4751 Mark Scheme June Mark Scheme 4751 June 2005

Transcription

1 475 Mark Scheme June 005 Mark Scheme 475 June 005

2 475 Mark Scheme June 005 Section A 40 subst of for x or attempt at long divn with x x seen in working; 0 for attempt at factors by inspection 6y [ x = ] as final answer + m n + and n + both seen n + =(n + ) o.e o.e. for x + mx = y + 5y o.e. and for x( + m) or ft sign error condone e.g. a instead of n for last marks or starting again with full method for middle number = y etc or a factor of both terms so divisible by for 0.6 or 0.6x o.e. or rearrangement to y = form [need not be correct] condone values of x and y given (4, 0) (0, /5) o.e x + 6x x isw 4 B for terms correct or all correct except for signs; B for two terms correct including at least one of x and 6x ; B for soi or for 8 and x 4 6 (i) (ii) a 8 cao (iii) a b or a b isw 7 (i) 6 or 54 isw M for two terms correct or for a b or or ; ignore ± 6 6 (9 ab) 9ab for (4 6) or 6 or seen 5 (ii) x (0 x) = x(5 x) = 56 or 0x x = for attempt to multiply num. and denom. by and for 8 or 5 7 seen 5 allow for length = 0 x soi NB answer given (x 7)(x 8) x = 7 or 8 7 by 6 or 8 by 4 9 [y=] x + = x 7x + [0 =] x 0x or x + 0x + 0 ± 00 + x = 6 0 ± 5 ± 8 = or o.e. isw 6 A 0 for formula or completing sq etc must be explicit; both values required allow for 6 and 4 found following 7 and 8; both required or rearrangement of linear and subst for x in quadratic attempted condone one error; dep on first attempt at formula [dep. on first and quadratic = 0]; M for whole method for completing square or to stage before taking roots for two of three terms correct [with correct fraction line] or for one root 5 5

3 475 Mark Scheme June 005 Section B 0 i (x 4) + 9 B for 4, B for 9 or for 5 6 ii (4, 9) or ft + parabola right way up 5 at intersection on y-axis (mark intent) G G condone stopping at y axis ignore posn of min: can ft theirs 4 iii x > 7 or x < for x 8x + 7 [> 0] and for (x 7)(x ) [>0] or for (x 4) [>] 9 and for x 4 > and x 4 < or B for and 7 iv [y =] x 8x + 5 or [y =] (x 4) i (6 0) + (0 ) AC = 0 AB = 98 and BC = clear correct use of Pythagoras s or for grad AB = and grad BC = and for comment/ showing theorem m m = o.e. 4 ii [angle in a semicircle so ]AC diameter [so radius = 5] midpt of AC = (6/, [0+]/) d or diameter needed; NB ans given method must be shown; NB ans givn (x ) + (y 6) = 5 o.e. isw iii [grad AC =] 8/6 or 4/ grad tgt = /4 y 0 = [ /4](x 6) o.e. [e.g. x + 4y = 58] or ft (58/, 0) and (0, 58/4) o.e. isw i (x + )(x )(x 5) (x + )(x 7x + 0) correct step shown towards completion [answer given] ii iii cubic the right way up, and 5 indicated on x axis 0 indicated at intn on y axis f(4) attempted = attempt at long division of x 6x + x + 0 by x 4 as far as x 4x in working x x 5 = 0 A G G G M A B for one side correct 4 for grad tgt = /their grad AC or for y = their m x + c then subst (6, 0) to find c each cao; condone not as coords 5 o.e. with two other factors; condone missing brackets if expanded correctly; A for x 5x x + x + 0x 5x x + 0 must extend beyond x = and 5 at intersections of curve and axis or f(4) + 0; or 4 a root implies (x 4) a factor or vv or 5 etc or correct long division if first earned or M for (x 4)(x +.. 5) or (x 4)(x x + k) seen; for realising long divn by x 4 needed but not doing it for x x 5 SC for finding f(x) (x 4) = x x 5 rem 0 without further explanation 6

4 Mark Scheme 475 January 006

5 Section A n (n + ) seen = odd even and/or even odd = even (i) translation of 0 or B for n odd n odd, and comment eg odd + odd = even B for n even n even, and comment eg even + even = even allow for any number multiplied by the consecutive number is even or to the right or x x + or all x values are increased by (ii) y = f(x ) for y = f(x + ) 6 + x + 4x + 8x + x 4 isw 4 for 4 terms correct, for terms correct, or for s.o.i. and for expansion with correct powers of 4 x > 4.5 o.e. isw www 4 accept 7/6 or better; for x = [ for etc expand brackets or divide by or Ms for each of the four steps carried out correctly with subtract constant from LHS inequality [ if working with divide to find x] equation] (ft from earlier errors if of comparable difficulty) 5 4P 4P 4 for PC + 4P = C [ C = ] or o.e. P P 6 f() used + + k = 6 k = 7 grad BC = ¼ soi y = ¼ (x ) o.e. cao 4 or ft from their BC 8 (i) 0 (ii) + or + 6 or mixture of these for 4P = C PC or ft for 4P = C ( P) or ft B for 4 [ C = ] P o.e. unsimplified or division by x as far as x + x or remainder = 4 + k B for k = www for m m = soi or for grad AB = 4 or grad BC = /4 e.g. y = 0.5x +.5 for subst y = 0 in their BC for 8= or 50 = 5 soi B for 6 50 or other correct a b for mult num and denom by 6+ and for denom = or

6 9 (i) k 5/4 (ii).5 B for + 6 or + M for 5 4k 0 or B for 5/4 obtained isw or for b 4ac soi or completing square accept 0/8 or better, isw; for attempt to express quadratic as (x + a) or for attempt at quadratic formula 5

7 Section B 0 i (0, 0), 45 isw or 5 + ii x = y or y = x seen or used subst in eqn of circle to eliminate variable 9 6y + y + y = 45 y 6y 6 = 0 or y y 8 = 0 (y 6)(y + )= 0 y = 6 or x = or 6 (6 ) + ( 6) for correct expn of ( y) seen oe condone one error if quadratic or quad. formula attempted [complete sq attempt earns last Ms] or for (6, ) and for (, 6) no ft from wrong points (A.G.) i (x.5) 6.5 B for a = 7/ o.e, B for b = 5/4 o.e. or for 6 (7/) or 6 (their a) 8 ii (.5, 6.5) o.e. or ft from their (i) iii (0, 6) (, 0) (6, 0) curve of correct shape fully correct intns and min in 4th quadrant iv x 7x + 6 = x x + 4 = 4x x = ½ or 0.5 or /4 cao i sketch of cubic the correct way up curve passing through (0, 0) curve touching x axis at (, 0) ii x(x 6x + 9) = iii x 6x + 9x = subst x = in LHS of their eqn or in x(x ) = o.e. working to show consistent + allow x =.5 and y = 6.5 or ft; allow shown on graph each [stated or numbers shown on graph] G G or 4x = 0 (simple linear form; condone one error) condone no comment re only one intn G G G or (x x)(x ) = [for one step in expanding brackets] for nd step, dep on first or for division of their eqn by (x ) and showing no remainder 5 division of their eqn by (x ) attempted x 4x + or inspection attempted with (x + kx + c) seen

8 soln of their quadratic by formula or completing square attempted x = ± or (4 ± )/ isw locating the roots on intersection of their curve and y = A G condone ignoring remainder if they have gone wrong for one correct must be intns; condone x = not marked; mark this when marking sketch graph in (i) 7 G

9 Mark Scheme 475 June 006

10 Section A V V V [] r = [ ± ] o.e. double-decker for r or π h r π h π h for correct constructive first step or for r = k ft their r = k a = ¼ for subst of or for 8 + 4a + 7 = 0 o.e. obtained eg by division by (x + ) x + y = 6 or y =.5x + isw for x + y = c or y =.5x + c for subst (, 0) to find c or for or for y 0 = their gradient (x ) 4 (i) P Q (ii) P Q 5 x + (x + ) = 6 o.e. 0x = or 0y = 9 o.e. (0.,.9) or x = 0. and y =.9 o.e. 6 < x < [condone x <, x > ] 7 (i) 8 6 condone omission of P and Q for subst or for rearrangement and multn to make one pair of coefficients the same or for both eqns in form y = (condone one error) graphical soln: (must be on graph paper) for each line, for (0.,.9) o.e cao; allow B for (0.,.9) o.e. 4 B for and or for x + x [< 0]or (x + ) < / = 4 and for (x + )(x ) or x= ( ±4)/ or for (x + ) and ± on opp. sides of eqn or inequality; if 0, then SC for one of x <, x > for 0 6 or 6 or or (ii) 49 5 isw 60 or ft for 8 their 0 9 (i) 4/7 (ii) a 0 b 8 c - or 0 x + 9x = 5 0x = 5 a b c 0 8 x= ±( 0)/ or.± (5/) or ±5/ 0 oe y = [±] (5/) o.e. eg y = [±].5 A B for 49 and for 5 or for correct terms from for just 0 seen in second part; for 6!/(!!) or better condone 60x ; for [ ] [their] 0 seen or for [their] 0 ( x) ; allow B 4 for 60 for 4 or 7 for elements correct, for elements correct, for any adding of elements; mark final answer; condone correct but unnecessary brackets 5 for subst for x or y attempted or x =.5 o.e.; condone one error from start [allow 0x 5 = 0 + correct substn in correct formula] allow ±.5; for one value ft their x value(s) if irrational; condone not written as coords. 5

11 Section B i grad AB = 8/4 or or y = x 0 grad BC = / or ½ or y = ½ x +.5 product of grads = [so perp] (allow seen or used) ii midpt E of AC = (6, 4.5) AC = (9 ) + (8 ) or 85 rad = ½ 85 o.e. (x 6) + (y 4.5) = 85/4 o.e. (5 6) + (0 4.5) = + 8/4 [= 85/4] iii uuur uuur BE = ED = 4.5 D has coords (6 +, ) ft or (5 +, 0 + 9) = (7, 9) i f( ) used = 0 B or for AB = or 80 and BC = + or 5 and AC = or 85; for AC = AB + BC and for [Pythag.] true so AB perp to BC; if 0, allow G for graph of A, B, C allow seen in (i) only if used in (ii); or AE = (9 their 6) + (8 their 4.5) or rad. = 85/4 o.e. e.g. in circle eqn for (x a) + (y b) = r soi or for lhs correct some working shown; or angle in semicircle [=90 ] 6 uuur o.e. ft their centre; or for BC = or (9, 8 + ); condone mixtures of vectors and coords. throughout part iii allow B for (7,9) or for division by (x + ) attempted as far as x + x then for x + 7x + 6 with no remainder or inspection with b = or c = found; B for correct answer ii divn attempted as far as x + x x + x + or (x + )(x + ) iii (x + )(x + 6)(x + ) allow seen earlier; for (x + )(x + ) iv sketch of cubic the right way up G with turning pts; no rd tp through marked on y axis G curve must extend to x > 0 intercepts 6,, on x axis G condone no graph for x < 6 v [x](x + 9x + 0) [x](x + 4)(x + 5) x = 0, 4, 5 i y = x + drawn on graph x = 0. to 0.4 and.7 to.9 ii = x + x x + x [= 0] attempt at formula or completing square ± 7 x = 4 iii branch through (,), branch through (,),approaching y = from below iv and ½ or ft intersection of their curve and line [tolerance mm] A A or other partial factorisation or B for each root found e.g. using factor theorem each; condone coords; must have line drawn for multiplying by x correctly for correctly rearranging to zero (may be earned first) or suitable step re completing square if they go on ft, but no ft for factorising for one soln 5 and approaching y = from above and extending below x axis each; may be found algebraically; ignore y coords.

