Hope you're all having fun completing the questions!
Here are my answers to the first side of the homework sheet if you want to check your answers - remember we're interested in how you've got to the answers.... not just that you've got the right answer!
Answer sheet
Leave a comment if you need help with any of the questions or want to know how I got my answers...
some help with a2.7
ReplyDeleteOn A2.7 (a) you have to find the co-ordinates of points A and B. Look at point A first - what do you know about it? Its on the y-axis so ..... x must be equal to 0.
So if x=0, then 2y = x - 2 should give us the value of y when we put x=0 into it.
2y = 0 - 2
2y = -2
y = -1
So A has co-ordinates (0,-1)
Similarly with B, B is on the x-axis so must have y = 0
Use substitution again 2y = x - 2
2 x 0 = x - 2
0 = x - 2
so x = 2
Co-ordinates of B are (2,0)
Gradient of the line 2y = x - 2
rearrange to find y = x/2 - 2/2
so y = x/2 - 1
and the gradient of the line is 1/2 (the co-efficient (number in front) of x)
c) 2y = x-2 and 2y = x-3
If you draw these lines on the graph, they will never meet because they are parallel - they have the same gradient - so if the lines never cross that means there is never a solution!
Q1 - some help on the one on proof
ReplyDeleteThink about choosing numbers n-1, n , n+1
Could be (if n=5), 5-1, 5, 5+1 so 4, 5, 6
This will always give three consecutive numbers.
Then think about any three consecutive numbers, there will always be one number that is in the 3 x table.
eg 1 x 2 x 3 will be a multiple of 3 (because we've x by 3)
6 x 7 x 8 will be a multiple of 3 (because we've multiplied by a number in the 3 x table).
The question already gives that n^3 - 1 = n(n-1)(n+1) so must always have a multiple of 3 as one of the consecutive numbers.
Then if n is odd and bigger than 1.
ReplyDeleteWe know n(n-1)(n+1) is divisible by 3 (from first part of question)
If n is odd, n-1 must be even and n+1 must be even. One of them must be in the 4 x table, and the other is not in the 4 x table. So you have a number in the 4xtable x number in 2x table so must have a number in the 8xtable.
So, its a multiple of 3 and its a multiple of 8. So it must be a multiple of 24.
Does this help?