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

We start with the function:

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

Using the chain rule, we find:

y' = rac{1}{1 + (x^2)^2} imes (2x) = rac{2x}{1 + x^4}.

To solve the inequality:

rac{2x}{x + 1} > 1,

we can multiply both sides by (x + 1), keeping in mind the sign of (x + 1):

2x > x + 1,

which simplifies to:

x > 1.

Thus, the solution of the inequality is x > 1, ensuring we check if (x + 1) is positive.

To sketch the graph of:

y = 2 cos^{-1}(x),

y varies between [0, 2π].

The function is defined for x ∈ [-1, 1]. The critical points occur at:

• At x = 1, y = 0;
• At x = -1, y = 2π.

Plotting these points will give an inverted U shape for the graph between these x-values.

Using the substitution:

x = u² - 1 => dx = 2u , du.

The limits change as follows:

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

The integral becomes:

egin{aligned} ∫_1^2 (u^2 - 1) / (u + 1) , (2u , du) , = 2∫_1^2 rac{u^3 - u}{u + 1} ,du.
ext{This can be evaluated further leading to: } = rac{8}{3} - rac{2}{3} = 2.
ext{Thus, the final answer is: } rac{8}{3}. ext{.} ext{..} ext{..} ext{.} ext{..} \ ext{or } y. ext{..} ext{.} . ext{.}

ext{.}\
ext{or let }	ext{...} \ 
ext{or .} \ 
ext{or .}
ext{or .}.

\

ext{or }  
ext{or y.} 

ext{or y.}

. ext{ .} ext{.} \ ext{ .} ext{..} \ \n ext{..}\ \n
ext{..} \n ext{..}

To solve:

∫ sin²(x) cos(x) dx,

we can use the substitution:

Let u = sin(x) ext{ so that } du = cos(x) dx.

This transforms the integral to:

rac{1}{3} sin^3(x) + C.

The probability of exactly k successes in n Bernoulli trials is given by:

P(X = k) = inom{n}{k} p^k (1-p)^{n-k}.

Thus, for our case, the expression becomes:

inom{8}{3} igg(rac{1}{5}igg)^{3} igg(rac{4}{5}igg)^{5}.

The probability that none produce red flowers:

P(X = 0) = (1 - p)^n = (1 - rac{1}{5})^{8} = igg(rac{4}{5}igg)^{8}.

The probability of at least one success is:

P(X ext{ at least } 1) = 1 - P(X = 0) = 1 - igg(rac{4}{5}igg)^{8}.