1.05l - OCR Spec

WJEC Further Unit 4 2023 June Q6

16 marks Challenging +1.8

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}

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]
Hence, find the polar coordinates of the points on $C$ where the tangent is perpendicular to the initial line. [6]
Calculate the area of the region enclosed by the curve $C$ and the initial line for $0 \leqslant \theta \leqslant \pi$. [5]

WJEC Further Unit 4 2023 June Q12

6 marks Challenging +1.2

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

WJEC Further Unit 4 2024 June Q4

21 marks Challenging +1.8

Given that $z^n + \frac{1}{z^n} = 2\cos n\theta$, where $z = \cos\theta + \mathrm{i}\sin\theta$, express $16\cos^4\theta$ in the form $$a\cos 4\theta + b\cos 2\theta + c,$$ where $a$, $b$, $c$ are integers whose values are to be determined. [5]

The diagram below shows a sketch of the curve C with polar equation $$r = 3 - 4\cos^2\theta, \quad \text{where } \frac{\pi}{6} \leq \theta \leq \frac{5\pi}{6}.$$ \includegraphics{figure_4}

Calculate the area of the region enclosed by the curve C. [8]
Find the exact polar coordinates of the points on C at which the tangent is perpendicular to the initial line. [8]

WJEC Further Unit 4 2024 June Q9

9 marks Challenging +1.8

Find the general solution of the equation $$\sin 6\theta + 2\cos^2\theta = 3\cos 2\theta - \sin 2\theta + 1.$$ [9]

WJEC Further Unit 4 Specimen Q5

8 marks Challenging +1.2

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]

SPS SPS SM Pure 2021 May Q4

3 marks Standard +0.3

Prove that $\sqrt{2}\cos(2\theta + 45°) \equiv \cos^2\theta - 2\sin\theta\cos\theta - \sin^2\theta$, where $\theta$ is measured in degrees. [3]

SPS SPS SM Pure 2020 October Q8

12 marks Challenging +1.3

1. Sketch the graph of $y = \cos \sec x$ for $0 < x < 4\pi$. [3]
2. It is given that $\cos \sec \alpha = \cos \sec \beta$, where $\frac{1}{2}\pi < \alpha < \pi$ and $2\pi < \beta < \frac{5}{2}\pi$. By using your sketch, or otherwise, express $\beta$ in terms of $\alpha$. [2]
1. Write down the identity giving $\tan 2\theta$ in terms of $\tan \theta$. [1]
2. Given that $\cot \phi = 4$, find the exact value of $\tan \phi \cot 2\phi \tan 4\phi$, showing all your working. [6]

SPS SPS SM Pure 2020 October Q10

7 marks Standard +0.3

Prove that $$\cos^2(\theta + 45°) - \frac{1}{2}(\cos 2\theta - \sin 2\theta) \equiv \sin^2 \theta.$$ [4]
Hence solve the equation $$6\cos^2(\frac{1}{2}\theta + 45°) - 3(\cos \theta - \sin \theta) = 2$$ for $-90° < \theta < 90°$. [3]

SPS SPS SM 2022 February Q3

8 marks Moderate -0.3

Solve each of the following equations, for $0° \leqslant x \leqslant 180°$.

$2\sin^2 x = 1 + \cos x$. [4]
$\sin 2x = -\cos 2x$. [4]

SPS SPS FM Pure 2022 June Q13

8 marks Standard +0.3

Show that $\sin(2\theta + \frac{1}{2}\pi) = \cos 2\theta$. [2]
Hence solve the equation $\sin 3\theta = \cos 2\theta$ for $0 \leq \theta \leq 2\pi$. [6]

SPS SPS SM Mechanics 2022 February Q7

7 marks Standard +0.3

In this question you must show all stages of your working. Solutions relying entirely on calculator technology are not acceptable.

Show that $$\frac{1 - \cos 2\theta}{\sin^2 2\theta} = k \sec^2 \theta \quad \theta = \frac{n\pi}{2} \quad n \in \mathbb{Z}$$ where $k$ is a constant to be found. [3]
Hence solve, for $-\frac{\pi}{2} < x < \frac{\pi}{2}$ $$\frac{1 - \cos 2x}{\sin^2 2x} = (1 + 2\tan x)^2$$ Give your answers to 3 significant figures where appropriate. [4]

SPS SPS SM 2021 November Q9

7 marks Moderate -0.3

1. Show that $\cos^2 x \equiv \frac{1}{2} + \frac{1}{2}\cos 2x$ [1]
2. Hence find $\int 2\cos^2 4x \, dx$ [3]
Find $\int \sin^3 x \, dx$ [3]

SPS SPS SM Pure 2023 October Q3

12 marks Moderate -0.3

Given that $\cos A = \frac{3}{4}$, where $270° < A < 360°$, find the exact value of $\sin 2A$. [5]
1. Show that $\cos\left(2x + \frac{\pi}{3}\right) + \cos\left(2x - \frac{\pi}{3}\right) = \cos 2x$. [3] Given that $$y = 3\sin^2 x + \cos\left(2x + \frac{\pi}{3}\right) + \cos\left(2x - \frac{\pi}{3}\right),$$
2. show that $\frac{dy}{dx} = \sin 2x$. [4]

