January Further Pure Mathematics FP1 Mark Scheme

Transcription

1 January 6667 Further Pure Mathematics FP Mar Q z + 8i + i (a) z i + i + i + 8i 8 + 5i z (b) ( ) + 5 z (or awrt 5.8) () ft () (c) 5 5 tan α or z arg π.... z + i (a) and attempt to multiply out for + i - for first, +5i for second (b) Square root required without i for z award for attempt at Pythagoras for both numerator and denominator z 5 (c) tan or tan, ± or ± seen with their and 5 award 5. correct answer only award () [7] GCE Further Pure Mathematics FP (6667) January

2 Q (a) f (.). 9 and f (.). 68 (allow awrt) B () (b) f (.5) < ( ).5 < α <. f (.75) < (.6...).75 < α <. () (c) f ( ) 6 + f ( ).68., f ( ) 6.7.8, (a) Both answers required for B. Accept anything that rounds to dp values above. (b) f(.5) or awrt -.6 (f(.5) and awrt -.6) AND (f(.75) and awrt -.) for first.75 < α <. or epression using bracets or equivalent in words for second (c) One term correct for, both correct for Correct formula seen or implied and attempt to substitute for awrt 6. for second which can be implied by correct final answer awrt.8 correct answer only [9] GCE Further Pure Mathematics FP (6667) January

3 Q For n : u, u 5 + Assume true for n : u 5 5(5 + ) u True for n + if true for n. True for n, true for all n. B cso [] Accept u + or above B 5(5 + ) seen award ( + ) 5 + or 5 + award first All three elements stated somewhere in the solution award final GCE Further Pure Mathematics FP (6667) January 5

4 Q (a) (, ) cao B () (b) P: y A and B lie on PB PS or a correct method to find both PB and PS Perimeter (b) Both B mars can be implied by correct diagram with lengths labelled or coordinates of vertices stated. Second for their four values added together. or awrt.7 for final B B [6] GCE Further Pure Mathematics FP (6667) January 6

5 Q5 (a) det A a ( a + ) ( 5 ) a + a + (b) a + a + ( a + ) + 6 Positive for all values of a, so A is non-singular () ft cso () 5 (c) A B for B () [8] (a) Correct use of ad bc for (b) Attempt to complete square for Alt Attempt to establish turning point (e.g. calculus, graph) Minimum value 6 for ft Positive for all values of a, so A is non-singular for cso Alt Attempt at b ac for. Can be part of quadratic formula Their correct - for first No real roots or equivalent, so A is non-singular for final cso (c) Swap leading diagonal, and change sign of other diagonal, with numbers or a for Correct matri independent of their award final GCE Further Pure Mathematics FP (6667) January 7

6 Q6 (a) 5 i is a root B () (b) ( (5 + i) )( (5 i) ) c + d ( + 9)( ) c 9, d 58 (c), y 5 6 Conjugate pair in st and th quadrants (symmetrical about real ais) Fully correct, labelled B B () (b) st M: Form bracets using ( α )( β ) and epand. nd M: Achieve a -term quadratic with no i's. [8] (b) Alternative: Substitute a comple root (usually 5+i) and epand bracets (5 + i) (5 + i) + c (5 + i) + d ( 5 + 5i 6 8i) (5 + i ) + (5c + ci) + d ( nd M for achieving an epression with no powers of i) Equate real and imaginary parts c 9, d 58, GCE Further Pure Mathematics FP (6667) January 8

7 Q7 (a) c dy y c d dy c d ( ct) t without or y c y ( ct) t y + ct t t (*) B cso () (b) Substitute ( 5c, c) : ct + 5c ct t + t 5 ( t + 5)( t ) t 5 t c c Points are 5c, and c, both 5 (a) Use of y y m( ) where m is their gradient epression in terms of c and / or t only for second. Accept y m + and attempt to find for second. (b) Correct absolute factors for their constant for second. Accept correct use of quadratic formula for second. Alternatives: d dy (a) c and ct dt dt B dy dy d d dt dt t, then as in main scheme. d y (a) y + d B dy y d t, then as in main scheme. [9] GCE Further Pure Mathematics FP (6667) January 9

8 Q8 (a) r and r Assume true for n : B + r r ( + ) + ( + ) [ + ( + ) ] ( + ) ( + ) ( True for n + if true for n. True for n, true for all n. + ) B cso (b) r + r + n ( n + ) + n( n + ), + n [ n( n + ) + 6( + ) + 8] n n [ n + n + 7n + ] n( n + )( + 7) n n (*) B, B cso (c) with attempt to sub in answer to part (b) (5 7 6) ( 6 ) () [] (a) Correct method to identify ( +) as a factor award ( + ) ( + ) award first All three elements stated somewhere in the solution award final (b) Attempt to factorise by n for and n + n + 7 n + for first (c) no woring / GCE Further Pure Mathematics FP (6667) January

9 GCE Further Pure Mathematics FP (6667) January Q9 (a) 5 or π rotation (anticlocwise), about the origin B, B () (b) q p 6 q p and 8 + q p or equivalent p 7 and q both correct () (c) Length of OA ( length of OB) 5 5, 7 +, () (d) M () (e) so coordinates are ), ( () [] Order of matri multiplication needs to be correct to award Ms (a) More than one transformation / (b) Second for correct matri multiplication to give two equations Alternative: (b) M First 7 Second (c) Correct use of their p and their q award (e) Accept column vector for final.