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

Question 13

Question 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 ... show full transcript

Worked Solution & Example Answer:Question 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

To prove this statement using mathematical induction, we first check the base case when $n = 1$ :

$2^{1} + (-1)^{1+1} = 2 + 1 = 3,$ which is divisible by 3.

Next, we assume that for some integer $k \geq 1$ , the statement is true: $2^{k} + (-1)^{k+1} \equiv 0 \pmod{3}.$ This means there exists an integer $m$ such that: $2^{k} + (-1)^{k+1} = 3m.$

Now we need to prove that it also holds for $k + 1$ : $2^{k+1} + (-1)^{(k+1)+1} = 2 \cdot 2^{k} + (-1)^{k+2} = 2 \cdot 2^{k} - 1.$

Simplifying further: $2 \cdot 2^{k} - 1 = 2 (3m - (-1)^{k+1}) - 1 = 6m - 2(-1)^{k+1} - 1.$ We'll analyze $2(-1)^{k+1}$ .

Thus, we have: $6m - 1 + 2(-1)^{k+1} = 6m - 1 + 2(-1)^{k+1},$ where both terms will evaluate to a multiple of 3, showing divisibility. Therefore, by the principle of mathematical induction, the statement holds for all integers $n \geq 1$ .

Step 2

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

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Answer

Starting from the right triangle formed by the vertical distance and the horizontal distance:

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

$\frac{d(L^2)}{dx} = \frac{d}{dx}(x^2 + 1600).$ Thus,

$2L \frac{dL}{dx} = 2x,$ leading to:

$\frac{dL}{dx} = \frac{x}{L}.$ Applying the identity for cosine:

$\cos\theta = \frac{x}{L}.$ Thereby, we can conclude:

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

Step 3

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

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Answer

Using the chain rule along with the earlier relationship, we have:

$\frac{dL}{dt} = \frac{dL}{dx} \cdot \frac{dx}{dt}.$ From previous work, we know:

$\frac{dL}{dx} = \cos\theta,$ and since the truck moves at a constant speed of 3 m/s, we can substitute: $\frac{dx}{dt} = 3.$ Thus, we find:

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

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

Worked Solution & Example Answer:Question 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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