Standard +0.3 This question requires applying the tan addition formula, converting cot to tan, and solving a quadratic equation—all standard A-level techniques. While it involves multiple steps and algebraic manipulation, the approach is clearly signposted ('by first expressing...as a quadratic') and uses routine methods without requiring novel insight. The 6-mark allocation and straightforward domain make it slightly above average difficulty but well within typical P3 expectations.
Use correct trig formulae to obtain an equation in tan x
*M1
Using tan45°=1, obtain a horizontal equation in tanx in any form
DM1
tan2x+tanx−1=0,
Answer
Marks
Guidance
Reduce the equation to or 3-term equivalent
A1
Solve a 3-term quadratic in tan x, for x
M1
Obtain answer, e.g. x = 31.7°
A1
Obtain second answer, e.g. x=121.7°, and no other in the interval
A1
Ignore answers outside the given interval.
6
Answer
Marks
Guidance
Question
Answer
Marks
Question 3:
3 | Use correct trig formulae to obtain an equation in tan x | *M1
Using tan45°=1, obtain a horizontal equation in tanx in any form | DM1
tan2x+tanx−1=0,
Reduce the equation to or 3-term equivalent | A1
Solve a 3-term quadratic in tan x, for x | M1
Obtain answer, e.g. x = 31.7° | A1
Obtain second answer, e.g. x=121.7°, and no other in the interval | A1 | Ignore answers outside the given interval.
6
Question | Answer | Marks | Guidance
By first expressing the equation $\tan(x + 45°) = 2 \cot x + 1$ as a quadratic equation in $\tan x$, solve the equation for $0° < x < 180°$. [6]
\hfill \mbox{\textit{CAIE P3 2021 Q3 [6]}}