12 Mark Scheme 475 January 007

13 475 Mark Scheme Jan 007 Section A y = x + 4 for m = stated [M0 if go on to use m = ½ ] or for y = x + k, k 7 and indep for y 0 = m(x ) or (, 0) subst in y = mx + c; allow for y = x + k and k = 4 neg quadratic curve intercept (0, 9) through (, 0) and (, 0) condone (0, 9) seen eg in table [ a = ] c c or as final answer f f 4 f() = seen or used + k + 5 = o.e. k = 5 for attempt to collect as and cs on different sides and ft for a ( f) or dividing by f; allow M for 7 c 5 c f etc allow for divn by (x ) with x + x + (k + 4) or x + x obtained alt: for (x )(x + x ) + (may B be seen in division) then dep (and B) for x 5x + 5 alt divn of x + kx + by x with no rem allow 75x 4 ; for 5 or 5 used or seen with x 4 and for 5 or 6 5 6! oe eg or 6 5 4!! seen [ 6 C 4 not sufft] 6 (i) 5 for 5 = 5 soi or for 5 (ii) 9 as final answer 49 7 showing a + b + c = 6 o.e 9 7 bc = 6 for a = soi eg by /7 or /49 4 a simple equiv fraction eg 9/ or 4/4 correct expansion of numerator; may be unsimplified 4 term expansion; M0 if get no further than ( ) 7 ; M0 if no evidence before 64/6 o.e. =64/6 o.e. correctly obtained completion showing abc = 6 o.e. may be implicit in use of factors in completion 4

14 475 Mark Scheme January b 4ac soi use of b 4ac < 0 k < 6 [may be implied by k < 4] 4 < k < 4 or k > 4 and k < 4 isw 9 (i) a 5 b as final answer may be implied by k < 6 deduct one mark in qn for instead of <; allow equalities earlier if final inequalities correct; condone b instead of k; if M not earned, give SC for qn [or SC] for k [=] 4 and 4 as answer] 4 for terms correct in final answer ( x+ )( x ) (ii) ( x )( x ) x + as final answer x M for each of numerator or denom. correct or, for correct factors seen separately 5 0 correct expansion of both brackets seen (may be unsimplified), or difference of squares used 4m correctly obtained [p =] [±]m cao M for one bracket expanded correctly; for M, condone done together and lack of brackets round second expression if correct when we insert the pair of brackets 4 Section B ia 0. to 0. and.7 to.8 + [tol. mm or 0.05 throughout qn]; if 0, allow for drawing down lines at both values ib x + = 4 x x their y = 4 x drawn 0. to 0.5 and.65 to.8 B condone one error allow M for plotting positive branch of y = x + /x [plots at (,) and (,4.5) and above other graph] or for plot of y = x 4x + each 4 ii (0, ± ) condone y = ± isw; each or for + y = 4 or y = o.e. iii centre (, 0) radius touches at (, ) [which is distance from centre] all points on other branch > from centre + allow seen in (ii) allow ft for both these marks for centre at (, 0), rad ; allow for good sketch or compassdrawn circle of rad centre (±, 0) 4

15 475 Mark Scheme January 007 i (, 6) grad AB = (8 4)/(7 ) or 4/8 grad normal = or ft perp bisector is y 6 = (x ) or ft their grad. of normal (not AB) and/or midpoint correct step towards completion ii Bisector crosses y axis at C (0, ) seen or used AB crosses y axis at D (0, 4.5) seen or used ½ ( their 4.5) (may be two triangles each) 45/4 o.e. without surds, isw y alt allow integration used: ( x+ )d x [ = 7] 0 obtaining AB is y 8 = their ½ (x 7) oe [y = ½ x + 4.5] C D A (, 4) ( x + 4.5)dx 0 = 6/4 o.e. cao their area under CB their area under AB = 45/4 o.e. cao i x is factor soi attempt at divn by x as far as x x seen in working x + x obtained attempt at quad formula or comp square ± as final answer M B (7, 8) 0 x B M A each coord indep obtained for use of m m = ; condone stated/used as with no working only if 4/8 seen or for showing grad given line = and for showing (, 6) fits given line 6 may be implicit in their area calcn for 4 + their grad AB or for eqn AB is y 8 = their ½ (x 7) oe with coords of A or their M used or for [MC] = + 6 or 45 or [MD] = +.5 or.5 oe and for ½ their MC MD; all ft their M MR: AMC used not DMC: lose B for D but then allow ft for MC or MA [=4 + ] and for ½ MA MC and for 5 MR: intn used as D(0, 4) can score a max of, B0, M (eg for their DM = ), A0 condone poor notation allow if seen, with correct line and limits seen/used as above ft from their AB allow only if at least some valid integration/area calculations for these trapezia seen if combined integration, so 6/4 not found separately, mark equivalently for Ms and allow A for final answer eg may be implied by divn or other factor (x. ) or (x + x...) or B www ft their quadratic for ± 8 seen; or B www 6 6 4

16 475 Mark Scheme January 007 ii f(x ) = (x ) 5(x ) + (x )(x 6x + 9) or other constructive attempt at expanding (x ) eg soi x 9x + 7x 7 5x + 5 [+] iii 5 ± or ft B B B B or (x 5)(x + )(x ) soi or ft from their (i) for attempt at multiplying out brackets or valid attempt at multiplying all alt: A for correct full unsimplified expansion or for correct bracket expansion eg (x 5)(x 4x + ) condone factors here, not roots if B0 in this part, allow SC for their roots in (i) 4 5

17 ADVANCED SUBSIDIARY GCE UNIT 475/0 MATHEMATICS (MEI) Introduction to Advanced Mathematics (C) INSERT TUESDAY 6 JANUARY 007 Morning Time: hour 0 minutes Candidate Name Centre Number Candidate Number INSTRUCTIONS TO CANDIDATES This insert should be used in Question. Write your name, centre number and candidate number in the spaces provided above and attach the page to your answer booklet. This insert consists of printed pages. HN/ OCR 007 [L/0/657] OCR is an exempt Charity [Turn over

18 (i) y y = x + x x OCR /0 Insert Jan 07

19 Mark Scheme 475 January 007

20 475 Mark Scheme Jan 007 Section A y = x + 4 for m = stated [M0 if go on to use m = ½ ] or for y = x + k, k 7 and indep for y 0 = m(x ) or (, 0) subst in y = mx + c; allow for y = x + k and k = 4 neg quadratic curve intercept (0, 9) through (, 0) and (, 0) condone (0, 9) seen eg in table [ a = ] c c or as final answer f f 4 f() = seen or used + k + 5 = o.e. k = 5 for attempt to collect as and cs on different sides and ft for a ( f) or dividing by f; allow M for 7 c 5 c f etc allow for divn by (x ) with x + x + (k + 4) or x + x obtained alt: for (x )(x + x ) + (may B be seen in division) then dep (and B) for x 5x + 5 alt divn of x + kx + by x with no rem allow 75x 4 ; for 5 or 5 used or seen with x 4 and for 5 or 6 5 6! oe eg or 6 5 4!! seen [ 6 C 4 not sufft] 6 (i) 5 for 5 = 5 soi or for 5 (ii) 9 as final answer 49 7 showing a + b + c = 6 o.e 9 7 bc = 6 for a = soi eg by /7 or /49 4 a simple equiv fraction eg 9/ or 4/4 correct expansion of numerator; may be unsimplified 4 term expansion; M0 if get no further than ( ) 7 ; M0 if no evidence before 64/6 o.e. =64/6 o.e. correctly obtained completion showing abc = 6 o.e. may be implicit in use of factors in completion 4

21 475 Mark Scheme January (C) Introduction to Advanced Mathematics Section A E [ v = ][ ± ] www m x 4 x + or 7 x + www as final answer M for = or for [ v ][ ] E v m E = ± or m for a correct constructive first step and for v= [ ± ] k ft their v = k; if M0 then SC for E/ ½ m or E/m etc for (x 4)(x ) and for (x + )(x ) (i) (ii) /64 www for dealing correctly with each of reciprocal, square root and cubing (allow only for /64) eg M for 64 or 64 or / 4096 or ¼ or for /6 / or 4 or 4 or 4 - etc 4 4 6x + (x 5) = 7 0x = 7 x =.7 o.e. isw y =.6 o.e.isw 5 (i) 4/5 or 0.8 o.e. (ii) (5, 0) or 5 found www for subst or multn of eqns so one pair of coeffts equal (condone one error) simplification (condone one error) or appropriate addn/subtn to eliminate variable allow as separate or coordinates as requested graphical soln: M0 4 for 4/5 or 4/ 5 or 0.8 or 4.8/6 or correct method using two points on the line (at least one correct) (may be graphical) or for 0.8x o.e. for y = their (i) x + o.e. or 4x + 5y = k and (0, ) subst and for using y = 0 eg = 0.8x or ft their eqn or for given line goes through (0, 4.8) and (6, 0) and for 6 /4.8 graphical soln: allow for correct required line drawn and for answer within mm of (5, 0) 5 9

22 475 Mark Scheme January f() used + k + 7 = k = 6 or division by x as far as x + x obtained correctly or remainder = (4 + k) + 7 o.e. nd dep on first 7 (i) 56 for or more simplified (ii) 7 or ft from their (i)/8 for 7 or ft their (i)/8 or for 56 ( /) o.e. or ft; condone x in answer or in expression; 0 in qn for just Pascal s triangle seen 4 8 (i) 5 for 48 = 4 (ii) common denominator = (5 )(5 + ) = numerator = 0 9 (i) n = m n + 6n = m + m or = m(m + ) (ii) showing false when n is odd e.g. n + 6n = odd + even = odd B M B allow for allow only for 0/ 5 or any attempt at generalising; M0 for just trying numbers or for n + 6n = n (n + ) = even even and for explaining that 4 is a factor of even even or for is a factor of 6n when n is even and for 4 is a factor of n so is a factor of n or n (n + ) = odd odd = odd or counterexample showing not always true; for false with partial explanation or incorrect calculation 5 7

23 475 Mark Scheme January 008 Section B 0 i correct graph with clear asymptote x = (though need not be marked) G G for one branch correct; condone (0, ½ ) not shown SC for both sections of graph shifted two to left G allow seen calculated (0, ½ ) shown ii /5 or. o.e. isw for correct first step iii x = x x(x ) = o.e. x x [= 0]; ft their equiv eqn attempt at quadratic formula ± cao position of points shown B or equivs with ys or (x ) = o.e. or (x ) = ± (condone one error) on their curve with y = x (line drawn or y = x indicated by both coords); condone intent of diagonal line with gradient approx through origin as y = x if unlabelled 6 i (x.5) o.e (x.5) + 7/4 o.e. for clear attempt at.5 allow MA0 for (x.5) + 7/4 o.e. with no (x.5) seen min y = 7/4 o.e. [so above x axis] or commenting (x.5) 0 B ft, dep on (x a) + b with b positive; condone starting again, showing b 4ac < 0 or using calculus 4 ii correct symmetrical quadratic shape 8 marked as intercept on y axis tp (5/, 7/4) o.e. or ft from (i) G G G or (0, 8) seen in table iii x 5x 6 seen or used and 6 obtained x < and x > 6 isw or ft their solns or (x.5) [> or =].5 or ft 4 b also implies first if M0, allow B for one of x < and x > 6 iv min = (.5, 8.5) or ft from (i) so yes, crosses or for other clear comment re translated 0 down and for referring to min in (i) or graph in (ii); or for correct method for solving x 5x = 0 or using b 4ac with this and for showing real solns eg b 4ac = ; allow A0 for valid comment but error in 8.5 ft; allow for showing y can be neg eg (0, ) found and for correct conclusion

24 475 Mark Scheme January 008 i (x 4) 6 + (y ) 4 = 9 o.e. rad = 9 M B for one completing square or for (x 4) or (y ) expanded correctly or starting with (x 4) + (y ) = r : for correct expn of at least one bracket and for = r o.e. or using x gx + y fy+ c = 0 for using centre is (g, f) [must be quoted] and for r = g + f c ii 4 + o.e = 0 which is less than 9 allow for showing circle crosses x axis at and 9 or equiv for y (or showing one positive; one negative); 0 for graphical solutions (often using A and B from (iii) to draw circle) iii showing midpt of AB = (4, ) and showing AB = 9 or showing AC or BC = 9 or that A or B lie on circle iv or showing both A and B lie on circle (or AC = BC = 9), and showing AB = 9 or that C is midpt of AB or that C is on AB or that gradients of AB and AC are the same or equiv. or showing C is on AB and showing both A and B are on circle or AC = BC = 9 grad AC or AB or BC = 5/ o.e. grad tgt = /their grad AC tgt is y 7 = their m (x ) o.e. y = /5 x + /5 o.e. in each method, two things need to be established. Allow for the concept of what should be shown and for correct completion with method shown allow A0 for AB just shown as 6 not 9 allow A0 for stating mid point of AB = (4,) without working/method shown NB showing AB = 9 and C lies on AB is not sufficient earns marks only if M0, allow SC for accurate graph of circle drawn with compasses and AB joined with ruled line through C. may be seen in (iii) but only allow this if they go on to use in this part allow for m m = used eg y = their mx + c then (, 7) subst; M0 if grad AC used condone y = /5x + c and c = /5 o.e

