# Thread: Cambridge HSC MX1 Textbook Marathon/Q&A

1. ## Re: Year 12 Mathematics 3 Unit Cambridge Question & Answer Thread

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By the way, thanks for making this thread David

2. ## Re: Year 12 Mathematics 3 Unit Cambridge Question & Answer Thread

Originally Posted by wu345
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By the way, thanks for making this thread David

13 (c) In how many ways can one yellow, two red and four green beads be placed on a bracelet if the beads are identical apart from colour? [Hint : This will require a listing of patterns to see if they are identical when turned over.]

To arrange 1 yellow, 2 red, and 4 green objects in a line, the number of possible ways is:
7! / (1! * 2! * 4!)
= (7*6*5*4*3*2*1) / (1 * 2*1 * 4*3*2*1)
= 105

However, since the objects are not in a line but in a circlular way (e.g. on a bracelet), the circle can be rotated to that what appears to be 7 possibilities is actually just 1, i.e. ABCDEFG = BCDEFGA = CDEFGAB = DEFGABC = EFGABCD = FGABCDE = GABCDEF. So instead of 105 possibilities, there are actually 105 divide 7 = 15 ways.

But since you can also FLIP the bracelet, not just rotate it, you'll find that of those 15 possibilities, 3 of them are symmetrical (so flipping them has no effect), and the other 12 are actually just 6 pairs of equivalent possibilities. So the actual answer is 15 - 6 = 3 + (12/2) = 3 + 6 = 9

3. ## Re: Year 12 Mathematics 3 Unit Cambridge Question & Answer Thread

I had a problem with this question tonight EX 3A, Q 5 (b)

$Michael the mailman rides 1km up a hill at a constant speed of 10 km/hr, and then rides 1 km down the other side of the hill at a constant speed of 30 km/hr.$

$(a) How many minutes does he take to ride (i) up the hill, (ii) down the hill ? (i) Ans: 6 minutes (ii) 2 minutes$

$(b) Draw a displacement-time graph, with the time axis in minutes$

Here is my effort of drawing the displacement-time graph:

What confused me was the answer should the graph distances being accumulated? Is this supposed to be? Im treating the base of the hill at its initial position, the top of the hill as its maximum, and returning back down the hill as back to where it initially started.

4. ## Re: Year 12 Mathematics 3 Unit Cambridge Question & Answer Thread

Originally Posted by davidgoes4wce
I had a problem with this question tonight EX 3A, Q 5 (b)

$Michael the mailman rides 1km up a hill at a constant speed of 10 km/hr, and then rides 1 km down the other side of the hill at a constant speed of 30 km/hr.$

$(a) How many minutes does he take to ride (i) up the hill, (ii) down the hill ? (i) Ans: 6 minutes (ii) 2 minutes$

$(b) Draw a displacement-time graph, with the time axis in minutes$

Here is my effort of drawing the displacement-time graph:

What confused me was the answer should the graph distances being accumulated? Is this supposed to be? Im treating the base of the hill at its initial position, the top of the hill as its maximum, and returning back down the hill as back to where it initially started.
It says that he goes down the other side of the hill, so he doesn't come back to where he starts; his final displacement should be 2 m.

5. ## Re: Year 12 Mathematics 3 Unit Cambridge Question & Answer Thread

Originally Posted by InteGrand
It says that he goes down the other side of the hill, so he doesn't come back to where he starts; his final displacement should be 2 m.
ok thanks, personally felt that question was pretty badly worded but maybe its just me. Could accept something along the lines of 'total distance-time' graph.

6. ## Re: Year 12 Mathematics 3 Unit Cambridge Question & Answer Thread

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7. ## Re: Year 12 Mathematics 3 Unit Cambridge Question & Answer Thread

http://imgur.com/uMddJgF

That's part a. Not entirely sure about part b though

8. ## Re: Year 12 Mathematics 3 Unit Cambridge Question & Answer Thread

part b... I think...
I'm not sure that you can just assume the altitudes meet at E
http://i.imgur.com/9qgFLtr.jpg

9. ## Year 12 Mathematics 3 Unit Cambridge Question &amp; Answer Thread

Originally Posted by wu345
part b... I think...
I'm not sure that you can just assume the altitudes meet at E
http://i.imgur.com/9qgFLtr.jpg
But isn't AC 5x^2 not 4x^2 in part a

10. ## Re: Year 12 Mathematics 3 Unit Cambridge Question & Answer Thread

It is not obvious at all to me that the altitudes intersect on the edge.

