Questions — WJEC Further Unit 4 (61 questions)

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WJEC Further Unit 4 2019 June Q1
8 marks Standard +0.3
A complex number is defined by \(z = 3 + 4\mathrm{i}\).
  1. Express \(z\) in the form \(z = re^{i\theta}\), where \(-\pi \leqslant \theta \leqslant \pi\). [3]
    1. Find the Cartesian coordinates of the vertices of the triangle formed by the cube roots of \(z\) when plotted in an Argand diagram. Give your answers correct to two decimal places.
    2. Write down the geometrical name of the triangle. [5]
WJEC Further Unit 4 2019 June Q2
9 marks Challenging +1.2
  1. Show that \(3\sin x + 4\cos x - 2\) can be written as \(\frac{6t + 2 - 6t^2}{1 + t^2}\), where \(t = \tan\left(\frac{x}{2}\right)\). [2]
  2. Hence, find the general solution of the equation \(3\sin x + 4\cos x - 2 = 3\). [7]
WJEC Further Unit 4 2019 June Q3
8 marks Standard +0.3
  1. Determine whether or not the following set of equations $$\begin{pmatrix} 2 & -7 & 2 \\ 0 & 3 & -2 \\ -7 & 8 & 4 \end{pmatrix} \begin{pmatrix} x \\ y \\ z \end{pmatrix} = \begin{pmatrix} a \\ b \\ c \end{pmatrix}$$ has a unique solution, where \(a\), \(b\), \(c\) are constants. [3]
  2. Solve the set of equations \begin{align} x + 8y - 6z &= 5,
    2x + 4y + 6z &= -3,
    -5x - 4y + 9z &= -7. \end{align} Show all your working. [5]
WJEC Further Unit 4 2019 June Q4
16 marks Standard +0.3
  1. Given that \(y = \cot^{-1} x\), show that \(\frac{\mathrm{d}y}{\mathrm{d}x} = \frac{-1}{x^2 + 1}\). [5]
  2. Express \(\frac{6x^2 - 10x - 9}{(2x + 3)(x^2 + 1)}\) in terms of partial fractions. [5]
  3. Hence find \(\int \frac{6x^2 - 8x - 6}{(2x + 3)(x^2 + 1)} \mathrm{d}x\). [5]
  4. Explain why \(\int_{-2}^{5} \frac{6x^2 - 8x - 6}{(2x + 3)(x^2 + 1)} \mathrm{d}x\) cannot be evaluated. [1]
WJEC Further Unit 4 2019 June Q5
8 marks Standard +0.8
  1. Show that \(\sin \theta - \sin 3\theta\) can be expressed in the form \(a\cos b\theta \sin \theta\), where \(a\), \(b\) are integers whose values are to be determined. [3]
  2. Find the mean value of \(y = 2\cos 2\theta \sin \theta + 7\) between \(\theta = 1\) and \(\theta = 3\), giving your answer correct to two decimal places. [5]
WJEC Further Unit 4 2019 June Q6
10 marks Standard +0.3
Solve the differential equation $$\frac{\mathrm{d}^2 y}{\mathrm{d}x^2} - 7\frac{\mathrm{d}y}{\mathrm{d}x} + 10y = 0,$$ where \(\frac{\mathrm{d}y}{\mathrm{d}x} = 1\) and \(\frac{\mathrm{d}^2 y}{\mathrm{d}x^2} = 8\) when \(x = 0\). [10]
WJEC Further Unit 4 2019 June Q7
6 marks Moderate -0.3
  1. Write down the Maclaurin series expansion for \(\ln(1 - x)\) as far as the term in \(x^3\). [2]
  2. Show that \(-2\ln\left(\frac{1-x}{(1+x)^2}\right)\) can be expressed in the form \(ax + bx^2 + cx^3 + \ldots\), where \(a\), \(b\), \(c\) are integers whose values are to be determined. [4]
WJEC Further Unit 4 2019 June Q8
10 marks Challenging +1.8
The curve \(C\) has polar equation $$r = \sin 2\theta, \quad \text{where} \quad 0 < \theta \leqslant \frac{\pi}{2}.$$
  1. Find the polar coordinates of the point on \(C\) at which the tangent is parallel to the initial line. Give your answers correct to three decimal places. [9]
  2. Write the coordinates of this point in Cartesian form. [1]
WJEC Further Unit 4 2019 June Q9
14 marks Standard +0.8
