Question 2 - A-Level Maths

CAIE P3 2006 November — Question 2 4 marks

Exam Board	CAIE
Module	P3 (Pure Mathematics 3)
Year	2006
Session	November
Marks	4
Paper	Download PDF ↗
Mark scheme	Download PDF ↗
Topic	Addition & Double Angle Formulae
Type	Solve equation with tan2x or mixed tan/double angle
Difficulty	Standard +0.3 This is a straightforward double angle equation requiring the identity tan(2x) = 2tan(x)/(1-tan²(x)), leading to a simple quadratic in tan(x). The solution involves standard algebraic manipulation and finding angles in a specified interval, making it slightly easier than average but requiring more than just recall.
Spec	1.05l Double angle formulae: and compound angle formulae 1.05o Trigonometric equations: solve in given intervals

2 Solve the equation $$\tan x \tan 2 x = 1 ,$$ giving all solutions in the interval $0 ^ { \circ } < x < 180 ^ { \circ }$.

Show mark scheme Show mark scheme source

Answer	Marks
EITHER: Use $\tan 2A$ formula and obtain a horizontal equation in $\tan x$	M1
Simplify the equation to the form $3\tan^2 x = 1$, or equivalents	A1
Obtain answer $30°$	A1
Obtain second answer $150°$ and no others in the range	A1
OR: Use $\sin 2A$ and $\cos 2A$ formulae and obtain a horizontal equation in $\sin x$ or $\cos x$	M1
Simplify the equation to $4\sin^2 x = 1.4\cos^2 x = 3$, or equivalent	A1
Obtain answer $30°$	A1
Obtain second answer $150°$ and no others in the range	A1
[Ignore answers outside the given range]
[Treat answers in radians as a MR and deduct one mark from the marks for the angles.]
[Methods leading to an equation in $\cos 3x$ or $\cos 2x$, or to the equality of two tangents can also earn M1A1, and then A1 + A1 for $30°$ and $150°$ only.]
[SR: If the answer $30°$ is found by inspection or from a graph, and is exactly verified, award B2. If a second answer $150°$ is found and verified, and no others stated, award B2]
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EITHER: Use $\tan 2A$ formula and obtain a horizontal equation in $\tan x$ | M1 |
Simplify the equation to the form $3\tan^2 x = 1$, or equivalents | A1 |
Obtain answer $30°$ | A1 |
Obtain second answer $150°$ and no others in the range | A1 |
OR: Use $\sin 2A$ and $\cos 2A$ formulae and obtain a horizontal equation in $\sin x$ or $\cos x$ | M1 |
Simplify the equation to $4\sin^2 x = 1.4\cos^2 x = 3$, or equivalent | A1 |
Obtain answer $30°$ | A1 |
Obtain second answer $150°$ and no others in the range | A1 |
[Ignore answers outside the given range] | | |
[Treat answers in radians as a MR and deduct one mark from the marks for the angles.] | | |
[Methods leading to an equation in $\cos 3x$ or $\cos 2x$, or to the equality of two tangents can also earn M1A1, and then A1 + A1 for $30°$ and $150°$ only.] | | |
[SR: If the answer $30°$ is found by inspection or from a graph, and is exactly verified, award B2. If a second answer $150°$ is found and verified, and no others stated, award B2] | | |
| | 4 |

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2 Solve the equation

$$\tan x \tan 2 x = 1 ,$$

giving all solutions in the interval $0 ^ { \circ } < x < 180 ^ { \circ }$.

\hfill \mbox{\textit{CAIE P3 2006 Q2 [4]}}

This paper (10 questions)

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Q1 4 Q2 4 Q3 6 Q4 6 Q5 6 Q6 9 Q7 9 Q8 10 Q9 10 Q10 11

Answer	Marks
EITHER: Use \(\tan 2A\) formula and obtain a horizontal equation in \(\tan x\)	M1
Simplify the equation to the form \(3\tan^2 x = 1\), or equivalents	A1
Obtain answer \(30°\)	A1
Obtain second answer \(150°\) and no others in the range	A1
OR: Use \(\sin 2A\) and \(\cos 2A\) formulae and obtain a horizontal equation in \(\sin x\) or \(\cos x\)	M1
Simplify the equation to \(4\sin^2 x = 1.4\cos^2 x = 3\), or equivalent	A1
Obtain answer \(30°\)	A1
Obtain second answer \(150°\) and no others in the range	A1
[Ignore answers outside the given range]
[Treat answers in radians as a MR and deduct one mark from the marks for the angles.]
[Methods leading to an equation in \(\cos 3x\) or \(\cos 2x\), or to the equality of two tangents can also earn M1A1, and then A1 + A1 for \(30°\) and \(150°\) only.]
[SR: If the answer \(30°\) is found by inspection or from a graph, and is exactly verified, award B2. If a second answer \(150°\) is found and verified, and no others stated, award B2]
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