OCR C4 (Core Mathematics 4) 2014 June

1 Express \(x + \frac { 1 } { 1 - x } + \frac { 2 } { 1 + x }\) as a single fraction, simplifying your answer.

2 The points \(O ( 0,0,0 ) , A ( 2,8,2 ) , B ( 5,5,8 )\) and \(C ( 3 , - 3,6 )\) form a parallelogram \(O A B C\). Use a scalar product to find the acute angle between the diagonals of this parallelogram.

3

1. Find the first three terms in the expansion of \(( 1 - 2 x ) ^ { - \frac { 1 } { 2 } }\) in ascending powers of \(x\), where \(| x | < \frac { 1 } { 2 }\).
2. Hence find the coefficient of \(x ^ { 2 }\) in the expansion of \(\frac { x + 3 } { \sqrt { 1 - 2 x } }\).

4 Show that \(\int _ { 0 } ^ { \frac { 1 } { 4 } \pi } \frac { 1 - 2 \sin ^ { 2 } x } { 1 + 2 \sin x \cos x } \mathrm {~d} x = \frac { 1 } { 2 } \ln 2\).

5 The equations of three lines are as follows. $$\begin{array} { l l } \text { Line } A : & \mathbf { r } = \mathbf { i } + 4 \mathbf { j } + \mathbf { k } + s ( - \mathbf { i } + 2 \mathbf { j } + 2 \mathbf { k } ) \\ \text { Line } B : & \mathbf { r } = 2 \mathbf { i } + 8 \mathbf { j } + 2 \mathbf { k } + t ( \mathbf { i } + 3 \mathbf { j } + 5 \mathbf { k } ) \\ \text { Line } C : & \mathbf { r } = - \mathbf { i } + 19 \mathbf { j } + 15 \mathbf { k } + u ( 2 \mathbf { i } - 4 \mathbf { j } - 4 \mathbf { k } ) \end{array}$$

1. Show that lines \(A\) and \(B\) are skew.
2. Determine, giving reasons, the geometrical relationship between lines \(A\) and \(C\).

6 \includegraphics[max width=\textwidth, alt={}, center]{02e31b5d-10dd-42b1-885a-6db610d788c3-2_570_1191_1509_420} The diagram shows the curve with equation \(x ^ { 2 } + y ^ { 3 } - 8 x - 12 y = 4\). At each of the points \(P\) and \(Q\) the tangent to the curve is parallel to the \(y\)-axis. Find the coordinates of \(P\) and \(Q\).

7 A curve has parametric equations $$x = 2 \sin t , \quad y = \cos 2 t + 2 \sin t$$ for \(- \frac { 1 } { 2 } \pi \leqslant t \leqslant \frac { 1 } { 2 } \pi\).

1. Show that \(\frac { \mathrm { d } y } { \mathrm {~d} x } = 1 - 2 \sin t\) and hence find the coordinates of the stationary point.
2. Find the cartesian equation of the curve.
3. State the set of values that \(x\) can take and hence sketch the curve.

8

1. Use division to show that \(\frac { t ^ { 3 } } { t + 2 } \equiv t ^ { 2 } - 2 t + 4 - \frac { 8 } { t + 2 }\).
2. Find \(\int _ { 1 } ^ { 2 } 6 t ^ { 2 } \ln ( t + 2 ) \mathrm { d } t\). Give your answer in the form \(A + B \ln 3 + C \ln 4\).

9 Express \(\frac { 2 + x ^ { 2 } } { ( 1 + 2 x ) ( 1 - x ) ^ { 2 } }\) in partial fractions and hence show that \(\int _ { 0 } ^ { \frac { 1 } { 4 } } \frac { 2 + x ^ { 2 } } { ( 1 + 2 x ) ( 1 - x ) ^ { 2 } } \mathrm {~d} x = \frac { 1 } { 2 } \ln \frac { 3 } { 2 } + \frac { 1 } { 3 }\).

10 A container in the shape of an inverted cone of radius 3 metres and vertical height 4.5 metres is initially filled with liquid fertiliser. This fertiliser is released through a hole in the bottom of the container at a rate of \(0.01 \mathrm {~m} ^ { 3 }\) per second. At time \(t\) seconds the fertiliser remaining in the container forms an inverted cone of height \(h\) metres.
[0pt] [The volume of a cone is \(V = \frac { 1 } { 3 } \pi r ^ { 2 } h\).]

1. Show that \(h ^ { 2 } \frac { \mathrm {~d} h } { \mathrm {~d} t } = - \frac { 9 } { 400 \pi }\).
2. Express \(h\) in terms of \(t\).
3. Find the time it takes to empty the container, giving your answer to the nearest minute.