Paper Reference(s) 666/0 Edecel GCE Core Mathematics C Advanced Subsidiary Monday May 00 Afternoon Time: hour 0 minutes Materials required for eamination papers Mathematical Formulae (Pink) Items included with question Nil Calculators may NOT be used in this eamination. Instructions to Candidates Write the name of the eamining body (Edecel), your centre number, candidate number, the unit title (Core Mathematics C), the paper reference (666), your surname, initials and signature. Information for Candidates A booklet Mathematical Formulae and Statistical Tables is provided. Full marks may be obtained for answers to ALL questions. There are questions in this question paper. The total mark for this paper is 75. Advice to Candidates You must ensure that your answers to parts of questions are clearly labelled. You must show sufficient working to make your methods clear to the Eaminer. Answers without working may not gain full credit. H58A This publication may only be reproduced in accordance with Edecel Limited copyright policy. 00 Edecel Limited.
. Write (75) (7) in the form k, where k and are integers.. Find ( 8 6 5) d, giving each term in its simplest form. (). Find the set of values of for which ( ) < 8, (b) ( 7)( + ) < 0, (c) both ( ) < 8 and ( 7)( + ) < 0. () (). Show that + 6 + can be written as where p and q are integers to be found. ( + p) + q, (b) Sketch the curve with equation y = + 6 +, showing clearly any intersections with the coordinate aes. (c) Find the value of the discriminant of + 6 +. N609A
5. A sequence of positive numbers is defined by a = ( a + ), n, n a =. n Find a and a, leaving your answers in surd form. (b) Show that a 5 =. 6. Figure Figure shows a sketch of the curve with equation y = f(). The curve has a maimum point A at (, ) and a minimum point B at (, 5). On separate diagrams sketch the curve with equation y = f ( + ), (b) y = f(). () () On each diagram show clearly the coordinates of the maimum and minimum points. The graph of y = f() + a has a minimum at (, 0), where a is a constant. (c) Write down the value of a. () H58A
7. Given that d y find. d y = 8 +, > 0, (6) 8. Find an equation of the line joining A(7, ) and B(, 0), giving your answer in the form a + by + c = 0, where a, b and c are integers. () (b) Find the length of AB, leaving your answer in surd form. The point C has coordinates (, t), where t > 0, and AC = AB. (c) Find the value of t. (d) Find the area of triangle ABC. () 9. A farmer has a pay scheme to keep fruit pickers working throughout the 0 day season. He pays a for their first day, (a + d ) for their second day, (a + d ) for their third day, and so on, thus increasing the daily payment by d for each etra day they work. A picker who works for all 0 days will earn 0.75 on the final day. Use this information to form an equation in a and d. A picker who works for all 0 days will earn a total of 005. (b) Show that 5(a + 0.75) = 005. (c) Hence find the value of a and the value of d. () H58A
0. On the aes below sketch the graphs of (i) y = ( ), (ii) y = (7 ), showing clearly the coordinates of the points where the curves cross the coordinate aes. (5) (b) Show that the -coordinates of the points of intersection of y = ( ) and y = (7 ) are given by the solutions to the equation ( 8 + ) = 0. () The point A lies on both of the curves and the and y coordinates of A are both positive. (c) Find the eact coordinates of A, leaving your answer in the form (p + q, r + s ), where p, q, r and s are integers. (7). The curve C has equation y = f(), > 0, where Given that the point P (, 5) lies on C, find f(), d y 5 =. d (b) an equation of the tangent to C at the point P, giving your answer in the form a + by + c = 0, where a, b and c are integers. () (5) END TOTAL FOR PAPER: 75 MARKS H58A 5
June 00 Core Mathematics C 666 Mark. ( 75 7 ) = 5 M = A for 5 from 75 or from 7 seen anywhere M or k =, = A for ; allow or allow k =, = Some Common errors 75 7 = 8 leading to is M0A0 5 9 = 6 is M0A0 GCE Core Mathematics C (666) Summer 00
