A solid is formed by rotating the region bounded by the curve $y=x(x-1)^2$ and the line $y=0$ about the y-axis - HSC - SSCE Mathematics Extension 2 - Question 5 - 2006 - Paper 1

To show the identity, we start from the left-hand side:

$\cos(\alpha + \beta) + \cos(\alpha - \beta)$

Using the cosine addition formulas:

$= \cos \alpha \cos \beta - \sin \alpha \sin \beta + \cos \alpha \cos \beta + \sin \alpha \sin \beta$

Combining like terms gives:

$= 2 \cos \alpha \cos \beta$

Using the result from part (i), we can substitute into the equation:

$\cos^4 \theta + \cos 2\theta + \cos 3\theta + \cos 4\theta = 0$

This could be simplified using known identities. We convert each term into cosines of related angles:

Relevant identities include:

• $\cos 2\theta = 2\cos^2 \theta - 1$
• $\cos 4\theta = 2\cos^2 2\theta - 1$

From here, we can solve for $\theta$ based on the simplifications.

To resolve the forces acting on particle P, we analyze the string tensions:

• In the vertical direction: $T_1 \cos(\alpha) + T_2 \cos(\beta) = mg$
• In the horizontal direction: $T_1 \sin(\alpha) = T_2 \sin(\beta)$

This gives us a system of equations representing the forces on P.

Given that $T_2 = 0$, we simplify the horizontal force equation:

$T_1 \sin(\alpha) = 0$

Thus, in terms of angular velocity:

$T_1 = mg \Rightarrow T_1 = m\ell\omega^2 \sin(\alpha)\Rightarrow \omega = \sqrt{\frac{g}{\ell \sin(\alpha)}}$

Each board can yield three outcomes: Win (W), Draw (D), or Lose (L). For four boards, the total number of recordings possible is:

$3^4 = 81.$

To calculate the probability of the specific result WDLD, we compute:

• Probability of Win on board 1: 0.2
• Probability of Draw on board 2: 0.6
• Probability of Loss on board 3: 0.2
• Probability of Draw on board 4: 0.6

Hence, $P(WDLD) = 0.2 \times 0.6 \times 0.2 \times 0.6 = 0.0144.$

To find the probability that the Home team scores more points than the Away team, we calculate:

• Points from a Win = 1
• Points from a Draw = 0.5

Considering every possible combination of outcomes, we sum the probabilities of arrangements where the Home team has more points than the Away team based on the scoring rules, analyzing all scoring scenarios.