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 the value of $k$, we must calculate the volume of the solid of revolution formed by the rotation of the curve around the x-axis.

The volume $V$ is given by:

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

Using the formula for $\\sin^2$:

$\sin^2(kx) = \frac{1 - \cos(2kx)}{2}$

Thus,

$V = \frac{\pi(k + 1)}{2} \left( \frac{\pi}{2k} - \frac{1}{2k} \sin(k \cdot \frac{\pi}{2k})\right)$

Setting $V = \pi^2$, we equate and solve for $k$.

After simplifying this equality, we find that: $k = 2$

To determine if $g$ is the inverse of $f^2$, we need to analyze the domains and ranges of both functions.

The function $f(x) = \sin(x)$ has a range of $[-1, 1]$. When considering $f^2(x)$, the range becomes $[0, 1]$. The function $g(x) = \arcsin(x)$ is defined on $[-1, 1]$ and returns values in $[0, \frac{\pi}{2}]$.

Since $g(x)$ is not defined for the entire range of $f^2$, it follows that $g(x)$ cannot be the inverse of $f^2$.

Given:

• $\alpha^2 + \beta^2 + \gamma^2 = 85$ (1)
• $P'(\alpha) + P'(\beta) + P'(\gamma) = 87$ (2)

From results about the roots of polynomials, we know:

$P'(x) = 3x^2 + bx + c$

Substituting from (2) into the relationship: $87 = 3(\alpha + \beta + \gamma) + 3\alpha\beta + 3\beta\gamma + 3\gamma\alpha$

Setting $s_1 = \alpha + \beta + \gamma$ and substituting back, we derive a set of equations to replace.

Finally, working through the simplifications yields: $\alpha\beta + \beta\gamma + \gamma\alpha = 23$

The method used by the inspectors may not be valid due to the sample size being potentially insufficient to represent the population accurately. The normal approximation to the binomial distribution assumes a large number of trials (typically $n \geq 30$) to apply the Central Limit Theorem.

In this case, with only 16 bars sampled, the assumption of normality may not hold, leading to potential miscalculations of probabilities. Moreover, if the true proportion of bars weighing less than 150 grams differs significantly from the claimed 80%, the inspectors' results could be skewed.