }\) in partial fractions.

2 Find the first three terms in the binomial expansion of \(\sqrt [ 3 ] { 1 + 3 x }\) in ascending powers of \(x\). State the set of values of \(x\) for which the expansion is valid.

3 Express \(2 \sin \theta - 3 \cos \theta\) in the form \(R \sin ( \theta - \alpha )\), where \(R\) and \(\alpha\) are constants to be determined, and \(0 < \alpha < \frac { 1 } { 2 } \pi\). Hence write down the greatest and least possible values of \(1 + 2 \sin \theta - 3 \cos \theta\).

4 A curve has parametric equations $$x = 2 \sin \theta , \quad y = \cos 2 \theta$$

1. Find the exact coordinates and the gradient of the curve at the point with parameter \(\theta = \frac { 1 } { 3 } \pi\).
2. Find \(y\) in terms of \(x\).

6 Fig. 6 shows the region enclosed by part of the curve \(y = 2 x ^ { 2 }\), the straight line \(x + y = 3\), and the \(y\)-axis. The curve and the straight line meet at \(\mathrm { P } ( 1,2 )\). \begin{figure}[h] \end{figure} The shaded region is rotated through \(360 ^ { \circ }\) about the \(y\)-axis. Find, in terms of \(\pi\), the volume of the solid of revolution formed.
[0pt] [You may use the formula \(V = \frac { 1 } { 3 } \pi r ^ { 2 } h\) for the volume of a cone.] Section B (36 marks)

7 A piece of cloth ABDC is attached to the tops of vertical poles \(\mathrm { AE } , \mathrm { BF } , \mathrm { DG }\) and CH , where \(\mathrm { E } , \mathrm { F } , \mathrm { G }\) and H are at ground level (see Fig. 7). Coordinates are as shown, with lengths in metres. The length of pole DG is \(k\) metres. \begin{figure}[h] \end{figure}

1. Write down the vectors \(\overrightarrow { \mathrm { AB } }\) and \(\overrightarrow { \mathrm { AC } }\). Hence calculate the angle BAC .
2. Verify that the equation of the plane ABC is \(x + y - 2 z + d = 0\), where \(d\) is a constant to be determined. Calculate the acute angle the plane makes with the horizontal plane.
3. Given that \(\mathrm { A } , \mathrm { B } , \mathrm { D }\) and C are coplanar, show that \(k = 3\). Hence show that ABDC is a trapezium, and find the ratio of CD to AB .

8 Water is leaking from a container. After \(t\) seconds, the depth of water in the container is \(x \mathrm {~cm}\), and the volume of water is \(V \mathrm {~cm} ^ { 3 }\), where \(V = \frac { 1 } { 3 } x ^ { 3 }\). The rate at which water is lost is proportional to \(x\), so that \(\frac { \mathrm { d } V } { \mathrm {~d} t } = - k x\), where \(k\) is a constant.

1. Show that \(x \frac { \mathrm {~d} x } { \mathrm {~d} t } = - k\). Initially, the depth of water in the container is 10 cm .
2. Show by integration that \(x = \sqrt { 100 - 2 k t }\).
3. Given that the container empties after 50 seconds, find \(k\). Once the container is empty, water is poured into it at a constant rate of \(1 \mathrm {~cm} ^ { 3 }\) per second. The container continues to lose water as before.
4. Show that, \(t\) seconds after starting to pour the water in, \(\frac { \mathrm { d } x } { \mathrm {~d} t } = \frac { 1 - x } { x ^ { 2 } }\).
5. Show that \(\frac { 1 } { 1 - x } - x - 1 = \frac { x ^ { 2 } } { 1 - x }\). Hence solve the differential equation in part (iv) to show that $$t = \ln \left( \frac { 1 } { 1 - x } \right) - \frac { 1 } { 2 } x ^ { 2 } - x .$$
6. Show that the depth cannot reach 1 cm .