25 475 Mark Scheme June (C) Introduction to Advanced Mathematics Section A x > 6/4 o.e. isw for 4x > 6 or for 6/4 o.e. found or for their final ans ft their 4x > k or kx > 6 (i) (0, 4) and (6, 0) (ii) 4/6 o.e. or ft their (i) isw each; allow x = 0, y = 4 etc; condone x = 6, y = 4 isw but 0 for (6, 4) with no working 4 for x 6 or 4/ 6 or 4/6 o.e. or ft (accept 0.67 or better) 0 for just rearranging to y = x+ 4 4 (i) 0 or / o.e. each 4 (i) T (ii) E (iii) T (iv) F (ii) k < 9/8 o.e. www M for ( )( 8k) < 0 o.e. or 9/8 found or for attempted use of b 4ac (may be in quadratic formula); SC: allow for 9 8k < 0 and ft for k > 9/8 for all correct, for correct. for correct 5 5 y (x ) = (x + ) xy y = x + or ft [ft from earlier errors if of comparable difficulty no ft if there are no xy terms] for multiplying by x ; condone missing brackets for expanding bracket and being at stage ready to collect x terms xy x = y + or ft [ ] y + x = y o.e. or ft for collecting x and other terms on opposite sides of eqn for factorising and division alt method: 5 y = + x 5 y = x 5 x = y 5 x = + y for either method: award 4 marks only if fully correct 4 8

26 475 Mark Scheme June (i) 5 www (ii) 8x 0 y z 4 or x 0 y z 4 allow for ±5; for 5 / seen or for /5 seen or for using 5 / = 5 with another error (ie for coping correctly with fraction and negative index or with square root) mark final answer; B for elements correct, B for elements correct; condone multn signs included, but from total earned if addn signs 5 7 (i) 5 or 5 + ( ) or 5 or a = 5, b =, c = ; for attempt to multiply numerator and denominator by 5 (ii) 7 7 isw www for 7 and for 7 or for correct terms from or or o.e. eg using 6 or M for or or or o.e.; for but not other equivs www 4 M for 0 5 ( [x]) o.e. or M for two of these elements or for 0 or (5 4 )/( ) o.e. used [ 5 C is not sufficient] or for seen; 5 9 (y )(y 4) [= 0] y = or 4 cao x =± or ± cao B or B for 000; condone x in ans; equivs: M for e.g [ ] x o.e. [5 5 may be outside a bracket for whole expansion of all terms], M for two of these elements etc similarly for factor of taken out at start 4 for factors giving two terms correct or attempt at quadratic formula or completing square or B (both roots needed) B for roots correct or ft their y (condone and 4 for B) 4 8

27 475 Mark Scheme June 008 Section B 0 i (x ) 7 mark final answer; for a =, for b = 7 or for + ; bod for (x ) 7 ii (, 7) or ft from (i) + iii sketch of quadratic correct way up and through (0, ) G accept (0, ) o.e. seen in this part [eg in table] if not marked as intercept on graph t.p. correct or ft from (ii) G accept and 7 marked on axes level with turning pt., or better; no ft for (0, ) as min iv x 6x + = x 4 o.e. or their (i) = x 4 x 8x + 6 [= 0] dep on first ; condone one error (x 4) [= 0] or correct use of formula, giving equal roots; allow (x + 4) o.e. ft x + 8x + 6 x = 4, y = 6 if M0M0M0, allow SC for showing (4, 6) is on both graphs (need to go on to show line is tgt to earn more) equal/repeated roots [implies tgt] - must be explicitly stated; condone only one root [so tgt] or line meets curve only once, so tgt or line touches curve only once etc] or for use of calculus to show grad of line and curve are same when x = 4 5

28 475 Mark Scheme June 008 i f( 4) used [= 0] ii division of f(x) by (x + 4) x x (x + )(x ) [f(x) =] (x + 4) (x + )(x ) or B for (x + 4)(x x ) here; or correct division with no remainder as far as x + 8x in working, or two terms of x x obtained by inspection etc (may be earned in (i)), or f( ) = 0 found x x seen implies or B4; allow final ft their factors if A0 earned 4 iii sketch of cubic correct way up G ignore any graph of y = f(x 4) through shown on y axis G or coords stated near graph roots 4,,.5 or ft shown on x axis G or coords stated near graph if no curve drawn, but intercepts marked on axes, can earn max of G0GG iv x (x )([x 4] ) o.e. or x (x )(x 5.5) or ft their factors or (x 4) + 7(x 4) 7(x 4) or stating roots are 0, and 5.5 or ft; condone one error eg x 7 not x correct expansion of one pair of brackets ft from their factors or for correct expn of (x 4) [allow unsimplified]; or for showing g(0) = g() = g(5.5) = 0 in given ans g(x) correct completion to given answer allow M for working backwards from given answer to x(x )(x ) and for full completion with factors or roots 4

29 475 Mark Scheme June 008 i grad AB = 9 or y 9 = (x ) or y = (x + ) y = x + o.e. ii mid pt of AB = (, 5) grad perp = /grad AB y 5 = ½ (x ) o.e. or ft [no ft for just grad AB used] at least one correct interim step towards given answer y + x =, and correct completion NB ans y + x = given alt method working back from ans: x y = o.e. ft their m, or subst coords of A or B in y = their m x + c or B condone not stated explicitly, but used in eqn soi by use eg in eqn ft their grad and/or midpt, but M0 if their midpt not used; allow for y = ½x + c and then their midpt subst no ft; correct eqn only mark one method or the other, to benefit of cand, not a mixture grad perp = /grad AB and showing/stating same as given line finding intn of their y = x + and y + x = [= (, 5)] showing midpt of AB is (, 5) eg stating ½ = or showing that (, 5) is on y + x =, having found (, 5) first [for both methods: for M4 must be fully correct] 4 iii showing ( 5) + ( ) = 40 showing B to centre = 40 or verifying that (, 9) fits given circle iv (x 5) + = 40 at least one interim step needed for each mark; M0 for just 6 + = 40 with no other evidence such as a first line of working or a diagram; condone marks earned in reverse order for subst y = 0 in circle eqn (x 5) = x = 5± or 0 ± 4 isw condone slip on rhs; or for rearrangement to zero (condone one error) and attempt at quad. formula [allow M0 for (x 5) = 40 or for (x 5) + = 0 ] 4 or 5 ± 5

30 475 Mark Scheme January (C) Introduction to Advanced Mathematics Section A (i) 0.5 or /8 (ii) as final answer y = 5x 4 www y x M for = o.e. 9 ( 9) or for grad = or 5 eg in y ( ) = 5x + k and for y = their m(x ) o.e. or subst (, ) or (, 9) in y = their mx + c or for y = kx 4 (eg may be found by drawing) x > 9/6 o.e. or 9/6 < x o.e. www isw M for 9 < 6x or for 6x < 9 or k < 6x or 9 < kx or 7 + < 5x + x [condone for Ms]; if 0, allow SC for 9/6 o.e found 4 a = 5 www for f() = 0 used and for 0 + a = 0 or better long division used: for reaching (8 + a)x 6 in working and for 8 + a = equating coeffts method: M for obtaining x + x + 4x + as other factor 5 (i) 4[x ] (ii) 84[x ] www ignore any other terms in expansion for [x ] and 7[x ] soi; for 7 6 or or for Pascal s triangle seen with 7 row and for or 4 or {x} 5 6

31 475 Mark Scheme January /5 or 0. o.e. www for x + = x 4 and for 5x = o.e. or 7 (i) 5.5 or k =.5 or 7/ o.e. (ii) 6a 6 b 0 8 b 4ac soi for + = and.5 4 x for.5 x = o.e. for 5 = 5 or 5 = 5 SC for 5 o.e. as answer without working for two terms correct and multiplied; mark final answer only allow in quadratic formula or clearly looking for perfect square 4 k 4 8 < 0 o.e. < k < 9 y + 5 = xy + x y xy = x 5 oe or ft y ( x) = x 5 oe or ft x 5 [ y = ] oe or ft as final answer x 0 (i) 9 A condone ; or for identified as boundary may be two separate inequalities; for used or for one end correct if two separate correct inequalities seen, isw for then wrongly combining them into one statement; condone b instead of k; if no working, SC for k < and SC for k > (ie SC for each end correct) for expansion for collecting terms for taking out y factor; dep on xy term for division and no wrong work after ft earlier errors for equivalent steps if error does not simplify problem 4 for 5 or 4 seen 4 (ii) 6 + www for attempt to multiply num. and denom. by + and for denom. 7 or 9 soi from denom. mult by + 5 0

32 475 Mark Scheme January 009 Section B i C, mid pt of AB = = (5, ) ( ) + 4, B evidence of method required may be on diagram, showing equal steps, or start at A or B and go half the difference towards the other [AB =] + 4 [= 60] oe or [CB =] 6 + [=40] oe with AC B or square root of these; accept unsimplified quote of (x a) + (y b) = r o.e with different letters B or (5, ) clearly identified as centre and 40 as r (or 40 as r ) www or quote of gfc formula and finding c = completion (ans given) B dependent on centre (or midpt) and radius (or radius ) found independently and correctly 4 ii correct subst of x = 0 in circle eqn soi (y ) = 5 or y 4y [= 0] y = ± 5 or ft [ y = ]± 5cao condone one error or use of quad formula (condone one error in formula); ft only for term quadratic in y if y = 0 subst, allow SC for (, 0) found alt method: for y values are ± a for a + 5 = 40 soi for a = 40 5 soi for [ y = ]± 5cao 4 iii grad AB = 4 ( ) or / o.e. so grad tgt = eqn of tgt is y 4 = (x ) y = x + 7 or x + y = 7 (0, 7) and (7/, 0) o.e. ft isw B or grad AC (or BC) or ft /their gradient of AB or subst (, 4) in y = x + c or ft (no ft for their grad AB used) accept other simplified versions B each, ft their tgt for grad or /; accept x = 0, y = 7 etc NB alt method: intercepts may be found first by proportion then used to find eqn 6 4

33 475 Mark Scheme January 009 i x + 6x + 0 = 4x x + 0x + 8 [= 0] (x + 4)(x + ) [=0] x = or 4/ o.e. y = 0 or / o.e. for subst for x or y or subtraction attempted or y 5y + 0 [=0]; for rearranging to zero (condone one error) or (y )(y 0); for sensible attempt at factorising or formula or completing square or for each of (, 0) and ( 4/, /) o.e. 5 ii (x + ) for a =, for b =, for c = 7 or for 0 their b soi or for 7/ or for 0/ their b soi 4 iii min at y = 7 or ft from (ii) for positive c (ft for (ii) only if in correct form) B may be obtained from (ii) or from good symmetrical graph or identified from table of values showing symmetry condone error in x value in stated min ft from (iii) [getting confused with factor] B if say turning pt at y = 7 or ft without identifying min or for min at x = [e.g. may start again and use calculus to obtain x = ] or min when (x + ) [] = 0; and for showing y positive at min or for showing discriminant neg. so no real roots and for showing above axis not below eg positive x term or goes though (0, 0) or for stating bracket squared must be positive [or zero] and for saying other term is positive 4

34 475 Mark Scheme January 009 i any correct y value calculated from quadratic seen or implied by plots B for x 0 or ; may be for neg x or eg min.at (.5,.5) (0, 5)(, )(, )(, )(4,) and (5,5) plotted P tol mm; P for 4 correct [including (.5,.5) if plotted]; plots may be implied by curve within mm of correct position good quality smooth parabola within mm of their points ii x 5x+ 5= x x 5x + 5x = and completion to given answer C allow for correct points only [accept graph on graph paper, not insert] 4 iii divn of x 5x + 5x by x as far as x x used in working x 4x + obtained or inspection eg (x )(x..+ ) or equating coeffts with two correct coeffts found use of b 4ac or formula with quadratic factor obtained and comment re shows other roots (real and) irrational or for ± or 4 ± obtained isw A or (x ) = ; may be implied by correct roots or obtained [ for and for comment] NB A is available only for correct quadratic factor used; if wrong factor used, allow ft for obtaining two irrational roots or for their discriminant and comment re irrational [no ft if their discriminant is negative] 5 5

