Edexcel C4 (Core Mathematics 4) 2014 January

1. (a) Find the binomial expansion of

$$\frac { 1 } { ( 4 + 3 x ) ^ { 3 } } , \quad | x | < \frac { 4 } { 3 }$$ in ascending powers of \(x\), up to and including the term in \(x ^ { 3 }\).
Give each coefficient as a simplified fraction. In the binomial expansion of $$\frac { 1 } { ( 4 - 9 x ) ^ { 3 } } , \quad | x | < \frac { 4 } { 9 }$$ the coefficient of \(x ^ { 2 }\) is \(A\).
(b) Using your answer to part (a), or otherwise, find the value of \(A\). Give your answer as a simplified fraction.

2. (i) Find $$\int x \cos \left( \frac { x } { 2 } \right) \mathrm { d } x$$ (ii) (a) Express \(\frac { 1 } { x ^ { 2 } ( 1 - 3 x ) }\) in partial fractions.
(b) Hence find, for \(0 < x < \frac { 1 } { 3 }\) $$\int \frac { 1 } { x ^ { 2 } ( 1 - 3 x ) } \mathrm { d } x$$

1. The number of bacteria, \(N\), present in a liquid culture at time \(t\) hours after the start of a scientific study is modelled by the equation

$$N = 5000 ( 1.04 ) ^ { t } , \quad t \geqslant 0$$ where \(N\) is a continuous function of \(t\).

1. Find the number of bacteria present at the start of the scientific study.
2. Find the percentage increase in the number of bacteria present from \(t = 0\) to \(t = 2\) Given that \(N = 15000\) when \(t = T\),
3. find the value of \(\frac { \mathrm { d } N } { \mathrm {~d} t }\) when \(t = T\), giving your answer to 3 significant figures.

4. \begin{figure}[h] \end{figure} Figure 1 shows a sketch of part of the curve with equation \(y = \frac { 4 \mathrm { e } ^ { - x } } { 3 \sqrt { } \left( 1 + 3 \mathrm { e } ^ { - x } \right) }\)
The finite region \(R\), shown shaded in Figure 1, is bounded by the curve, the \(x\)-axis, the line \(x = - 3 \ln 2\) and the \(y\)-axis. The table below shows corresponding values of \(x\) and \(y\) for \(y = \frac { 4 \mathrm { e } ^ { - x } } { 3 \sqrt { } \left( 1 + 3 \mathrm { e } ^ { - x } \right) }\)

1. Complete the table above by giving the missing value of \(y\) to 4 decimal places.
2. Use the trapezium rule, with all the values of \(y\) in the completed table, to obtain an estimate for the area of \(R\), giving your answer to 2 decimal places.
3. 1. Using the substitution \(u = 1 + 3 \mathrm { e } ^ { - x }\), or otherwise, find $$\int \frac { 4 \mathrm { e } ^ { - x } } { 3 \sqrt { } \left( 1 + 3 \mathrm { e } ^ { - x } \right) } \mathrm { d } x$$
   2. Hence find the value of the area of \(R\).

1. Given that \(y = 2\) at \(x = \frac { \pi } { 8 }\), solve the differential equation

$$\frac { \mathrm { d } y } { \mathrm {~d} x } = \frac { 3 y ^ { 2 } } { 2 \sin ^ { 2 } 2 x }$$ giving your answer in the form \(y = \mathrm { f } ( x )\).
\includegraphics[max width=\textwidth, alt={}, center]{245bbe52-3a14-4494-af17-7711caf79b22-17_81_102_2649_1779}

6. Oil is leaking from a storage container onto a flat section of concrete at a rate of \(0.48 \mathrm {~cm} ^ { 3 } \mathrm {~s} ^ { - 1 }\). The leaking oil spreads to form a pool with an increasing circular cross-section. The pool has a constant uniform thickness of 3 mm . Find the rate at which the radius \(r\) of the pool of oil is increasing at the instant when \(r = 5 \mathrm {~cm}\). Give your answer, in \(\mathrm { cm } \mathrm { s } ^ { - 1 }\), to 3 significant figures. \includegraphics[max width=\textwidth, alt={}, center]{245bbe52-3a14-4494-af17-7711caf79b22-19_104_95_2617_1786}

7. The curve \(C\) has parametric equations $$x = 2 \cos t , \quad y = \sqrt { 3 } \cos 2 t , \quad 0 \leqslant t \leqslant \pi$$ where \(t\) is a parameter.

1. Find an expression for \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) in terms of \(t\). The point \(P\) lies on \(C\) where \(t = \frac { 2 \pi } { 3 }\)
   The line \(l\) is a normal to \(C\) at \(P\).
2. Show that an equation for \(l\) is $$2 x - 2 \sqrt { 3 } y - 1 = 0$$ The line \(l\) intersects the curve \(C\) again at the point \(Q\).
3. Find the exact coordinates of \(Q\). You must show clearly how you obtained your answers.
   \includegraphics[max width=\textwidth, alt={}, center]{245bbe52-3a14-4494-af17-7711caf79b22-23_106_63_2595_1882}

8. With respect to a fixed origin \(O\), the lines \(l _ { 1 }\) and \(l _ { 2 }\) are given by the equations $$l _ { 1 } : \mathbf { r } = \left( \begin{array} { r } 2
- 3
4 \end{array} \right) + \lambda \left( \begin{array} { r } - 1
2
1 \end{array} \right) , \quad l _ { 2 } : \mathbf { r } = \left( \begin{array} { r } 2
- 3
4 \end{array} \right) + \mu \left( \begin{array} { r } 5
- 2
5 \end{array} \right)$$ where \(\lambda\) and \(\mu\) are scalar parameters.

1. Find, to the nearest \(0.1 ^ { \circ }\), the acute angle between \(l _ { 1 }\) and \(l _ { 2 }\) The point \(A\) has position vector \(\left( \begin{array} { l } 0
   1
   6 \end{array} \right)\).
2. Show that \(A\) lies on \(l _ { 1 }\) The lines \(l _ { 1 }\) and \(l _ { 2 }\) intersect at the point \(X\).
3. Write down the coordinates of \(X\).
4. Find the exact value of the distance \(A X\). The distinct points \(B _ { 1 }\) and \(B _ { 2 }\) both lie on the line \(l _ { 2 }\)
   Given that \(A X = X B _ { 1 } = X B _ { 2 }\)
5. find the area of the triangle \(A B _ { 1 } B _ { 2 }\) giving your answer to 3 significant figures. Given that the \(x\) coordinate of \(B _ { 1 }\) is positive,
6. find the exact coordinates of \(B _ { 1 }\) and the exact coordinates of \(B _ { 2 }\)
   \includegraphics[max width=\textwidth, alt={}, center]{245bbe52-3a14-4494-af17-7711caf79b22-28_96_59_2478_1834}