Edexcel C4 (Core Mathematics 4) Specimen

Use the binomial theorem to expand \(( 4 - 3 x ) ^ { - \frac { 1 } { 2 } }\), in ascending powers of \(x\), up to and including the term in \(x ^ { 3 }\). Give each coefficient as a simplified fraction.

The curve \(C\) has equation $$13 x ^ { 2 } + 13 y ^ { 2 } - 10 x y = 52$$ Find an expression for \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) as a function of \(x\) and \(y\), simplifying your answer.
(6)

3. Use the substitution \(x = \tan \theta\) to show that $$\int _ { 0 } ^ { 1 } \frac { 1 } { \left( 1 + x ^ { 2 } \right) ^ { 2 } } \mathrm {~d} x = \frac { \pi } { 8 } + \frac { 1 } { 4 }$$ (8)

4. \begin{figure}[h] \end{figure} Figure 1 shows part of the curve with parametric equations $$x = \tan t , \quad y = \sin 2 t , \quad - \frac { \pi } { 2 } < t < \frac { \pi } { 2 } .$$

1. Find the gradient of the curve at the point \(P\) where \(t = \frac { \pi } { 3 }\).
2. Find an equation of the normal to the curve at \(P\).
3. Find an equation of the normal to the curve at the point \(Q\) where \(t = \frac { \pi } { 4 }\).

5. The vector equations of two straight lines are $$\begin{aligned} & \mathbf { r } = 5 \mathbf { i } + 3 \mathbf { j } - 2 \mathbf { k } + \lambda ( \mathbf { i } - 2 \mathbf { j } + 2 \mathbf { k } ) \quad \text { and }
& \mathbf { r } = 2 \mathbf { i } - 11 \mathbf { j } + a \mathbf { k } + \mu ( - 3 \mathbf { i } - 4 \mathbf { j } + 5 \mathbf { k } ) . \end{aligned}$$ Given that the two lines intersect, find

1. the coordinates of the point of intersection,
2. the value of the constant \(a\),
3. the acute angle between the two lines.

6. Given that $$\frac { 11 x - 1 } { ( 1 - x ) ^ { 2 } ( 2 + 3 x ) } \equiv \frac { A } { ( 1 - x ) ^ { 2 } } + \frac { B } { ( 1 - x ) } + \frac { C } { ( 2 + 3 x ) }$$

1. find the values of \(A , B\) and \(C\).
2. Find the exact value of \(\int _ { 0 } ^ { \frac { 1 } { 2 } } \frac { 11 x - 1 } { ( 1 - x ) ^ { 2 } ( 2 + 3 x ) } \mathrm { d } x\), giving your answer in the form \(k + \ln a\), where \(k\) is an integer and \(a\) is a simplified fraction.

7. (a) Given that \(u = \frac { x } { 2 } - \frac { 1 } { 8 } \sin 4 x\), show that \(\frac { \mathrm { d } u } { \mathrm {~d} x } = \sin ^ { 2 } 2 x\). \begin{figure}[h] \end{figure} Figure 2 shows the finite region bounded by the curve \(y = x ^ { \frac { 1 } { 2 } } \sin 2 x\), the line \(x = \frac { \pi } { 4 }\) and the \(x\)-axis. This region is rotated through \(2 \pi\) radians about the \(x\)-axis.
(b) Using the result in part (a), or otherwise, find the exact value of the volume generated.
(8)

8. A circular stain grows in such a way that the rate of increase of its radius is inversely proportional to the square of the radius. Given that the area of the stain at time \(t\) seconds is \(A \mathrm {~cm} ^ { 2 }\),

1. show that \(\frac { \mathrm { d } A } { \mathrm {~d} t } \propto \frac { 1 } { \sqrt { A } }\).
   (6) Another stain, which is growing more quickly, has area \(S \mathrm {~cm} ^ { 2 }\) at time \(t\) seconds. It is given that $$\frac { \mathrm { d } S } { \mathrm {~d} t } = \frac { 2 \mathrm { e } ^ { 2 t } } { \sqrt { S } }$$ Given that, for this second stain, \(S = 9\) at time \(t = 0\),
2. solve the differential equation to find the time at which \(S = 16\). Give your answer to 2 significant figures. \section*{END}