35 475 Mark Scheme June (C) Introduction to Advanced Mathematics Section A (0, 4) and (4/4, 0) o.e. isw 4 M for evidence of correct use of gradient with (, 6) eg sketch with stepping or y 6 = 4(x ) seen or y = 4x + 4 o.e. or for y = 4x + c [accept any letter or number] and for 6 = 4 + c; for (0, 4) [c = 4 is not sufficient for ] and for (4/4, 0) o.e.; allow when x = 0, y = 4 etc isw 4 ( s ut) [ a = ] o.e. as final answer t ( s ut) [condone [ a = ] ] 0.5 t for each of complete correct steps, ft from previous error if equivalent difficulty [eg dividing by t does not count as step needs to be by t ] ( ) [ ] s a = ut gets M only (similarly t other triple-deckers) 0 www for f() = soi and for k = or 7 k = o.e. [a correct -term or -term equation] long division used: for reaching (9 k)x + 4 in working and for 4 + (9 k) = o.e. equating coeffts method: M for (x )(x + x ) [+ ] o.e. (from inspection or division) 4 x < 0 or x > 6 (both required) B each; if B0 then for 0 and 6 identified; 5 (i) 0 www (ii) 80 www or ft 8 their (i) for 5 4 ( ) or 5 4 ( ) or for seen B for 80x ; for or (x) seen 4 6

36 475 Mark Scheme June any general attempt at n being odd and n being even even 7 (i) n odd implies n odd and odd odd = even n even implies n even and even even = even M0 for just trying numbers, even if some odd, some even or n(n ) used with n odd implies n even and odd even = even etc [allow even odd = even] or A for n(n )(n + ) = product of consecutive integers; at least one even so product even; odd odd = odd etc is not sufft for SC for complete general method for only one of odd or even eg n = m leading to (4m m) B for 5 0 or for 5 /5 o.e. (ii) (i) / www (ii) 4 0 answer 9 (i) (x + ) 4 www as final (ii) ft their ( a, b); if error in (i), accept (, 4) if evidence of being independently obtained 0 (x 9)(x + 4) M for 4/6 or for 48 = or 4 or 7 = or 08 = or for 4 9 M for terms correct of soi, for terms correct B for a =, B for b = 4 or for 5 soi B each coord.; allow x =, y = 4; or for 4 o.e. oe for sketch with and 4 marked on axes but no coords given or correct use of quad formula or comp sq reaching 9 and 4; allow for attempt with correct eqn at factorising with factors giving two terms correct, or sign error, or attempt at formula or comp sq [no more than two errors in formula/substn]; for this first M or allow use of y etc or of x instead of x 5 5 x = 9 [or 4] or ft for integers /fractions if first earned x =± cao must have x ; or for (x + )(x ); this may be implied by x =± A0 if extra roots if M0 then allow SC for use of factor theorem to obtain both and as roots or (x + ) and (x ) found as factors and SC for x + 4 found as other factor using factor theorem [ie max SC] 4 0

37 475 Mark Scheme June 009 Section B i y = x ii iii eqn AB is y = -/ x + o.e. or ft x = -/x + or ft x = 9/0 or 0.9 o.e. cao y = 7/0 oe ft their their x 9 ( + ) o.e and 0 completion to given answer for grad AB = or / o.e. 6 need not be simplified; no ft from midpt used in (i); may be seen in (i) but do not give mark unless used in (ii) eliminating x or y, ft their eqns if find y first, cao for y then ft for x ft dep on both Ms earned 4 or square root of this; for or soi or ft 0 0 their coords (inc midpt) or for distance = cos θ and tan θ = and for showing sinθ = and completion 0 iv 0 for 6 + or 40 or square roots of these v 9 www or ft their a 0 for ½ 6 or 9 their 0 0 0

38 475 Mark Scheme June 009 ia expansion of one pair of brackets correct 6 term expansion eg [(x + )](x 6x + 8); need not be simplified eg x 6x + 8x + x 6x + 8; or M for correct 8 term expansion: x 4x + x x + 8x 4x x + 8, if one error allow equivalent marks working backwards to factorisation, by long division or factor theorem etc or for all three roots checked by factor theorem and for comparing coeffts of x ib cubic the correct way up x-axis:,, 4 shown y-axis 8 shown G G G with two tps and extending beyond the axes at ends ignore a second graph which is a translation of the correct graph ic [y=](x )(x 5)(x 7) isw or (x ) 5(x ) + (x ) + 8 isw or x 4x + 59x 70 if one slip or for [y =] f(x ) or for roots identified at, 5, 7 or for translation to the left allow for complete attempt: (x + 4)( x + )(x ) isw or (x + ) 5(x + ) + (x + ) + 8 isw (0, 70) or y = 70 allow for (0, 4) or y = 4 after f(x + ) used ii = 4 or = 0 long division of f(x) or their f(x) + 4 by (x ) attempted as far as x x in working x x 4 obtained ( ) ( ) ± 4 4 [ x = ] or (x ) = 5 B or correct long division of x 5x + x + by (x ) with no remainder or of x 5x + x + 8 with rem 4 or inspection with two terms correct eg (x )(x.. 4) dep on previous earned; for attempt at formula or comp square on their other factor ± 0 o.e. isw or ± 5 5 4

39 475 Mark Scheme June 009 i (5, ) 0 or 5 ii no, since 0 < 5 or showing roots of y 4y + 9 = 0 o.e. are not real 0 for ± 0 etc or ft from their centre and radius for attempt (no and mentioning 0 or 5) or sketch or solving by formula or comp sq ( 5) + (y ) = 0 [condone one error] or SC for fully comparing distance from x axis with radius and saying yes iii y = x 8 or simplified alternative for y = (x 5) or ft from (i) or for y = x + c and subst their (i) or for ans y = x + k, k 0 or 8 iv (x 5) + (x) = 0 o.e. subst x + for y [oe for x] 5x 0x + 5[ = 0] or better equiv. obtaining x = (with no other roots) or showing roots equal expanding brackets and rearranging to 0; condone one error; dep on first one intersection [so tangent] (, 4) cao alt method y = ½ (x 5) o.e. x + = ½ (x 5) o.e. x = y = 4 cao showing (, 4) is on circle alt method perp dist between y = x 8 and y = x + = 0 cos θ where tan θ = showing this is 0 so tgt B o.e.; must be explicit; or showing line joining (,4) to centre is perp to y = x + allow y = 4 line through centre perp to y = x + dep; subst to find intn with y = x + by subst in circle eqn or finding dist from centre = 0 [a similar method earns first for eqn of diameter, nd for intn of diameter and circle each for x and y coords and last B for showing (, 4) on line award only if (, 4) and (9, 0) found without (, 4) being identified as the soln] x = 5 0 sin θ x = (, 4) cao or other valid method for obtaining x allow y = 4 5 5

40 475 Mark Scheme 475 (C) Introduction to Advanced Mathematics [ a = ]c b www o.e. for each of complete correct steps, ft from previous error if equivalent difficulty condone = used for first two Ms 5x < x+ 0 M0 for just 5x < (x + 5) x < or < x or ft x < o.e. or ft; isw further simplification of /; M0 for just x < 4. (i) (4, 0) allow y = 0, x = 4 bod B for x = 4 but do not isw: 0 for (0, 4) seen 0 for (4, 0) and (0, 0) both given (choice) unless (4, 0) clearly identified as the x-axis intercept (ii) 5x + (5 x) = 0 o.e. (0/, 5/) www isw A for subst or for multn to make coeffts same and appropriate addn/subtn; condone one error or for x = 0/ and for y = 5/ o.e. isw; condone. or better and.67 or better for (.,.7) 4 (i) translation 4 by or 4 [units] to left 0 4 (ii) sketch of parabola right way up and with minimum on negative y-axis min at (0, 4) and graph through and on x-axis B B B B 0 for shift/move or 4 units in negative x direction o.e. mark intent for both marks must be labelled or shown nearby 5 (i) or ± for 44 o.e. or for 44 = soi 5 (ii) denominator = 8 B B0 if 6 after addition numerator = ( 5+ 7) = as final answer for, allow in separate fractions allow B for as final answer 8 www

41 475 Mark Scheme January 00 6 (i) cubic correct way up and with two turning pts B touching x-axis at, and through it at.5 and no other intersections y- axis intersection at 5 B B intns must be shown labelled or worked out nearby 6 (ii) x x 8x 5 B for terms correct or for correct expansion of product of two of the given factors 7 attempt at f( ) k = 6 8 k = x + 5x+ + www isw x x or x + 5x + 75x - + 5x - www isw 4 or for long division by (x + ) as far as obtaining x x and for obtaining remainder as k 4 (but see below) equating coefficients method: M for (x + )(x x + 8) [+6] o.e. (from inspection or division) eg M for obtaining x x + 8 as quotient in division 5 B for both of x and x and or 5x - isw for soi; for each of 5x and 75 x or 75x- isw or SC for completely correct unsimplified answer

42 475 Mark Scheme January 00 9 x 5x + 7 = x 0 or attempt to subst (y + 0)/ for x x 8x + 7 [= 0] o.e or y 4y + [= 0] o.e use of b 4ac with numbers subst (condone one error in substitution) (may be in quadratic formula) b 4ac = or 4 cao [or 6 5 or 6 if y used] condone one error; allow for x 8x = 7 [oe for y] only if they go on to completing square method or (x 4) = 6 7 or (x 4) + = 0 (condone one error) or (x 4) = or x = 4± [or (y ) = 9 or y = ± 9 ] [< 0] so no [real] roots [so line and curve do not intersect] or conclusion from comp. square; needs to be explicit correct conclusion and correct ft; allow < 0 so no intersection o.e.; allow 4 so no roots etc allow A for full argument from sum of two squares = 0; for weaker correct conclusion some may use the condition b < 4ac for no real roots; allow equivalent marks, with first for 64 < 68 o.e. 0 (i) grad CD = 5 = o.e. 4 isw ( ) NB needs to be obtained independently of grad AB ( ) ( ) grad AB = or isw same gradient so parallel www 0 (ii) [BC =] + [BC =] showing AD = + 4 [=7] [ BC ] isw must be explicit conclusion mentioning same gradient or parallel if M0, allow B for 'parallel lines have same gradient' o.e. accept (6 ) + ( 5) o.e. or [BC =] or [AD =] 7 or equivalent marks for finding AD or AD first alt method: showing AC BD mark equivalently

43 475 Mark Scheme January 00 0 (iii) [BD eqn is] y = eg allow for at M, y = or for subst in eqn of AC eqn of AC is y 5 =6/5 (x ) o.e [ y =.x +.4 o.e.] M is (4/, ) o.e. isw 0 (iv) midpt of BD = (5/, ) or equivalent simplified form cao midpt AC = (/, ) or equivalent simplified form cao or M is / of way from A to C conclusion neither diagonal bisects the other M or for grad AC = 6/5 o.e. (accept unsimplified) and for using their grad of AC with coords of A(, ) or C (, 5) in eqn of line or for stepping method to reach M allow : at M, x = 6/ o.e. [eg =4/] isw A0 for. without a fraction answer seen or showing BM MD oe [BM = 4/, MD = 7/] or showing AM MC or AM MC in these methods is dependent on coords of M having been obtained in part (iii) or in this part; the coordinates of M need not be correct; it is also dependent on midpts of both AC and BD attempted, at least one correct alt method: show that mid point of BD does not lie on AC () and vice-versa (), for both and conclusion 4

44 475 Mark Scheme January 00 (i) centre C' = (, ) radius 5 0 for ± 5 or 5 (ii) showing (6 ) + ( 6 + ) = 5 B interim step needed showing that AC = C B = 4 o.e. B or B each for two of: showing midpoint of AB = (, ); showing B (0, ) is on circle; showing AB = 0 or B for showing midpoint of AB = (, ) and saying this is centre of circle or B for finding eqn of AB as y = 4/ x + o.e. and B for finding one of its intersections with the circle is (0, ) or B for showing C B = 5 and B for showing AB = 0 or that AC and BC have the same gradient or B for showing that AC and BC have the same gradient and B for showing that B (0, ) is on the circle (iii) grad AC' or AB = 4/ o.e. grad tgt = /their AC' grad y ( 6) = their m(x 6) o.e. y = 0.75x 0.5 o.e. isw (iv) centre C is at (, 4) cao circle is (x ) + (y + 4) = 00 B B or ft from their C', must be evaluated may be seen in eqn for tgt; allow M for grad tgt = ¾ oe soi as first step or for y = their m x + c then subst (6, 6) eg for 4y = x 4 allow B4 for correct equation www isw B for each coord ft their C if at least one coord correct 5

