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**ganesh****Moderator**- Registered: 2005-06-28
- Posts: 21,084

Probelm # k + 85

If u and v are the roots of the equation x² + ax + b = 0, what are roots of the equation x² -ax + b = 0?

Nothing is better than reading and gaining more and more knowledge - Stephen William Hawking.

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**mathsyperson****Moderator**- Registered: 2005-06-22
- Posts: 4,900

That's an interesting one. I think it's something like this:

Why did the vector cross the road?

It wanted to be normal.

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**Ricky****Moderator**- Registered: 2005-12-04
- Posts: 3,791

I'm not sure, this is what I get:

*Last edited by Ricky (2006-01-11 04:16:58)*

"In the real world, this would be a problem. But in mathematics, we can just define a place where this problem doesn't exist. So we'll go ahead and do that now..."

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**mathsyperson****Moderator**- Registered: 2005-06-22
- Posts: 4,900

I think our answers are the same, and I've just taken a long way round.

Why did the vector cross the road?

It wanted to be normal.

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**Ricky****Moderator**- Registered: 2005-12-04
- Posts: 3,791

Yea, I guess they are. They looked completely different at first glance.

"In the real world, this would be a problem. But in mathematics, we can just define a place where this problem doesn't exist. So we'll go ahead and do that now..."

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**irspow****Member**- Registered: 2005-11-24
- Posts: 457

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**irspow****Member**- Registered: 2005-11-24
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**irspow****Member**- Registered: 2005-11-24
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**irspow****Member**- Registered: 2005-11-24
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**irspow****Member**- Registered: 2005-11-24
- Posts: 457

darn that k+42, I just cant figure it. Please someone, put me out of my misery. I think that you have to incorporate a geometric series somehow, but everything that I try turns to nonsense.

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**mathsyperson****Moderator**- Registered: 2005-06-22
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Yes, k+42 is an incredibly difficult one. I think it could probably be solved brutally by getting excel to do all the calculations for you, but it's still tough.

Also, I think your answer to k+40 is wrong. I remember it being *much* smaller.

Why did the vector cross the road?

It wanted to be normal.

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**ryos****Member**- Registered: 2005-08-04
- Posts: 394

I didn't check if this one has already been solved, but since irspow asked about it, I gave it a go.

*Last edited by ryos (2006-01-14 16:15:25)*

El que pega primero pega dos veces.

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**ganesh****Moderator**- Registered: 2005-06-28
- Posts: 21,084

Four days Pongal break, and so many solutions posted! I shall reply to all of them after I return from Holiday on Jan 16. mathsyperson is right, irspow's solution to problem # k + 40 isn't correct. It *is much smaller. *. Same about ryos' solution to problem # k + 42.

Nothing is better than reading and gaining more and more knowledge - Stephen William Hawking.

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**ganesh****Moderator**- Registered: 2005-06-28
- Posts: 21,084

irspow's solution to Problem # k + 59 is correct. I shall wait for some more time before posting the solutions to unanswered problems.

Nothing is better than reading and gaining more and more knowledge - Stephen William Hawking.

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**ganesh****Moderator**- Registered: 2005-06-28
- Posts: 21,084

Problem # k + 86

Out of the total 390 students studying in a college of Arts and Science, boys and girls are in the ratio of 7 : 6 respectively and the number of students studying Arts and Science are in the ratio of 3 : 7 respectively. The boys and girls studying Arts are in the ratio of 4 : 5 respectively. How many boys are studying Science?

Nothing is better than reading and gaining more and more knowledge - Stephen William Hawking.

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**ganesh****Moderator**- Registered: 2005-06-28
- Posts: 21,084

Excellent, irspow's solution to problem # k + 48 is correct too!

Nothing is better than reading and gaining more and more knowledge - Stephen William Hawking.

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**irspow****Member**- Registered: 2005-11-24
- Posts: 457

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**ganesh****Moderator**- Registered: 2005-06-28
- Posts: 21,084

Well done, irspow

Nothing is better than reading and gaining more and more knowledge - Stephen William Hawking.

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**ganesh****Moderator**- Registered: 2005-06-28
- Posts: 21,084

Problem # k + 87

Prove that there exists atleast one multiple of 5 between 10^k and 10^k+1 where k is a Natural number.

Nothing is better than reading and gaining more and more knowledge - Stephen William Hawking.

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**Ricky****Moderator**- Registered: 2005-12-04
- Posts: 3,791

"In the real world, this would be a problem. But in mathematics, we can just define a place where this problem doesn't exist. So we'll go ahead and do that now..."

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**ganesh****Moderator**- Registered: 2005-06-28
- Posts: 21,084

I made a mistake while posting problem # k + 87.

The problem should have read 'Prove that there exists atleast one **power** of 5 between 10^k and 10^k+1 where k is a Natural number.

Problem # k + 88

Show that the sum of any number of terms of the series

1/1*2, 1/2*3, 1/3*4, 1/4*5, 1/5*6,................ is less than 1.

Nothing is better than reading and gaining more and more knowledge - Stephen William Hawking.

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**ganesh****Moderator**- Registered: 2005-06-28
- Posts: 21,084

Problem # k + 89

In a bolt factory, machines A, B, and C manufacture respectively 25%, 35% and 40% of the total bolts. Of their output, 5, 4 and 2 percent are respectively defective bolts. A bolt is drawn at random. If the bolt drawn is found to be defective, what is the probability that it is manufactured by machine B?

Nothing is better than reading and gaining more and more knowledge - Stephen William Hawking.

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**John E. Franklin****Member**- Registered: 2005-08-29
- Posts: 3,588

**igloo** **myrtilles** **fourmis**

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**mathsyperson****Moderator**- Registered: 2005-06-22
- Posts: 4,900

ganesh wrote:

I made a mistake while posting problem # k + 87.

The problem should have read 'Prove that there exists atleast onepowerof 5 between 10^k and 10^k+1 where k is a Natural number.

Ah. That makes more sense. I thought there was a mistake somewhere because as it was, the proof was really obvious, so it didn't seem sensible.

I've got a vague idea for k+88, but I'll wait for it to mature a bit more before posting it.

Why did the vector cross the road?

It wanted to be normal.

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**irspow****Member**- Registered: 2005-11-24
- Posts: 457

*Last edited by irspow (2006-02-15 11:49:14)*

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