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(x²).

To differentiate, we apply the chain rule:

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

$= \frac{1}{1 + x^4} \cdot 2x$

Thus, the derivative is:

$\frac{dy}{dx} = \frac{2x}{1 + x^4}.$

To solve the inequality:

$\frac{2x}{x + 1} > 1,$

we first clear the fraction by multiplying both sides by (x + 1), assuming x + 1 ≠ 0:

$2x > x + 1$

This simplifies to:

$x > 1.$

However, we must check the values for x + 1 > 0 and x + 1 < 0 separately. Thus the solution is:

$x > 1.$

To sketch the graph of the function y = 2 cos⁻¹(x), we note:

• The range of cos⁻¹(x) is [0, π].
• Therefore, the range of y is [0, 2π].
• The domain of cos⁻¹(x) is [-1, 1].
• Thus, the domain of y is also [-1, 1].

At x = -1, y = 2π; at x = 0, y = π; and at x = 1, y = 0. Connect these points smoothly to represent the curve.

Using the substitution x = u² - 1, we have:

$dx = 2u \, du.$

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

Thus, we rewrite the integral as:

$\int_{1}^{2} \frac{u^2 - 1}{\sqrt{u^2}} 2udu = 2\int_{1}^{2} u \, du.$

Calculating this gives:

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

Using the identity sin²(x) = 1 - cos²(x), we rewrite the integral as:

$\int sin^2(x) cos(x) \, dx = \int (1 - cos^2(x)) cos(x) \, dx.$

Substituting u = sin(x), so that du = cos(x) dx gives:

$\int (1 - u^2) \, du = u - \frac{u^3}{3} + C = sin(x) - \frac{sin^3(x)}{3} + C.$

Let the probability of a seedling producing a red flower be p = 1/5. Therefore, the probability of not producing a red flower is q = 1 - p = 4/5.

The required expression can be represented using the binomial probability formula:

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

where n = 8 and k = 3:

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

Using the same probabilities as before (p = 1/5, q = 4/5), the expression for none producing red flowers can be calculated using:

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

The probability that at least one seedling produces red flowers can be calculated as:

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

Using the earlier expression: $= 1 - \left(\frac{4}{5}\right)^{8}.$