Author Topic: Extension 2 Topic Tests (Read 1103 times) Tweet Share

Ali_Abbas · « **on:** August 01, 2017, 10:59:19 pm »

Hi all,

As you may know, the Topic Tests version for the Extension 2 course is available on the ATAR Notes website (the link is https://atarnotes.com/product/hsc-mathematics-extension-2-topic-tests/). The author is myself, however, during the final stage of editing, I was never given a chance to review the final draft (for reasons I do not know). As such, I have now had a look at the book in its entirety and thought I'd open this new thread simply for the purpose of amending any typos or issues with the actual content. They are as follows:

For Integration Test 2, Question 1, an integral sign is missing. It should read:

$\text{Find } \int\frac{\sin{x}}{\cos{x}}dx \space \space \text{ not } \space \space \text{ Find } \frac{\sin{x}}{\cos{x}}dx.$

For Complex Numbers Test 1, Question 1, the question reads:

$\text{On a separate diagram, plot the complex number } \frac{z-\overline{x}}{2}i. \\ \text{The } \overline{x} \text{ should really be a } \overline{z}.$

For Complex Numbers Test 1, Question 5, part a) reads:

$\text{Explain why } z^{n}\frac{1}{z^{n}} = 2\cos{n\theta}. \\ \text{ There should be a plus } (+) \text{ in between } z^{n} \text{ and } \frac{1}{z^{n}}.$

For Polynomials Test 2, Question 1 part (c), it reads:

$\text{Show that } \alpha \text{ lies outside the unit circle when } k < -1.$
$\text{It should be corrected to read:}$
$\text{Show that } \alpha \text{ does not lie on the unit circle when } k < -1.$
$\text{The difference between the two is that the former statement is} \\ \text{basically saying that all roots satisfy } |\alpha| > 1 \text{ when } k < -1 \\ \text{when in reality, exactly two roots will satisfy} \\ \text{this while the remaining two will satisfy } |\alpha|<1.\\ \text{The solution for this question also needs a slight modification } \\ \text{and I will see that it happens. I apologize for the error.}$

For Mechanics Test 2, Question 1, I am not completely sure but I believe the question is not worded correctly. It is my understanding that the terminal velocity of a particle can never be reached as it only gets approached asymptotically (as t --> infinity). So I believe the question should rather say "Under what conditions will the terminal velocity exist?". Furthermore, the solution does not take h into account. So even if the terminal velocity can be reached in finite time, the value of h must be sufficiently large to allow the motion to take place for
at least that amount of time. If I can get a second opinion on this it will be much appreciated (I'm not the one who wrote the question).

Lastly, I want to talk about the solution to Conics Test 1, Question 1. The method given in the book is certainly not the intended approach. Again, had I been given a chance to look at the manuscript before printing, I would have amended this right away. The 'nicer' solution is as follows:

$\text{The point } z \text{ satisfies } |z-(1+3i)| + |z-(9+3i)| = 10. \\ \text{If we denote the points corresponding to the complex numbers} \\ 1+3i \text{ and } 9+3i \text{ by } S \text{ and } S' \text{ respectively, and denote} \\ \text{the position of } z \text{ by } P, \text{ then the equation of the locus can be re-written as} $
$PS+PS' = 2a$
$\text{where } 2a = 10. \text{This immediately gives } a = 5 \text{ and this is clearly the}\\ \text{two-foci definition of an ellipse (with foci } S \text{ and } S'). \text{ The centre will be the} \\ \text{midpoint of the foci, namely the point } (5, 3). \text{ So the ellipse will} \\ \text{have equation } \frac{(x-5)^{2}}{a^{2}} + \frac{(y-3)^{2}}{b^{2}} = 1 \text{ (where } a^{2} = 25). \\ \text{We also know that the distance between the two foci will be } 2ae. \\ \text{This gives } 2ae = \sqrt{(9-1)^{2}+(3-3)^{2}} = 8 \\ \Longrightarrow e = \frac{4}{5}. \\ \text{Lastly, we find } b \text{ by using the relationship } b^{2} = a^{2}(1-e^{2}) = ... = 9.$
$\therefore \frac{(x-5)^{2}}{25} + \frac{(y-3)^{2}}{9} = 1 \text{ is the equation of the ellipse.}$

In future, I will post any further issues that I find and I welcome you to share any problems you find yourself. Above all, I hope this book is of tremendous benefit to your high school endeavours. Happy reading

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Author Topic: Extension 2 Topic Tests (Read 1103 times) Tweet Share

Ali_Abbas

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