WJEC Further Unit 4 (Further Unit 4) 2023 June

The functions \(f\) and \(g\) have domains \((-1, \infty)\) and \((0, \infty)\) respectively and are defined by $$f(x) = \cosh x, \qquad g(x) = x^2 - 1.$$

1. State the domain and range of \(fg\). [2]
2. Solve the equation \(fg(x) = 3\). Give your answer correct to three decimal places. [3]

The matrix \(\mathbf{A}\) is given by \(\mathbf{A} = \begin{pmatrix} \lambda & 1 & 14 \\ -1 & 2 & 8 \\ -3 & 2 & \lambda \end{pmatrix}\), where \(\lambda\) is a real constant.

1. Find an expression for the determinant of \(\mathbf{A}\) in terms of \(\lambda\). Give your answer in the form \(a\lambda^2 + b\lambda + c\), where \(a\), \(b\), \(c\) are integers whose values are to be determined. [3]
2. Show that \(\mathbf{A}\) is non-singular for all values of \(\lambda\). [4]

1. Given that \(z = \cos\theta + i\sin\theta\), use de Moivre's theorem to show that $$z^n + \frac{1}{z^n} = 2\cos n\theta .$$ [3]
2. Express \(32\cos^6\theta\) in the form \(a\cos 6\theta + b\cos 4\theta + c\cos 2\theta + d\), where \(a\), \(b\), \(c\), \(d\) are integers whose values are to be determined. [6]

Solve the simultaneous equations \begin{align} 4x - 2y + 3z &= 8,
2x - 3y + 8z &= -1,
2x + 4y - z &= 0. \end{align} [5]

1. Write down and simplify the Maclaurin series for \(\sin 2x\) as far as the term in \(x^5\). [2]
2. Using your answer to part (a), determine the Maclaurin series for \(\cos^2 x\) as far as the term in \(x^4\). [5]

1. Show that \(\tan\theta\) may be expressed as \(\frac{2t}{1-t^2}\), where \(t = \tan\left(\frac{\theta}{2}\right)\). [1]

The diagram below shows a sketch of the curve \(C\) with polar equation $$r = \cos\left(\frac{\theta}{2}\right), \quad \text{where } -\pi < \theta \leqslant \pi.$$ \includegraphics{figure_6}

1. Show that the \(\theta\)-coordinate of the points at which the tangent to \(C\) is perpendicular to the initial line satisfies the equation $$\tan\theta = -\frac{1}{2}\tan\left(\frac{\theta}{2}\right).$$ [4]
2. Hence, find the polar coordinates of the points on \(C\) where the tangent is perpendicular to the initial line. [6]
3. Calculate the area of the region enclosed by the curve \(C\) and the initial line for \(0 \leqslant \theta \leqslant \pi\). [5]

Find the cube roots of the complex number \(z = 11 - 2i\), giving your answers in the form \(x + iy\), where \(x\) and \(y\) are real and correct to three decimal places. [7]

The function \(f\) is defined by $$f(x) = \frac{1}{\sqrt{x^2 + 4x + 3}}.$$

1. Find the mean value of the function \(f\) for \(0 \leqslant x \leqslant 2\), giving your answer correct to three decimal places. [5]
2. The region \(R\) is bounded by the curve \(y = f(x)\), the \(x\)-axis and the lines \(x = 0\) and \(x = 2\). Find the exact value of the volume of the solid generated when \(R\) is rotated through four right angles about the \(x\)-axis. [6]

Consider the differential equation $$\left(x+1\right)\frac{\mathrm{d}y}{\mathrm{d}x} + 5y = (x+1)^2, \quad x > -1.$$ Given that \(y = \frac{1}{4}\) when \(x = 1\), find the value of \(y\) when \(x = 0\). [8]

1. By writing \(y = \sin^{-1}(2x + 5)\) as \(\sin y = 2x + 5\), show that \(\frac{\mathrm{d}y}{\mathrm{d}x} = \frac{2}{\sqrt{1-(2x+5)^2}}\). [5]
2. Deduce the range of values of \(x\) for which \(\frac{\mathrm{d}}{\mathrm{d}x}\left(\sin^{-1}(2x+5)\right)\) is valid. [3]

Evaluate the integrals

1. \(\int_{-2}^{0} e^{2x} \sinh x \, \mathrm{d}x\), [5]
2. \(\int_{\frac{1}{2}}^{3} \frac{5}{(x-1)(x^2+9)} \, \mathrm{d}x\). [9]

Find the general solution of the equation $$\cos 4\theta + \cos 2\theta = \cos\theta.$$ [6]

Two species of insects, \(X\) and \(Y\), co-exist on an island. The populations of the species at time \(t\) years are \(x\) and \(y\) respectively, where \(x\) and \(y\) are measured in millions. The situation can be modelled by the differential equations $$\frac{\mathrm{d}x}{\mathrm{d}t} = 3x + 10y,$$ $$\frac{\mathrm{d}y}{\mathrm{d}t} = x + 6y.$$

1. 1. Show that \(\frac{\mathrm{d}^2x}{\mathrm{d}t^2} - 9\frac{\mathrm{d}x}{\mathrm{d}t} + 8x = 0\).
   2. Find the general solution for \(x\) in terms of \(t\). [7]
2. Find the corresponding general solution for \(y\). [4]
3. When \(t = 0\), \(\frac{\mathrm{d}x}{\mathrm{d}t} = 5\) and the population of species \(Y\) is 4 times the population of species \(X\). Find the particular solution for \(x\) in terms of \(t\). [6]