mark scheme from old p4, p5, p6 and fp1, fp2, fp3...
TRANSCRIPT
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FP1 Mark Schemes from old P4, P5, P6 and FP1, FP2, FP3 papers (back to June 2002)
Please note that the following pages contain mark schemes for questions from past papers which were not written at an AS standard and may be less accessible than those you will find on future AS FP1 papers from Edexcel. Some questions would certainly worth more marks at AS level.
The standard of the mark schemes is variable, depending on what we still have – many are scanned, some are handwritten and some are typed.
The questions are available on a separate document, originally sent with this one.
FP1 question mark schemes from old P4, P5, P6 and FP1, FP2, FP3 papers – Version 2 – March 2009 1
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[P4 January 2002 Qn 1]
[P4 January 2002 Qn 3]
FP1 question mark schemes from old P4, P5, P6 and FP1, FP2, FP3 papers – Version 2 – March 2009 2
2.
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[P4 January 2002 Qn 4]
5. 6r2 – 6 = n(n + 1)(2n + 1) , – 6n M1, A1
= n(2n2 + 3n – 5) M1
= n(n – 1)(2n – 5) (*) A1
(4 marks)[P4 June 2002 Qn 1]
FP1 question mark schemes from old P4, P5, P6 and FP1, FP2, FP3 papers – Version 2 – March 2009 3
4.
[*P4 January 2002 Qn 5]
3.
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6. (a) |w| = 50 (or equivalent) B1 (1)
(b)
B ×
× AB1 (1)
(c) B1
, isoscelesM1, A1
52 + 52 = (50)2, right-angled (or gradient method) M1, A1 (5)
(d) M1
= (–)AOB = M1, A1 (3)
(10 marks)[P4 June 2002 Qn 5]
7. ; at x = 2p M1, A1
Equation of tangent at P, M1
etc)
At Q q2y + x + 4q Two correct equations in any form A1
(p2 – q2)y = 4(p – q) M1
() A1
() M1, A1 (8)
[*P5 June 2002 Qn 7]
8. (a) For n = 1 25 + 52 = 57, which is divisible by 3 M1, A1
FP1 question mark schemes from old P4, P5, P6 and FP1, FP2, FP3 papers – Version 2 – March 2009 4
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Assume true for n = k (k + 1)th term is 23k +5 + 5k+2 B1
(k + 1)th term kth term = 23k +5 + 5k+2 23k +2 + 5k+1 M1
= 23k +2(23 1) + 5k+1(5 1) M1, A1
= 6(23k +2 + 5k+1) + 3. 23k +2 or = 4(23k +2 + 5k+1) + 3. 23k +2 M1
which is divisible by 3 (k + 1)th term is divisible by 3 A1
Thus by induction true for all n cso B1 (9)
(b) For n = 1 RHS = B1
Assume true for n = k
M1 A3/2/1/0
(-1 each error)
B1
If true for k then true for k + 1 by induction true for all n B1 (7)
(16 marks)
[P6 June 2002 Qn 6]
FP1 question mark schemes from old P4, P5, P6 and FP1, FP2, FP3 papers – Version 2 – March 2009 5
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3.
9. f(2) = 1.514
B1
f() = 1.142 B1
M1
1.514 + 2 1.142 = (1.142 + 1.514)
= 2.65 A1 (4)[*P4 January 2003 Qn 4]
10. (a) z2 = (3 – 3i)(3 – 3i) = 18i M1 A1 (2)
(b) = = = M1 A1 (2)
(c) z = (9 + 9) = 18 = 32
z = 18 two correct M1
= = = all three correct A1 (2)
(d) C D O
two correct A
four correct
B
B1
B1 (2)
(e) = 18 M1 A1
AOB = COD = 45 similar B1 (3)
(11 marks)[P4 January 2003 Qn 6]
11. (a) x3 – 27 = (x – 3)(x2 + 3x + 9) M1
FP1 question mark schemes from old P4, P5, P6 and FP1, FP2, FP3 papers – Version 2 – March 2009 6
2 1.142
1.514
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(x = 3 is one root). Others satisfy (x2 + 3x + 9) = 0 (*) A1 (2)
(b) Roots are x = 3 B1
and x = M1
= A1 (3)
(c)
3
3 and one other root in
correct quad
Root in complex
conjugate posn.
B1
B1 ft (2)
(7 marks)[#P4 June 2003 Qn 3]
12. (a) f(1) = –4 f(2) = 1 M1
Change of sign (and continuity) implies (1, 2) A1 (2)
(b) f(1.5) = –2.3… B1
f(1.75) = –0.9…
f(1.875) = –0.03… B1 (2)
NB Exact answer is 1.8789…
Alt to (a) y y = 3x
98 y = x + 67
3
O 1 2 x
Two graphs with single point of intersection (x > 0)
Two calculations at both x = 1 and x = 2
M1
A1
[*P4 June 2003 Qn 4]
13. (a) = = ( = 1 – i ) M1
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| z | = , = awrt 1.12, accept exact equivalentsM1, A1
(3)
(b) = accept in (a) if clearly applied to (b)
M1
1 = obtaining quadratic or equivalent M1 A1
a2 + 5a – 6 = (a + 6)(a – 1)
a = –6, 1 M1 A1
Reject a = –6, wrong quadrant/– , one value A1 (6)
(9 marks)
Alt. (a) | 4 + 3i | = 5, | 2 + 4i | = 20 M1
| z | = ( = )M1 A1
(3)
(b) arg z = arg (a + 3i) – arg (2 + ai)
M1
1 = M1 A1
(3)
leading to a2 + 5a – 6 = 0, then as before
[P4 June 2003 Qn 5]
FP1 question mark schemes from old P4, P5, P6 and FP1, FP2, FP3 papers – Version 2 – March 2009 8
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[P6 June 2003 Qn 2]
15. (a) | z | = 22 | w | = 2 M1, A1
| wz2 | = (22)2 × 2 = 16 M1, A1
arg z = – arg w = ; arg wz2 = – – + = , 60° M1, A1 (6)
ALT z2 = –8i; z2w = 8 + 83i M1, A1
| z2w | = M1
= 16 A1
arg z2w = tan–13 M1
= A1
(b) C
B
A
Points A and B B1
Point C B1ft
angle BOC = – M1
= , 90° A1 (4)
(10 marks)[P4 January 2004 Qn 3]
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14.