45 475 Mark Scheme January 00 (i) 0 (ii) [x =] 5 or ft their (i) ht = 5[m] cao (iii) d = 7/ o.e. [y =] /5.5 (0.5) o.e. or ft = 9/0 o.e. cao isw (iv) 4.5 = /5 x(0 x) o.e..5 = x(0 x) o.e. x 0x + 45 [= 0] o.e. eg x 0x +.5 [=0] or (x 5) =.5 0 ± 40 [ x = ] or 5± 0 o.e. 4 width = 0 o.e. eg.5 cao not necessarily ft from (i) eg they may start again with calculus to get x = 5 or ft their (ii).5 or their (i).5 o.e. or 7 /5.5 or ft or showing y 4 = /0 o.e. cao eg 4.5 = x( 0.x) etc cao; accept versions with fractional coefficients of x, isw or x 5= [ ± ].5 o.e.; ft their quadratic eqn provided at least gained already; condone one error in formula or substitution; need not be simplified or be real accept simple equivalents only 6

46 GCE Mathematics (MEI) Advanced Subsidiary GCE 475 Introduction to Advanced Mathematics (C) Mark Scheme for June 00 Oxford Cambridge and RSA Examinations

47 475 Mark Scheme June 00 SECTION A y = x + c or y y = (x x ) y 5 = their m(x 4) o.e. y = x 7 or simplified equiv. (i) 50a 6 b 7 (ii) 6 cao allow for clearly stated/ used as gradient of required line or (4, 5) subst in their y = mx + c; allow for y 5 = m(x 4) o.e. condone y = x + c and c = 7 or B www B for two elements correct; condone multiplication signs left in SC for eg 50 + a 6 + b 7 (iii) 64 condone ±64 for [±]4 or for only or for ac = y 5 o.e. ac + 5 = y o.e. ( ac ) [ y = ] + 5 o.e. isw for each of correct or ft correct steps s.o.i. leading to y as subject or some/all steps may be combined; allow B for ( ) [ y = ] ac+ 5 o.e. isw or B if one error 4 (i) x > 6x + 5 > 8x o.e. or ft x < /8 o.e. or ft isw or x > x +.5 for collecting terms of their inequality correctly on opposite sides eg 8x > allow B for correct inequality found after working with equation allow SC for /8 o.e. found with equation or wrong inequality 4 (ii) 4 < x < ½ o.e. accept as two inequalities for one end correct or for 4 and ½ 5 (i) 7 for 48 = 4 or 7 =

48 475 Mark Scheme June 00 5 (ii) www isw B for 7 [B0 for 7 wrongly obtained] and B for or B for one term of numerator correct; if B0, then for attempt to multiply num and denom by k soi k = attempt at f() m = 59 o.e. m = 4 cao allow for expansion with 5x + kx and no other x terms or for (9 5) / soi must substitute for x in cubic not product or long division as far as obtaining x + x in quotient or from division m ( 6) = 59 o.e. or for 7 + k + m = 59 or ft their k 7 4 x x x 6 x oe (must be simplified) isw 4 B for 4 terms correct, or B for terms correct or for all correct but unsimplified (may be at an earlier stage, but factorial or n C r notation must be expanded/worked out) or B for, 4, 6, 4, soi or for x [must have at least one other term] 8 5(x + ) 4 4 B for a = 5, and B for b = and B for c = 4 or for c = 6 their ab or for [their a](6/their a their b ) [no ft for a = ] 9 mention of 5 as a square root of 5 or ( 5) = o.e. or x + 5 = 0 condone 5 = 5 or, dep on first being obtained, allow for showing that 5 is the only soln of x 5 = 0 Section A Total: 6 allow M for x 5 = 0 (x + 5)(x 5) [= 0] so x 5 = 0 or x + 5 = 0

49 475 Mark Scheme June 00 SECTION B 0 (i) (x )(x + ) x = / and obtained 0 (ii) graph of quadratic the correct way up and crossing both axes crossing x-axis only at / and or ft from their roots in (i), or their factors if roots not given crossing y-axis at 0 (iii) use of b 4ac with numbers subst (condone one error in substitution) (may be in quadratic formula) 5 40 < 0 or 5 obtained 0 (iv) x x = x 5x + 0 o.e. x + 4x [= 0] M B B B B for factors with one sign error or giving two terms correct allow for (x.5)(x + ) with no better factors seen or ft their factors for x = / condone and marked on axis and crossing roughly halfway between; intns must be shown labelled or worked out nearby may be in formula or (x.5) =6.5 0 or (x.5) +.75 = 0 oe (condone one error) or 5 seen in formula or (x.5) =.75 oe or x =.5 ±.75 oe attempt at eliminating y by subst or subtraction or (x + ) = 7; for rearranging to form ax + bx + c [ = 0] or to completing square form condone one error for each of nd and rd s use of quad. formula on resulting eqn (do not allow for original quadratics used) ± 7 cao or x + =± 7 o.e. nd and rd s may be earned for good attempt at completing square as far as roots obtained

50 475 Mark Scheme June 00 (i) grad AB = [= /] 5 ( ) y = their grad (x ( )) or y = their grad (x 5) or use of y = their gradient x + c with coords of A or B y x ( ) or M for = o.e. 5 ( ) y = /x + 8/ or y = x + 8 o.e isw (ii) when y = 0, x = 8; when x = 0, y = 8/ or ft their (i) [Area =] ½ 8/ 8 o.e. cao isw (iii) grad perp = /grad AB stated, or used after their grad AB stated in this part midpoint [of AB] = (, ) y = their grad perp (x ) or ft their midpoint alt method working back from ans: grad perp = /grad AB and showing/stating same as given line finding intn of their y = /x 8/and y = x 4 is (, ) showing midpt of AB is (, ) or o.e. eg x + y 8 = 0 or 6y = 6 x allow B for correct eqn www allow y = 8/ used without explanation if already seen in eqn in (i) NB answer / given; allow 4 8/ if first earned; or for 8 ( ) ( ) x dx= 8 x x 0 64 [ 0] and dep for ( ) or showing / = if (i) is wrong, allow the first here ft, provided the answer is correct ft must state midpoint or show working for M this must be correct, starting from grad AB = /, and also needs correct completion to given ans y = x 4 mark one method or the other, to benefit of candidate, not a mixture eg stating / = or showing that (, ) is on y = x 4, having found (, ) first [for both methods: for M must be fully correct]

51 475 Mark Scheme June 00 (iv) subst x = into y = x 4 and obtaining centre = (, 5) or using ( ) + ( b) = (5 ) + ( b) and finding (, 5) r = (5 ) + ( 5) o.e. r = 0 o.e. cao eqn is (x ) + (y 5) = 0 or ft their r and y-coord of centre (i) trials of at calculating f(x) for at least one factor of 0 details of calculation for f() or f( ) or f( 5) attempt at division by (x ) as far as x x in working correctly obtaining x + 8x + 5 factorising a correct quadratic factor (x )(x + )(x + 5) (ii) sketch of cubic right way up, with two turning points values of intns on x axis shown, correct ( 5,, and ) or ft from their factors/ roots in (i) y-axis intersection at 0 B B B B or ( ) + ( 5) or ft their centre using A or B condone (x ) + (y b) = r o.e. or (x ) + (y their 5) = r o.e. (may be seen earlier) M0 for division or inspection used or equiv for (x + ) or (x + 5); or inspection with at least two terms of quadratic factor correct or B for another factor found by factor theorem for factors giving two terms of quadratic correct; M0 for formula without factors found condone omission of first factor found; ignore = 0 seen allow last four marks for (x )(x + )(x + 5) obtained; for all 6 marks must see factor theorem use first 0 if stops at x-axis on graph or nearby in this part mark intent for intersections with both axes or x = 0, y = 0 seen in this part if consistent with graph drawn

52 475 Mark Scheme June 00 (iii) (x ) substituted for x in either form of eqn for y = f(x) correct or ft their (i) or (ii) for factorised form; condone one error; allow for new roots stated as 4, and or ft Section B Total: 6 (x ) expanded correctly (need not be simplified) or two of their factors multiplied correctly correct completion to given answer [condone omission of y = ] dep or for correct or correct ft multiplying out of all brackets at once, condoning one error [x x + 4x + x + 8x 6x x 4] unless all brackets already expanded, must show at least one further interim step allow SC for (x + ) subst and correct exp of (x + ) or two of their factors ft or, for those using given answer: for roots stated or used as 4, and or ft for showing all roots satisfy given eqn B for comment re coefft of x or product of roots to show that eqn of translated graph is not a multiple of RHS of given eqn

53 GCE Mathematics (MEI) Advanced Subsidiary GCE Unit 475: Introduction to Advanced Mathematics Mark Scheme for January 0 Oxford Cambridge and RSA Examinations

54 475 Mark Scheme January 0 Marking instructions for GCE Mathematics (MEI): Pure strand. You are advised to work through the paper yourself first. Ensure you familiarise yourself with the mark scheme before you tackle the practice scripts.. You will be required to mark ten practice scripts. This will help you to understand the mark scheme and will not be used to assess the quality of your marking. Mark the scripts yourself first, using the annotations. Turn on the comments box and make sure you understand the comments. You must also look at the definitive marks to check your marking. If you are unsure why the marks for the practice scripts have been awarded in the way they have, please contact your Team Leader.. When you are confident with the mark scheme, mark the ten standardisation scripts. Your Team Leader will give you feedback on these scripts and approve you for marking. (If your marking is not of an acceptable standard your Team Leader will give you advice and you will be required to do further work. You will only be approved for marking if your Team Leader is confident that you will be able to mark candidate scripts to an acceptable standard.) 4. Mark strictly to the mark scheme. If in doubt, consult your Team Leader using the messaging system within scoris, by or by telephone. Your Team Leader will be monitoring your marking and giving you feedback throughout the marking period. An element of professional judgement is required in the marking of any written paper. Remember that the mark scheme is designed to assist in marking incorrect solutions. Correct solutions leading to correct answers are awarded full marks but work must not be judged on the answer alone, and answers that are given in the question, especially, must be validly obtained; key steps in the working must always be looked at and anything unfamiliar must be investigated thoroughly. Correct but unfamiliar or unexpected methods are often signalled by a correct result following an apparently incorrect method. Such work must be carefully assessed. When a candidate adopts a method which does not correspond to the mark scheme, award marks according to the spirit of the basic scheme; if you are in any doubt whatsoever (especially if several marks or candidates are involved) you should contact your Team Leader. 5. The following types of marks are available. M A suitable method has been selected and applied in a manner which shows that the method is essentially understood. Method marks are not usually lost for numerical errors, algebraic slips or errors in units. However, it is not usually sufficient for a candidate just to indicate an intention of using some method or just to quote a formula; the formula or idea must be applied to the specific problem in hand, eg by substituting the relevant quantities into the formula. In some cases the nature of the errors allowed for the award of an M mark may be specified. A Accuracy mark, awarded for a correct answer or intermediate step correctly obtained. Accuracy marks cannot be given unless the associated Method mark is earned (or implied). Therefore M0 cannot ever be awarded. B Mark for a correct result or statement independent of Method marks.

55 475 Mark Scheme January 0 E A given result is to be established or a result has to be explained. This usually requires more working or explanation than the establishment of an unknown result. Unless otherwise indicated, marks once gained cannot subsequently be lost, eg wrong working following a correct form of answer is ignored. Sometimes this is reinforced in the mark scheme by the abbreviation isw. However, this would not apply to a case where a candidate passes through the correct answer as part of a wrong argument. 6. When a part of a question has two or more method steps, the M marks are in principle independent unless the scheme specifically says otherwise; and similarly where there are several B marks allocated. (The notation dep * is used to indicate that a particular mark is dependent on an earlier, asterisked, mark in the scheme.) Of course, in practice it may happen that when a candidate has once gone wrong in a part of a question, the work from there on is worthless so that no more marks can sensibly be given. On the other hand, when two or more steps are successfully run together by the candidate, the earlier marks are implied and full credit must be given. 7. The abbreviation ft implies that the A or B mark indicated is allowed for work correctly following on from previously incorrect results. Otherwise, A and B marks are given for correct work only differences in notation are of course permitted. A (accuracy) marks are not given for answers obtained from incorrect working. When A or B marks are awarded for work at an intermediate stage of a solution, there may be various alternatives that are equally acceptable. In such cases, exactly what is acceptable will be detailed in the mark scheme rationale. If this is not the case please consult your Team Leader. Sometimes the answer to one part of a question is used in a later part of the same question. In this case, A marks will often be follow through. In such cases you must ensure that you refer back to the answer of the previous part question even if this is not shown within the image zone. You may find it easier to mark follow through questions candidate-by-candidate rather than question-by-question. 8. Wrong or missing units in an answer should not lead to the loss of a mark unless the scheme specifically indicates otherwise. Candidates are expected to give numerical answers to an appropriate degree of accuracy, with significant figures often being the norm. Small variations in the degree of accuracy to which an answer is given (eg or 4 significant figures where is expected) should not normally be penalised, while answers which are grossly over- or under-specified should normally result in the loss of a mark. The situation regarding any particular cases where the accuracy of the answer may be a marking issue should be detailed in the mark scheme rationale. If in doubt, contact your Team Leader. 9. Rules for crossed out and/or replaced work If work is crossed out and not replaced, examiners should mark the crossed out work if it is legible. If a candidate attempts a question more than once, and indicates which attempt he/she wishes to be marked, then examiners should do as the candidate requests. If two or more attempts are made at a question, and just one is not crossed out, examiners should ignore the crossed out work and mark the work that is not crossed out. If there are two or more attempts at a question which have not been crossed out, examiners should mark what appears to be the last (complete) attempt and ignore the others.

