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:

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

To solve the inequality:

1. Start with the inequality: ( \frac{2x}{x+1} - 1 > 0 )

2. Combine into a single fraction: ( \frac{2x - (x+1)}{x+1} > 0 ) Simplifying gives: ( \frac{x-1}{x+1} > 0 )

3. Find critical points: ( x = 1 ) and ( x = -1 )

4. Test intervals:

   • For ( x < -1 ): negative
   • For ( -1 < x < 1 ): positive
   • For ( x > 1 ): positive

Thus, the solution is: ( x < -1 ) or ( x > 1 ).

The function ( y = 2 \cos^{-1}(x) ) is defined for ( -1 \leq x \leq 1 ).

1. Determine the range:

   • The range of ( \cos^{-1}(x) ) is ( [0, \pi] ), so the range of our function is ( [0, 2\pi] ).

2. Key points:

   • At ( x = -1, y = 2\pi )
   • At ( x = 0, y = \pi )
   • At ( x = 1, y = 0 )

3. Graph these points and draw a continuous curve between them.

Using the substitution ( x = u^2 - 1 ), we express dx as:

1. Change the limits:

   • When ( x = 0 ), ( u^2 = 1 \Rightarrow u = 1 )
   • When ( x = 3 ), ( u^2 = 4 \Rightarrow u = 2 )

2. Substitute in the integral: $\int_1^2 \frac{u^2 - 1}{u} (2u) \, du = 2 \int_1^2 (u^2 - 1) \, du$

3. Finding the primitive: $2 \left[ \frac{u^3}{3} - u \right]_1^2 = 2 \left[ \frac{8}{3} - 2 - \left( \frac{1}{3} - 1 \right) \right] = 2 \left[ \frac{8}{3} - 2 + \frac{2}{3} \right] = 2 \left[ \frac{10}{3} - 2 \right] = \frac{8}{3}$

To solve:

1. Use substitution: let ( u = \sin x \Rightarrow du = \cos x , dx ).

2. The integral becomes: $\int u^2 \, du = \frac{u^3}{3} + C$ and reverting: $\frac{\sin^3 x}{3} + C$

This can be expressed using the binomial probability formula:

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

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

Thus:

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

Using the binomial probability formula:

$P(X = 0) = \binom{8}{0} p^0 (1-p)^8$

Thus:

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

This can be computed using the complement rule:

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

Thus:

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