Use a SEPARATE writing booklet - HSC - SSCE Mathematics Extension 1 - Question 11 - 2014 - Paper 1

The number of days it rains can be modeled by a binomial distribution with parameters ( n = 30 ) and ( p = 0.1 ). The expression for the probability that it rains on fewer than 3 days is:

$$P(X < 3) = P(X = 0) + P(X = 1) + P(X = 2)$$

where ( P(X = k) = \binom{n}{k} p^k (1-p)^{n-k}. )

To sketch the graph of ( y = 6\tan^{-1}(x) ):

1. The function (\tan^{-1}(x)) approaches ( \frac{\pi}{2} ) as ( x ) increases. Therefore, ( y ) approaches ( 6 \cdot \frac{\pi}{2} = 3\pi \approx 9.42 )
2. For ( x \to -\infty ), ( y \to -3\pi
3. The range of the graph is ( (-3\pi, 3\pi) ).

Using the substitution ( x = u^2 + 1 ):

1. Then ( dx = 2u , du ) and the limits change as follows:
   • For ( x = 2 ), ( u = 1 )
   • For ( x = 5 ), ( u = 2 ).
2. The integral becomes:

$$\int_{1}^{2} \frac{u^2 + 1}{\sqrt{u^2}} (2u) \, du = 2\int_{1}^{2} (u + \frac{1}{u}) \,du$$

3. Evaluating gives:

= 2\left[ \frac{u^2}{2} + \ln|u| \right]_{1}^{2} = 2\left( 2 + \ln(2) - \frac{1}{2} - 0 \right) = 3 + 2\ln(2)\.

Squaring both sides gives:

$$\sqrt{x^2 + 5} > 6 \\ \implies x^2 + 5 > 36 \\ \implies x^2 > 31 \\ \implies x > \sqrt{31} \, \text{ or } \, x < -\sqrt{31}.$$

Using the product rule for differentiation,

$$\frac{d}{dx}\left( e^{x} \ln x \right) = e^{x} \ln x + e^{x} \cdot \frac{1}{x}.$$

Thus, the derivative is ( e^{x} \ln x + \frac{e^{x}}{x} ).