Edexcel C4 (Core Mathematics 4)

1. The region bounded by the curve \(y = x ^ { 2 } - 2 x\) and the \(x\)-axis is rotated through \(2 \pi\) radians about the \(x\)-axis.

Find the volume of the solid formed, giving your answer in terms of \(\pi\).

2. Use the substitution \(u = 1 - x ^ { \frac { 1 } { 2 } }\) to find $$\int \frac { 1 } { 1 - x ^ { \frac { 1 } { 2 } } } \mathrm {~d} x$$

1. A curve has the equation

$$2 \sin 2 x - \tan y = 0$$

1. Find an expression for \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) in its simplest form in terms of \(x\) and \(y\).
2. Show that the tangent to the curve at the point \(\left( \frac { \pi } { 6 } , \frac { \pi } { 3 } \right)\) has the equation $$y = \frac { 1 } { 2 } x + \frac { \pi } { 4 }$$
   1. continued
   \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{3cf64017-e982-4165-9885-8524aaabdf84-06_433_812_246_479} \captionsetup{labelformat=empty} \caption{Figure 1}
   \end{figure} Figure 1 shows the curve with parametric equations $$x = a \sqrt { t } , \quad y = a t ( 1 - t ) , \quad t \geq 0$$ where \(a\) is a positive constant.
3. Find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) in terms of \(t\). The curve meets the \(x\)-axis at the origin, \(O\), and at the point \(A\). The tangent to the curve at \(A\) meets the \(y\)-axis at the point \(B\) as shown.
4. Show that the area of triangle \(O A B\) is \(a ^ { 2 }\).

5. The gradient at any point \(( x , y )\) on a curve is proportional to \(\sqrt { y }\). Given that the curve passes through the point with coordinates \(( 0,4 )\),

1. show that the equation of the curve can be written in the form $$2 \sqrt { y } = k x + 4$$ where \(k\) is a positive constant. Given also that the curve passes through the point with coordinates ( 2,9 ),
2. find the equation of the curve in the form \(y = \mathrm { f } ( x )\).
   5. continued

6. \begin{figure}[h] \end{figure} Figure 2 shows a vertical cross-section of a vase.
The inside of the vase is in the shape of a right-circular cone with the angle between the sides in the cross-section being \(60 ^ { \circ }\). When the depth of water in the vase is \(h \mathrm {~cm}\), the volume of water in the vase is \(V \mathrm {~cm} ^ { 3 }\).

1. Show that \(V = \frac { 1 } { 9 } \pi h ^ { 3 }\). The vase is initially empty and water is poured in at a constant rate of \(120 \mathrm {~cm} ^ { 3 } \mathrm {~s} ^ { - 1 }\).
2. Find, to 2 decimal places, the rate at which \(h\) is increasing
   1. when \(h = 6\),
   2. after water has been poured in for 8 seconds.
      6. continued

7. Relative to a fixed origin, the points \(A\) and \(B\) have position vectors \(\left( \begin{array} { c } - 4
1
3 \end{array} \right)\) and \(\left( \begin{array} { c } - 3
6
1 \end{array} \right)\) respectively.

1. Find a vector equation for the line \(l _ { 1 }\) which passes through \(A\) and \(B\). The line \(l _ { 2 }\) has vector equation $$\mathbf { r } = \left( \begin{array} { c } 3
   - 7
   9 \end{array} \right) + \mu \left( \begin{array} { c } 2
   - 3
   1 \end{array} \right)$$
2. Show that lines \(l _ { 1 }\) and \(l _ { 2 }\) do not intersect.
3. Find the position vector of the point \(C\) on \(l _ { 2 }\) such that \(\angle A B C = 90 ^ { \circ }\).
   7. continued

9 \end{array} \right) + \mu \left( \begin{array} { c } 2
- 3
1 \end{array} \right)$$ (b) Show that lines \(l _ { 1 }\) and \(l _ { 2 }\) do not intersect.
(c) Find the position vector of the point \(C\) on \(l _ { 2 }\) such that \(\angle A B C = 90 ^ { \circ }\).
7. continued
8. $$f ( x ) = \frac { x ( 3 x - 7 ) } { ( 1 - x ) ( 1 - 3 x ) } , | x | < \frac { 1 } { 3 }$$ (a) Find the values of the constants \(A , B\) and \(C\) such that $$\mathrm { f } ( x ) = A + \frac { B } { 1 - x } + \frac { C } { 1 - 3 x }$$ (b) Evaluate $$\int _ { 0 } ^ { \frac { 1 } { 4 } } f ( x ) d x$$ giving your answer in the form \(p + \ln q\), where \(p\) and \(q\) are rational.
(c) Find the series expansion of \(\mathrm { f } ( x )\) in ascending powers of \(x\) up to and including the term in \(x ^ { 3 }\), simplifying each coefficient.
8. continued
8. continued