CAIE P3 2013 June — Question 3 5 marks

Exam BoardCAIE
ModuleP3 (Pure Mathematics 3)
Year2013
SessionJune
Marks5
PaperDownload PDF ↗
Mark schemeDownload PDF ↗
TopicAddition & Double Angle Formulae
TypeSolve equation with tan2x or mixed tan/double angle
DifficultyStandard +0.3 This question requires applying the double angle formula for tan(2x) and converting cot(x) to 1/tan(x), then solving a quadratic equation in tan(x). It's a straightforward application of standard formulae with algebraic manipulation, slightly above average difficulty due to the multiple steps involved, but follows a predictable pattern for this topic.
Spec1.05l Double angle formulae: and compound angle formulae1.05o Trigonometric equations: solve in given intervals

3 Solve the equation \(\tan 2 x = 5 \cot x\), for \(0 ^ { \circ } < x < 180 ^ { \circ }\).

AnswerMarks Guidance
Use correct \(\tan 2A\) formula and \(\cot x = 1/\tan x\) to form an equation in \(\tan x\)M1
Obtain a correct horizontal equation in any formA1
Solve an equation in \(\tan^2 x\) for \(x\)M1
Obtain answer, e.g. \(40.2°\)A1
Obtain second answer, e.g. \(139.8°\), and no other in the given intervalA1 [5 marks total]
Guidance: Ignore answers outside the given interval. Treat answers in radians as a misread and deduct A1 from the marks for the angles. [SR: For the answer \(x = 90°\) give B1 and A1 for one of the other angles.]
Use correct $\tan 2A$ formula and $\cot x = 1/\tan x$ to form an equation in $\tan x$ | M1 |
Obtain a correct horizontal equation in any form | A1 |
Solve an equation in $\tan^2 x$ for $x$ | M1 |
Obtain answer, e.g. $40.2°$ | A1 |
Obtain second answer, e.g. $139.8°$, and no other in the given interval | A1 | [5 marks total]

**Guidance:** Ignore answers outside the given interval. Treat answers in radians as a misread and deduct A1 from the marks for the angles. [SR: For the answer $x = 90°$ give B1 and A1 for one of the other angles.]

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3 Solve the equation $\tan 2 x = 5 \cot x$, for $0 ^ { \circ } < x < 180 ^ { \circ }$.

\hfill \mbox{\textit{CAIE P3 2013 Q3 [5]}}