Fun Questions 2 Watch

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Trevor 12345
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rounds 3 and 4
No calcs.
10 mins per round

Mrm.
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dvs
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#2
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Answers in white.


Round 3:
1. b
2. b
3. d
4. a
5. e

Round 4:
1. c
2. a
3. a
4. d (left this for last and ran out of time -- damn factorization! :P)
5. b

Reasoning:

2. x^2000 - 1 = (x-1)(x^1999 + x^1998 + ... + 1)
3. (sin30cos10 - sin10cos30)/(sin10cos10) = sin(30-10)/(0.5sin(2*10)) = 2sin20/sin20 = 2
4. We only consider x-2y+2=x+y=0.

2. Decimal representation of N = a + 10b + 100b + ... + 10^2004 = a + k = 0 mod 7. Hence a=0,7(=0 mod 7) will make the number divisible by 7.
3. 2^2001(2^4 - 2^3 - 2^2 - 2^1 - 1)
4. 2004=2².3.167
5. 1/(1+e^x) = 1 - e^x/(1+e^x), and ln(1/e)=-1.
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Trevor 12345
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#3
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(Original post by dvs)
Answers in white.


Round 3:
1. b
2. b
3. d
4. a
5. e

Round 4:
1. c
2. a
3. a
4. d (left this for last and ran out of time -- damn factorization! :P)
5. b



9/10

excellent stuff !!
Set 3 Question 4: hint do a sketch

also Set 2 Question 5

shortcut: from the 2nd summation consider adding 2P !!!!

dave


Reasoning:

2. x^2000 - 1 = (x-1)(x^1999 + x^1998 + ... + 1)
3. (sin30cos10 - sin10cos30)/(sin10cos10) = sin(30-10)/(0.5sin(2*10)) = 2sin20/sin20 = 2
4. We only consider x-2y+2=x+y=0.

2. Decimal representation of N = a + 10b + 100b + ... + 10^2004 = a + k = 0 mod 7. Hence a=0,7(=0 mod 7) will make the number divisible by 7.
3. 2^2001(2^4 - 2^3 - 2^2 - 2^1 - 1)
4. 2004=2².3.167
5. 1/(1+e^x) = 1 - e^x/(1+e^x), and ln(1/e)=-1.



9/10

excellent stuff !!
Set 3 Question 4: hint do a sketch

also Set 2 Question 5

shortcut: from the 2nd summation consider adding 2P !!!!

dave
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dvs
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(Original post by Mrm.)
9/10

excellent stuff !!
Set 3 Question 4: hint do a sketch

also Set 2 Question 5

shortcut: from the 2nd summation consider adding 2P !!!!

dave
That's how I did q5. It still uses the summation formula for P, doesn't it?

And: 5 solutions for q4?
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Trevor 12345
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#5
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(Original post by dvs)
That's how I did q5. It still uses the summation formula for P, doesn't it?

And: 5 solutions for q4?
Yes 5 solutions:

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