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[P4 June 2004 Qn 1]
17. and B1
B1
(2)
[*P4 June 2004 Qn 2]
[P4 June 2004 Qn 3]
FP1 question mark schemes from old P4, P5, P6 and FP1, FP2, FP3 papers – Version 2 – March 2009 10
16.
18.
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[P4 June 2004 Qn 5]
FP1 question mark schemes from old P4, P5, P6 and FP1, FP2, FP3 papers – Version 2 – March 2009 11
19.
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20.
The normal to the curve has gradient .
The equation of the normal is
The equation may be written
M1 A1
B1
M1A1 (5)
[*P5 June 2004 Qn 8]
21. (a) , M1
M1 A1 (3)
(b) M1
M1
In correct quadrant A1 (3)
(c) A B1 B B1 C ft B1ft (3)
(d) B1 Correct method for M1 A1 (3)
(e) awrt 3.46 M1 A1 (2)
[#FP1/P4 January 2005 Qn 8]
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A
B
C
22. (a) Long division or any complete method M1
M1 A1 (3)
(b)
In correct quadrants and on negative B1ft (1) - axis only required
(c) Full method M1
triangle is right angled A1 (2) (6)
Alternative to (c) Result follows by (converse of) Pythagoras, or any complete method. M1 A1
[#FP1/P4 June 2005 Qn 2]
23. … to 1sf or better B1
… Attempt at , M1A1 (3)
[*FP1/P4 June 2005 Qn 4]
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[FP1/P4 June 2005 Qn 5]
[*FP2/P5 June 2005 Qn 5]
FP1 question mark schemes from old P4, P5, P6 and FP1, FP2, FP3 papers – Version 2 – March 2009 14
24.
25.
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[FP3/P6 June 2005 Qn 1]
[FP1/P4 January 2006 Qn 1]
FP1 question mark schemes from old P4, P5, P6 and FP1, FP2, FP3 papers – Version 2 – March 2009 15
26.
27.
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[FP1/P4 January 2006 Qn 3]
29. (a) … awrt B1
… awrt B1
or equivalent M1
cao A1 (4)
(b) 112 (min) (1 hr 52 min) B1 (1) (5)
[#*FP1/P4 January 2006 Qn 5]
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28.
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[*FP2/P5 Jan 2006 Qn 9]
[*FP3/P6 Jan 2006 Qn 3]
FP1 question mark schemes from old P4, P5, P6 and FP1, FP2, FP3 papers – Version 2 – March 2009 17
30.
31.
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33 . (a)
Adding Eliminating either variable M1
A1
M1
= A1 (4)
(b) arctan 2 M1
cao A1 (2)
(6 marks)[FP1 June 2006 Qn 1]
FP1 question mark schemes from old P4, P5, P6 and FP1, FP2, FP3 papers – Version 2 – March 2009 18
32.
[*FP3/P6 January 2006 Qn 5]
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34. (a) accept 1sf M1
Change of sign (and continuity) A1 (2)
(b) accept 1sf M1
M1 A1 (3)
(5)[*FP1 June 2006 Qn 6]
[FP1 Jan 2007 Qn 1]
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35.
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[FP1 Jan 2007 Qn 3]
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36.
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[FP1 June 2007 Qn 4]
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37.
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[FP1 June 2007 Qn 6]
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38.
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39. (a) Can be implied B1
Use of any correct method or formula to obtain an equation of PQ in any form. M1
Leading to A1 (3)
(b) Gradient of normal at P is . Given or implied at any stage B1 Obtaining any correct form for normal at either point. Allow if just written down. M1 A1
Using both normal equations and eliminating x or y. Allow in any unsimplified form. M1
Any correct form for x or y A1 Leading to cso A1 cso A1 (7)
[*FP2 June 2007 Qn8]
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40. : B1
(Hence result is true for .)
, by induction
hypothesisM1
M1 A1
(Hence, if result is true for , then it is true for )
By Mathematical Induction, above implies the result is true for all
cso A1 (5)
(5 marks) [FP3 June 2007 Qn5]
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41. (a)
M1
can be implied A1
M1
Hence cso A1 (4)
Note: is divisible by 240 and other appropriate multiples of 15 lead to the required result.
(b) : B1
(Hence result is true for .)
From (a) say. By induction hypothesis say.
M1
(Hence, if result is true for , then it is true for )
By Mathematical Induction, above implies the result is true for all
Accept equivalent arguments cso A1 (3)
(7 marks)[*FP3 June 2007 Qn6]
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[FP1 January 2008 Qn 2]
43. 3 dp or better B1, B1
Using to obtain M1
3dp or better A1 (4)
[*FP1 January 2008 Qn 4]
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42.
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[FP1 January 2008 Qn 6]
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44.
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[FP1 June 2008 Qn 1]
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45.
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[FP1 June 2008 Qn 3]
FP1 question mark schemes from old P4, P5, P6 and FP1, FP2, FP3 papers – Version 2 – March 2009 29
46.
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[FP3 June 2008 Qn 5]
FP1 question mark schemes from old P4, P5, P6 and FP1, FP2, FP3 papers – Version 2 – March 2009 30
47.