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# Further maths question help- sum of roots and argand diagrams watch

1. Could someone explain what they did in the mark scheme for these two questions

Question 6c

https://snag.gy/BKMw7d.jpg

https://snag.gy/A7aI6Z.jpg

Question 6c

https://snag.gy/VK6bZ3.jpg

https://snag.gy/84M6CR.jpg
2. (Original post by YAH_BOY_ASH)
Could someone explain what they did in the mark scheme for these two questions

Question 6c

https://snag.gy/BKMw7d.jpg

https://snag.gy/A7aI6Z.jpg

If a,b,c are the roots of a cubic (f(x)), whose coeff of x^3 is 1, then

f(x) = (x-a)(x-b)(x-c)
= x^3-(a+b+c)x^2 + ...

So, the sum of the roots is minus the coeff of x^2, i.e. 3 in this case.
Hence the first line of the markscheme.

Also, tan 9pi/12 = tan 3pi/4 = - tan pi/4 = -1
Sub in and rearrange to get the second line.

The maximum value of |z1-z2| will occur when z1,z2 are on the line connecting the centres of the two circles, extended. Hence the maximum distance is the distance between the two centres plus the two radii.
3. (Original post by ghostwalker)
If a,b,c are the roots of a cubic (f(x)), whose coeff of x^3 is 1, then

f(x) = (x-a)(x-b)(x-c)
= x^3-(a+b+c)x^2 + ...

So, the sum of the roots is minus the coeff of x^2, i.e. 3 in this case.
Hence the first line of the markscheme.

Also, tan 9pi/12 = tan 3pi/4 = - tan pi/4 = -1
Sub in and rearrange to get the second line.

The maximum value of |z1-z2| will occur when z1,z2 are on the line connecting the centres of the two circles, extended. Hence the maximum distance is the distance between the two centres plus the two radii.
Ah thank you, that makes sense, I just have a small query about the second question. I am having trouble picturing what |z1-z2| would look like in general. I understand how the length was calculated but not what |z1-z2| looks like, let me know if what i said is not clear.
4. (Original post by YAH_BOY_ASH)
Ah thank you, that makes sense, I just have a small query about the second question. I am having trouble picturing what |z1-z2| would look like in general. I understand how the length was calculated but not what |z1-z2| looks like, let me know if what i said is not clear.
|z1-z2| is the distance between z1 and z2.

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Updated: June 19, 2017
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