The point P divides the interval from A(−4,−4) to B(1, 6) internally in the ratio 2:3 - HSC - SSCE Mathematics Extension 1 - Question 11 - 2017 - Paper 1

Let y = \tan^{-1}(x^2).

Using the chain rule to differentiate:

$\frac{dy}{dx} = \frac{1}{1 + (x^2)^2} \cdot \frac{d}{dx}(x^2) = \frac{2x}{1 + x^4}$.

Hence, the derivative of tan^−1(x^2) is (\frac{2x}{1 + x^4}).

To solve the inequality (\frac{2x}{x + 1} > 1), first rewrite it:

$2x > x + 1$

Subtracting x from both sides gives:

$x > 1$

Next, consider the point where the expression is undefined, which is at x = -1. Thus, the solution is:

$x > 1$.

The function (y = 2 , \cos^{-1}(x)) has a range of [0, 2\pi] and is only defined for x in the interval [−1, 1]. The graph will have concave segments, centered at (y = \pi) for x = 0, and approach the values 0 and 2\pi at x = −1 and x = 1, respectively. A rough sketch shows a downward-opening curve with these features.

Using the substitution (x = u^2 - 1), we find:

1. When x = 0, u = 1.
2. When x = 3, u = 2.

Next, compute:

$dx = 2u \, du$

and the integral becomes:

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

This simplifies to:

$\int_{1}^{2}(2u)(u) \, du$.

Evaluating gives:

$\left[\frac{2}{3}u^3\right]_{1}^{2} = \frac{2}{3}(8 - 1) = \frac{14}{3}.$

Thus, the evaluation of the integral is (\frac{14}{3}).

Using the identity for sin^2, we solve the integral:

$\int \sin^2(x) \cos(x) \; dx$.

Letting u = sin(x), then du = cos(x) dx:

$\int u^2 \, du = \frac{u^3}{3} + C = \frac{\sin^3(x)}{3} + C.$

Thus, the result is (\frac{\sin^3(x)}{3} + C).

This problem follows the binomial probability formula:

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

where n = 8, k = 3, and p = \frac{1}{5}.

The probability expression is:

$P(X = 3) = \binom{8}{3} \left(\frac{1}{5}\right)^3 \left(\frac{4}{5}\right)^{5}.$

Calculating gives the probability of exactly three seedlings producing red flowers.

Using the same binomial formula, with k = 0:

$P(X = 0) = \binom{8}{0} \left(\frac{1}{5}\right)^0 \left(\frac{4}{5}\right)^{8} = \left(\frac{4}{5}\right)^{8}.$

This is the expression for the probability that none produce red flowers.

The probability of at least one is given by:

$P(X \geq 1) = 1 - P(X = 0)$

Thus:

$P(X \geq 1) = 1 - \left(\frac{4}{5}\right)^{8}.$

This represents the desired probability.