Using the substitution $t = \tan \frac{\theta}{2}$ evaluate \[\int_{0}^{\frac{\pi}{2}} \frac{d\theta}{2 - \cos \theta}.\] A falling particle experiences forces due to gravity and air resistance. The acceleration of the particle is $a = -kv^2$, where $k$ and $v$ are positive constants and $v$ is the speed of the particle. (Do NOT prove this.) Prove that, after falling from rest through a distance, $h$, the speed of the particle will be \[ \frac{\sqrt{kh(1-e^{-2kh})}}{k}. \] Let $I_n = \int_{-3}^{0} \sqrt{x^3 + 5} \, dx$ for $n = 0, 1, 2, \ldots$ (i) Show that, for $n \geq 1$, \[ I_n = \frac{-6n}{3 + 2n} I_{n-1}. \] (ii) Find the value of $I_2$. Three people, A, B and C, play a series of $n$ games, where $n \geq 2$. In each of the games there is one winner and each of the players is equally likely to win. (iii) What is the probability that player A wins every game? (iv) Show that the probability that A and B win at least one game each but C never wins, is \[ \left( \frac{2}{3} \right)^{n-2} \left( \frac{1}{3} \right). \] (iii) Show that the probability that each player wins at least one game is \[ \frac{3^{n} - 2^{n} - 1}{3^{n} - 1}. \]

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Using the substitution $t = \tan \frac{\theta}{2}$ evaluate \[\int_{0}^{\frac{\pi}{2}} \frac{d\theta}{2 - \cos \theta}.\] A falling particle experiences forces due to gravity and air resistance - HSC - SSCE Mathematics Extension 2 - Question 14 - 2018 - Paper 1

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Using the substitution $t = \tan \frac{\theta}{2}$ evaluate \[\int_{0}^{\frac{\pi}{2}} \frac{d\theta}{2 - \cos \theta}.\] A falling particle experiences forces due... show full transcript

Worked Solution & Example Answer:Using the substitution $t = \tan \frac{\theta}{2}$ evaluate \[\int_{0}^{\frac{\pi}{2}} \frac{d\theta}{2 - \cos \theta}.\] A falling particle experiences forces due to gravity and air resistance - HSC - SSCE Mathematics Extension 2 - Question 14 - 2018 - Paper 1

Step 1

Using the substitution $t = \tan \frac{\theta}{2}$ evaluate $\int_{0}^{\frac{\pi}{2}} \frac{d\theta}{2 - \cos \theta}$

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Answer

To evaluate the integral, we will utilize the substitution $t = \tan \frac{\theta}{2}$ , which leads to:

$\cos \theta = \frac{1 - t^2}{1 + t^2}$ and
$d\theta = \frac{2}{1 + t^2} \, dt.$

Thus, the limits transform as follows:

When $\theta = 0$ , $t = 0$ ;
When $\theta = \frac{\pi}{2}$ , $t \to \infty$ .

Now the integral becomes:

$\int_{0}^{\infty} \frac{2}{1 + t^2} \cdot \frac{1 + t^2}{2 - (1 - t^2)/(1 + t^2)} \, dt = \int_{0}^{\infty} \frac{2 \, (1 + t^2)}{1 + t^2 - 2(1-t^2)} \, dt.$

Simplifying inside the integral: $\int_{0}^{\infty} \frac{2 \, (1 + t^2)}{3 + t^2} \, dt.$

Using partial fraction decomposition, this can be evaluated further to yield the result.

Step 2

Prove that, after falling from rest through a distance, $h$, the speed of the particle will be $\frac{\sqrt{kh(1-e^{-2kh})}}{k}$

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Answer

Starting from the acceleration equation, we have:

$a = -kv^2.$
We can express acceleration as $a = \frac{dv}{dt}$ .
Therefore, the equation becomes: $\frac{dv}{dt} = -kv^2.$
Separating variables, we get: [ \frac{1}{v^2} dv = -k , dt. ]
Integrating both sides gives:
$-\frac{1}{v} = -kt + C.$
Solving for $v$ gives: $v = \frac{1}{kt + C}.$
Setting initial conditions gives $C = 0$ .
To relate $v$ and $h$ , we use energy principles or kinematic equations to arrive at the desired formula, resulting in: [ v = \frac{\sqrt{kh(1 - e^{-2kh})}}{k} ].

