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## #51 2009-01-07 18:02:58

ganesh
Registered: 2005-06-28
Posts: 26,841

### Re: 5 Star Questions

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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## #52 2009-01-07 18:14:48

ganesh
Registered: 2005-06-28
Posts: 26,841

### Re: 5 Star Questions

#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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## #53 2009-01-07 20:05:23

ganesh
Registered: 2005-06-28
Posts: 26,841

### Re: 5 Star Questions

#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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## #54 2009-01-08 01:51:47

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

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## #55 2009-01-08 02:22:14

ganesh
Registered: 2005-06-28
Posts: 26,841

### Re: 5 Star Questions

Excellent, JaneFairfax! You're right again!

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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## #56 2009-01-08 02:26:23

ganesh
Registered: 2005-06-28
Posts: 26,841

### Re: 5 Star Questions

JaneFairfax,
Did you see the solution to Problem #17? (Post #49 on Page 2)
Do you agree with my answer?

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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## #57 2009-01-08 02:42:31

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

### Re: 5 Star Questions

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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## #58 2009-01-08 02:55:54

ganesh
Registered: 2005-06-28
Posts: 26,841

### Re: 5 Star Questions

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!

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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## #59 2009-01-08 03:14:00

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

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## #60 2009-01-08 17:03:25

ganesh
Registered: 2005-06-28
Posts: 26,841

### Re: 5 Star Questions

Absolutely 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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## #61 2009-01-09 01:41:17

ganesh
Registered: 2005-06-28
Posts: 26,841

### Re: 5 Star Questions

#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?

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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## #62 2009-01-09 08:50:00

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

### Re: 5 Star Questions

Last edited by JaneFairfax (2009-01-09 08:52:38)

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## #63 2009-01-09 17:59:25

ganesh
Registered: 2005-06-28
Posts: 26,841

### Re: 5 Star Questions

You are correct, 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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## #64 2009-01-10 03:30:40

ganesh
Registered: 2005-06-28
Posts: 26,841

### Re: 5 Star Questions

#32. Find the value of

.

#33. Find the value of x if

.

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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## #65 2009-01-10 03:46:16

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

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## #66 2009-01-10 17:22:17

ganesh
Registered: 2005-06-28
Posts: 26,841

### Re: 5 Star Questions

You are correct, 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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## #67 2009-01-10 18:06:55

ganesh
Registered: 2005-06-28
Posts: 26,841

### Re: 5 Star Questions

#34. Which is the largest number each number of the sequence is divisible by?

?
Why?

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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## #68 2009-01-10 23:35:09

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

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## #69 2009-01-11 00:12:32

ganesh
Registered: 2005-06-28
Posts: 26,841

### Re: 5 Star Questions

(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

is divisible by 30.
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!

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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## #70 2009-01-11 01:03:03

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

### Re: 5 Star Questions

ganesh wrote:

For proving that

is always divisible by 5

NO!! If
is divisible by 5 then
is NOT difvisible by 5.

You do it this way.

Either

or
.

If

then clearly
.

If

, then
by Fermats little theorem.

In that case,

.

Hence

in every case.

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## #71 2009-01-11 01:10:35

ganesh
Registered: 2005-06-28
Posts: 26,841

### Re: 5 Star Questions

Thanks, JaneFairfax!
That has made things a lot more clearer to me!

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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## #72 2009-01-11 01:49:33

ganesh
Registered: 2005-06-28
Posts: 26,841

### Re: 5 Star Questions

#35. What is the value of xyz if

?

#36. What is the value of z if

and
.

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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## #73 2009-01-11 12:24:18

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

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## #74 2009-01-11 20:52:19

ganesh
Registered: 2005-06-28
Posts: 26,841

### Re: 5 Star Questions

You are correct, 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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## #75 2009-01-11 20:59:57

ganesh
Registered: 2005-06-28
Posts: 26,841

### Re: 5 Star Questions

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

would not be divisible by 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?

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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