Question 4 - A-Level Maths

CAIE P3 2004 November — Question 4 7 marks

Exam Board	CAIE
Module	P3 (Pure Mathematics 3)
Year	2004
Session	November
Marks	7
Paper	Download PDF ↗
Mark scheme	Download PDF ↗
Topic	Addition & Double Angle Formulae
Type	Solve equation with tan(θ ± α)
Difficulty	Standard +0.3 This is a straightforward application of the tan(A±B) addition formula followed by algebraic manipulation and solving a quadratic. Part (i) is guided ('show that'), requiring routine expansion and simplification. Part (ii) is direct quadratic formula application with domain restriction. Slightly easier than average due to the scaffolding and standard technique.
Spec	1.05l Double angle formulae: and compound angle formulae 1.05o Trigonometric equations: solve in given intervals

Show that the equation $$\tan \left( 45 ^ { \circ } + x \right) = 2 \tan \left( 45 ^ { \circ } - x \right)$$ can be written in the form $$\tan ^ { 2 } x - 6 \tan x + 1 = 0$$
Hence solve the equation $\tan \left( 45 ^ { \circ } + x \right) = 2 \tan \left( 45 ^ { \circ } - x \right)$, for $0 ^ { \circ } < x < 90 ^ { \circ }$.

Show mark scheme Show mark scheme source

Answer	Marks	Guidance
(i) EITHER: Use $\tan(A \pm B)$ formula correctly to obtain an equation in $\tan x$	M1
State or imply the equation $\frac{1+\tan x}{1-\tan x} = \frac{2(1-\tan x)}{1+\tan x}$ or equivalent	A1
Transform to an expanded horizontal quadratic equation in $\tan x$	M1
Obtain given answer correctly	A1
OR: Use $\sin(A \pm B)$ and $\cos(A \pm B)$ formulae correctly to obtain an equation in $\sin x$ and $\cos x$	M1
Using values of $\sin 45°$ and $\cos 45°$, or their equality, obtain an expanded horizontal equation in $\sin x$ and $\cos x$	A1
Transform to a quadratic equation in $\tan x$	M1
Obtain given answer correctly	A1	Total: 4 marks
(ii) Solve the given quadratic and calculate an angle in degrees or radians	M1
Obtain one answer e.g. $80.3°$	A1
Obtain second answer $9.7°$ and no others in the range	A1	[Ignore answers outside the given range.]
Total: 3 marks

**(i)** **EITHER:** Use $\tan(A \pm B)$ formula correctly to obtain an equation in $\tan x$ | M1 | |
State or imply the equation $\frac{1+\tan x}{1-\tan x} = \frac{2(1-\tan x)}{1+\tan x}$ or equivalent | A1 | |
Transform to an expanded horizontal quadratic equation in $\tan x$ | M1 | |
Obtain given answer correctly | A1 | |

**OR:** Use $\sin(A \pm B)$ and $\cos(A \pm B)$ formulae correctly to obtain an equation in $\sin x$ and $\cos x$ | M1 | |
Using values of $\sin 45°$ and $\cos 45°$, or their equality, obtain an expanded horizontal equation in $\sin x$ and $\cos x$ | A1 | |
Transform to a quadratic equation in $\tan x$ | M1 | |
Obtain given answer correctly | A1 | **Total: 4 marks** |

**(ii)** Solve the given quadratic and calculate an angle in degrees or radians | M1 | |
Obtain one answer e.g. $80.3°$ | A1 | |
Obtain second answer $9.7°$ and no others in the range | A1 | [Ignore answers outside the given range.] |
| | | **Total: 3 marks** |

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4 (i) Show that the equation

$$\tan \left( 45 ^ { \circ } + x \right) = 2 \tan \left( 45 ^ { \circ } - x \right)$$

can be written in the form

$$\tan ^ { 2 } x - 6 \tan x + 1 = 0$$

(ii) Hence solve the equation $\tan \left( 45 ^ { \circ } + x \right) = 2 \tan \left( 45 ^ { \circ } - x \right)$, for $0 ^ { \circ } < x < 90 ^ { \circ }$.

\hfill \mbox{\textit{CAIE P3 2004 Q4 [7]}}

This paper (10 questions)

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Q1 4 Q2 4 Q3 6 Q4 7 Q5 8 Q6 8 Q7 9 Q8 9 Q9 10 Q10 10

Answer	Marks	Guidance
(i) EITHER: Use \(\tan(A \pm B)\) formula correctly to obtain an equation in \(\tan x\)	M1
State or imply the equation \(\frac{1+\tan x}{1-\tan x} = \frac{2(1-\tan x)}{1+\tan x}\) or equivalent	A1
Transform to an expanded horizontal quadratic equation in \(\tan x\)	M1
Obtain given answer correctly	A1
OR: Use \(\sin(A \pm B)\) and \(\cos(A \pm B)\) formulae correctly to obtain an equation in \(\sin x\) and \(\cos x\)	M1
Using values of \(\sin 45°\) and \(\cos 45°\), or their equality, obtain an expanded horizontal equation in \(\sin x\) and \(\cos x\)	A1
Transform to a quadratic equation in \(\tan x\)	M1
Obtain given answer correctly	A1	Total: 4 marks
(ii) Solve the given quadratic and calculate an angle in degrees or radians	M1
Obtain one answer e.g. \(80.3°\)	A1
Obtain second answer \(9.7°\) and no others in the range	A1	[Ignore answers outside the given range.]
Total: 3 marks