SPS SPS FM Pure 2025 June Q6

9 marks Standard +0.3

Prove that $$1 - \cos 2\theta = \tan \theta \sin 2\theta, \quad \theta \neq \frac{(2n + 1)\pi}{2}, \quad n \in \mathbb{Z}$$ [3]
Hence solve, for $-\frac{\pi}{2} < x < \frac{\pi}{2}$, the equation $$(\sec^2 x - 5)(1 - \cos 2x) = 3\tan^2 x \sin 2x$$ Give any non-exact answer to 3 decimal places where appropriate. [6]

OCR H240/02 2018 December Q4

10 marks Standard +0.3

In this question you must show detailed reasoning.

Show that $\cos A + \sin A \tan A = \sec A$. [3]
Solve the equation $\tan 2\theta = 3 \tan \theta$ for $0° \leqslant \theta \leqslant 180°$. [7]

OCR H240/01 2017 Specimen Q8

6 marks Standard +0.3

Show that $\frac{2\tan\theta}{1 + \tan^2\theta} = \sin 2\theta$. [3]

In this question you must show detailed reasoning. Solve $\frac{2\tan\theta}{1 + \tan^2\theta} = 3\cos 2\theta$ for $0 \leq \theta \leq \pi$. [3]

OCR H240/03 2017 Specimen Q3

4 marks Standard +0.8

In this question you must show detailed reasoning. Given that $5\sin 2x = 3\cos x$, where $0° < x < 90°$, find the exact value of $\sin x$. [4]

Pre-U Pre-U 9794/1 2010 June Q11

11 marks Challenging +1.2

Write down an identity for $\tan 2\theta$ in terms of $\tan \theta$ and use this result to show that $$\tan 3\theta = \frac{3 \tan \theta - \tan^3 \theta}{1 - 3 \tan^2 \theta}.$$ [4]
Given that $0 < \theta < \frac{1}{2}\pi$ and $\theta = \sin^{-1}\left(\frac{1}{\sqrt{10}}\right)$, show that $\tan 3\theta = \frac{13}{3}$. [3]
Show that the solutions of the equation $$\tan(3 \sin^{-1} x) = \frac{13}{3}$$ for $0 < x < 2\pi$ are $$x = \frac{\sqrt{10}}{10} \quad \text{and} \quad x = \frac{\sqrt{10(1 + 3\sqrt{3})}}{20}.$$ [4]

Pre-U Pre-U 9794/1 2011 June Q9

9 marks Standard +0.8

Prove that $\sin 3\theta = 3 \sin \theta - 4 \sin^3 \theta$ and deduce that $$\sin \theta + \sin 3\theta = 4 \sin \theta \cos^2 \theta.$$ [5]
Hence find the values of $\theta$ such that $0° < \theta < 180°$ that satisfy the equation $$\cot^2 \theta = \sin \theta + \sin 3\theta.$$ [4]

Pre-U Pre-U 9794/2 2011 June Q7

9 marks Moderate -0.3

Functions f, g and h are defined for $x \in \mathbb{R}$ by $$f : x \mapsto x^2 - 2x,$$ $$g : x \mapsto x^2,$$ $$h : x \mapsto \sin x.$$

1. State whether or not f has an inverse, giving a reason. [2]
2. Determine the range of the function f. [2]
1. Show that gh(x) can be expressed as $\frac{1}{2}(1 - \cos 2x)$. [2]
2. Sketch the curve C defined by $y = \text{gh}(x)$ for $0 \leqslant x \leqslant 2\pi$. [3]

Pre-U Pre-U 9794/2 2012 June Q11

15 marks Challenging +1.2

The function f is defined by $f : t \mapsto 2 \sin t + \cos 2t$ for $0 \leqslant t < 2\pi$.

Show that $\frac{df}{dt} = 2 \cos t(1 - 2 \sin t)$. [2]
Determine the range of f. [5]

A curve $C$ is given parametrically by $x = 2 \cos t + \sin 2t$, $y = f(t)$ for $0 \leqslant t < 2\pi$.

Show that $x^2 + y^2 = 5 + 4 \sin 3t$. [3]
Deduce that $C$ lies between two circles centred at the origin, and touches both. [2]
Find the gradient of the tangent to $C$ at the point at which $t = 0$. [3]

Pre-U Pre-U 9795/1 2013 November Q2

5 marks Standard +0.3

Use de Moivre's theorem to express $\cos 3\theta$ in terms of powers of $\cos \theta$ only, and deduce the identity $\cos 6x \equiv \cos 2x(2\cos 4x - 1)$. [5]

Pre-U Pre-U 9794/2 Specimen Q7