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**ganesh****Moderator**- Registered: 2005-06-28
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Answer to #24:-

You're right, JaneFairfax!

It is no good to try to stop knowledge from going forward. Ignorance is never better than knowledge - Enrico Fermi.

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

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**ganesh****Moderator**- Registered: 2005-06-28
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#26. What is the value of

#27. What is the value of log tan1° + log tan2° + log tan3° + log tan4° + ....log tan 89°

#28. What is the value of

It is no good to try to stop knowledge from going forward. Ignorance is never better than knowledge - Enrico Fermi.

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

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**ganesh****Moderator**- Registered: 2005-06-28
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#29. Decompose into partial fractions:-

.#30. If p and q are the roots of the equation x²+8x = -15, form the equation whose roots are (p+q) and 3pq.

It is no good to try to stop knowledge from going forward. Ignorance is never better than knowledge - Enrico Fermi.

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

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**JaneFairfax****Member**- Registered: 2007-02-23
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**ganesh****Moderator**- Registered: 2005-06-28
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Answer to #30:-**Excellent, JaneFairfax! You're right again!**

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

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**ganesh****Moderator**- Registered: 2005-06-28
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JaneFairfax,

Did you see the solution to Problem #17? (Post #49 on Page 2)

Do you agree with my answer?

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

When you worked out the area of the parallelogram, you use its slant height instead of its perpendicular height and so your answer is larger than the actual area.

I get the area of the trapezium to be the same as Jane's 3rd attempt.

Why did the vector cross the road?

It wanted to be normal.

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**ganesh****Moderator**- Registered: 2005-06-28
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Oops!

That was a stupid mistake!

Thanks, mathsyperson, for correcting me!

I am sorry Jane, I said your last attempt was wrong too!

I shall be more careful while checking the answers!

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

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**JaneFairfax****Member**- Registered: 2007-02-23
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**ganesh****Moderator**- Registered: 2005-06-28
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Answer to #27:-

Absolutely right, JaneFairfax!

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

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**ganesh****Moderator**- Registered: 2005-06-28
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#31. A person covers the distance from P to Q at the speed of 3 kilometers per hour. From Q to P, he covers the distance at 6 kilometers per hour. What is the average speed per hour?

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

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**JaneFairfax****Member**- Registered: 2007-02-23
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*Last edited by JaneFairfax (2009-01-09 08:52:38)*

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**ganesh****Moderator**- Registered: 2005-06-28
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Answer to #31:-

You are correct, JaneFairfax!

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

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**ganesh****Moderator**- Registered: 2005-06-28
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#32. Find the value of

#33. Find the value of x if

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

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**JaneFairfax****Member**- Registered: 2007-02-23
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**ganesh****Moderator**- Registered: 2005-06-28
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Answer to #33:-

You are correct, JaneFairfax!

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

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**ganesh****Moderator**- Registered: 2005-06-28
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#34. Which is the largest number each number of the sequence is divisible by?

Why?

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

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**JaneFairfax****Member**- Registered: 2007-02-23
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**ganesh****Moderator**- Registered: 2005-06-28
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Answer to #34:

(1) The question has been modified. I didn't notice what meaning the question conveyed while posting, sorry!

(2) I have a proof that every number of the form

I wanted to know if I am right.

This product is certainly divisible by 6.

For proving that is always divisible by 5, I use this logic.

Last digit of

0 0

1 1

2 6

3 1

4 6

5 5

6 6

7 1

8 6

9 1

It can be seen that the last digit of

is always 0 or 5, which means it is a multiple of 5.I hope my reasoning is acceptable!

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

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**JaneFairfax****Member**- Registered: 2007-02-23
- Posts: 6,868

ganesh wrote:

For proving that

is always divisible by 5

You do it this way.

Either

or .If

then clearly .If

, then by Fermats little theorem.In that case,

.Hence

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**ganesh****Moderator**- Registered: 2005-06-28
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Thanks, JaneFairfax!

That has made things a lot more clearer to me!

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

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**ganesh****Moderator**- Registered: 2005-06-28
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#35. What is the value of xyz if

#36. What is the value of z if

and.

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

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**JaneFairfax****Member**- Registered: 2007-02-23
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**ganesh****Moderator**- Registered: 2005-06-28
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Answer to #35:-

You are correct, JaneFairfax!

Your point is fully acceptable.

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

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**ganesh****Moderator**- Registered: 2005-06-28
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To JaneFairfax:

It occured to me that, as mentioned by you in post #70, if n is a multiple of of 5,

That isn't going to affect the proof.

The idea is to prove that is divisible by 30, hence, if isn't divisible by 5, n would be. We already know that

is divisible by 2 and 3 (because (n-1)(n)(n+1) is a factor). Now that is has been shown that it is also divisible by 5, can we say decisively that a number of the form (n>0, n is a Natural number)

is divisible by 30 with the help of posts #69 and #75 alone, without the help of Fermat's little theorem?

Am I right?

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

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