. 8 6 + 5 + c M A = +, 5+ c A A M for some attempt to integrate a term in : n n + st A for correct, possibly un-simplified nd A for both 8 6 or term. e.g. or and terms correct and simplified on the same line N.B. some candidates write or which are, of course, fine for A rd A for 5+ c. Accept 5 + c. The +c must appear on the same line as the 5 N.B. We do not need to see one line with a fully correct integral Ignore ISW (ignore incorrect subsequent working) if a correct answer is followed by an incorrect version. Condone poor use of notation e.g. + 5 + c will score full marks. GCE Core Mathematics C (666) Summer 00
. 6< 8 5< (Accept 5 < 0 (o.e.)) M <.8 or or 5 5 (condone < ) A (b) Critical values are Choosing inside 7 = and B 7 < < M A () (c) < <.8 Bft () Accept any eact equivalents to -,.8,.5 6 M for attempt to rearrange to k < m (o.e.) Either k = 5 or m = should be correct Allow 5 = or even 5 > (b) B for both correct critical values. (May be implied by a correct inequality) M ft their values and choose the inside region A for fully correct inequality (Must be in part (b): do not give marks if only seen in (c)) Condone seeing < in working provided < is in the final answer. 7 7 7 e.g. >, < or > "or" < or > "blank space" < score MA0 7 BUT allow > and < to score MA (the and must be seen) 7, will score MA Also ( ) 7 NB <, < is of course M0A0 and a number line even with open ends is M0A0 Allow.5 instead of 7 (c) Bft for < <.8(ignoring their previous answers) or ft their answers to part and part (b) provided both answers were regions and not single values. Allow use of and between inequalities as in part (b) If their set is empty allow a suitable description in words or the symbol. Common error: If is correct and in (b) they simply leave their answer as <, <.5 then in (c) < would get Bft as this is a correct follow through of these inequalities. Penalise use of only on the A in part (b). [i.e. condone in part ] GCE Core Mathematics C (666) Summer 00
. ( + ) + or 6 p = or q = B B (b) y U shape with min in nd quad (Must be above -ais and not on y=ais) B U shape crossing y-ais at (0, ) only (Condone (,0) marked on y-ais) B (c) b ac= 6 M = 8 A 6 Ignore an = 0 so ( + ) + = 0 can score both marks (b) (c) The U shape can be interpreted fairly generously. Penalise an obvious V on st B only. The U needn t have equal arms as long as there is a clear min that holds water st B for U shape with minimum in nd quad. Curve need not cross the y-ais but minimum should NOT touch -ais and should be left of (not on) y-ais nd B for U shaped curve crossing at (0, ). Just marked on y-ais is fine. The point must be marked on the sketch (do not allow from a table of values) Condone stopping at (0, ) M for some correct substitution intob ac. This may be as part of the quadratic formula but must be in part (c) and must be only numbers (no terms present). Substitution into b < ac or b = ac or b > ac is M0 A for 8 only. If they write 8 < 0 treat the < 0 as ISW and award A If they write 8 > 0 then score A0 A substitution in the quadratic formula leading to 8 inside the square root is A0. So substituting into b ac< 0 leading to 8 < 0 can score MA. Only award marks for use of the discriminant in part (c) GCE Core Mathematics C (666) Summer 00
5. a = ( + ) = 7 B a = "their 7" + = 0 Bft (b) a = 0 + ( = ) M a 5 = + = * A cso st B for 7 only nd Bft follow through their 7 in correct formula provided they have n,where n is an integer. (b) M for an attempt to find a. Should see ( ) "their" a +. Must see evidence for M. a = provided this follows from their a working or answer is sufficient Acso for a correct solution (M eplicit) must include the =. Ending at 6 only is A0 and ending with + is A0. Ignore any incorrect statements that are not used e.g. common difference = Listing: A full list: ( = ), 7, 0,, 6 = is fine for MA ALT Formula: Some may state (or use) an = n+ leading to a5 = 5 + =. This will get marks in [if correct values are seen] and can score the M in (b) if an = n+ or a = are seen. ± If ± appear any where ignore in part and withhold the final A mark only GCE Core Mathematics C (666) Summer 00
