Questions — WJEC (504 questions)

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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]
WJEC Further Unit 4 2022 June Q14
10 marks Challenging +1.8
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]
WJEC Further Unit 4 2023 June Q1
5 marks Standard +0.3
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]
WJEC Further Unit 4 2023 June Q2
7 marks Standard +0.3
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]
WJEC Further Unit 4 2023 June Q3
9 marks Standard +0.8
  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]