  1. Given that \(y = \sin^{-1}(\cos \theta)\), where \(0 \leqslant \theta \leqslant \pi\), show that \(\frac{\mathrm{d}y}{\mathrm{d}\theta} = k\), where the value of \(k\) is to be determined. [4]
  2. Find the value of the gradient of the curve \(y = x^3 \tan^{-1} 4x\) when \(x = \frac{\pi}{2}\). [4]
  3. Find the equation of the normal to the curve \(y = \tanh^{-1}(1 - x)\) when \(x = 1.7\). [6]
WJEC Further Unit 4 2019 June Q10
8 marks Challenging +1.8
Given the differential equation $$\sec x \frac{\mathrm{d}y}{\mathrm{d}x} + y\cos \sec x = 2$$ and \(x = \frac{\pi}{2}\) when \(y = 3\), find the value of \(y\) when \(x = \frac{\pi}{4}\). [8]
WJEC Further Unit 4 2019 June Q11
9 marks Standard +0.3
  1. Find the area of the region enclosed by the curve \(y = x\sinh x\), the \(x\)-axis and the lines \(x = 0\) and \(x = 1\). [4]
  2. The region \(R\) is bounded by the curve \(y = \cosh 2x\), the \(x\)-axis and the lines \(x = 0\) and \(x = 1\). Find the volume of the solid generated when \(R\) is rotated through four right-angles about the \(x\)-axis. [4]
  3. Using your answer to part (b), find the total volume of the solid generated by rotating the region bounded by the curve \(y = \cosh 2x\) and the lines \(x = -1\) and \(x = 1\). [1]
WJEC Further Unit 4 2019 June Q12
14 marks Challenging +1.2
  1. Evaluate \(\int_3^4 \frac{1}{\sqrt{x^2 - 4}} \mathrm{d}x\), giving your answer correct to three decimal places. [3]
  2. Given that \(\int_1^2 \frac{k}{9 - x^2} \mathrm{d}x = \ln \frac{25}{4}\), find the value of \(k\). [5]
  3. Show that \(\int \frac{(\cosh x - \sinh x)^3}{\cosh^2 x + \sinh^2 x - \sinh 2x} \mathrm{d}x\) can be expressed as \(-e^{-x} + c\), where \(c\) is a constant. [6]
WJEC Further Unit 4 2022 June Q1
8 marks Standard +0.8
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]
WJEC Further Unit 4 2022 June Q2
4 marks Standard +0.8
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]
WJEC Further Unit 4 2022 June Q3
9 marks Standard +0.8
  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]
WJEC Further Unit 4 2022 June Q4
5 marks Standard +0.3
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]
WJEC Further Unit 4 2022 June Q5
5 marks Standard +0.8
  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]
WJEC Further Unit 4 2022 June Q6
6 marks Challenging +1.8
Solve the equation $$\cos 2\theta - \cos 4\theta = \sin 3\theta \quad \text{for} \quad 0 \leq \theta \leq \pi$$ [6]
WJEC Further Unit 4 2022 June Q7
8 marks Challenging +1.2
  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]
WJEC Further Unit 4 2022 June Q8
6 marks Standard +0.8
By writing \(x = \sinh y\), show that \(\sinh^{-1} x = \ln\left(x + \sqrt{x^2 + 1}\right)\). [6]
WJEC Further Unit 4 2022 June Q9
12 marks Challenging +1.3
    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]
WJEC Further Unit 4 2022 June Q10
9 marks Standard +0.3
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]
WJEC Further Unit 4 2022 June Q11
15 marks Standard +0.8
  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]
WJEC Further Unit 4 2022 June Q12
12 marks Challenging +1.2
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]
WJEC Further Unit 4 2022 June Q13
11 marks Challenging +1.3
The curve C has polar equation \(r = 2 - \cos\theta\) for \(0 \leq \theta \leq 2\pi\).
  1. Sketch the curve C. [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]