You are Here: Home >< Maths

# A couple more P6 questions watch

1. 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.
2. (Original post 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

For 88 see

http://www.thestudentroom.co.uk/show...37&postcount=4
3. Thanks for that.

4. 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.
5. (Original post 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.

### Related university courses

TSR Support Team

We have a brilliant team of more than 60 Support Team members looking after discussions on The Student Room, helping to make it a fun, safe and useful place to hang out.

This forum is supported by:
Updated: June 24, 2005
The home of Results and Clearing

### 2,802

people online now

### 1,567,000

students helped last year
Today on TSR

### University open days

1. Sheffield Hallam University
Tue, 21 Aug '18
2. Bournemouth University
Wed, 22 Aug '18
3. University of Buckingham
Thu, 23 Aug '18
Poll
Useful resources

### Maths Forum posting guidelines

Not sure where to post? Read the updated guidelines here

### How to use LaTex

Writing equations the easy way

### Study habits of A* students

Top tips from students who have already aced their exams