56 475 Mark Scheme January 0 NB Follow these maths-specific instructions rather than those in the assessor handbook. 0. For a genuine misreading (of numbers or symbols) which is such that the object and the difficulty of the question remain unaltered, mark according to the scheme but following through from the candidate s data. A penalty is then applied; mark is generally appropriate, though this may differ for some units. This is achieved by withholding one A mark in the question. Note that a miscopy of the candidate s own working is not a misread but an accuracy error.. Annotations should be used whenever appropriate during your marking. The A, M and B annotations must be used on your standardisation scripts for responses that are not awarded either 0 or full marks. It is vital that you annotate standardisation scripts fully to show how the marks have been awarded. For subsequent marking you must make it clear how you have arrived at the mark you have awarded.. For answers scoring no marks, you must either award NR (no response) or 0, as follows: Award NR (no response) if: Nothing is written at all in the answer space There is a comment which does not in any way relate to the question being asked ( can t do, don t know, etc.) There is any sort of mark that is not an attempt at the question (a dash, a question mark, etc.) The hash key [#] on your keyboard will enter NR. Award 0 if: There is an attempt that earns no credit. This could, for example, include the candidate copying all or some of the question, or any working that does not earn any marks, whether crossed out or not.. The following abbreviations may be used in this mark scheme. method mark (M, etc, is also used) accuracy mark B independent mark E mark for explaining U mark for correct units G mark for a correct feature on a graph dep* method mark dependent on a previous mark, indicated by * cao correct answer only ft follow through isw ignore subsequent working oe or equivalent rot rounded or truncated sc special case soi seen or implied www without wrong working

57 475 Mark Scheme January 0 4. Annotating scripts. The following annotations are available: and BOD Benefit of doubt FT Follow through ISW Ignore subsequent working (after correct answer obtained) M0, Method mark awarded 0, A0, Accuracy mark awarded 0, B0, B Independent mark awarded 0, SC Special case ^ Omission sign MR Misread Highlighting is also available to highlight any particular points on a script. 5. The comments box will be used by the Principal Examiner to explain his or her marking of the practice scripts for your information. Please refer to these comments when checking your practice scripts. Please do not type in the comments box yourself. Any questions or comments you have for your Team Leader should be communicated by the scoris messaging system, e- mail or by telephone. 6. Write a brief report on the performance of the candidates. Your Team Leader will tell you when this is required. The Assistant Examiner s Report Form (AERF) can be found on the Cambridge Assessment Support Portal. This should contain notes on particular strengths displayed, as well as common errors or weaknesses. Constructive criticisms of the question paper/mark scheme are also appreciated. 7. Link Additional Objects with work relating to a question to those questions (a chain link appears by the relevant question number) see scoris assessor Quick Reference Guide page 9-0 for instructions as to how to do this this guide is on the Cambridge Assessment Support Portal and new users may like to download it with a shortcut on your desktop so you can open it easily! For AOs containing just formulae or rough working not attributed to a question, tick at the top to indicate seen but not linked. When you submit the script, scoris asks you to confirm that you have looked at all the additional objects. Please ensure that you have checked all Additional Objects thoroughly. 8. The schedule of dates for the marking of this paper is displayed under OCR Subject Specific Details on the Cambridge Assessment Support Portal. It is vitally important that you meet these requirements. If you experience problems that mean you may not be able to meet the deadline then you must contact your Team Leader without delay. 4

58 475 Mark Scheme January 0 SECTION A y = 5x + M for y = 5(x ) oe or for y = 5x [+ k] [k = letter or number other than 4] and for = their m + k (i)(a) /6 isw attempted conversion of /6 to decimals or for y b = 5(x a) with wrong a, b or for y = their 5(x ) oe M0 for first M if /5 used as gradient even if 5 seen first; second M still available if earned accept (i)(b) set image fit to height so that in marking this question you also check that there is no working on the back page attached to the image (ii) 56/65 for num or denom correct or for 4/5 or 0.8 accept y x oe as final answer for each term ; 7/6 gets 0 for first term allow eg 4.5x - y 0 if 0, allow B for (xy 4 ) = 7x y 4 x > 5/ oe ( 5/ oe not sufft) for 5 < x or for 5/ oe obtained with equation or wrong inequality M0 for just x < 5 (not sufft) ; for x > 5/ 5

59 475 Mark Scheme January 0 5 V l r r V l r r l V r [ l ] r V r r for correctly getting non- l r terms on other side[m0 for triple decker fraction] oe or ft; for squaring correctly oe or ft; for getting l term as subject oe. or ft; mark final answer; for finding square root ( and dealing correctly with coefficient of l term if needed at this stage); condone ± etc may be done in several steps, if so, condone omission of brackets in eg 9V = π r 4 l r if they recover if not, do not give st [but can earn the nd ] for combined steps, allow credit for correct process where possible; eg π r 4 l as the term on one side For M4, the final expression must be totally correct, [condoning omission of l and insertion of ±] 6 9V r eg M4 for r 6 40x + 70x isw 4 B for all correct except for sign error(s) B for terms correct numerically, ignoring any sign error or for, 40 and 70 found or B for all correct, including signs, but unsimplified B for binomial coeffts, 5, 0 used or seen accept terms listed separately; condone 40x expressions left in n C r form or with factorials not sufft SC for 40x + 70x 080x isw or for 4x x 4 080x or SC for these terms with sign error(s) 6

60 475 Mark Scheme January 0 7 (i) 7/ oe or k = 7/ oe 4 for or 8 / or 8 or / or 7 / or better or for 8 = 4 or = / / or or (following correct rationalisation of denominator) for 7 = isw conversion of 7/ oe M0 for just 8 = oe indices needed allow an M mark for partially correct work still seen in 4 fraction form eg gets mark for 8 = 4 / 7 (ii) 4 5 or 8 0 www isw for multiplying num and denom by 5 + and for num or denom correct in final answer (M0 if wrongly obtained) nd is not dependent on st 8 (7/, 4/) oe www B for one coord correct; condone not expressed as coords, isw or for subst or elimination; eg x + (5x ) = 5 oe; condone one error SC for mixed fractions and decimals eg (.5/5.5, /5.5) 9 (i) ½ x (x + + x + 6) oe x(4x + 8) = 40 oe and given ans x + x 5 = 0 obtained correctly with at least one further interim step correct statement of area of trap; may be rectangle ± triangle, or two triangles eg x(x + ) + ½ x (x + 4) condone missing brackets for ; condone also for if expansion is treated as if they were there 7

61 475 Mark Scheme January 0 (ii) [AB =] www or B for x = [ 7 or] 5 cao www or for AB = or 5 or for (x + 7)(x 5) [= 0]or formula or completing square used eg (x + ) 6 [= 0]; condone one error eg factors with sign wrong or which give two terms correct when expanded or for showing f(5) = 0 without stating x = 5 may be done in (i) if not here allow the marks if seen in either part of the image some candidates are omitting the request in (i) and going straight to solving the equation (in which case give 0 [not NR] for (i), but annotate when the image appears again in (ii)) 5 on its own or AB = 5 with no working scores 0; we need to see x = 5 0 (i) P Q or or Q P Condone single arrows (ii) none [of the above] (iii) P Q or Section A Total: 6 8

62 475 Mark Scheme January 0 SECTION B (i) grad AB = 0 6 oe [= ] isw for full marks, it should be clear that grads are independently obtained eg grads of and / without earlier working earn M0 grad BC = 0 4 oe [= /] isw product of grads = [so lines perp] stated or shown numerically or one grad is neg. reciprocal of other or for length of one side (or square of it) for length of other two sides (or their squares) found independently for showing or stating that Pythag holds [so triangle rt angled] for M, must be fully correct, with gradients evaluated at least to 6/ and 4/ stage AB = 6 + = 40, BC = 4 + = 60, AC = 4 + = 00 (ii) AB = 40 or BC = 60 allow for ( ) (6 0) or for ( ) (4 0) ½ oe or ft their AB, BC 40 or for one of area under AC (=70), under AB (=6) under BC (=4) (accept unsimplified) and for their trap. two triangles or for rectangle triangles method, [ 6 4 ()(6) (4)() ()(4) = ] for two of the 4 areas correct and for the subtraction 9

63 475 Mark Scheme January 0 (iii) angle subtended by diameter = 90 soi B or angle at centre = twice angle at circumf = 90 = 80 soi or showing BM = AM or CM, where M is midpt of AC; or showing that BM = ½ AC condone AB and BC are perpendicular or ABC is right angled triangle provided no spurious extra reasoning mid point M of AC = (6, 5) B allow if seen in circle equation ; for correct working seen for both coords rad of circle = 4 00 oe or equiv using r (x a) + (y b) = r seen or (x their 6) + (y their 5) = k used, with k > 0 accept unsimplified; or eg r = 7 + or ; may be implied by correct equation for circle or by correct method for AM, BM or CM ft their M allow bod intent for AC = 00 followed by r = 00 (x 6) + (y 5) = 50 cao or x y x y 0 0 must be simplified (no surds) (iv) (, 0) cao (i)(a) sketch of cubic correct way up and with two tps, crossing x-axis in distinct points B 0 if stops at x-axis; condone not crossing y-axis No section to be ruled; no curving back; condone slight flicking out at ends; condone some doubling (eg erased curves may continue to show) crossing x axis at,.5 and 4 B intersections labelled on graph or shown nearby in this part; mark intent for intersections with both axes (eg condone graphs stopping at axes) allow.5 indicated by graph crossing halfway between their marked and on scale; allow if no graph but 0 if graph inconsistent with values crossing y axis at 0 B or x = 0, y = 0 seen in this part if consistent with graph drawn allow if no graph, but eg B0 for graph with intn on +ve y-axis or nowhere near their indicated 0 0

64 475 Mark Scheme January 0 (i)(b) correct expansion of two brackets correct interim step(s) multiplying out linear and quadratic factors before given answer or M for all brackets multiplied at once, showing all 8 terms ( if error in one term): x 8x x 5x + 8x + 5x + 0x 0 eg for (x 5)(x 5x + 4) condone missing brackets if intent clear /used correctly or showing that,.5 and 4 all satisfy f(x) = 0 for cubic in x... form comparing coeffts of eg x in the two forms or or for dividing x... form by one of the linear factors and for factorising the resultant quadratic factor (ii)(a) isw B or showing that x 5 is a factor by eg division and then stating that x = 5 is root or that g(5) = is not sufft or [g(5) =] f(5) 0 = [= 0] (ii) (B) (x 5) seen or used as linear factor may be in attempt at division allow if seen in (ii)(a) division by (x 5) as far as x 0x seen in working or inspection/equating coefficients with two terms correct eg (x.. + 8) for division: condone signs of x 0x changed for subtraction, or subtraction sign in front of first term x 5x + 8 obtained isw eg may be seen in grid; condone g(x) not expressed as product

65 475 Mark Scheme January 0 (ii)(c) b 4ac used on their quadratic factor may be in formula ( 5) 4 8 oe and negative [or 9] so no [real] root [may say only one [real] root, thinking of x = 5] [or allow marks for complete correct attempt at completing square and conclusion, or using calculus to show min value is above x-axis and comment re curve all above x-axis] no ft for A mark from wrong quadratic factor condone error in working out 9 if correct unsimplified expression seen and neg result obtained evaluated correctly with comment is eligible for, otherwise bod for the only (iii) translation 0 0 B B NB Moves not sufficient for this first mark or 0 down; B0 for second mark if choice of one wrong, one right description (i) (0, ) or crosses y-axis at oe isw ( 4, 0) oe isw B B or [when y = 0], 4 [ x ] or or B for one root correct 4 isw condone y =

