Edexcel C4 (Core Mathematics 4)

1. (a) Expand \(( 1 + 4 x ) ^ { \frac { 3 } { 2 } }\) in ascending powers of \(x\) up to and including the term in \(x ^ { 3 }\), simplifying each coefficient.
   (b) State the set of values of \(x\) for which your expansion is valid.
   
2. Use the substitution \(u = 1 + \sin x\) to find the value of

$$\int _ { 0 } ^ { \frac { \pi } { 2 } } \cos x ( 1 + \sin x ) ^ { 3 } d x$$

1. (a) Express \(\frac { x + 11 } { ( x + 4 ) ( x - 3 ) }\) as a sum of partial fractions.
   (b) Evaluate

$$\int _ { 0 } ^ { 2 } \frac { x + 11 } { ( x + 4 ) ( x - 3 ) } d x$$ giving your answer in the form \(\ln k\), where \(k\) is an exact simplified fraction. (5)
3. continued

4. \begin{figure}[h] \end{figure} Figure 1 shows the curve with equation \(y = 2 \sin x + \operatorname { cosec } x , 0 < x < \pi\). The shaded region bounded by the curve, the \(x\)-axis and the lines \(x = \frac { \pi } { 6 }\) and \(x = \frac { \pi } { 2 }\) is rotated through \(360 ^ { \circ }\) about the \(x\)-axis. Show that the volume of the solid formed is \(\frac { 1 } { 2 } \pi ( 4 \pi + 3 \sqrt { 3 } )\).
4. continued

5. A curve has the equation $$x ^ { 2 } - 3 x y - y ^ { 2 } = 12$$

1. Find an expression for \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) in terms of \(x\) and \(y\).
2. Find an equation for the tangent to the curve at the point \(( 2 , - 2 )\).
   5. continued

6. Relative to a fixed origin, \(O\), the points \(A\) and \(B\) have position vectors \(\left( \begin{array} { c } 1
5
- 1 \end{array} \right)\) and \(\left( \begin{array} { c } 6
3
- 6 \end{array} \right)\) respectively. Find, in exact, simplified form,

1. the cosine of \(\angle A O B\),
2. the area of triangle \(O A B\),
3. the shortest distance from \(A\) to the line \(O B\).
   6. continued

7. A curve has parametric equations $$x = t ( t - 1 ) , \quad y = \frac { 4 t } { 1 - t } , \quad t \neq 1$$

1. Find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) in terms of \(t\). The point \(P\) on the curve has parameter \(t = - 1\).
2. Show that the tangent to the curve at \(P\) has the equation $$x + 3 y + 4 = 0$$ The tangent to the curve at \(P\) meets the curve again at the point \(Q\).
3. Find the coordinates of \(Q\).
   7. continued

8. 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 model. 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.
   8. continued
   8. continued