6. (-5, ) y Horizontal translation of ± M ( 5, ) marked on sketch or in tet B (0, -5) (0, 5) and min intentionally on y-ais Condone ( 5, 0) if correctly placed on negative y-ais A () (-, 6) y Correct shape and intentionally through (0,0) between the ma and min B (b) (, 6) marked on graph or in tet B (, -0) (, 0) marked on graph or in tet B () (c) (a = ) 5 B () Turning points (not on aes) should have both co-ordinates given in form(,y). Do not accept points marked on aes e.g. 5on -ais and on y-ais is not sufficient. For repeated offenders apply this penalty once only at first offence and condone elsewhere. In and (b) no graphs means no marks. In and (b) the ends of the graphs do not need to cross the aes provided ma and min are clear y M for a horizontal translation of ± so accept i.e ma in st quad and coordinates of (, ) or (6, 5) seen. [Horizontal translation to the left should have a min on the y-ais] If curve passes through (0,0) then M0 (and A0) but they could score the B mark. A for minimum clearly on negative y-ais and at least 5 marked on y-ais. Allow this mark if the minimum is very close and the point (0, 5) clearly indicated (b) st B Ignore coordinates for this mark Coordinates or points on sketch override coordinates given in the tet. Condone (y, ) confusion for points on aes only. So ( 5,0) for (0, 5) is OK if the point is marked correctly but (,0) is B0 even if in th quadrant. (c) This may be at the bottom of a page or in the question...make sure you scroll up and down! GCE Core Mathematics C (666) Summer 00
7. + = + M A ( ) y =,, + M A AA + st M for attempting to divide(one term correct) st A for both terms correct on the same line, accept for or for These first two marks may be implied by a correct differentiation at the end. nd n n M for an attempt to differentiate for at least one term of their epression + 6 Differentiating and getting is M0 nd A for only rd A for allow. Must be simplified to this, not e.g. th A for allow. Both terms needed. Condone + ( ) If +c is included then they lose this final mark They do not need one line with all terms correct for full marks. Award marks when first seen in this question and apply ISW. Condone a mied line of some differentiation and some division e.g. + + can score st MA and nd MA 6 Quotient /Product Rule 6 + st P Q M for an attempt: or R + ( S) with or 6 ( ) ( + + )( ) one of P,Q or R,S correct. st A for a correct epression or (o.e.) th A same rules as above ( ) ( ) GCE Core Mathematics C (666) Summer 00
8. m AB 0 = = 7 5 Equation of AB is: y M 0 ( ) y = 7 M 5 5 (o.e.) - 5y - 8 = 0 (o.e.) A () = or ( ) AB = 7 + 0 M (b) ( ) ( ) ( ) = A (c) Using isos triangle with AB = AC then t = y A = = 8 B () (d) Area of triangle = t (7 ) M = 0 A Apply the usual rules for quoting formulae here. For a correctly quoted formula with some correct substitution award M If no formula is quoted then a fully correct epression is needed for the M mark st M for attempt at gradient of AB. Some correct substitution in correct formula. nd M for an attempt at equation of AB. Follow through their gradient, not e.g. m Using y = m + c scores this mark when c is found. y y Use of = scores st M for denominator, nd M for use of a correct point y y A requires integer form but allow 5y + 8 = etc. Must have an = or A0 8 (b) M for an epression for AB or AB. Ignore what is left of the equals sign (c) B for t = 8. May be implied by correct coordinates (, 8) or the value appearing in (d) (d) M for an epression for the area of the triangle, follow through their t ( 0) but must have the (7 ) or 5 and the. 7 Area = 8 7 0 0 8 + t+ + + t 0 t 0 Must have the for M DET e.g. ( ) GCE Core Mathematics C (666) Summer 00