(If it did, I would have already used my signature phrase)

11. ## Re: Year 12 Mathematics 3 Unit Cambridge Question & Answer Thread

$Ex 10J Q18$

$\\During winter, it rains on average 18 out of 30 days. Five winter days are selected at random. Find the probability that:\\a) The first two days chosen will be fine.\\b) more rainy days than fine days have been chosen.$

I've forgotten how to binomial probability completely

12. ## Re: Year 12 Mathematics 3 Unit Cambridge Question & Answer Thread

Originally Posted by leehuan
$Ex 10J Q18$

$\\During winter, it rains on average 18 out of 30 days. Five winter days are selected at random. Find the probability that:\\a) The first two days chosen will be fine.\\b) more rainy days than fine days have been chosen.$

I've forgotten how to binomial probability completely
Use standard binomial probability formulas with p = 18/30 if you want to define 'success' as a rainy day, or p = 1 – 18/30 = 12/30, if you want to define 'success' as a fine day.

13. ## Re: Year 12 Mathematics 3 Unit Cambridge Question & Answer Thread

Originally Posted by InteGrand
Use standard binomial probability formulas with p = 18/30 if you want to define 'success' as a rainy day, or p = 1 – 18/30 = 12/30, if you want to define 'success' as a fine day.
Didn't work when I tried because I forgot what nCr is

14. ## Re: Year 12 Mathematics 3 Unit Cambridge Question & Answer Thread

Originally Posted by leehuan
Didn't work when I tried because I forgot what nCr is
n!/(r!.(n-r)!)

n will be 5 because we have 5 days total.

15. ## Re: Year 12 Mathematics 3 Unit Cambridge Question & Answer Thread

Oops, my bad on the wording. I meant what to use as n and r.

Basically, two clumsy attempts whilst being tired weren't correct.

30C3.(3/5)^3.(2/5)^27
+30C4.(3/5)^4.(2/5)^26
+30C5.(3/5)^5.(2/5)^25 which I realised was obviously wrong but

5C3(3/5)^3 (2/5)^2 + 5C4 (3/5)^4 (2/5)^1 + 5C5 (3/5)^5 (2/5)^0. still didn't work

16. ## Re: Year 12 Mathematics 3 Unit Cambridge Question & Answer Thread

Originally Posted by leehuan
Oops, my bad on the wording. I meant what to use as n and r.

Basically, two clumsy attempts whilst being tired weren't correct.

30C3.(3/5)^3.(2/5)^27
+30C4.(3/5)^4.(2/5)^26
+30C5.(3/5)^5.(2/5)^25 which I realised was obviously wrong but

5C3(3/5)^3 (2/5)^2 + 5C4 (3/5)^4 (2/5)^1 + 5C5 (3/5)^5 (2/5)^0. still didn't work
For the first two days chosen to be fine, the probability is: (prob. a day is fine)^2 = (12/30)^2

This Q. is like a coin toss Q. basically.

(We are assuming for this Q. that the weather of the days are independent of each other (which is probably not a valid assumption, but oh well).)

17. ## Re: Year 12 Mathematics 3 Unit Cambridge Question & Answer Thread

Oh right I was staring at part b) though I did get stuck on both

The answers were a) 0.0124, b) 0.7102 (at 4d.p.)
_______________________________

But I'm feeling really dumb right now. I still can't see anything. Might need a more specific sketchout...

For a) If I assume the same day can get chosen twice I used (2/5)^2(3/5)^3 but that just gives 0.03456.

I don't know what I'm doing

18. ## Re: Year 12 Mathematics 3 Unit Cambridge Question & Answer Thread

Originally Posted by leehuan
Oh right I was staring at part b) though I did get stuck on both

The answers were a) 0.0124, b) 0.7102 (at 4d.p.)
_______________________________

But I'm feeling really dumb right now. I still can't see anything. Might need a more specific sketchout...

For a) If I assume the same day can get chosen twice I used (2/5)^2(3/5)^3 but that just gives 0.03456.

