WJEC Further Unit 4 (Further Unit 4) 2024 June

1. (a) Express the three cube roots of \(5 + 10 \mathrm { i }\) in the form \(r \mathrm { e } ^ { \mathrm { i } \theta }\), where \(0 \leqslant \theta < 2 \pi\).
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   (b) The three cube roots of \(5 + 10 \mathrm { i }\) are plotted in an Argand diagram. The points are joined by straight lines to form a triangle. Find the area of this triangle, giving your answer correct to two significant figures.
   
2. The function \(f\) is defined by \(f ( x ) = \cosh \left( \frac { x } { 2 } \right)\).
   (a) State the Maclaurin series expansion for \(\cosh \left( \frac { x } { 2 } \right)\) up to and including the term in \(x ^ { 4 }\).
   

Another function \(g\) is defined by \(g ( x ) = x ^ { 2 } - 2\). The diagram below shows parts of the graphs of \(y = f ( x )\) and \(y = g ( x )\).
\includegraphics[max width=\textwidth, alt={}, center]{7316672a-ae33-4f5b-9c59-51ef43af8ff1-04_894_940_1471_552}
(b) The two graphs intersect at the point \(A\), as shown in the diagram. Use your answer from part (a) to find an approximation for the \(x\)-coordinate of \(A\), giving your answer correct to two decimal places.
(c) Using your answer to part (b), find an approximation for the area of the shaded region enclosed by the two graphs, the \(x\)-axis and the \(y\)-axis.
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1. Given the differential equation

$$\cos x \frac { \mathrm {~d} y } { \mathrm {~d} x } + y \sin x = 4 \cos ^ { 3 } x \sin x + 5$$ and \(y = 3 \sqrt { 2 }\) when \(x = \frac { \pi } { 4 }\), find an equation for \(y\) in terms of \(x\).
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4. (a) Given that \(z ^ { n } + \frac { 1 } { z ^ { n } } = 2 \cos n \theta\), where \(z = \cos \theta + \mathrm { i } \sin \theta\), express \(16 \cos ^ { 4 } \theta\) in the form \(a \cos 4 \theta + b \cos 2 \theta + c\), where \(a , b , c\) are integers whose values are to be determined. [5]
The diagram below shows a sketch of the curve C with polar equation $$r = 3 - 4 \cos ^ { 2 } \theta , \quad \text { where } \frac { \pi } { 6 } \leqslant \theta \leqslant \frac { 5 \pi } { 6 }$$ Initial line
(b) Calculate the area of the region enclosed by the curve \(C\).
(c) Find the exact polar coordinates of the points on \(C\) at which the tangent is perpendicular to the initial line.
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1. Find each of the following integrals.
   1. \(\quad \int \frac { 3 - x } { x \left( x ^ { 2 } + 1 \right) } \mathrm { d } x\)
   2. \(\quad \int \frac { \sinh 2 x } { \sqrt { \cosh ^ { 4 } x - 9 \cosh ^ { 2 } x } } \mathrm {~d} x\)
      
   3. The matrix \(\mathbf { M }\) is defined by
   $$\mathbf { M } = \left( \begin{array} { c c c } 12 & 30 & 8
   18 & 25 & 20
   19 & 50 & 16 \end{array} \right)$$
2. Given that \(\operatorname { det } \mathbf { M } = - 1040\), give a geometrical interpretation of the solution to the following equation. $$\left( \begin{array} { c c c } 12 & 30 & 8
   18 & 25 & 20
   19 & 50 & 16 \end{array} \right) \left( \begin{array} { l } x
   y
   z \end{array} \right) = \left( \begin{array} { l } 2668
   3402
   4581 \end{array} \right)$$
3. Three hotels \(\mathrm { A } , \mathrm { B } , \mathrm { C }\) each have different types of room available to book: single, double and family rooms. For each type of room, the price per night is the same in each of the three hotels. The table below gives, for each hotel, details of the number of each type of room and the total revenue per night when the hotel is full.

   	Types of room
   Hotel	Single	Double	Family	Total revenue
   A	12	30	8	£2,668
   B	18	25	20	£3,402
   C	19	50	16	£4,581

   Find the price per night of each type of room.

7. (a) A curve \(C\) is defined by the equation \(y = \frac { 1 } { \sqrt { 16 - 6 x - x ^ { 2 } } }\) for \(- 3 \leqslant x \leqslant 1\).

1. Find the mean value of \(y = \frac { 1 } { \sqrt { 16 - 6 x - x ^ { 2 } } }\) between \(x = - 3\) and \(x = 1\).
2. The region \(R\) is bounded by the curve \(C\), the \(x\)-axis and the lines \(x = - 3\) and \(x = 1\). Find the volume of the solid generated when \(R\) is rotated through four right-angles about the \(x\)-axis.
   (b) Evaluate the improper integral $$\int _ { 1 } ^ { \infty } \frac { - 8 \mathrm { e } ^ { - 2 x } } { 4 \mathrm { e } ^ { - 2 x } - 5 } \mathrm {~d} x$$ giving your answer correct to three decimal places.
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1. (a) By writing \(y = \sinh ^ { - 1 } ( 4 x + 3 )\) as \(\sinh y = 4 x + 3\), show that \(\frac { \mathrm { d } y } { \mathrm {~d} x } = \frac { 4 } { \sqrt { 16 x ^ { 2 } + 24 x + 10 } }\).
   (b) Show that the graph of \(\mathrm { e } ^ { - 3 x } y = \sinh 2 x\) has only one stationary point.

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9. Find the general solution of the equation $$\sin 6 \theta + 2 \cos ^ { 2 } \theta = 3 \cos 2 \theta - \sin 2 \theta + 1 .$$

10. The following simultaneous equations are to be solved. $$\begin{aligned} & \frac { \mathrm { d } x } { \mathrm {~d} t } = 4 x + 2 y + 6 \mathrm { e } ^ { 3 t }
& \frac { \mathrm {~d} y } { \mathrm {~d} t } = 6 x + 8 y + 15 \mathrm { e } ^ { 3 t } \end{aligned}$$

1. Show that \(\frac { \mathrm { d } ^ { 2 } x } { \mathrm {~d} t ^ { 2 } } - 12 \frac { \mathrm {~d} x } { \mathrm {~d} t } + 20 x = 0\).
2. Given that \(\frac { \mathrm { d } x } { \mathrm {~d} t } = 9\) and \(\frac { \mathrm { d } ^ { 2 } x } { \mathrm {~d} t ^ { 2 } } = 10\) when \(t = 0\), find the particular solution for \(x\) in terms of \(t\). Additional page, if required. Write the question number(s) in the left-hand margin. \section*{PLEASE DO NOT WRITE ON THIS PAGE} \section*{PLEASE DO NOT WRITE ON THIS PAGE} \section*{PLEASE DO NOT WRITE ON THIS PAGE}