(a) (i) Write $\sqrt{3}\cos x - \sin x$ in the form $2\cos(\alpha + \theta)$, where $0 < \alpha < \frac{\pi}{2}$ - HSC - SSCE Mathematics Extension 1 - Question 12 - 2013 - Paper 1

We start with the equation $\sqrt{3}\cos x = 1 + \sin x$. Substituting the expression from part (i):

1. Rearranging gives: $\sqrt{3}\cos x - \sin x - 1 = 0.$
2. Knowing the range for $\alpha$ is $(0, \frac{\pi}{2})$, we can analyze the critical points and periodicity of the trigonometric functions involved.
3. Using the Pythagorean identity: $\sin^2 x + \cos^2 x = 1$, we can solve this for specific values within the interval $0 < x < 2\pi$; alternative solutions can include numerical methods or graphical solutions as needed.

To find the volume of the solid formed by rotating the area under $y = \frac{3}{2}\sin x$ from $x = 0$ to $x = \frac{3\pi}{2}$ about the x-axis, we use the washer method:

1. The volume V is given by: $V = \pi \int_{0}^{\frac{3\pi}{2}} \left(\frac{3}{2}\sin x\right)^2 dx = \pi \int_{0}^{\frac{3\pi}{2}} \frac{9}{4}\sin^2 x \,dx.$
2. Using $\sin^2 x = \frac{1 - \cos(2x)}{2}$: $V = \frac{9\pi}{8} \int_{0}^{\frac{3\pi}{2}} (1 - \cos(2x)) \,dx.$
3. Evaluate the integral to find the volume.$$

To find how long it takes for the coffee to drop to 40°C using Newton's law of cooling:

1. The temperature equation is: $T = A + Be^{-kt}.$
2. Given:
   • Initial Temp: $80^{\circ}C \rightarrow A = 22$.
   • After 10 minutes, $T = 60^{\circ}C$ gives:
   • Setup the equation to find B and k based on the constants.
   • Solve for $t$ when $T = 40^{\circ}C$, calculating using logarithms as appropriate.

To show that the perpendicular distance from point P to the line l is:

1. The distance $D(t)$ can be derived using the formula for the distance from a point to a line: $D = \frac{|Ax_1 + By_1 + C|}{\sqrt{A^2 + B^2}},$ where for the line is $y = 2x - 1 \implies A=2, B=-1, C=1$. Here $(x_1, y_1) = (t, t^2+3)$ gives:
2. Substitute values to find D(t): $D(t) = \frac{|2t - (t^2 + 3) + 1|}{\sqrt{2^2 + (-1)^2}}.$
3. Simplifying leads to proving the expression.

The value of t when point P is closest to line l can be found by minimizing D(t):

1. Differentiate $D(t)$.
2. Set the derivative to 0 and solve for t gives the critical points.
3. Verify minimal distance by analyzing the second derivative or comparing distances.

To show that the tangent line at P is parallel to l:

1. Find the derivative of the curve at P to find the slope of the tangent;
2. Compare with the slope of the line l. When they are equal, the tangent is parallel.

The equation $v^2 + 9x^2 = k$ resembles the form of simple harmonic motion.

1. By substituting $k$ and rearranging: $\frac{dx}{dt} = \sqrt{k - 9x^2}.$
2. Recognizing that the motion has harmonics, we define periodicity related back to $\omega = 3$:
3. Therefore, the period is given by: $T = \frac{2\pi}{\omega} = \frac{2\pi}{3}.$