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# AQA June 2016 Further Pure 2 Unofficial Mark Scheme

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1. These are my answers to todays FP2 exam. Not all of them are necassarily correct, please feel free to correct me on any you think are wrong. Not sure about the question numbers.
FP2 June 2016
1.
a) (2 marks)
b) (4 Marks?)

2.
a) i) (1 Mark)
ii) (1/2 marks)
b)
c) (3 Marks)

3. a) Show that s is the required integral
b) (5 marks?)

4. a) (2 Marks)
b) (4 Marks)

5. a) (2 Marks)
b)i) Sketch with a circle with centre radius (3 Marks)
ii) (2 Marks)
c) Roots are where (5 Marks)

6.a) Show that Q
b)

7. Induction Q. (6 marks?)
Trivial in the case since clearly
Assume true for

as
Thus true for so true for all positive integers n by induction.

8. a) Show that Q (3/4 Marks?)
b) Roots were where (2 Marks)
c) i) (4 Marks)
ii) (4 marks)
3. I thought the second last part of the last question you had to use the product of roots of the quartic equation so would equal -2?
4. (Original post by Barbecuetime123)
Probably same as last year. Maybe slightly lower.

so 59-61 for A* and about 53-56 for an A
5. (Original post by girlruinedme)
I thought the second last part of the last question you had to use the product of roots of the quartic equation so would equal -2?
That's what I did... It seemed a bit wishy washy as I was doing it though
6. (Original post by jjsnyder)
These are my answers to todays FP2 exam. Not all of them are necassarily correct, please feel free to correct me on any you think are wrong. Not sure about the question numbers.
FP2 June 2016
1.
a) A=4
b) 50/609
2.
a) i) B=1-2i
ii) ab = 5
b) Gamma is 1/3
c) p=-5 q=-7
3. a) Show that s is the required integral
b) ln7-3/4
4. a) (sqrt(3)/2)/(1+3x)sqrt(x)
b) (sqrt3)pi/18
5. Induction Q.
6.a) Show that Q
b) 6sqrt3 + 3arcosh2 + 5
7. a) 8
b)i) Sketch with a circle with centre -4root3 + 4i radius 4
ii) 2pi/3
c) Roots are 2e^i(xpi) where x=-7,5,17
8. a) Show that Q
b) Roots were itan(xpi) where x= 1,3,5,7
c) i) 1
ii) 6
Yep this is all correct.
7. (Original post by Barbecuetime123)
That's what I did... It seemed a bit wishy washy as I was doing it though
Considering it was the product of two squares it has to be positive. You used the product of the roots which for a quartic is the positive of the last coefficient not the negative. And the last coefficient was 1 (you had to remember to divide by 2 as the term in z^4 had coefficient 2).

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8. (Original post by girlruinedme)
I thought the second last part of the last question you had to use the product of roots of the quartic equation so would equal -2?
Nope, leading coefficient of the quartic was in fact 2, not 1. Plus it's a quartic so the constant term is equal to the product of the roots (same sign). So you get 2/2=1.
9. I got (sqrt3)pi/9 for Q4b

(Original post by jjsnyder)
These are my answers to todays FP2 exam. Not all of them are necassarily correct, please feel free to correct me on any you think are wrong. Not sure about the question numbers.
FP2 June 2016
1.
a) A=4
b) 50/609
2.
a) i) B=1-2i
ii) ab = 5
b) Gamma is 1/3
c) p=-5 q=-7
3. a) Show that s is the required integral
b) ln7-3/4
4. a) (sqrt(3)/2)/(1+3x)sqrt(x)
b) (sqrt3)pi/18
5. Induction Q.
6.a) Show that Q
b) 6sqrt3 + 3arcosh2 + 5
7. a) 8
b)i) Sketch with a circle with centre -4root3 + 4i radius 4
ii) 2pi/3
c) Roots are 2e^i(xpi) where x=-7,5,17
8. a) Show that Q
b) Roots were itan(xpi) where x= 1,3,5,7
c) i) 1
ii) 6
10. (Original post by wallpaperpaste)
I got (sqrt3)pi/9 for Q4b
Was 18 unfortunately, must have missed a factor of 1/2 somewhere.
11. (Original post by wallpaperpaste)
I got (sqrt3)pi/9 for Q4b
Anyone else get this? Was fairly sure it was a denominator of 18

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12. Can anyone explain the proof by induction question?
13. (Original post by wallpaperpaste)
Can anyone explain the proof by induction question?
I'll put an explanation in the MS now

EDIT: Done it
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14. (Original post by jjsnyder)
Considering it was the product of two squares it has to be positive. You used the product of the roots which for a quartic is the positive of the last coefficient not the negative. And the last coefficient was 1 (you had to remember to divide by 2 as the term in z^4 had coefficient 2).

Posted from TSR Mobile
Last coefficient was one and first coefficient was 2, so shouldnt the product of the roots be 1/2 ? Basically I'm wondering why you multiply by 2
15. (Original post by jjsnyder)
Considering it was the product of two squares it has to be positive. You used the product of the roots which for a quartic is the positive of the last coefficient not the negative. And the last coefficient was 1 (you had to remember to divide by 2 as the term in z^4 had coefficient 2).

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Ohh i remember putting 2. So ill probably lose a mark for forgetting to divide by 2?
16. (Original post by Iridocyclytis)
Last coefficient was one and first coefficient was 2, so shouldnt the product of the roots be 1/2 ? Basically I'm wondering why you multiply by 2
Last and first coefficients were 2 (this can be seen by symmetry of the expansion). Thus 2/2=1

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17. (Original post by jjsnyder)
These are my answers to todays FP2 exam. Not all of them are necassarily correct, please feel free to correct me on any you think are wrong. Not sure about the question numbers.
FP2 June 2016
1.
a) A=4
b) 50/609
2.
a) i) B=1-2i
ii) ab = 5
b) Gamma is 1/3
c) p=-5 q=-7
3. a) Show that s is the required integral
b) ln7-3/4
4. a) (sqrt(3)/2)/(1+3x)sqrt(x)
b) (sqrt3)pi/18
5. Induction Q.
Trivial in the case n=1 since clearly 1+p >= 1+p
Assume true for n=k
(1+p)^k+1 = (1+p)^k(1+p) = (1+kp)(1+p)
= 1 + kp + p + kp^2 >= 1 + (k+1)p as kp^2>=0
Thus true for n=k+1 so true for all positive integers n by induction.

6.a) Show that Q
b) 6sqrt3 + 3arcosh2 + 5
7. a) 8
b)i) Sketch with a circle with centre -4root3 + 4i radius 4
ii) 2pi/3
c) Roots are 2e^i(xpi) where x=-7,5,17
8. a) Show that Q
b) Roots were itan(xpi) where x= 1,3,5,7
c) i) 1
ii) 6
In 7c) aren't the roots 2e^(xpi) where x= -7/18, 5/18 and 17/18?
18. (Original post by jjsnyder)
I'll put an explanation in the MS now

EDIT: Done it
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Thanks!
19. (Original post by tanyapotter)
In 7c) aren't the roots 2e^(xpi) where x= -7/18, 5/18 and 17/18?
Sorry meant that :P forgot to put a /18 by the X, will change now, thanks!

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20. (Original post by jjsnyder)
Sorry meant that :P forgot to put a /18 by the X, will change now, thanks!

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thank god! i panicked so much

what would 61/75 be? A?

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Updated: June 25, 2016
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