WJEC Further Unit 4 (Further Unit 4) Specimen

1. Evaluate the integral $$\int_0^{\infty} \frac{dx}{(1+x)^5}.$$ [3]
2. By putting \(u = \ln x\), determine whether or not the following integral has a finite value. $$\int_2^{\infty} \frac{dx}{x \ln x}.$$ [4]

Evaluate the integral $$\int_0^1 \frac{dx}{\sqrt{2x^2 + 4x + 6}}.$$ [6]

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. [5]

Find the three cube roots of the complex number \(2 + 3i\), giving your answers in Cartesian form. [9]

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 \(\pi\). [8]

The matrix \(\mathbf{M}\) is given by $$\mathbf{M} = \begin{pmatrix} 2 & 1 & 3 \\ 1 & 3 & 2 \\ 3 & 2 & 5 \end{pmatrix}.$$

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

The function \(f\) is defined by $$f(x) = \frac{8x^2 + x + 5}{(2x + 1)(x^2 + 3)}.$$

1. Express \(f(x)\) in partial fractions. [4]
2. Hence evaluate $$\int_2^5 f(x)dx,$$ giving your answer correct to three decimal places. [6]

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. [5]
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 \(a\). [5]

1. Use mathematical induction to prove de Moivre's Theorem, namely that $$(\cos \theta + i \sin \theta)^n = \cos n\theta + i \sin n\theta,$$ where \(n\) is a positive integer. [7]
2. 1. 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.
   2. Hence determine the value of \(\lim_{\theta \to 0} \frac{\sin 5\theta}{\sin \theta}\). [7]

Consider the differential equation $$\frac{dy}{dx} + 2y \tan x = \sin x, \quad 0 < x < \frac{\pi}{2}.$$

1. Find an integrating factor for this differential equation. [4]
2. Solve the differential equation given that \(y = 0\) when \(x = \frac{\pi}{4}\), giving your answer in the form \(y = f(x)\). [7]

1. Show that $$\tanh^{-1} x = \frac{1}{2} \ln \left(\frac{1+x}{1-x}\right), \quad \text{where } -1 < x < 1.$$ [4]
2. Given that $$a \cosh x + b \sinh x \equiv r \cosh(x + \alpha), \quad \text{where } a > b > 0,$$ 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 \(b\). [7]
3. Hence solve the equation $$5 \cosh x + 4 \sinh x = 10,$$ giving your answers correct to three significant figures. [6]

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

1. Show that \(f''(x) = -2e^x \sin x\). [2]
2. Determine the Maclaurin series for \(f(x)\) as far as the \(x^4\) term. [6]
3. Hence, by differentiating your series, determine the Maclaurin series for \(e^x \sin x\) as far as the \(x^3\) term. [4]
4. The equation $$10e^x \sin x - 11x = 0$$ has a small positive root. Determine its approximate value, giving your answer correct to three decimal places. [4]