Question 11 (15 marks) Use a SEPARATE writing booklet - HSC - SSCE Mathematics Extension 1 - Question 11 - 2016 - Paper 1

Using the substitution:

Let $u = x - 4$, then:

$x = u + 4$

Also, we have:

$dx = du$

The integral becomes:

$\int \sqrt{x-4} \, dx = \int \sqrt{u} \, du$

Evaluating this:

$\int \sqrt{u} \, du = \frac{2}{3} u^{3/2} + C = \frac{2}{3} (x - 4)^{3/2} + C$

To differentiate $3 \tan^{-1}(2x)$, we use the chain rule. The derivative of ( \tan^{-1}(u) ) is ( \frac{1}{1 + u^2} ) and we substitute $u = 2x$:

$\frac{d}{dx}[3 \tan^{-1}(2x)] = 3 \cdot \frac{1}{1 + (2x)^2} \cdot 2 = \frac{6}{1 + 4x^2}$

We use the fact that ( \lim_{x \to 0} \frac{\sin x}{x} = 1 ). Rewrite the limit:

$\lim_{x \to 0} \frac{2 \sin x \cos x}{3x} = \lim_{x \to 0} \frac{2 \sin x}{3x} \cdot \cos x$

Evaluating:

$= \frac{2}{3} \cdot 1 \cdot 1 = \frac{2}{3}$

First, isolate the fraction:

$\frac{3}{2x + 5} > x \implies 3 > x(2x + 5)$

This leads to:

$3 > 2x^2 + 5x$

Rearranging gives:

$2x^2 + 5x + 3 < 0$

We can use the quadratic formula to find the roots:

$x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} = \frac{-5 \pm \sqrt{5^2 - 4 \cdot 2 \cdot 3}}{2 \cdot 2} = \frac{-5 \pm 1}{4}$

Thus, the roots are $x = -1$ and $x = -3/2$. Testing intervals:

The solution is:

$-3/2 < x < -1$

Let $P(B) = \frac{2}{5}$ and $P(N) = \frac{3}{5}$. The probability of hitting the bullseye exactly once in three attempts (BNN, NBN, NNB) is:

$P(1B) = \binom{3}{1} P(B) P(N)^2 = 3 \cdot \frac{2}{5} \cdot \left( \frac{3}{5} \right)^2 = 3 \cdot \frac{2}{5} \cdot \frac{9}{25} = \frac{54}{125}$

Calculating the complementary event, we find the probabilities for 0 and 1 bullseyes:

• Probability of hitting 0:

$P(0B) = \left( \frac{3}{5} \right)^6 = \frac{729}{15625}$

• Probability of hitting 1:

$P(1B) = \binom{6}{1} \cdot P(B) \cdot P(N)^5 = 6 \cdot \frac{2}{5} \cdot \left( \frac{3}{5} \right)^5 = 6 \cdot \frac{2}{5} \cdot \frac{243}{3125} = \frac{2916}{15625}$

Adding these:

$P(0B) + P(1B) = \frac{729 + 2916}{15625} = \frac{3645}{15625}$

Thus the probability of at least 2 bullseyes is:

$1 - \frac{3645}{15625} = \frac{11980}{15625}$