WJEC Further Unit 4 (Further Unit 4) Specimen

1. (a) Evaluate the integral

$$\int _ { 0 } ^ { \infty } \frac { \mathrm { d } x } { ( 1 + x ) ^ { 5 } }$$ (b) By putting \(u = \ln x\), determine whether or not the following integral has a finite value. $$\int _ { 2 } ^ { \infty } \frac { \mathrm { d } x } { x \ln x }$$

1. Evaluate the integral

$$\int _ { 0 } ^ { 1 } \frac { d x } { \sqrt { 2 x ^ { 2 } + 4 x + 6 } }$$

1. The curve \(C\) has polar equation \(r = 3 ( 2 + \cos \theta ) , 0 \leq \theta \leq \pi\). Determine the area enclosed between \(C\) and the initial line. Give your answer in the form \(\frac { a } { b } \pi\), where \(a\) and \(b\) are positive integers whose values are to be found.
2. Find the three cube roots of the complex number \(2 + 3 \mathrm { i }\), giving your answers in Cartesian form.
3. Find all the roots of the equation

$$\cos \theta + \cos 3 \theta + \cos 5 \theta = 0$$ lying in the interval \([ 0 , \pi ]\). Give all the roots in radians in terms of \(\pi\).

6. The matrix \(\mathbf { M }\) is given by $$\mathbf { M } = \left[ \begin{array} { l l l } 2 & 1 & 3
1 & 3 & 2
3 & 2 & 5 \end{array} \right]$$

1. Find
   1. the adjugate matrix of \(\mathbf { M }\),
   2. hence determine the inverse matrix \(\mathbf { M } ^ { - 1 }\).
2. Use your result to solve the simultaneous equations $$\begin{aligned} & 2 x + y + 3 z = 13
   & x + 3 y + 2 z = 13
   & 3 x + 2 y + 5 z = 22 \end{aligned}$$

1. The function \(f\) is defined by

$$f ( x ) = \frac { 8 x ^ { 2 } + x + 5 } { ( 2 x + 1 ) \left( x ^ { 2 } + 3 \right) }$$

1. Express \(f ( x )\) in partial fractions.
2. Hence evaluate $$\int _ { 2 } ^ { 3 } f ( x ) \mathrm { d } x$$ giving your answer correct to three decimal places.

8. The curve \(y = 1 + x ^ { 3 }\) is denoted by \(C\).

1. A bowl is designed by rotating the arc of \(C\) joining the points \(( 0,1 )\) and \(( 2,9 )\) through four right angles about the \(y\)-axis. Calculate the capacity of the bowl.
2. Another bowl with capacity 25 is to be designed by rotating the arc of \(C\) joining the points with \(y\) coordinates 1 and \(a\) through four right angles about the \(y\)-axis. Calculate the value of \(a\).

9. (a) Use mathematical induction to prove de Moivre's Theorem, namely that $$( \cos \theta + \mathrm { i } \sin \theta ) ^ { n } = \cos n \theta + \mathrm { i } \sin n \theta$$ where \(n\) is a positive integer.
(b) (i) Use this result to show that $$\sin 5 \theta = a \sin ^ { 5 } \theta - b \sin ^ { 3 } \theta + c \sin \theta$$ where \(a , b\) and \(c\) are positive integers to be found.
(ii) Hence determine the value of \(\lim _ { \theta \rightarrow 0 } \frac { \sin 5 \theta } { \sin \theta }\)

10. Consider the differential equation $$\frac { \mathrm { d } y } { \mathrm {~d} x } + 2 y \tan x = \sin x , \quad 0 < x < \frac { \pi } { 2 }$$

1. Find an integrating factor for this differential equation.
2. Solve the differential equation given that \(y = 0\) when \(x = \frac { \pi } { 4 }\), giving your answer in the form \(y = f ( x )\).

11. (a) Show that $$\tanh ^ { - 1 } x = \frac { 1 } { 2 } \ln \left( \frac { 1 + x } { 1 - x } \right) , \quad \text { where } - 1 < x < 1$$ (b) Given that $$a \cosh x + b \sinh x \equiv \operatorname { rcosh } ( x + \alpha ) , \quad \text { where } a > b > 0$$ show that $$\alpha = \frac { 1 } { 2 } \ln \left( \frac { a + b } { a - b } \right)$$ and find an expression for \(r\) in terms of \(a\) and \(b\).
(c) Hence solve the equation $$5 \cosh x + 4 \sinh x = 10$$ giving your answers correct to three significant figures.

12. The function \(f\) is given by $$f ( x ) = \mathrm { e } ^ { x } \cos x$$

1. Show that \(f ^ { \prime \prime } ( x ) = - 2 \mathrm { e } ^ { x } \sin x\).
2. Determine the Maclaurin series for \(f ( x )\) as far as the \(x ^ { 4 }\) term.
3. Hence, by differentiating your series, determine the Maclaurin series for \(\mathrm { e } ^ { x } \sin x\) as far as the \(x ^ { 3 }\) term.
4. The equation $$10 \mathrm { e } ^ { x } \sin x - 11 x = 0$$ has a small positive root. Determine its approximate value, giving your answer correct to three decimal places.