The following diagram shows the graph of $y = g(x)$ - HSC - SSCE Mathematics Extension 2 - Question 3 - 2008 - Paper 1

For $y = \frac{1}{g(x)}$, identify where $g(x) = 0$ since this will create vertical asymptotes in the graph. For regions where $g(x)$ is positive, plot $y$ values that are the reciprocals of $g(x)$. For regions where $g(x)$ is negative, the graph will also be plotted but will be negative.

For the graph of $y = f(x)$, for $x \geq 1$, use the graph of $g(x)$ directly. For $x < 1$, calculate $f(x)$ as $\frac{g(2 - x)}{g(x)}$. This will create a piecewise function with the characteristics of both parts of $g(x)$.

To show that the polynomial $p(z) = 1 + z^2 + z^3$ has no real zeros, note that the first derivative, $p'(z) = 2z + 3z^2$, has real roots that can be computed to identify critical points. Examining $p(z)$ at these points and at the boundaries shows that it does not cross the x-axis, hence confirming no real zeros.

Assuming $\beta$ is a root of $p(z)$, apply the roots of unity concept. As $p(z)$ relates to the roots of unity, show algebraically that raising $\beta$ to the sixth power results in $1$. This can involve showing that multiplying out the roots leads back to the initial polynomial, thereby confirming $\beta^6 = 1$.

With the established relation that $\beta^6 = 1$, compute $p(\beta^2)$. Through polynomial evaluation, confirm that substituting $\beta^2$ results in $0$, proving that $\beta^2$ is indeed another root of $p(z)$.

Using integration by parts, let $u = \tan^{n-1} \theta$ and $dv = \tan \theta \, d\theta$. Differentiate and integrate appropriately, while simplifying the integral to isolate $I_n$ and relate it back to $I_{n-1}$. This verifies the required relationship.

Starting from the relationship derived in part (i), recursively calculate $I_3$ using the values computed from $I_2$, $I_1$, and the known value of $I_0$. This step-by-step calculation will yield the numeric value of $I_3$.

Applying Newton's second law in both horizontal and vertical directions leads to two equations. Use the sum of forces to express tension and gravitational force in terms of $\,\ell$ and angles involved. Conclusively, derive the expression for $\omega^2$ through simplification and substitution of forces.