13 (15 marks) Use a SEPARATE writing booklet. (a) Use mathematical induction to prove that $2^n + (-1)^{n+1}$ is divisible by 3 for all integers $n \geq 1$. (b) One end of a rope is attached to a truck and the other end to a weight. The rope passes over a small wheel located at a vertical distance of 40 m above the point where the rope is attached to the truck. The distance from the truck to the small wheel is $L$ m, and the horizontal distance between them is $m$ m. The rope makes an angle $\theta$ with the horizontal at the point where it is attached to the truck. The truck moves to the right at a constant speed of 3 m s$^{-1}$, as shown in the diagram. (i) Using Pythagoras’ Theorem, or otherwise, show that $\frac{dL}{dx} = \cos \theta$. (ii) Show that $\frac{dL}{dt} = 3\cos \theta$.

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13 (15 marks) Use a SEPARATE writing booklet - HSC - SSCE Mathematics Extension 1 - Question 13 - 2014 - Paper 1

Question 13

13 (15 marks) Use a SEPARATE writing booklet. (a) Use mathematical induction to prove that $2^n + (-1)^{n+1}$ is divisible by 3 for all integers $n \geq 1$. (b) ... show full transcript

Worked Solution & Example Answer:13 (15 marks) Use a SEPARATE writing booklet - HSC - SSCE Mathematics Extension 1 - Question 13 - 2014 - Paper 1

Step 1

Use mathematical induction to prove that $2^n + (-1)^{n+1}$ is divisible by 3 for all integers $n \geq 1$

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Answer

We utilize the principle of mathematical induction.

Base Case: For $n = 1$ , we have:

$2^1 + (-1)^{1+1} = 2 + 1 = 3,$

which is divisible by 3.
Inductive Step: Assume it holds for some integer $k$ , i.e., assume $2^k + (-1)^{k+1}$ is divisible by 3.
We need to show that it also holds for $k + 1$ :

$2^{k+1} + (-1)^{(k+1)+1} = 2 imes 2^k + (-1)^{k+1}.$

Substituting the inductive hypothesis in:

$= 2(2^k + (-1)^{k+1}) - 2(-1)^{k+1}.$

Since $2^k + (-1)^{k+1}$ is divisible by 3, we can express it as:

$= 3m - 2(-1)^{k+1}$ for some integer $m$ .

Depending on whether $k$ is even or odd, this reduces to showing $-2(-1)^{k+1}$ is also divisible by 3. Hence, the statement is true for $k + 1$ , completing the induction.

Step 2

Using Pythagoras’ Theorem, or otherwise, show that $\frac{dL}{dx} = \cos \theta$

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Answer

In the given scenario, we apply Pythagoras’ Theorem:

The relationship between $L$ , $x$ , and the height of the wheel is:

$L^2 = x^2 + 40^2.$
Taking the derivative of both sides with respect to $x$ :

$2L \frac{dL}{dx} = 2x.$

Thus, we find:

$\frac{dL}{dx} = \frac{x}{L}.$
From the triangle formed, we know that:

$\cos \theta = \frac{x}{L}.$ Therefore, we can write:

$\frac{dL}{dx} = \cos \theta.$

Step 3

Show that $\frac{dL}{dt} = 3\cos \theta$

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Answer

From the previous step we have:

We use the chain rule on $L$ with respect to time $t$ :

$\frac{dL}{dt} = \frac{dL}{dx} \cdot \frac{dx}{dt}.$
Substituting the expression we derived:

$\frac{dL}{dt} = \cos \theta \cdot \frac{dx}{dt}.$
Given that the truck moves at 3 m/s:

$\frac{dx}{dt} = 3,$ thus gives:

$\frac{dL}{dt} = 3\cos \theta.$

13 (15 marks) Use a SEPARATE writing booklet - HSC - SSCE Mathematics Extension 1 - Question 13 - 2014 - Paper 1

Worked Solution & Example Answer:13 (15 marks) Use a SEPARATE writing booklet - HSC - SSCE Mathematics Extension 1 - Question 13 - 2014 - Paper 1

Use mathematical induction to prove that $2^n + (-1)^{n+1}$ is divisible by 3 for all integers $n \geq 1$

Using Pythagoras’ Theorem, or otherwise, show that $\frac{dL}{dx} = \cos \theta$

Show that $\frac{dL}{dt} = 3\cos \theta$

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