Three different points A, B and C are chosen on a circle centred at O - HSC - SSCE Mathematics Extension 1 - Question 13 - 2022 - Paper 1

To find $k$, we begin with the volume formula for a solid of revolution. The volume $V$ is given by:

$V = \pi \int_0^{\frac{\pi}{2k}} (k + 1)\sin(kx)^2 \, dx.$

Using the double angle identity, we can simplify this integral. Set up the appropriate integral and solve for $k$ such that $V = \pi^2$. Now, use numerical or analytical methods to get the specific value of $k$.

The function $f^2$ is defined as $f^2(x) = \sin^2(x)$. Since $g(x) = \arcsin(x)$, we must check if $g(f^2(x)) = x$ is true.

1. Compute $g(f^2(x)) = \arcsin(\sin^2(x))$.
2. However, $ext{arcsin}$ is not defined for values outside of $[-1, 1]$ and $ext{sin}^2(x)$ will not cover the full range continuously over all $x$. Thus, $g$ does not serve as an inverse function for $f^2$.

From Vieta’s formulas for the polynomial $P_t$, we know:

1. $\alpha + \beta + \gamma = s$ (where $s$ is the sum of the roots),
2. To find $\alpha\beta + \beta\gamma + \gamma\alpha$, use the identity: $\alpha\beta + \beta\gamma + \gamma\alpha = \frac{s^2 - 85}{2}.$
3. Substitute the value from the identity and solve for the desired sum.

Using the normal approximation:

1. Let $p = 0.2$ (the proportion of bars weighing less than 150g). We calculate: $P_p = P(X \geq 8) = 1 - P(X < 8) = 1 - \text{binom}(16,8) \left(0.2\right)^8 \left(0.8\right)^{16-8}.$

2. Find the values and evaluate.

In point (ii), the method might not be valid because the normal approximation assumes a large sample size. Here, with only 16 trials, the approximation may lead to inaccuracies.