I don't know what I'm doing
I found how they got their answer to part a): they did (12/30)^5 (but this should be the probability that all days are fine, not the probability that the first two days are fine).

19. ## Re: Year 12 Mathematics 3 Unit Cambridge Question & Answer Thread

Originally Posted by leehuan
Oops, my bad on the wording. I meant what to use as n and r.

Basically, two clumsy attempts whilst being tired weren't correct.

30C3.(3/5)^3.(2/5)^27
+30C4.(3/5)^4.(2/5)^26
+30C5.(3/5)^5.(2/5)^25 which I realised was obviously wrong but

5C3(3/5)^3 (2/5)^2 + 5C4 (3/5)^4 (2/5)^1 + 5C5 (3/5)^5 (2/5)^0. still didn't work
$Cambridge Ex 10J Q18. During winter it rains on average 18 out of 30 days. Five winter days are selected at random. Find, correct to four decimal places, the probability that:$

$a) The first 2 days chosen will be fine and the remainder wet$

$b) More rainy days than fine days have been chosen$

$I reckon part a) and b) of the answers is wrong. Turns out to be 0.03456, obviously conflicting with the answers cambridge provided$

$a) P(2 fine then 3 wet) = (12/30)^{2}.(18/30)^{3}= 0.03456$

$(b)$

20. ## Re: Year 12 Mathematics 3 Unit Cambridge Question & Answer Thread

Reckon they did a mistake in the book for both parts a) and b)

21. ## Re: Year 12 Mathematics 3 Unit Cambridge Question & Answer Thread

Originally Posted by davidgoes4wce
$Cambridge Ex 10J Q18. During winter it rains on average 18 out of 30 days. Five winter days are selected at random. Find, correct to four decimal places, the probability that:$

$a) The first 2 days chosen will be fine and the remainder wet$

$b) More rainy days than fine days have been chosen$

$I reckon part a) and b) of the answers is wrong. Turns out to be 0.03456, obviously conflicting with the answers cambridge provided$

$a) P(2 wet then 3 fine) = (12/30)^{2}.(18/30)^{3}= 0.03456$

$(b)$

In leehuan's version of the question (the one he typed up some posts ago), part a) just asked for the probability that the first two days are wet, which is different to your part a), which also requires the remaining days to be fine. Is leehuan's edition of the textbook different to yours?

22. ## Re: Year 12 Mathematics 3 Unit Cambridge Question & Answer Thread

Facepalm... I didn't type a bit of the question...

23. ## Re: Year 12 Mathematics 3 Unit Cambridge Question & Answer Thread

Originally Posted by leehuan
Facepalm... I didn't type a bit of the question...
Ah OK, that would explain the difference. Though if davidgoes4wce's version of the Q. is correct, then yes, the answers had a mistake, they did the probability all days were fine. This mistake could be added to the textbook mistakes thread if anyone wants to add it in.

24. ## Re: Year 12 Mathematics 3 Unit Cambridge Question & Answer Thread

Originally Posted by InteGrand
Ah OK, that would explain the difference. Though if davidgoes4wce's version of the Q. is correct, then yes, the answers had a mistake, they did the probability all days were fine. This mistake could be added to the textbook mistakes thread if anyone wants to add it in.

This disheartened my morning lol.

25. ## Re: Year 12 Mathematics 3 Unit Cambridge Question & Answer Thread

$Cambridge Exercise 3E Q4 (a)$

$A stone is dropped from a lookout 500 metres above the valley floor. Take g =10 m/s^{2}, ignore air resistance, take downwards as positive, and use the lookout as the origin of displacement- the equation of motion is then \ddot{x}=10$

$(a) Replace \ \ddot{x} \ by \frac{d}{dx} (\frac{1}{2} v^2) and show that v^{2}=20x. Hence find the impact speed.$

$(b) \ Explain why, during the fall, v=\sqrt{20x} rather than v=-\sqrt{20x}$

$(c) Integrate to find the displacement-time function, and find how long it takes to fall.$

$Ans a) 100 m/s b) Downwards is positive, so while the stone is falling, its velocity is positive c) x=5t^{2}, 10 seconds$

Just had a bit of trouble with part (a)

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