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

The volume $V$ of the solid of revolution is given by the formula:

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

Using integration techniques, we find the value of $k$ that satisfies $V = \pi^2$. After determining the integral and simplifying, we set the equation to solve for $k$.

The function $f(x) = \sin(x)$ is not injective across the entire real line as it oscillates. Therefore, $f^2(x) = \sin^2(x)$ does not have a unique inverse on $\mathbb{R}$, which means $g(x) = \arcsin(x)$ cannot be the inverse of $f^2$. We conclude that they are not inverses.

We use the identity for the sum of squares:

$\alpha^2 + \beta^2 + \gamma^2 = 85$

and the relation for the derivatives gives us a second equation. With these, we can express the desired sum using the relationship between the roots of the polynomial. Solving gives us the total as needed.

Using the binomial approximation, we assume the proportion is $p = 0.2$. Therefore, using the binomial distribution, we find

$P_p = P(X \geq 8)$

We then utilize the normal approximation and standard calculations to arrive at the result.

The normal approximation may not be valid if the sample size is small or if the probability of success ($p$) is not moderate, which could lead to inaccuracies in the estimation. The inspectors' approach assumes a larger sample distribution which may not reflect the factory's actual production.