[hello]

A couple of questions from the Review Exercise of the Heinemann P6 book. Help would be much appreciated! Cheers.

Q. 82:
n
Given that ∑ r³ = ¼n²(n+1)²
r=1

...show that the sum of the cubes of all the numbers between 100 and 200 which are divisible by 3 is a multiple of 4200.

Q. 88:

Given that x²+y²<3x+4y-5, show that y/x lies between 1/2 and 11/2.

[RichE]

Originally Posted by hello

A couple of questions from the Review Exercise of the Heinemann P6 book. Help would be much appreciated! Cheers.

Q. 82:
n
Given that ∑ r³ = ¼n²(n+1)²
r=1

...show that the sum of the cubes of all the numbers between 100 and 200 which are divisible by 3 is a multiple of 4200.

Q. 88:

Given that x²+y²<3x+4y-5, show that y/x lies between 1/2 and 11/2.

For 82

r^3 is divisible by 3 only if r is divisible by 3

So work out

[Mimetex cannot convert this formula]

For 88 see

http://www.thestudentroom.co.uk/show...37&postcount=4

[hello]

Thanks for that.

[hello]

Sub y = ax in

(a^2+1) x^2 - (3 + 4a) + 5 = 0

We need this to have repeated roots

Just a question about the solution...

Why are repeated roots required? Isn't b²-4ac a solution for 'a' and hence two (real) distinct values are needed since 'a' represents the gradient of the two tangents from the origin - which are clearly different?

I suppose it has something to do with the fact tangents hit the circle only once and this enables two solutions for 'a' anyway, but a proper explanation would be much appreciated.

Cheers.

[Gaz031]

Originally Posted by hello

Just a question about the solution...

Why are repeated roots required? Isn't b²-4ac a solution for 'a' and hence two (real) distinct values are needed since 'a' represents the gradient of the two tangents from the origin - which are clearly different?

I suppose it has something to do with the fact tangents hit the circle only once and this enables two solutions for 'a' anyway, but a proper explanation would be much appreciated.

Cheers.

You want repeated roots as the lines are tangent and only intersects the circle once.
You don't want multiple roots because while it's true that 'a' can change, both values give lines which have the property that they only intersect the circle once.