a) Find the inverse of the function $y=x^3 - 2$. b) Use the substitution $u=x-4$ to find $\int \sqrt{x-4} \, dx$. c) Differentiate $3\tan^{-1}(2x)$. d) Evaluate $\lim_{x \to 0} \frac{2\sin x \cos x}{3x}$. e) Solve $\frac{3}{2x+5} > 0$. f) A darts player calculates that when she aims for the bullseye the probability of her hitting the bullseye is $\frac{2}{5}$ with each throw. i) Find the probability that she hits the bullseye with exactly one of her first three throws. ii) Find the probability that she hits the bullseye with at least two of her first six throws.

SSCE HSC Mathematics Extension 1 Indefinite integrals and substitution: a) Find the inverse of the funct

a) Find the inverse of the function $y=x^3 - 2$ - HSC - SSCE Mathematics Extension 1 - Question 11 - 2016 - Paper 1

Question 11

a) Find the inverse of the function $y=x^3 - 2$. b) Use the substitution $u=x-4$ to find $\int \sqrt{x-4} \, dx$. c) Differentiate $3\tan^{-1}(2x)$. d) Evaluat... show full transcript

Worked Solution & Example Answer:a) Find the inverse of the function $y=x^3 - 2$ - HSC - SSCE Mathematics Extension 1 - Question 11 - 2016 - Paper 1

Step 1

Find the inverse of the function $y=x^3 - 2$

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Answer

To find the inverse, we first interchange $x$ and $y$ .

Start with: $y = x^3 - 2$
Interchange $x$ and $y$ : $x = y^3 - 2$
Solve for $y$ : $y^3 = x + 2$ $y = \sqrt[3]{x + 2}$ Thus, the inverse function is: $f^{-1}(x) = \sqrt[3]{x + 2}$

Step 2

Use the substitution $u=x-4$ to find $\int \sqrt{x-4} \, dx$

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Answer

Let $u = x - 4$ , then $dx = du$ and $x = u + 4$ .
Rewrite the integral: $\int \sqrt{x-4} \, dx = \int \sqrt{u} \, du$
Now, compute the integral: $= \int u^{1/2} \, du = \frac{u^{3/2}}{3/2} + C = \frac{2}{3} u^{3/2} + C$
Substitute back to get: $= \frac{2}{3} (x-4)^{3/2} + C$

Step 3

Differentiate $3\tan^{-1}(2x)$

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Answer

Use the chain rule for differentiation: $\frac{d}{dx}[3\tan^{-1}(2x)] = 3 \cdot \frac{1}{1+(2x)^2} \cdot 2$
This simplifies to: $= \frac{6}{1 + 4x^2}$

Step 4

Evaluate $\lim_{x \to 0} \frac{2\sin x \cos x}{3x}$

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Answer

Recognize that $\lim_{x \to 0} \frac{\sin x}{x} = 1$ .
Rewrite the limit: $\lim_{x \to 0} \frac{2\sin x \cos x}{3x} = \frac{2}{3} \cdot \lim_{x \to 0} \frac{\sin x}{x} \cdot \cos x$
Now substituting: $= \frac{2}{3} \cdot 1 \cdot 1 = \frac{2}{3}$

Step 5

Solve $\frac{3}{2x+5} > 0$

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Answer

The fraction is positive when its numerator and denominator have the same sign.
Since the numerator $3$ is always positive, we need to solve: $2x + 5 > 0$
This simplifies to: $x > -\frac{5}{2}$$$

Step 6

Find the probability that she hits the bullseye with exactly one of her first three throws

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Answer

We use the binomial probability formula: $P(X=k) = {n \choose k} p^k (1-p)^{n-k}$
In this case, $n=3$ , $k=1$ , and $p=\frac{2}{5}$ : $P(X=1) = {3 \choose 1}\left(\frac{2}{5}\right)^1 \left(\frac{3}{5}\right)^{2}$ $= 3 \cdot \frac{2}{5} \cdot \frac{9}{25} = \frac{54}{125}$

Step 7

Find the probability that she hits the bullseye with at least two of her first six throws

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Answer

We find $P(X \geq 2)$ : $P(X \geq 2) = 1 - P(X=0) - P(X=1)$
Calculate:
- For $X=0$ : $P(X=0) = {6 \choose 0}\left(\frac{2}{5}\right)^{0}\left(\frac{3}{5}\right)^{6} = \frac{729}{15625}$
- For $X=1$ : $P(X=1) = {6 \choose 1}\left(\frac{2}{5}\right)^{1}\left(\frac{3}{5}\right)^{5} = 6 \cdot \frac{2}{5} \cdot \frac{243}{3125} = \frac{2916}{15625}$
Thus: $P(X \geq 2) = 1 - \frac{729 + 2916}{15625} = 1 - \frac{3645}{15625} = \frac{11980}{15625}$

a) Find the inverse of the function $y=x^3 - 2$ - HSC - SSCE Mathematics Extension 1 - Question 11 - 2016 - Paper 1

Worked Solution & Example Answer:a) Find the inverse of the function $y=x^3 - 2$ - HSC - SSCE Mathematics Extension 1 - Question 11 - 2016 - Paper 1

Find the inverse of the function $y=x^3 - 2$

Use the substitution $u=x-4$ to find $\int \sqrt{x-4} \, dx$

Differentiate $3\tan^{-1}(2x)$

Evaluate $\lim_{x \to 0} \frac{2\sin x \cos x}{3x}$

Solve $\frac{3}{2x+5} > 0$

Find the probability that she hits the bullseye with exactly one of her first three throws

Find the probability that she hits the bullseye with at least two of her first six throws

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