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Further Mathematics: Post your doubts here!

Thought blocker

Thought blocker

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randomcod said:
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.
Click to expand...
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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Alina Rahman said:
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
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Thanks for your help ;)
 
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randomcod

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Thought blocker said:
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.
Click to expand...
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.
 
scouserlfc

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Thought blocker said:
May june session. You gave it? How was it?
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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. :)
 
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scouserlfc said:
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. :)
Click to expand...
Congratulations to you and your friend. Thanks for the tips. ;)
 
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