CAIE P3 2010 June — Question 3 5 marks

Exam BoardCAIE
ModuleP3 (Pure Mathematics 3)
Year2010
SessionJune
Marks5
PaperDownload PDF ↗
Mark schemeDownload PDF ↗
TopicStandard trigonometric equations
TypeConvert to quadratic in tan
DifficultyStandard +0.3 This requires applying the tangent subtraction formula, then solving a quadratic equation in tan x. It's a standard multi-step trigonometric equation with straightforward algebraic manipulation, slightly above average due to the formula application and quadratic solving, but still routine for P3 level.
Spec1.05l Double angle formulae: and compound angle formulae1.05o Trigonometric equations: solve in given intervals

Solve the equation $$\tan(45° - x) = 2\tan x,$$ giving all solutions in the interval \(0° < x < 180°\). [5]

AnswerMarks Guidance
Attempt to use \(\tan(A \pm B)\) formula and obtain an equation in \(\tan x\)M1
Obtain 3-term quadratic \(2\tan^2 x + 3\tan x - 1 = 0\), or equivalentA1
Solve a 3-term quadratic and find a numerical value of \(x\)M1
Obtain answer \(15.7°\)A1
Obtain answer \(119.3°\) and no others in the given intervalA1 [5]
[Ignore answers outside the given interval. Treat answers in radians, 0.274 and 2.08, as a misread.]
Attempt to use $\tan(A \pm B)$ formula and obtain an equation in $\tan x$ | M1 | —

Obtain 3-term quadratic $2\tan^2 x + 3\tan x - 1 = 0$, or equivalent | A1 | —

Solve a 3-term quadratic and find a numerical value of $x$ | M1 | —

Obtain answer $15.7°$ | A1 | —

Obtain answer $119.3°$ and no others in the given interval | A1 | [5]

| [Ignore answers outside the given interval. Treat answers in radians, 0.274 and 2.08, as a misread.] | — | —

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Solve the equation
$$\tan(45° - x) = 2\tan x,$$
giving all solutions in the interval $0° < x < 180°$. [5]

\hfill \mbox{\textit{CAIE P3 2010 Q3 [5]}}