Question 5 (15 marks) Use a SEPARATE writing booklet - HSC - SSCE Mathematics Extension 2 - Question 5 - 2005 - Paper 1

To find the total number of sequences in which either Mary or Ferdinand can score three goals, we can use combinatorial counting again. If we denote the goals by 'M' and 'F', we would generally require the winner to score the third goal. For cases where Mary wins, we can have configurations of M and F where Mary scores last:

• The last goal must be M, hence the first two goals can be arranged in any order as long as they include 2 'F's and 1 'M':

$\frac{n!}{k!(n-k)!} = \frac{2!}{0!2!} = 1$ for fixed last as M.

The combinations for other sequences will yield different arrangements, ultimately giving a total combination of all arrangements leading to Mary's victory. Hence, the number of ways Mary wins is represented as:

$6$ arrangements. Repeat for Ferdinand:

Thus total: $10$ arrangements of wins (5 for each player in their configurations).

Given that $f(x)$ is an increasing function bounded between 0 and $b$. The area under the curve $f(x)$ from 0 to 'a' can be related to the area under the inverse function from 'b' back to 0 based on the properties of the function. The function's behavior can be leveraged via the Fundamental Theorem of Calculus where the switching of limits includes a change in sign, allowing us to express:

$\int_0^a f(x) dx = A - B = ab - \int_0^b f^{-1}(x) dx,$ where $A = ab$ and $B$ is the area under the inverse function.

To find $\int_0^2 sin^{-1}(x \frac{1}{4}) dx$, we can use integration techniques involving substitution and properties of arcsine. Letting:

$u = sin^{-1}(x \frac{1}{4}),$ we can differentiate and apply techniques involving substitution back into the integral, leading to simplifications through integration by parts if necessary, ultimately producing a solvable integral. Finding the exact bounds gives calculated results leading to:

$\left[ … \right]$ which can be solved to exact value, however requires careful evaluation within ranges through continuity of sine and cosine properties.

In the design of the wedge and following cylindrical geometry, we have set the equation of the circular base as given by the radius r. The area of rectangle ABCD can thus be derived using integration from the circular area, whereby considering the height as function described through the intersection of curve with defined bounds. The expressions yield:

$Area = Width * Height, \text{ where Height is given as } 2y = 2\sqrt{9-x^2}$ substituted to calculate the effective area with respect to distance x, yielding area formula as stated: $2x \sqrt{27 - 3x^2}$.

The volume of the wedge can be calculated by integrating the area of the cross-section along the height of the defined wedge shape. Taking $dx$ from our previous area expression, we can set up the volume integral:

$V = \int_A^{B} Area_{ABCD} \, dx$ by substituting back from the area derived in step (i)

$V = \int_{0}^{h} 2x \sqrt{27 - 3x^2} \; dx.$ This will require applying integration techniques to effectively evaluate from limits yielding the total volume.