Question 6 - A-Level Maths

CAIE P3 2015 November — Question 6 8 marks

Exam Board	CAIE
Module	P3 (Pure Mathematics 3)
Year	2015
Session	November
Marks	8
Paper	Download PDF ↗
Mark scheme	Download PDF ↗
Topic	Addition & Double Angle Formulae
Type	Two angles with tan relationships
Difficulty	Standard +0.8 This question requires applying addition formulae to derive tan A from the first equation, solving a quadratic in tan B from the second equation (using sec²B = 1 + tan²B), then applying the tan(A-B) formula. While the techniques are standard A-level content, the multi-step nature, algebraic manipulation with surds, and need to combine results from two independent parts makes this moderately challenging but still within typical P3 scope.
Spec	1.05h Reciprocal trig functions: sec, cosec, cot definitions and graphs 1.05j Trigonometric identities: tan=sin/cos and sin^2+cos^2=1 1.05l Double angle formulae: and compound angle formulae 1.05o Trigonometric equations: solve in given intervals

6 The angles $A$ and $B$ are such that $$\sin \left( A + 45 ^ { \circ } \right) = ( 2 \sqrt { } 2 ) \cos A \quad \text { and } \quad 4 \sec ^ { 2 } B + 5 = 12 \tan B$$ Without using a calculator, find the exact value of $\tan ( A - B )$.

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Answer	Marks	Guidance
State or imply $\sin A \times \cos 45° + \cos A \times \sin 45° = 2\sqrt{2}\cos A$	B1
Divide by $\cos A$ to find value of $\tan A$	M1
Obtain $\tan A = 3$	A1
Use identity $\sec^2 B = 1 + \tan^2 B$	B1
Solve three-term quadratic equation and find $\tan B$	M1
Obtain $\tan B = \frac{3}{2}$ only	A1
Substitute numerical values in $\frac{\tan A - \tan B}{1 + \tan A \tan B}$	M1
Obtain $\frac{3}{11}$	A1	[8]

State or imply $\sin A \times \cos 45° + \cos A \times \sin 45° = 2\sqrt{2}\cos A$ | B1 |
Divide by $\cos A$ to find value of $\tan A$ | M1 |
Obtain $\tan A = 3$ | A1 |
Use identity $\sec^2 B = 1 + \tan^2 B$ | B1 |
Solve three-term quadratic equation and find $\tan B$ | M1 |
Obtain $\tan B = \frac{3}{2}$ only | A1 |
Substitute numerical values in $\frac{\tan A - \tan B}{1 + \tan A \tan B}$ | M1 |
Obtain $\frac{3}{11}$ | A1 | [8]

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6 The angles $A$ and $B$ are such that

$$\sin \left( A + 45 ^ { \circ } \right) = ( 2 \sqrt { } 2 ) \cos A \quad \text { and } \quad 4 \sec ^ { 2 } B + 5 = 12 \tan B$$

Without using a calculator, find the exact value of $\tan ( A - B )$.

\hfill \mbox{\textit{CAIE P3 2015 Q6 [8]}}

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Q1 2 Q2 3 Q3 6 Q4 8 Q5 8 Q6 8 Q7 10 Q8 10 Q9 10 Q10 10

Answer	Marks	Guidance
State or imply \(\sin A \times \cos 45° + \cos A \times \sin 45° = 2\sqrt{2}\cos A\)	B1
Divide by \(\cos A\) to find value of \(\tan A\)	M1
Obtain \(\tan A = 3\)	A1
Use identity \(\sec^2 B = 1 + \tan^2 B\)	B1
Solve three-term quadratic equation and find \(\tan B\)	M1
Obtain \(\tan B = \frac{3}{2}\) only	A1
Substitute numerical values in \(\frac{\tan A - \tan B}{1 + \tan A \tan B}\)	M1
Obtain \(\frac{3}{11}\)	A1	[8]