Standard +0.3 This is a straightforward stationary point problem requiring differentiation of standard trigonometric functions (tan x and sin x), setting the derivative to zero, and solving a trigonometric equation. While it involves transcendental functions and numerical solving, it's a standard textbook exercise with no novel insight required, making it slightly easier than average.
1 A curve has equation \(\mathrm { y } = 2 \tan \mathrm { x } - 5 \sin \mathrm { x }\) for \(0 \leqslant x < \frac { 1 } { 2 } \pi\).
Find the \(x\)-coordinate of the stationary point of the curve. Give your answer correct to 3 significant figures.
Equate first derivative to zero and solve correctly at least as far as \(\cos x = \ldots\)
M1
B1 M1 for \(x = 42.5°\)
Obtain \(\cos x = \sqrt[3]{0.4}\) and hence \(x = 0.742\)
A1
Or greater accuracy \((0.74261\ldots)\)
Total
3
## Question 1:
| Answer | Marks | Guidance |
|--------|-------|----------|
| Differentiate to obtain $2\sec^2 x - 5\cos x$ | B1 | |
| Equate first derivative to zero and solve correctly at least as far as $\cos x = \ldots$ | M1 | B1 M1 for $x = 42.5°$ |
| Obtain $\cos x = \sqrt[3]{0.4}$ and hence $x = 0.742$ | A1 | Or greater accuracy $(0.74261\ldots)$ |
| **Total** | **3** | |
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1 A curve has equation $\mathrm { y } = 2 \tan \mathrm { x } - 5 \sin \mathrm { x }$ for $0 \leqslant x < \frac { 1 } { 2 } \pi$.\\
Find the $x$-coordinate of the stationary point of the curve. Give your answer correct to 3 significant figures.\\
\hfill \mbox{\textit{CAIE P2 2024 Q1 [3]}}