3 (15 marks) Use a SEPARATE writing booklet - HSC - SSCE Mathematics Extension 2 - Question 3 - 2010 - Paper 1

To sketch the graph of $y = \frac{1}{x^2 + 4x}$, we need to find its asymptotes. First, we observe that the denominator can be rewritten as:

$x^2 + 4x = x(x + 4)$

This function has vertical asymptotes where the denominator is zero. Thus, at $x = 0$ and $x = -4$, there are vertical asymptotes. The horizontal asymptote occurs as $x$ approaches infinity, where the value approaches $0$.

This graph will approach the x-axis without touching it, exhibiting typical hyperbolic behavior.

To find the volume generated by rotating the shaded area bounded by $y = 2x - x^2$ about the line $x = 4$, we apply the method of shells:

The volume of a shell is given by:

$V = 2 \pi \int_{a}^{b} (4 - x)(2x - x^2) \, dx$

In this case, the limits of integration $a$ and $b$ will be determined by the points of intersection of $y = 2x - x^2$ with the x-axis, which are $x = 0$ and $x = 2$. Thus:

$V = 2 \pi \int_{0}^{2} (4 - x)(2x - x^2) \, dx$

Integrating, we compute this to find the final volume.

Let the probability of the coin landing heads be $p$, and the probability of landing tails be $q = 1 - p$. The probability of one coin showing heads and the other showing tails is $2pq = 0.48$. Since they are identical coins, we can calculate:

$\text{P(both heads)} = p^2$

We can express $q$ in terms of $p$ using:

$q = 1 - p$

Thus, we have:

$2p(1 - p) = 0.48$

Solving this quadratic equation will give the value of $p$, from which we can find $p^2$, the probability that both coins land showing heads.