OCR MEI C4 (Core Mathematics 4) 2012 January

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

3 Express \(3 \sin x + 2 \cos x\) in the form \(R \sin ( x + \alpha )\), where \(R > 0\) and \(0 < \alpha < \frac { \pi } { 2 }\).
Hence find, correct to 2 decimal places, the coordinates of the maximum point on the curve \(y = \mathrm { f } ( x )\), where $$f ( x ) = 3 \sin x + 2 \cos x , 0 \leqslant x \leqslant \pi .$$

4

1. Complete the table of values for the curve \(y = \sqrt { \cos x }\).

   \(x\)	0	\(\frac { \pi } { 8 }\)	\(\frac { \pi } { 4 }\)	\(\frac { 3 \pi } { 8 }\)	\(\frac { \pi } { 2 }\)
   \(y\)		0.9612	0.8409

   Hence use the trapezium rule with strip width \(h = \frac { \pi } { 8 }\) to estimate the value of the integral \(\int _ { 0 } ^ { \frac { \pi } { 2 } } \sqrt { \cos x } \mathrm {~d} x\), giving your answer to 3 decimal places. Fig. 4 shows the curve \(y = \sqrt { \cos x }\) for \(0 \leqslant x \leqslant \frac { \pi } { 2 }\). \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{81433914-a56f-4765-af34-990a0127f98b-02_457_750_1446_653} \captionsetup{labelformat=empty} \caption{Fig. 4}
   \end{figure}
2. State, with a reason, whether the trapezium rule with a strip width of \(\frac { \pi } { 16 }\) would give a larger or smaller estimate of the integral.

5 Verify that the vector \(2 \mathbf { i } - \mathbf { j } + 4 \mathbf { k }\) is perpendicular to the plane through the points \(\mathrm { A } ( 2,0,1 ) , \mathrm { B } ( 1,2,2 )\) and \(\mathrm { C } ( 0 , - 4,1 )\). Hence find the cartesian equation of the plane.

6 Given the binomial expansion \(( 1 + q x ) ^ { p } = 1 - x + 2 x ^ { 2 } + \ldots\), find the values of \(p\) and \(q\). Hence state the set of values of \(x\) for which the expansion is valid.

7 Show that the straight lines with equations \(\mathbf { r } = \left( \begin{array} { l } 4
2
4 \end{array} \right) + \lambda \left( \begin{array} { l } 3
0
1 \end{array} \right)\) and \(\mathbf { r } = \left( \begin{array} { r } - 1
4
9 \end{array} \right) + \mu \left( \begin{array} { r } - 1
1
3 \end{array} \right)\) meet.
Find their point of intersection.

8 Fig. 8 shows a cross-section of a car headlight whose inside reflective surface is modelled, in suitable units, by the curve $$x = 2 t ^ { 2 } , y = 4 t , \quad - \sqrt { 2 } \leqslant t \leqslant \sqrt { 2 } .$$ \(\mathrm { P } \left( 2 t ^ { 2 } , 4 t \right)\) is a point on the curve with parameter \(t\). TS is the tangent to the curve at P , and PR is the line through P parallel to the \(x\)-axis. Q is the point \(( 2,0 )\). The angles that PS and QP make with the positive \(x\)-direction are \(\theta\) and \(\phi\) respectively. \begin{figure}[h] \end{figure}

1. By considering the gradient of the tangent TS, show that \(\tan \theta = \frac { 1 } { t }\).
2. Find the gradient of the line QP in terms of \(t\). Hence show that \(\phi = 2 \theta\), and that angle TPQ is equal to \(\theta\).
   [0pt] [The above result shows that if a lamp bulb is placed at Q , then the light from the bulb is reflected to produce a parallel beam of light.] The inside surface of the headlight has the shape produced by rotating the curve about the \(x\)-axis.
3. Show that the curve has cartesian equation \(y ^ { 2 } = 8 x\). Hence find the volume of revolution of the curve, giving your answer as a multiple of \(\pi\).

9 \begin{figure}[h] \end{figure} Fig. 9 shows a hemispherical bowl, of radius 10 cm , filled with water to a depth of \(x \mathrm {~cm}\). It can be shown that the volume of water, \(V \mathrm {~cm} ^ { 3 }\), is given by $$V = \pi \left( 10 x ^ { 2 } - \frac { 1 } { 3 } x ^ { 3 } \right) .$$ Water is poured into a leaking hemispherical bowl of radius 10 cm . Initially, the bowl is empty. After \(t\) seconds, the volume of water is changing at a rate, in \(\mathrm { cm } ^ { 3 } \mathrm {~s} ^ { - 1 }\), given by the equation $$\frac { \mathrm { d } V } { \mathrm {~d} t } = k ( 20 - x ) ,$$ where \(k\) is a constant.

1. Find \(\frac { \mathrm { d } V } { \mathrm {~d} x }\), and hence show that \(\pi x \frac { \mathrm {~d} x } { \mathrm {~d} t } = k\).
2. Solve this differential equation, and hence show that the bowl fills completely after \(T\) seconds, where $$T = \frac { 50 \pi } { k } .$$ Once the bowl is full, the supply of water to the bowl is switched off, and water then leaks out at a rate of \(k x \mathrm {~cm} ^ { 3 } \mathrm {~s} ^ { - 1 }\).
3. Show that, \(t\) seconds later, \(\pi ( 20 - x ) \frac { \mathrm { d } x } { \mathrm {~d} t } = - k\).
4. Solve this differential equation. Hence show that the bowl empties in \(3 T\) seconds.