OCR C4 (Core Mathematics 4)

1. Express

$$\frac { 5 x } { ( x - 4 ) ( x + 1 ) } + \frac { 3 } { ( x - 2 ) ( x + 1 ) }$$ as a single fraction in its simplest form.

2. A curve has the equation $$x ^ { 2 } + 2 x y ^ { 2 } + y = 4$$ Find an expression for \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) in terms of \(x\) and \(y\).

3. Evaluate $$\int _ { 0 } ^ { \frac { \pi } { 3 } } \sin 2 x \cos x d x$$

1. A curve has parametric equations

$$x = \cos 2 t , \quad y = \operatorname { cosec } t , \quad 0 < t < \frac { \pi } { 2 }$$ The point \(P\) on the curve has \(x\)-coordinate \(\frac { 1 } { 2 }\).

1. Find the value of the parameter \(t\) at \(P\).
2. Show that the tangent to the curve at \(P\) has the equation $$y = 2 x + 1$$

1. (i) Express \(\frac { 2 + 20 x } { 1 + 2 x - 8 x ^ { 2 } }\) as a sum of partial fractions.
   (ii) Hence find the series expansion of \(\frac { 2 + 20 x } { 1 + 2 x - 8 x ^ { 2 } } , | x | < \frac { 1 } { 4 }\), in ascending powers of \(x\) up to and including the term in \(x ^ { 3 }\), simplifying each coefficient.
2. Use the substitution \(x = 2 \tan u\) to show that

$$\int _ { 0 } ^ { 2 } \frac { x ^ { 2 } } { x ^ { 2 } + 4 } d x = \frac { 1 } { 2 } ( 4 - \pi )$$

1. A straight road passes through villages at the points \(A\) and \(B\) with position vectors \(( 9 \mathbf { i } - 8 \mathbf { j } + 2 \mathbf { k } )\) and ( \(4 \mathbf { j } + \mathbf { k }\) ) respectively, relative to a fixed origin.

The road ends at a junction at the point \(C\) with another straight road which lies along the line with equation $$\mathbf { r } = ( 2 \mathbf { i } + 16 \mathbf { j } - \mathbf { k } ) + t ( - 5 \mathbf { i } + 3 \mathbf { j } ) ,$$ where \(t\) is a scalar parameter.

1. Find the position vector of \(C\). Given that 1 unit on each coordinate axis represents 200 metres,
2. find the distance, in kilometres, from the village at \(A\) to the junction at \(C\).

8. (i) Find \(\int \tan ^ { 2 } x \mathrm {~d} x\).
(ii) Show that $$\int \tan x \mathrm {~d} x = \ln | \sec x | + c$$ where \(c\) is an arbitrary constant.
\includegraphics[max width=\textwidth, alt={}, center]{1e93a786-6105-4c69-a79a-a5f6e6c4aa0a-2_554_784_1484_507} The diagram shows part of the curve with equation \(y = x ^ { \frac { 1 } { 2 } } \tan x\).
The shaded region bounded by the curve, the \(x\)-axis and the line \(x = \frac { \pi } { 3 }\) is rotated through \(360 ^ { \circ }\) about the \(x\)-axis.
(iii) Show that the volume of the solid formed is \(\frac { 1 } { 18 } \pi ^ { 2 } ( 6 \sqrt { 3 } - \pi ) - \pi \ln 2\).

9. An entomologist is studying the population of insects in a colony. Initially there are 300 insects in the colony and in a model, the entomologist assumes that the population, \(P\), at time \(t\) weeks satisfies the differential equation $$\frac { \mathrm { d } P } { \mathrm {~d} t } = k P$$ where \(k\) is a constant.

1. Find an expression for \(P\) in terms of \(k\) and \(t\). Given that after one week there are 360 insects in the colony,
2. find the value of \(k\) to 3 significant figures. Given also that after two and three weeks there are 440 and 600 insects respectively,
3. comment on suitability of the modelling assumption. An alternative model assumes that $$\frac { \mathrm { d } P } { \mathrm {~d} t } = P ( 0.4 - 0.25 \cos 0.5 t )$$
4. Using the initial data, \(P = 300\) when \(t = 0\), solve this differential equation.
5. Compare the suitability of the two models.