Edexcel C4 (Core Mathematics 4) 2016 June

1. Use the binomial series to find the expansion of

$$\frac { 1 } { ( 2 + 5 x ) ^ { 3 } } , \quad | x | < \frac { 2 } { 5 }$$ in ascending powers of \(x\), up to and including the term in \(x ^ { 3 }\).
Give each coefficient as a fraction in its simplest form.
(6)

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

1. Complete the table above, 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 3 decimal places.
3. Use integration to find the exact value for the area of \(R\).

1. The curve \(C\) has equation

$$2 x ^ { 2 } y + 2 x + 4 y - \cos ( \pi y ) = 17$$

1. Use implicit differentiation to find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) in terms of \(x\) and \(y\). The point \(P\) with coordinates \(\left( 3 , \frac { 1 } { 2 } \right)\) lies on \(C\).
   The normal to \(C\) at \(P\) meets the \(x\)-axis at the point \(A\).
2. Find the \(x\) coordinate of \(A\), giving your answer in the form \(\frac { a \pi + b } { c \pi + d }\), where \(a , b , c\) and \(d\) are integers to be determined.

4. The rate of decay of the mass of a particular substance is modelled by the differential equation $$\frac { \mathrm { d } x } { \mathrm {~d} t } = - \frac { 5 } { 2 } x , \quad t \geqslant 0$$ where \(x\) is the mass of the substance measured in grams and \(t\) is the time measured in days.
Given that \(x = 60\) when \(t = 0\),

1. solve the differential equation, giving \(x\) in terms of \(t\). You should show all steps in your working and give your answer in its simplest form.
2. Find the time taken for the mass of the substance to decay from 60 grams to 20 grams. Give your answer to the nearest minute.

5. \begin{figure}[h] \end{figure} Figure 2 shows a sketch of the curve \(C\) with parametric equations $$x = 4 \tan t , \quad y = 5 \sqrt { 3 } \sin 2 t , \quad 0 \leqslant t < \frac { \pi } { 2 }$$ The point \(P\) lies on \(C\) and has coordinates \(\left( 4 \sqrt { 3 } , \frac { 15 } { 2 } \right)\).

1. Find the exact value of \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) at the point \(P\). Give your answer as a simplified surd. The point \(Q\) lies on the curve \(C\), where \(\frac { \mathrm { d } y } { \mathrm {~d} x } = 0\)
2. Find the exact coordinates of the point \(Q\).

6. (i) Given that \(y > 0\), find $$\int \frac { 3 y - 4 } { y ( 3 y + 2 ) } d y$$ (ii) (a) Use the substitution \(x = 4 \sin ^ { 2 } \theta\) to show that $$\int _ { 0 } ^ { 3 } \sqrt { \left( \frac { x } { 4 - x } \right) } \mathrm { d } x = \lambda \int _ { 0 } ^ { \frac { \pi } { 3 } } \sin ^ { 2 } \theta \mathrm {~d} \theta$$ where \(\lambda\) is a constant to be determined.
(b) Hence use integration to find $$\int _ { 0 } ^ { 3 } \sqrt { \left( \frac { x } { 4 - x } \right) } d x$$ giving your answer in the form \(a \pi + b\), where \(a\) and \(b\) are exact constants.

7. (a) Find $$\int ( 2 x - 1 ) ^ { \frac { 3 } { 2 } } d x$$ giving your answer in its simplest form. \begin{figure}[h] \end{figure} Figure 3 shows a sketch of part of the curve \(C\) with equation $$y = ( 2 x - 1 ) ^ { \frac { 3 } { 4 } } , \quad x \geqslant \frac { 1 } { 2 }$$ The curve \(C\) cuts the line \(y = 8\) at the point \(P\) with coordinates \(( k , 8 )\), where \(k\) is a constant.
(b) Find the value of \(k\). The finite region \(S\), shown shaded in Figure 3, is bounded by the curve \(C\), the \(x\)-axis, the \(y\)-axis and the line \(y = 8\). This region is rotated through \(2 \pi\) radians about the \(x\)-axis to form a solid of revolution.
(c) Find the exact value of the volume of the solid generated.

8. With respect to a fixed origin \(O\), the line \(l _ { 1 }\) is given by the equation $$\mathbf { r } = \left( \begin{array} { r } 8
1
- 3 \end{array} \right) + \mu \left( \begin{array} { r } - 5
4
3 \end{array} \right)$$ where \(\mu\) is a scalar parameter.
The point \(A\) lies on \(l _ { 1 }\) where \(\mu = 1\)

1. Find the coordinates of \(A\). The point \(P\) has position vector \(\left( \begin{array} { l } 1
   5
   2 \end{array} \right)\).
   The line \(l _ { 2 }\) passes through the point \(P\) and is parallel to the line \(l _ { 1 }\)
2. Write down a vector equation for the line \(l _ { 2 }\)
3. Find the exact value of the distance \(A P\). Give your answer in the form \(k \sqrt { 2 }\), where \(k\) is a constant to be determined. The acute angle between \(A P\) and \(l _ { 2 }\) is \(\theta\).
4. Find the value of \(\cos \theta\) A point \(E\) lies on the line \(l _ { 2 }\) Given that \(A P = P E\),
5. find the area of triangle \(A P E\),
6. find the coordinates of the two possible positions of \(E\).