Question
Find the least positive value of $x$ for which $\cos \left(\frac{x}{2}+\frac{\pi}{3}\right)=\frac{1}{\sqrt{2}}$.
Answer/Explanation
determines $\frac{\pi}{4}$ (or $45^{\circ}$ ) as the first quadrant (reference) angle attempts to solve $\frac{x}{2}+\frac{\pi}{3}=\frac{\pi}{4}$
Note: Award $\boldsymbol{M 1}$ for attempting to solve $\frac{x}{2}+\frac{\pi}{3}=\frac{\pi}{4}, \frac{7 \pi}{4}(, \ldots)$
$\frac{x}{2}+\frac{\pi}{3}=\frac{\pi}{4} \Rightarrow x<0$ and so $\frac{\pi}{4}$ is rejected
$\begin{aligned} & \frac{x}{2}+\frac{\pi}{3}=2 \pi-\frac{\pi}{4}\left(=\frac{7 \pi}{4}\right) \\ & x=\frac{17 \pi}{6} \quad \text { (must be in radians) }\end{aligned}$
Question
(a) The expression \(\frac{3\sqrt{x}-5}{\sqrt{x}}\) can be written as 3 – 5x p. Write down the value of p.
(b) Hence, find the value of \(\int_{1}^{9}\left ( \frac{3\sqrt{x}-5}{\sqrt{x}} \right )dx.\)
Answer/Explanation
Ans: (a) \(\frac{3\sqrt{x}-5}{\sqrt{x}} =3-5x^{\frac{1}{2}}\)
\(p = -\frac{1}{2}\)
(b) \(\int \frac{3\sqrt{x}-5}{\sqrt{x}}dx = 3x-10x^{\frac{1}{2}}(+c)\)
substituting limits into their integrated function and subtracting
\(3(9)-10(9)^{\frac{1}{2}}-\left ( 3(1)-10(1)^{\frac{1}{2}} \right )OR 27-10\times 3-(3-10)\)
= 4