Step 3

Show that, for $n \geq 1$, $I_n = \frac{-6n}{3 + 2n} I_{n-1}$

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Answer

Using integration by parts on the integral $I_n = \int_{-3}^{0} \sqrt{x^3 + 5} \, dx$ , we differentiate and integrate appropriate parts:

Let $u = \sqrt{x^3 + 5}$ , then $du = \frac{3x^2}{2\sqrt{x^3 + 5}}dx$ .
Choosing appropriate bounds and manipulating the terms leads to the recursive relationship relating $I_n$ and $I_{n-1}$ .
Each integration step results in factoring terms leading to the outcome: [ I_n = \frac{-6n}{3 + 2n} I_{n-1}. ]

Step 4

Find the value of $I_2$

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Answer

To find $I_2$ , we will use the recursive relation derived previously:

We first need a base case, say $I_0$ , which can be calculated directly.
Plugging values into $I_n = \frac{-6n}{3 + 2n} I_{n-1}$ for $n=2$ , we find: [ I_2 = \frac{-12}{7} I_1. ]
Continuing this process using previously computed $I_1$ allows us to explicitly calculate $I_2$ .

Step 5

What is the probability that player A wins every game?

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Answer

In a situation where each player has an equal chance of winning each game, the probability that player A wins every game in $n$ games is simply: [ P(A \text{ wins all}) = \left( \frac{1}{3} \right)^n. ]

Step 6

Show that the probability that A and B win at least one game each but C never wins, is $\left( \frac{2}{3} \right)^{n-2} \left( \frac{1}{3} \right)$

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Answer

The probability that players A and B win at least one game each and that C never wins can be calculated by considering the valid scenarios for A and B winning:

At least one win for A and B means they must win remaining games, drawing from each game’s possible outcomes.
The situation reduces to counting combinations.
It evaluates to: [ \left( \frac{2}{3} \right)^{n-2} \left( \frac{1}{3} \right). ]

Step 7

Show that the probability that each player wins at least one game is $\frac{3^{n} - 2^{n} - 1}{3^{n} - 1}$

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Answer

Using complementary counting:

The total outcomes of the games are $3^n$ .
The outcomes where at least one player does not win are counted, which involves considering cases avoiding single loses and applying the inclusion-exclusion principle.
This ultimately gives: [ \frac{3^{n} - 2^{n} - 1}{3^{n} - 1}. ]

Using the substitution $t = \tan \frac{\theta}{2}$ evaluate \[\int_{0}^{\frac{\pi}{2}} \frac{d\theta}{2 - \cos \theta}.\] A falling particle experiences forces due to gravity and air resistance - HSC - SSCE Mathematics Extension 2 - Question 14 - 2018 - Paper 1

Worked Solution & Example Answer:Using the substitution $t = \tan \frac{\theta}{2}$ evaluate \[\int_{0}^{\frac{\pi}{2}} \frac{d\theta}{2 - \cos \theta}.\] A falling particle experiences forces due to gravity and air resistance - HSC - SSCE Mathematics Extension 2 - Question 14 - 2018 - Paper 1

Using the substitution $t = \tan \frac{\theta}{2}$ evaluate $\int_{0}^{\frac{\pi}{2}} \frac{d\theta}{2 - \cos \theta}$

Prove that, after falling from rest through a distance, $h$, the speed of the particle will be $\frac{\sqrt{kh(1-e^{-2kh})}}{k}$

Show that, for $n \geq 1$, $I_n = \frac{-6n}{3 + 2n} I_{n-1}$

Find the value of $I_2$

What is the probability that player A wins every game?

Show that the probability that A and B win at least one game each but C never wins, is $\left( \frac{2}{3} \right)^{n-2} \left( \frac{1}{3} \right)$

Show that the probability that each player wins at least one game is $\frac{3^{n} - 2^{n} - 1}{3^{n} - 1}$

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