Edexcel C4 (Core Mathematics 4)

1. 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 \(y\).

2. Use integration by parts to find $$\int x ^ { 2 } \mathrm { e } ^ { - x } \mathrm {~d} x$$

1. The first four terms in the series expansion of \(( 1 + a x ) ^ { n }\) in ascending powers of \(x\) are

$$1 - 4 x + 24 x ^ { 2 } + k x ^ { 3 }$$ where \(a , n\) and \(k\) are constants and \(| a x | < 1\).

1. Find the values of \(a\) and \(n\).
2. Show that \(k = - 160\).
   3. continued

4. (a) Use the trapezium rule with two intervals of equal width to find an estimate for the value of the integral $$\int _ { 0 } ^ { 3 } e ^ { \cos x } d x$$ giving your answer to 3 significant figures.
(b) Use the trapezium rule with four intervals of equal width to find another estimate for the value of the integral to 3 significant figures.
(c) Given that the true value of the integral lies between the estimates made in parts (a) and (b), comment on the shape of the curve \(y = \mathrm { e } ^ { \cos x }\) in the interval \(0 \leq x \leq 3\) and explain your answer.
4. continued

5. 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 $$\mathbf { r } = ( 2 \mathbf { i } + 16 \mathbf { j } - \mathbf { k } ) + \mu ( - 5 \mathbf { i } + 3 \mathbf { j } ) ,$$ where \(\mu\) 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 \(C\).
   5. continued

6. A small town had a population of 9000 in the year 2001. In a model, it is assumed that the population of the town, \(P\), at time \(t\) years after 2001 satisfies the differential equation $$\frac { \mathrm { d } P } { \mathrm {~d} t } = 0.05 P \mathrm { e } ^ { - 0.05 t }$$

1. Show that, according to the model, the population of the town in 2011 will be 13300 to 3 significant figures.
2. Find the value which the population of the town will approach in the long term, according to the model.
   6. continued

7. \begin{figure}[h] \end{figure} Figure 1 shows the curve with parametric equations $$x = t ^ { 3 } + 1 , \quad y = \frac { 2 } { t } , \quad t > 0 .$$ The shaded region is bounded by the curve, the \(x\)-axis and the lines \(x = 2\) and \(x = 9\).

1. Find the area of the shaded region.
2. Show that the volume of the solid formed when the shaded region is rotated through \(2 \pi\) radians about the \(x\)-axis is \(12 \pi\).
3. Find a cartesian equation for the curve in the form \(y = \mathrm { f } ( x )\).
   7. continued

8. (a) Show that the substitution \(u = \sin x\) transforms the integral $$\int \frac { 6 } { \cos x ( 2 - \sin x ) } d x$$ into the integral $$\int \frac { 6 } { \left( 1 - u ^ { 2 } \right) ( 2 - u ) } \mathrm { d } u .$$ (b) Express \(\frac { 6 } { \left( 1 - u ^ { 2 } \right) ( 2 - u ) }\) in partial fractions.
(c) Hence, evaluate $$\int _ { 0 } ^ { \frac { \pi } { 6 } } \frac { 6 } { \cos x ( 2 - \sin x ) } d x$$ giving your answer in the form \(a \ln 2 + b \ln 3\), where \(a\) and \(b\) are integers.
8. continued
8. continued