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Further Mathematics: Post your doubts here!

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http://papers.xtremepapers.com/CIE/Cambridge International A and AS Level/Mathematics - Further (9231)/9231_s10_qp_12.pdf
Question 6. There has to be a formula for S6 right?
Please help theorem you use for these kind of summatories, in which you have to cube S2 and all that! I'm very lost.
As α, β and γ are the roots of the equation, (x - α)(x - β)(x - γ) = 0

Expanding the above, x³ - x²(α + β + γ) + x(αβ + βγ + γα) - αβγ = 0 ------- (1)
But given is: x³ + x - 1 = 0 --------- (2)

Comparing the corresponding coefficients of (1) & (2):
(α + β + γ) = 0; (αβ + βγ + γα) = 1 and αβγ = 1 --------------- (3)

As x = √y; x² = y; ==> y = α² or β² or γ² [Since, α, β and γ are the values of x as given]
==> (y - α²)(y - β²)(y - γ²) = 0

Expanding, y³ - (α² + β² + γ²)y² + (α²β² + β²γ² + α²γ²)y - α²*β²*γ² = 0 ------ (4)

By identity, (α + β + γ)² = (α² + β² + γ²) + 2(αβ + βγ + γα)
Substituting the values from (3) and simplifying, (α² + β² + γ²) = -2

(αβ + βγ + γα)² = (α²β² + β²γ² + α²γ²) + 2(αβγ)(αβ + βγ + γα)
Substituting the values from (3) and simplifying, (α²β² + β²γ² + α²γ²) = 1
and α²*β²*γ² = (αβγ)² = 1

Substituting all these values in (4):
y³ + 2y² + y - 1 = 0 [Proved]

Already it is shown that: (α² + β² + γ²) = -2; so S₂ = -2

Squaring the above, we can get S₄ = 2

For part (ii) you may try yourself in similar to the above. If you don't get, knock me anytime.
 
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Practising pastpapers is a very helpful technique if anyone of you need to get an A or A* in your exams! If you practise about 5-10 years of pastpapers before your exam, i can guarantee that there is no doubt you will get a good grade! You can find pastpapers on many different websites. I use the one mentioned in my Signature
Thanks for your help ;)
 
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As α, β and γ are the roots of the equation, (x - α)(x - β)(x - γ) = 0

Expanding the above, x³ - x²(α + β + γ) + x(αβ + βγ + γα) - αβγ = 0 ------- (1)
But given is: x³ + x - 1 = 0 --------- (2)

Comparing the corresponding coefficients of (1) & (2):
(α + β + γ) = 0; (αβ + βγ + γα) = 1 and αβγ = 1 --------------- (3)

As x = √y; x² = y; ==> y = α² or β² or γ² [Since, α, β and γ are the values of x as given]
==> (y - α²)(y - β²)(y - γ²) = 0

Expanding, y³ - (α² + β² + γ²)y² + (α²β² + β²γ² + α²γ²)y - α²*β²*γ² = 0 ------ (4)

By identity, (α + β + γ)² = (α² + β² + γ²) + 2(αβ + βγ + γα)
Substituting the values from (3) and simplifying, (α² + β² + γ²) = -2

(αβ + βγ + γα)² = (α²β² + β²γ² + α²γ²) + 2(αβγ)(αβ + βγ + γα)
Substituting the values from (3) and simplifying, (α²β² + β²γ² + α²γ²) = 1
and α²*β²*γ² = (αβγ)² = 1

Substituting all these values in (4):
y³ + 2y² + y - 1 = 0 [Proved]

Already it is shown that: (α² + β² + γ²) = -2; so S₂ = -2

Squaring the above, we can get S₄ = 2

For part (ii) you may try yourself in similar to the above. If you don't get, knock me anytime.
This has been a great help! Thank you so much.
Do you, by any chance, have a sheet with all the summatories (S2, S3, S4, S5, S6, S7, S8) identities? I would really appreciate that! Thanks.
 
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May june session. You gave it? How was it?
Oh I was sooooo nervous in both P1 and P2 but after both papers I felt pretty confident of getting at least an A . Just lost abt 6marks in p1 and just 2 or 3 in P2 . The key was the preparation doing each and every question particularly the old papers 2002 to 2007 these papers are very difficult if u can master the questions here then u can easily get very close to full marks. But remember if u are giving further maths u shud have this general uneasy feeling when u can't solve a question and the belief that u won't look in the er or ms and obviously at one point u wud need to look but don't lose hope immediately. Further just a tip to get you a good study environment get a close friend doing further as well to solve the same questions on the same topic or year u are and any problems he has u help him and any problem u have u ask him. I used this and my friend and I were the only ones in our school to get A*s ALHUMDULILLAH. :)
 
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Oh I was sooooo nervous in both P1 and P2 but after both papers I felt pretty confident of getting at least an A . Just lost abt 6marks in p1 and just 2 or 3 in P2 . The key was the preparation doing each and every question particularly the old papers 2002 to 2007 these papers are very difficult if u can master the questions here then u can easily get very close to full marks. But remember if u are giving further maths u shud have this general uneasy feeling when u can't solve a question and the belief that u won't look in the er or ms and obviously at one point u wud need to look but don't lose hope immediately. Further just a tip to get you a good study environment get a close friend doing further as well to solve the same questions on the same topic or year u are and any problems he has u help him and any problem u have u ask him. I used this and my friend and I were the only ones in our school to get A*s ALHUMDULILLAH. :)
Congratulations to you and your friend. Thanks for the tips. ;)
 
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