9. a + 9d = 0.75 or a= 0.75 9 d or 9d = 0.75 a M A 0 0 0 0 = l or ( a + 0.75) or (a + (0 ) d) or 5(a + 9d) M So 005 = 5[ a + 0.75] * A cso (b) ( S ) ( a + ) 05 (c) 67 = a +0.75 so a = ( ) 6.5 or 65p or 6 NOT M A 9d = 0.75-6.5 =.5 so d = ( )0.50 or 0.5 or 50p or M A () 8 M for attempt to use a + (n )d with n =0 to form an equation. So a + (0 )d = any number is OK A as written. Must see 9d not just (0 )d. Ignore any floating signs e.g. a+ 9 d = 0.75 is OK for MA These two marks must be scored in. Some may omit but get correct equation in (c) [or (b)] but we do not give the marks retrospectively. Parts (b) and (c) may run together (b) M for an attempt to use an S n formula with n =0. Must see one of the printed forms. ( S 0 = is not required) Acso for forming an equation with 005 and S n and simplifying to printed answer. Condone signs e.g. 5[a+ 0.75]=005 is OK for A (c) st M for an attempt to simplify the given linear equation for a. Correct processes. Must get to ka = or k = a + m i.e. one step (division or subtraction) from a = Commonly: 5a = 005 6.5 (= 9.75) st 05 A For a = 6.5 or 65p or 6 NOT or any other fraction nd M for correct attempt at a linear equation for d, follow through their a or equation in Equation just has to be linear in d, they don t have to simplify to d = nd A depends upon nd M and use of correct a. Do not penalise a second time if there were minor arithmetic errors in finding a provided a = 6.5 (o.e.) is used. Do not accept other fractions other than If answer is in pence a p must be seen. Sim Equ Use this scheme: st MA for a and nd MA for d. Typically solving: 005=0a + 5d and 0.75 = a + 9d. If they find d first then follow through use of their d when finding a. GCE Core Mathematics C (666) Summer 00
0. y O 7 (i) shape (anywhere on diagram) Passing through or stopping at (0, 0) and (,0) only(needn t be shape) (ii) correct shape (-ve cubic) with a ma and min drawn anywhere Minimum or maimum at (0,0) Passes through or stops at (7,0) but NOT touching. (7, 0) should be to right of (,0) or B0 Condone (0,) or (0, 7) marked correctly on -ais. Don t penalise poor overlap near origin. Points must be marked on the sketch...not in the tet B B B B B (5) (b) ( ) ( 7 ) = (0 = ) [7 ( )] M (0 = ) [7 ( )] (o.e.) Bft ( ) (c) ( 0 8 ) 0= 8 + * A cso () 8± 6 6 = + = or ( ) ( ) ± + ( = 0) = 8± = or ( ) =± B = ± A From sketch A is = M So y = ( )( [ ] ) (dependent on st M) M = + 8 A (7) (b) M for forming a suitable equation B for a common factor of taken out legitimately. Can treat this as an M mark. Can ft their cubic = 0 found from an attempt at solving their equations e.g. 8 = (... Acso no incorrect working seen. The = 0 is required but condone missing from some lines of working. Cancelling the scores B0A0. M A 5 (c) st M for some use of the correct formula or attempt to complete the square st A for a fully correct epression: condone + instead of + or for ( ) = B for simplifying 8 = or =.Can be scored independently of this epression nd A for correct solution of the form p + q : can be + or + or nd M for selecting their answer in the interval (0,). If they have no value in (0,) score M0 rd M for attempting y = using their in correct equation. An epression needed for MA0 rd A for correct answer. If answers are given A0. GCE Core Mathematics C (666) Summer 00
. (b) 5 ( y = ) ( + c) f() = 5 5= 6 0 8+ c f( ) 0 = + 9 5 5 m = = 7.5 or 5 y 5= Equation is: ( ) c = 9 y 5+ 50= 0 o.e. MAA M A (5) M MA A () (9marks) (b) st n n M for an attempt to integrate + st A for at least correct terms in (unsimplified) nd A for all terms in correct (condone missing +c at this point). Needn t be simplified nd M for using the point (, 5) to form a linear equation for c. Must use = and y = 5 and have no term and the function must have changed. rd A for c = 9. The final epression is not required. st M for an attempt to evaluate f(). Some correct use of = in f( ) but condone slips. 5 They must therefore have at least or and clearly be using f( ) with =. Award this mark wherever it is seen. nd M for using their value of m [or their ] (provided it clearly comes from using = in m f () ) to form an equation of the line through (,5)). Allow this mark for an attempt at a normal or tangent. Their m must be numerical. Use of y = m +c scores this mark when c is found. st A for any correct epression for the equation of the line nd A for any correct equation with integer coefficients. An = is required. e.g. y = 5 50 etc as long as the equation is correct and has integer coefficients. Normal Attempt at normal can score both M marks in (b) but A0A0 GCE Core Mathematics C (666) Summer 00