66 475 Mark Scheme January 0 (ii) [y = ] x = x 4 oe and rearrangement to x 4 x [= 0] or y y [=0] (x )(x + ) = 0 oe in y or formula or completing square; condone one error; condone replacement of x by another letter or by x for nd (but not the rd ) if completing square, and haven t arranged to zero, can earn first as well for an attempt such as (x 0.5) =.5 x = [or ] or y = or or ft or x or x or ft dep on nd ; allow inclusion of correct complex roots; M0 if any incorrect roots are included for x or x NB for second and third M: M0 for x = 0 or x = oe straight from quartic eqn some candidates probably thinking x 4 x simplifies to x ; last two marks for roots are available as B marks (, ) and (, ); with no other intersections given B or B for one of these two intersections (even if extra intersections given) or for x (and no other roots) or for y = (and no other roots), marking to candidates advantage some candidates having several attempts at solving this equation mark the best in this particular case

67 475 Mark Scheme January 0 (iii) from x 4 kx [= 0]: k + 8 > 0 oe B Allow x replaced by other letters or x or from y k y k [= 0] k 4 + 8k > 0 oe [alt methods: may use completing square to show similarly, or comment that at x = 0 the quadratic is above the quartic and that as x, x 4 > kx for all k] condone lack of brackets in ( k) k k 8 0 for all k B k k 8k > 0 oe for all k 4 [so there is a positive root for x and hence real root for x and so intersection] [so there is a positive root for y and hence real root for x and so intersection] if B0B0, allow SC for k k 8k simplified] 4 k k 8 obtained [need not be or Section B Total: 6 4

68 GCE Mathematics (MEI) Advanced Subsidiary GCE Unit 475: Introduction to Advanced Mathematics Mark Scheme for June 0 Oxford Cambridge and RSA Examinations

69 OCR (Oxford Cambridge and RSA) is a leading UK awarding body, providing a wide range of qualifications to meet the needs of pupils of all ages and abilities. OCR qualifications include AS/A Levels, Diplomas, GCSEs, OCR Nationals, Functional Skills, Key Skills, Entry Level qualifications, NVQs and vocational qualifications in areas such as IT, business, languages, teaching/training, administration and secretarial skills. It is also responsible for developing new specifications to meet national requirements and the needs of students and teachers. OCR is a not-for-profit organisation; any surplus made is invested back into the establishment to help towards the development of qualifications and support which keep pace with the changing needs of today s society. This mark scheme is published as an aid to teachers and students, to indicate the requirements of the examination. It shows the basis on which marks were awarded by Examiners. It does not indicate the details of the discussions which took place at an Examiners meeting before marking commenced. All Examiners are instructed that alternative correct answers and unexpected approaches in candidates scripts must be given marks that fairly reflect the relevant knowledge and skills demonstrated. Mark schemes should be read in conjunction with the published question papers and the Report on the Examination. OCR will not enter into any discussion or correspondence in connection with this mark scheme. OCR 0 Any enquiries about publications should be addressed to: OCR Publications PO Box 5050 Annesley NOTTINGHAM NG5 0DL Telephone: Facsimile: publications@ocr.org.uk

70 475 Mark Scheme June 0 SECTION A x > /4 o.e. isw www condone x > / 4 or / 4 < x; M for 4x > or for one side of this correct with correct inequality, and B for final step ft from their ax > b or c > dx for a and d ; if no working shown, allow SC for /4 oe with equals sign or wrong inequality for > 4x (may be followed by / 4 > x, which earns no further credit); 6x + > x + 5 is an error not an MR; can get for 4x >... following this, and then a possible B 7 condone y = 7 or (5, 7); (i) 4/ isw condone ±4/; k ( 5) for = or other correct 5 use of gradient eg triangle with 4 across, up for numerator or denominator correct or for or oe or for soi condone omission of brackets; or for correct method for eqn of line and x = 5 subst in their eqn and evaluated to find k; or for both of y k = (x 5) oe and y ( 5) = (x ) oe for just 4/; allow for 6 = 4 and 9 = soi; condone missing brackets

71 475 Mark Scheme June 0 a 5 B for each term correct; condone a ; (ii) or ac 5 c mark final answer; condone multiplication signs but 0 for addition signs if B0, then SC for (ac ) = 8a c 6 or 7a 5 c 7 seen 4 (i) (0, 4) 0 for (5, 4); otherwise for each coordinate ignore accompanying working / description of transformation; condone omission of brackets; (Image includes back page for examiners to check that there is no work there) 4 (ii) (5, ) 0 for (5, 4); otherwise for each coordinate ignore accompanying working / description of transformation; condone omission of brackets M for ; or M for two of these elements correct with multiplication or all three elements correct but without multiplication (e.g. in list or with addition signs); or for 5 soi or for seen in Pascal s triangle; SC for 0000[x ] condone inclusion of x 4 eg (x) 4 ; condone omission of brackets in x 4 if 6 used; allow M for correct term seen (often all terms written down) but then wrong term evaluated or all evaluated and correct term not identified; 5 5 (x) 4 earns M even if followed by 5 5 calculated; no MR for wrong power evaluated but SC for fourth term evaluated

72 475 Mark Scheme June 0 6 x + 9x + 4x 5 as final answer; ignore = 0 ; B for correct terms of answer seen or correct 8-term expansion: x + 6x x + 5x 6x + 5x 5x 5 correct 6-term expansions: for an 8-term or 6 term expansion with x + 4x + 5x 6x + 0x 5 at most one error: x + 6x + x + 9x 5x 5 x + x x + 5x x 5 or for correct quadratic expansion of one pair of brackets; or SC for a quadratic expansion with one error then a good attempt to multiply by the remaining bracket for, need not be simplified; ie SC for knowing what to do and making a reasonable attempt, even if an error at an early stage means more marks not available 7 b 4ac soi allow seen in formula; need not have numbers substituted but discriminant part must be correct; www or B clearly found as discriminant, or stated as b 4ac, not just seen in formula eg A0 for b 4ac = = ; [distinct real roots] B B0 for finding the roots but not saying how many there are condone discriminant not used; ignore incorrect roots found

73 475 Mark Scheme June 0 8 yx + y = x oe or ft for multiplying to eliminate each mark is for carrying out the operation correctly; ft denominator and for expanding earlier errors for equivalent steps if error does not brackets, simplify problem; or for correct division by y and writing x as separate fractions: x y y some common errors: yx + x = y oe or ft x (y + ) = y oe or ft for collecting terms; dep on having an ax term and an xy term, oe after division by y, for taking out x factor; dep on having an ax term and an xy term, oe after division by y, y (x + ) = x yx + x = x M0 yx + 5x = ft x (y + 5) = ft x = y + 5 ft yx + = x M0 yx + x = ft x (y + ) = ft x = y + ft y [ x = ] y + oe or ft as final answer for division with no wrong work after; dep on dividing by a two-term expression; last M not earned for tripledecker fraction as final answer for M4, must be completely correct; 4

74 475 Mark Scheme June 0 9 x + y = k (k 6) or y = ½ x + c (c ) for attempt to use gradients of parallel lines the same; M0 if just given line used; eg following an error in manipulation, getting original line as y = ½ x + then using y = ½ x + c earns and can then go on to get A0 for y = ½ x 4, for (0, 4) for (8, 0) and A0 for area of 6; x + y = or [y = ] ½ x + 6 oe or B; must be simplified; or evidence of correct stepping using (0, ) eg may be on diagram; allow bod B for a candidate who goes straight to y = ½ x + 6 from y = x + 6; NB the equation of the line is not required; correct intercepts obtained will imply this ; (, 0) or ft (0, 6)or ft or when y = 0, x = etc or using or ft as a limit of integration; intersections must ft from their line or stepping diagram using their gradient or integrating to give ¼ x + 6x or ft their line NB for intersections with axes, if both Ms are not gained, it must be clear which coord is being found eg M0 for intn with x axis = 6 from correct eqn;; if the intersections are not explicit, they may be implied by the area calculation eg use of ht = 6 or the correct ft area found; allow ft from the given line as well as others for both these intersection Ms; 6 [sq units] cao or B www NB A0 if 6 is incorrectly obtained eg after intersection x = seen (which earns M0 from correct line); 5

75 475 Mark Scheme June 0 0 n (n + )(n + ) condone division by n and then ignore = 0 ; (n + )(n + ) seen, or separate factors shown after factor theorem used; argument from general consecutive numbers leading to: an induction approach using the factors may also be used eg by those doing paper FP as well; at least one must be even [exactly] one must be multiple of or divisible by ; if M0: allow SC for showing given expression always even A0 for just substituting numbers for n and stating results; allow SC for a correct induction approach using the original cubic (SC for each of showing even and showing divisible by ) 6

76 475 Mark Scheme June 0 SECTION B (i) x + 4x + 4x + = 0 oe for subst of x or y or subtraction to eliminate variable; condone one error; 4x + 5x + [= 0] for collection of terms and rearrangement to zero; condone one error; (4x + )(x + ) for factors giving at least two terms of their quadratic correct or for subst into formula with no more than two errors [dependent on attempt to rearrange to zero]; x = or /4 oe isw or for (, ) and for ( /4, 6/4) oe or 4y 05y + 67 [= 0]; eg condone spurious y = 4x + 5x + as one error (and then count as eligible for rd ); or (y )(4y 6); [for full use of completing square with no more than two errors allow nd and rd s simultaneously]; from formula: accept x = or 4/8 oe isw y = or 6/4 oe isw (ii) 4(x + ) 5 isw 4 B for a = 4, B for b =, B for c = 5 or for 4 their b soi or for 5/4 or for /4 their b soi eg an answer of (x + ) 5 / 4 earns B0 B ; (x + 6) 5 earns B0 B0 B; 4( earns first B; condone omission of square symbol (iii)(a) x = or ft ( their b) from (ii) 0 for just or ft; 0 for x =, y = 5 or ft (iii)(b) 5 or ft their c from (ii) allow y = 5 or ft 0 for just (, 5); bod for x = stated then y = 5 or ft 7

77 475 Mark Scheme June 0 (i) y = x + 5 drawn condone unruled and some doubling; tolerance: must pass within/touch at least two circles on overlay; the line must be drawn long enough to intersect curve at least twice;,.4 to., 0.7 to 0.85 A for two of these correct condone coordinates or factors (ii) 4 = x + 5x 4 or x + 5 = 0 x and completion to given answer B condone omission of final = 0 ; f( ) = = 0 B or correct division / inspection showing that x + is factor; use of x + as factor in long division of given cubic as far as x + 4x in working or inspection or equating coefficients, with at least two terms correct; may be set out in grid format x + x obtained condone omission of + sign (eg in grid format) ± 4 [ x = ] oe dep on previous earned; for attempt at formula or full attempt at completing square, using their other factor not more than two errors in formula / substitution / completing square; allow even if their factor has a remainder shown in working; M0 for just an attempt to factorise ± 7 4 oe isw 8

78 475 Mark Scheme June 0 4 (iii) x = x + or y = x + soi eg is earned by correct line drawn condone intent for line; allow slightly out of tolerance; y = x + drawn condone unruled; need drawn for.5 x.; to pass through/touch relevant circle(s) on overlay real root (i) [radius = ] 4 B B0 for ± 4 [centre] (4, ) B condone omission of brackets 9

79 475 Mark Scheme June 0 (ii) (x 4) + ( ) = 6 oe for subst y = 0 in circle eqn; NB candidates may expand and rearrange eqn first, making errors they can still earn this when they subst y = 0 in their circle eqn; condone omission of ( ) for this first only; not for second and third s; do not allow substitution of x = 0 for any Ms in this part (x 4) = or x 8x + 4 [= 0] putting in form ready to solve by comp sq, or for rearrangement to zero; condone one error; eg allow for x + 4 = 0 [but this two-term quadratic is not eligible for rd ]; x 4=± or 8± [ x = ] for attempt at comp square or formula; dep on previous M earned and on three-term quadratic; not more than two errors in formula / substitution; allow for x 4= ; M0 for just an attempt to factorise [ x = ]4± or 4± or isw 8± 48 oe or or sketch showing centre (4, ) and triangle with hyp 4 and ht 4 = or the square root of this; implies previous if no sketch seen; [ x = ]4± oe A for one solution 0

