WJEC Further Unit 4 (Further Unit 4) 2022 June

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. [5]
2. Find the nature of this stationary point. [2]
3. State the largest possible range of \(f(x)\). [1]

When plotted on an Argand diagram, the four fourth roots of the complex number \(9 - 3\sqrt{3}i\) lie on a circle. Find the equation of this circle. [4]

1. 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 $$4t^2 - 4t - 1 = 0.$$ [3]
2. Hence find the general solution of the equation $$4\sin\theta + 5\cos\theta = 3.$$ [6]

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. [5]

1. Determine the number of solutions of the equations \begin{align} x + 2y &= 3,
   2x - 5y + 3z &= 8,
   6y - 2z &= 0. \end{align} [4]
2. Give a geometric interpretation of your answer in part (a). [1]

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

1. Express \(4x^2 + 10x - 24\) in the form \(a(x + b)^2 + c\), where \(a\), \(b\), \(c\) are constants whose values are to be found. [3]
2. Hence evaluate the integral $$\int_3^5 \frac{6}{\sqrt{4x^2 + 10x - 24}} dx.$$ Give your answer correct to 3 decimal places. [5]

By writing \(x = \sinh y\), show that \(\sinh^{-1} x = \ln\left(x + \sqrt{x^2 + 1}\right)\). [6]

1. 1. Expand \(\left(\cos\frac{\theta}{3} + i\sin\frac{\theta}{3}\right)^3\).
   2. Hence, by using de Moivre's theorem, show that \(\cos\theta\) can be expressed as $$4\cos^3\frac{\theta}{3} - 3\cos\frac{\theta}{3}.$$ [6]
2. Hence, or otherwise, find the general solution of the equation \(\frac{\cos\theta}{\cos\frac{\theta}{3}} = 1\). [6]

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

1. Show that there are two values of \(\lambda\) for which \(\mathbf{A}\) is singular. [4]
2. Given that \(\lambda = 3\),
   1. determine the adjugate matrix of \(\mathbf{A}\),
   2. determine the inverse matrix \(\mathbf{A}^{-1}\). [5]

1. Differentiate each of the following with respect to \(x\).
   1. \(y = e^{3x}\sin^{-1}x\)
   2. \(y = \ln\left(\cosh^2(2x^2 + 7x)\right)\) [7]
2. Find the equations of the tangents to the curve \(x = \sinh^{-1}(y^2)\) at the points where \(x = 1\). [8]

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

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

1. Sketch the curve C. [2]
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. [9]

Evaluate the integral $$\int_2^4 \frac{6x^2 + 2x + 16}{x^3 - x^2 + 3x - 3} dx,$$ giving your answer correct to three decimal places. [10]