WJEC Further Unit 4 (Further Unit 4) 2022 June

1. A function \(f\) has domain \(( - \infty , \infty )\) and is defined by \(f ( x ) = \cosh ^ { 3 } x - 3 \cosh x\).
   1. Show that the graph of \(y = f ( x )\) has only one stationary point.
   2. Find the nature of this stationary point.
   3. State the largest possible range of \(f ( x )\).
   4. When plotted on an Argand diagram, the four fourth roots of the complex number \(9 - 3 \sqrt { 3 } \mathrm { i }\) lie on a circle. Find the equation of this circle.
   5. (a) By putting \(t = \tan \left( \frac { \theta } { 2 } \right)\), show that the equation
   $$4 \sin \theta + 5 \cos \theta = 3$$ can be written in the form $$4 t ^ { 2 } - 4 t - 1 = 0$$
2. Hence find the general solution of the equation $$4 \sin \theta + 5 \cos \theta = 3$$

1. The region \(R\) is bounded by the curve \(x = \sin y\), the \(y\)-axis and the lines \(y = 1 , y = 3\). Find the volume of the solid generated when \(R\) is rotated through four right angles about the \(y\)-axis. Give your answer correct to two decimal places.
2. (a) Determine the number of solutions of the equations

$$\begin{array} { r } x + 2 y = 3
2 x - 5 y + 3 z = 8
6 y - 2 z = 0 \end{array}$$ (b) Give a geometric interpretation of your answer in part (a).

6. Solve the equation $$\cos 2 \theta - \cos 4 \theta = \sin 3 \theta \quad \text { for } \quad 0 \leqslant \theta \leqslant \pi$$

1. (a) Express \(4 x ^ { 2 } + 10 x - 24\) in the form \(a ( x + b ) ^ { 2 } + c\), where \(a , b , c\) are constants whose values are to be found.
   (b) Hence evaluate the integral

$$\int _ { 3 } ^ { 5 } \frac { 6 } { \sqrt { 4 x ^ { 2 } + 10 x - 24 } } \mathrm {~d} x$$ Give your answer correct to 3 decimal places.

8. By writing \(x = \sinh y\), show that \(\sinh ^ { - 1 } x = \ln \left( x + \sqrt { x ^ { 2 } + 1 } \right)\).

9. (a) (i) Expand \(\left( \cos \frac { \theta } { 3 } + i \sin \frac { \theta } { 3 } \right) ^ { 3 }\).
(ii) Hence, by using de Moivre's theorem, show that \(\cos \theta\) can be expressed as $$4 \cos ^ { 3 } \frac { \theta } { 3 } - 3 \cos \frac { \theta } { 3 }$$ (b) Hence, or otherwise, find the general solution of the equation \(\frac { \cos \theta } { \cos \frac { \theta } { 3 } } = 1\).

10. The matrix \(\mathbf { A }\) is defined by $$\mathbf { A } = \left( \begin{array} { r r r } 4 & 8 & 0
0 & \lambda & - 2
4 & 0 & \lambda \end{array} \right)$$

1. Show that there are two values of \(\lambda\) for which \(\mathbf { A }\) is singular.
2. Given that \(\lambda = 3\),
   1. determine the adjugate matrix of \(\mathbf { A }\),
   2. determine the inverse matrix \(\mathbf { A } ^ { - 1 }\).

11. (a) Differentiate each of the following with respect to \(x\).

1. \(y = \mathrm { e } ^ { 3 x } \sin ^ { - 1 } x\)
2. \(y = \ln \left( \cosh ^ { 2 } \left( 2 x ^ { 2 } + 7 x \right) \right)\)
   (b) Find the equations of the tangents to the curve \(x = \sinh ^ { - 1 } \left( y ^ { 2 } \right)\) at the points where \(x = 1\).

12. Find the solution of the differential equation $$3 \frac { \mathrm {~d} ^ { 2 } y } { \mathrm {~d} x ^ { 2 } } + 5 \frac { \mathrm {~d} y } { \mathrm {~d} x } - 2 y = 8 + 6 x - 2 x ^ { 2 }$$ where \(y = 6\) and \(\frac { \mathrm { d } y } { \mathrm {~d} x } = 5\) when \(x = 0\).

13. The curve \(C\) has polar equation \(r = 2 - \cos \theta\) for \(0 \leqslant \theta \leqslant 2 \pi\).

1. Sketch the curve \(C\).
2. 1. Show that the values of \(\theta\) at which the tangent to the curve \(r = 2 - \cos \theta\) is parallel to the initial line satisfy the equation $$2 \cos ^ { 2 } \theta - 2 \cos \theta - 1 = 0$$
   2. Find the polar coordinates of the points where the tangent to the curve \(r = 2 - \cos \theta\) is parallel to the initial line.

14. Evaluate the integral $$\int _ { 2 } ^ { 4 } \frac { 6 x ^ { 2 } + 2 x + 16 } { x ^ { 3 } - x ^ { 2 } + 3 x - 3 } \mathrm {~d} x$$ giving your answer correct to three decimal places.