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 generated by rotating the region is given by the integral:

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

To find k, we solve the equation by using the known formula for volume and setting it equal to $ext{π}^2$, leading us to the equation after simplification.

To determine if $g$ is the inverse of $f^2$, we evaluate:

$f(g(x)) = f( ext{arcsin}(x)) = x$

Since this holds for all $x$ in the range of $[-1, 1]$, we conclude that $g$ is not technically the inverse of $f^2$, as $f^2$ does not cover the entire domain required for such a relationship.

Using the relations given:

$P(a) + P(b) + P(c) = α^3 + 3α^2β + 3αβ^2 + β^3 + \cdots$ Substituting the known sums, we analyze:

$αβ + βγ + αγ = \frac{(α^2 + β^2 + γ^2)^2 - (α^4 + β^4 + γ^4)}{2}$ Determining values under the constraints leads to the derived result.

The inspectors' method assumes normality in the distribution of bar weights, which might not be justifiable if the sample is biased (e.g., if the 16 bars selected are not representative of the entire production). Additionally, the use of a binomial approximation may not hold if the probabilities do not adhere to the conditions necessary for such normal approximations to apply.