CAIE P3 2024 November — Question 7 6 marks

Exam BoardCAIE
ModuleP3 (Pure Mathematics 3)
Year2024
SessionNovember
Marks6
PaperDownload PDF ↗
Mark schemeDownload PDF ↗
TopicStandard trigonometric equations
TypeConvert to quadratic in tan
DifficultyStandard +0.3 Part (a) requires applying the tan(2x) double angle formula and algebraic manipulation—straightforward but multi-step. Part (b) applies the result with a substitution (let u=2θ) and solving a cubic that factors nicely, then finding angles in the given range. Slightly above average due to the algebraic manipulation and multiple solution finding, but uses standard techniques throughout.
Spec1.05l Double angle formulae: and compound angle formulae1.05o Trigonometric equations: solve in given intervals
  1. Show that the equation \(\tan^3 x + 2 \tan 2x - \tan x = 0\) may be expressed as $$\tan^3 x - 2 \tan^2 x - 3 = 0$$ for \(\tan x \neq 0\). [3]
  2. Hence solve the equation \(\tan^3 2\theta + 2 \tan 4\theta - \tan 2\theta = 0\) for \(0 < \theta < \pi\). Give your answers in exact form. [3]
Question 7:
AnswerMarks Guidance
7(a)Use correct double angle formula to obtain an equation in tan x M1
e.g. tan3x+ −tanx(=0).
1−tan2x
Allow if the correct formula is quoted but then
they lose the 2 from the numerator when they
use the formula.
AnswerMarks Guidance
Obtain a correct equation in tan x in any form without fractionsA1 E.g. tan3x−tan5x+4tanx−tanx+tan3x(=0).
Condone if ‘= 0’ is missing here.
AnswerMarks Guidance
Reduce to the given answer of tan4x−2tan2x−3=0correctlyA1 Obtain given answer from correct working but
condone if never mention tanx0.
Condone the right terms in a different order
‘Show that’ so each line must be correct.
3
AnswerMarks Guidance
7(b)A complete correct method to solve the equation to obtain a value for  M1
tan2= 3
Allow if they make a slip in copying the
equation but do have a complete method to
obtain a value of .
M0 if they get a value for 2but never halve it.
1 1 2 5
Obtain two of ( =) , ,  and 
AnswerMarks
6 3 3 6A1
1 1 2 5
Obtain the other two of(=) , ,  and  and no others in the interval
AnswerMarks Guidance
6 3 3 6A1 Exact, ignore any answers outside interval
2 1 4 2
Accept  for  and  for 
6 3 6 3
Do not need to see= (from tan2=0).
2
3
AnswerMarks Guidance
QuestionAnswer Marks 
Question 7:
--- 7(a) ---
7(a) | Use correct double angle formula to obtain an equation in tan x | M1 | 22tanx
e.g. tan3x+ −tanx(=0).
1−tan2x
Allow if the correct formula is quoted but then
they lose the 2 from the numerator when they
use the formula.
Obtain a correct equation in tan x in any form without fractions | A1 | E.g. tan3x−tan5x+4tanx−tanx+tan3x(=0).
Condone if ‘= 0’ is missing here.
Reduce to the given answer of tan4x−2tan2x−3=0correctly | A1 | Obtain given answer from correct working but
condone if never mention tanx0.
Condone the right terms in a different order
‘Show that’ so each line must be correct.
3
--- 7(b) ---
7(b) | A complete correct method to solve the equation to obtain a value for  | M1 | ( )
tan2= 3
Allow if they make a slip in copying the
equation but do have a complete method to
obtain a value of .
M0 if they get a value for 2but never halve it.
1 1 2 5
Obtain two of ( =) , ,  and 
6 3 3 6 | A1
1 1 2 5
Obtain the other two of(=) , ,  and  and no others in the interval
6 3 3 6 | A1 | Exact, ignore any answers outside interval
2 1 4 2
Accept  for  and  for 
6 3 6 3

Do not need to see= (from tan2=0).
2
3
Question | Answer | Marks | Guidance
\begin{enumerate}[label=(\alph*)]
\item Show that the equation $\tan^3 x + 2 \tan 2x - \tan x = 0$ may be expressed as
$$\tan^3 x - 2 \tan^2 x - 3 = 0$$
for $\tan x \neq 0$. [3]

\item Hence solve the equation $\tan^3 2\theta + 2 \tan 4\theta - \tan 2\theta = 0$ for $0 < \theta < \pi$. Give your answers in exact form. [3]
\end{enumerate}

\hfill \mbox{\textit{CAIE P3 2024 Q7 [6]}}
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