80 475 Mark Scheme June 0 B or showing sketch of centre C and A or subst the value for one coord in circle eqn and and using Pythag: correctly working out the other as a possible value; + = 8+ 8= 6; (iii) subst ( 4+, + ) into circle eqn and showing at least one step in correct completion ( ) ( ) Sketch of both tangents need not be ruled; must have negative gradients with tangents intended to be parallel and one touching above and to right of centre; mark intent to touch allow just missing or just crossing circle twice; condone A not labelled grad tgt = or /their grad CA allow ft after correct method seen for grad CA = + oe (may be on/ 4+ 4 near sketch); y ( + ) = their m( x (4 + )) or y = their mx + c and subst of 4+, + ; ( ) y = x oe isw accept simplified equivs eg x+ y = 6+ 4 ; allow ft from wrong centre found in (i); for intent; condone lack of brackets for ; independent of previous Ms; condone grad of CA used; A0 if obtained as eqn of other tangent instead of the tangent at A (eg after omission of brackets); parallel tgt goes through ( 4, ) or ft wrong centre; may be shown on diagram; may be implied by correct equation for the tangent (allow ft their gradient); no bod for just y = ( x 4 ) without first seeing correct coordinates; eqn is y = x+ 6 4 oe isw accept simplified equivs eg x+ y = 6 4 A0 if this is given as eqn of the tangent at A instead of other tangent (eg after omission of brackets) Section B Total: 6

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82 475 Mark Scheme June 0 Question Answer Marks Guidance y = x + 7 isw for y = (x ) or = + c oe (0, 7) and (.5, 0) oe or ft their y = x + c condone lack of brackets and eg y = 7, x =.5 or ft isw but 0 for poor notation such as (.5, 7) and no better answers seen [] a a [ b ] oe www M for [ b ] soi a eg M for [ b ] c c c [] (i) 5 (ii) x 5 or as final answer www x x [] or for other [ b ] ka or [ b ] a oe c kc and for correctly taking the square root of their b, including the ± sign; for 5 or 5 or 5 or 5 for 4 or 9 or 9 or or or seen [] B for correct answer seen and then spoilt for (x + )(x ) and for (x + )(x + ) [] allow for a triple-decker or quadruple-decker fraction or decimals eg.5a, if no recovery later c square root must extend below the fraction line 64 0 for just 79 5

83 475 Mark Scheme June 0 Question Answer Marks Guidance 5 (i) 0 M0 for ie cubing 0 as well 5 (ii) 8 for soi and for 4 6 soi or allow SC for final answer of 5 6 or5 6 or 0 9 etc [] for common denominator soi - may be in separate fractions or for a final answer with denominator, even if worked with only one fraction for those using indices: for both 0 6 / and 6 / oe then for 5 6 oe award SC for similar correct answer with no denominator condone lack of brackets [] 6 (i) 0 cao [] 6 (ii) 70 [x ] 4 B for 70 [x ] or for [x ] or M for 0 ( ) oe or ft from (i) or for two of these three elements correct or ft; condone x still included [4] condone 70 x etc allow equivalent marks for the x term as part of a longer expansion 5 eg M for or for 0 etc 6

84 475 Mark Scheme June 0 Question Answer Marks Guidance 7 4k 4 5 or k 5 [< 0] oe or [(x + k) +] 5 k [> 0] oe M allow =, >, etc instead of < or for b 4ac soi (may be in formula) or for attempt at completing square allow M for k < 0, k 0 = 0 etc but only for just k 0 ignore rest of quadratic formula ignore b 4ac 0 seen if b 4ac 0 then used, otherwise just for b 4ac 0 5 k 5 A may be two separate inequalities or for one end correct or B for endpoint = 5 [4] b + c = 0 oe need not be simplified; condone 8 or as first term if 4 not seen 8 b + c = 85 oe B for f( ) seen or used, condoning one error except +b need not be simplified or for long division as far as obtaining x x in quotient 0 + 5b = 0 oe for elimination of one variable, ft their equations in b and c, condoning one error in rearrangement of their original equations or in one term in the elimination b = 4 and c = 8 allow correct answers to imply last after correct earlier equations [5] allow SC for 0 k 0 following at least for k 0 oe in this question use annotation to indicate where part marks are earned eg for 8 b + c = 0 long division may be seen in grid or a mixture of methods may be used eg B for c (b 7) = 85 correct operation must be used in elimination for misread of x 4 as x or x or higher powers, allow all Ms equivalently 7

85 475 Mark Scheme June 0 Question Answer Marks Guidance 9 6n + 9 isw or (n + ) B 6n is even [but 9 is odd], even + odd = odd or n + is odd since even + odd = odd and odd odd = odd B dep this mark is dependent on the previous B accept equiv. general statements using either 6n + 9 or (n + ) n is a multiple of or n is divisible by without additional incorrect statement(s) B B for it is divisible by 9, so n is divisible by for 6n is divisible by 9 or n + is divisible by or for n is a multiple of oe with additional incorrect statement(s) B for just it is divisible by but for it is divisible by 9, so it is divisible by eg for n is divisible by 9, so n is divisible by [4] N.B. 0 for n is a factor of (but may be earned earlier) 8

86 475 Mark Scheme June 0 Question Answer Marks Guidance 0 (i) AB = ( ( )) + (5 ) oe, or square root of this; condone poor notation re roots; condone ( + ) instead of ( ( )) allow for vector AB = 4, condoning poor notation, or triangle with hyp AB and lengths and 4 correctly marked BC = ( ( )) + ( ) oe, or square root of this; condone poor notation re roots; condone ( + ) instead of ( ( )) oe 4 allow for vector BC = poor notation, or triangle with hyp BC and lengths 4 and correctly marked shown equal eg AB = + 4 [=0] and BC = 4 + [=0] with correct notation for final comparison or statement that AB and BC are each the hypotenuse of a right-angled triangle with sides and 4 so are equal SC for just AB = + 4 and BC = 4 + (or roots of these) with no clearer earlier working; condone poor notation eg A0 for AB = 0 etc [] 9

87 475 Mark Scheme June 0 Question Answer Marks Guidance 0 (ii) 5 [grad. of AC =] or 6 award at first step shown even if errors after oe 5 [grad. of BD =] or 4 oe if one or both of grad AC = - and grad BD = / seen without better working for both gradients, award one only. For it must be clear that they are obtained independently showing or stating product of gradients = or that one gradient is the negative reciprocal of the other oe B eg accept m m = or one gradient is negative reciprocal of the other B0 for opposite used instead of negative or reciprocal may be earned independently of correct gradients, but for all marks to be earned the work must be fully correct [] 0

88 475 Mark Scheme June 0 Question Answer Marks Guidance 0 (iii) midpoint E of AC = (, ) www B condone missing brackets for both Bs 0 for ((5+ )/, (+)/) = (, ) 4 eqn BD is y x oe accept any correct form isw or correct ft their gradients or their midpt F of BD this mark will often be gained on the first line of their working for BD eqn AC is y = x + 8 oe accept any correct form isw or correct ft their gradients or their midpt E of AC this mark will often be gained on the first line of their working for AC may be earned using (, ) but then must independently show that B or D or (5, ) is on this line to be eligible for if equation(s) of lines are seen in part ii, allow the s if seen/used in this part [see appendix for alternative methods instead showing E is on BD for this ] using both lines and obtaining intersection E is (, ) (NB must be independently obtained from midpt of AC) [see appendix for alternative ways of gaining these last two marks in different methods] midpoint F of BD = (5,) B this mark is often earned earlier see the appendix for some common alternative methods for this question; for all methods, for to be earned, all work for the 5 marks must be correct for all methods show annotations B etc then omission mark or A0 if that mark has not been earned [5]

89 475 Mark Scheme June 0 Question Answer Marks Guidance (i) (x + )(x + )(x 5) or (x + /)(x + )(x 5); need not be written as product throughout, ignore =0 correct expansion of two linear factors of their product of three linear factors for all Ms in this part condone missing brackets if used correctly expansion of their linear and quadratic factors dep on first ; ft one error in previous expansion; condone one error in this expansion or for direct expansion of all three factors, allow M for x 0x + 4x + x 0x 5x + x 0 [or half all these], or if one or two errors, dep on first [y =] x 5x x 0 or a = 5, b = and c = 0 condone poor notation when doubling to reach expression with x... for an attempt at setting up three simultaneous equations in a, b, and c: for at least two of the three equations a + 5b + c = a b + c = 0 ¼+ ¼ a ½ b + c = 0 oe then M for correctly eliminating any two variables or for correctly eliminating one variable to get two equations in two unknowns [4] and then for values.

90 475 Mark Scheme June 0 Question Answer Marks Guidance (ii) graph of cubic correct way up B must not be ruled; no curving back; condone slight flicking out at ends; allow min on y axis or in rd or 4th quadrants; condone some doubling or feathering (deleted work still may show in scans) B0 if stops at x-axis crossing x axis at, / and 5 B on graph or nearby in this part allow if no graph, but marked on x- axis mark intent for intersections with both axes crossing y axis at 0 or ft their cubic in (i) B or x = 0, y = 0 or ft in this part if consistent with graph drawn; allow if no graph, but eg B0 for graph nowhere near their indicated 0 or ft [] (iii) (0, 8); accept 8 or ft their constant 8 or ft their intn on y-axis 8 [] (iv) roots at.5,, 8 or attempt to substitute (x ) in (x + )(x + )(x 5) or in (x + /)(x + )(x 5) or in their unfactorised form of f(x) attempt need not be simplified (x 5)(x )(x 8) accept (x.5) oe instead of (x 5) M0 for use of (x + ) or roots.5, 5, but then allow SC for (x + 7)(x + 5)(x ) (0, 40); accept 40 B for 5 8 or ft or for f( ) attempted or g(0) attempted or for their answer ft from their factorised form eg for (0, 70) or 70 after (x + 7)(x + 5)(x ) after M0, allow SC for f() = 70 [4]

91 475 Mark Scheme June 0 Question Answer Marks Guidance (i) (, 6) (0,) (, ) (, ) (, ) (4, ) (5,6) seen plotted B or for a curve within mm of these points; B for correct plots or for at least of the pairs of values seen eg in table use overlay; scroll down to spare copy of graph to see if used [or click fit height also allow B for (, 0) and (, ) seen or plotted and curve not through other correct points smooth curve through all 7 points B dep dep on correct points; tolerance mm; condone some feathering/ doubling (deleted work still may show in scans); curve should not be flat-bottomed or go to a point at min. or curve back in at top; (0. to 0.5, 0. to 0.5) and (.5 to.7,.5 to.7) and (4, ) (ii) x 4x x B may be given in form x =..., y =... B for two intersections correct or for all the x values given correctly [5] = (x )( x 4x + ) condone omission of brackets only if used correctly afterwards, with at most one error; condone omission of = for this only if it reappears allow for terms expanded correctly with at most one error at least one further correct interim step with = or =0,as appropriate, leading to given answer, which must be stated correctly there may also be a previous step of expansion of terms without an equation, eg in grid NB mark method not answer - given answer is x 7x + x 4 = 0 if M0, allow SC for correct division of given cubic by quadratic to gain (x ) with remainder, or vice-versa [] 4

92 475 Mark Scheme June 0 Question Answer Marks Guidance (iii) quadratic factor is x x + B found by division or inspection; allow for division by x 4 as far as x 4x in the working, or for inspection with two terms correct substitution into quadratic formula or for completing the square used as far as x oe condone one error no ft from a wrong factor ; A [5] if one error in final numerical expression, but only if roots are real isw factors Appendix: alternative methods for 0(iii) [details of equations etc are in main scheme] for a mixture of methods, look for the method which gives most benefit to candidate, but take care not to award the second twice the final is not earned if there is wrong work leading to the required statements ignore wrong working which has not been used for the required statements for full marks to be earned in this part, there must be enough to show both the required statements find midpt E of AC B find midpt E of AC B find midpt E of AC B find midpt E of AC B find eqn BD find eqn BD find eqn BD use gradients or vectors to show E is on BD eg grad BE = and grad M ED = 5 [condone poor vector notation] show E on BD show E on BD show E on BD find midpt F of BD B find midpt F of BD B show BE = 0 and DE = B find midpt F of BD B 90 oe state so not E find eqn of AC and correctly show F not on AC (the correct eqn for AC earns the second as per the main scheme, if not already earned) showing BE = 0 and DE = 90 oe earns this A mark as well as the B if there are no errors elsewhere state so not E or show F not on AC [5] 5] 5

93 GCE Mathematics (MEI) Advanced Subsidiary GCE Unit 475: Introduction to Advanced Mathematics Mark Scheme for January 0 Oxford Cambridge and RSA Examinations