PMT GCE. Edexcel GCE Core Mathematics C1(6663) Summer Mark Scheme (Results) Core Mathematics C1 (6663) Edexcel GCE

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1 GCE Edexcel GCE Core Mathematics C(666) Summer 005 Mark (Results) Edexcel GCE Core Mathematics C (666)

2 June Core Mathematics C Mark. Penalise ± B () 8 = 64 or ( a) or 8 or Allow ± 8 = 4 or 0.5 A () () for understanding that - ower means recirocal 8 = 4 is M0A0 and - is A0 4. dy = 6 + 8x x n x n dx 6x ( 6x 4x ) dx = + 4x + c both A () A A () (5) st A for one correct term in x : 6x or + 4x (or better simlified versions) nd A for all terms as rinted or better in one line. 666 Core Mathematics C June 005 Advanced Subsidiary/Advanced Level in GCE Mathematics

3 . x 8x 9 ( x 4) 45 ( x ± 4) ( x 4) 6 + ( 9) ( x ± 4) 45 A A () ALT Comare coefficients 8 = a equation for a a = 4 AND a + b = 9 b = 45 A A () ( x 4) = 45 (follow through their a and b from ) x 4 = ± 45 c = 4 x = 4 ± 5 d = A A () (6) for ( x ± 4) or an equation for a. for a full method leading to x 4 =... or x =... A for c and A for d 8 ± 6 5 Note Use of formula that ends with scores A A0 (but must be 5) i.e. only enalise non-integers by one mark. 666 Core Mathematics C June 005 Advanced Subsidiary/Advanced Level in GCE Mathematics

4 4. Shae Points B B () for shae: grahs must have curved sides and round to. - and 4 max A A () (5) st B for shae through (0, 0) and ( (k,0) where k >0) nd B for max at (, 5) and 6 labelled or (6, 0) seen Condone (5,) if and 5 are correct on axes. Similarly (5,) in for shae NOT through (0, 0) but must cut x-axis twice. st A for - and 4 labelled or (-, 0) and (4, 0) seen nd A for max at (, 5). Must be clearly in st quadrant 5. x = + y and sub ( + y ) + y = 9 5y + 4y 8( = 0) i.e. ( 5y + 4)( y ) = 0 ( y =) or 4 (o.e.) (both) 5 A A y = x = + 4 = 5 ; 4 y = 5 x = (o.e) 5 A f.t. (6) st Attemt to sub leading to equation in variable st A Correct TQ (condone = 0 missing) nd Attemt to solve TQ leading to values for y. nd A Condone mislabelling x = for y = but then M0A0 in art (c). rd Attemt to find at least one x value r d A f.t. f.t. only in x = + y (sf if not exact) Both values N.B. False squaring (e.g. y = x + 4y = ) can only score the last marks. 666 Core Mathematics C June 005 Advanced Subsidiary/Advanced Level in GCE Mathematics

5 6. 6 x + > 5 x 8x > x > or 0.5 or 4 8 A () ( x )( x ) (> 0) Critical values x =, (both) A Choosing outside region x > or x < A f.t. (4) ( c ) x > or < x < 4 Bf.t. Bf.t. () (8) Multily out and collect terms (allow one sli and allow use of = here) st Attemting to factorise TQ x =... nd Choosing the outside region nd A f.t. f.t. their critical values N.B.(x>, x > is M0A0) For < x < q where > q enalise the final A in. (c) f.t. their answers to and st B a correct f.t. leading to an infinite region nd B a correct f.t. leading to a finite region Penalise or once only at first offence. e.g. (c) Mark x > 4 x > 4 <x < <x < B0 B x >, x > x > B B0 666 Core Mathematics C June 005 Advanced Subsidiary/Advanced Level in GCE Mathematics

6 7. ( x ) = 9 6 x + x by x 9x 6 + x A c.s.o. () 9x x (9x 6 + x ) dx = 6x + ( + c) A//0 use y = and x = : = c So y = 8x 6x + x - c = - A c.s..o. Af.t. (6) (8) Attemt to multily out ( x). Must have or 4 terms, allow one sign error A cso Fully correct solution to rinted answer. Penalise wrong working. n+ st n Some correct integration: x x A At least correct unsimlified terms A All terms correct (unsimlified) Ignore + c nd Use of y = and x = to find c. No + c is M0. Ac.s.o. for -. (o.e.) Award this mark if c = " stated i.e. not as art of an exression for y Af.t. for simlified x terms with y = and a numerical value for c. Follow through their value of c but it must be a number. 666 Core Mathematics C June 005 Advanced Subsidiary/Advanced Level in GCE Mathematics

7 8. y ( 4) = ( x 9) y x + = 0 (o.e.) (condone terms with integer coefficients e.g. y+=x) A A () Equation of l is: y = x (o.e.) Solving l and l : 6 x x + = 0 is oint where x =, = 6 y x or y y or x B A Af.t. (4) (c ) ( l is y = x 7 ) C is (0, -7) or OC = 7 Area of OCP = OC x, = 7 = 0. 5 or Bf.t. Ac.a.o. () (0) for full method to find equation of l sta any unsimlified form (c ) Attemt to solve two linear equations leading to linear equation in one variable nd A f.t. only f.t. their x or y in y = x Bf.t. Either a correct OC or f.t. from their l for correct attemt in letters or symbols for OCP A c.a.o. 7 scores A0 666 Core Mathematics C June 005 Advanced Subsidiary/Advanced Level in GCE Mathematics

8 9 ( = ) a + ( a + d) a + ( n ) d ( S = )[ a + ( n ) d] a S = [ a + ( n ) d] [ a + ( n ) d] } either S = n[ a + ( n ) d] n S = [ a + ( n ) d] S [ ] B d A c.s.o (4) ( a = 49, d = ) u = ( ) = 09 A () (c ) n S n = 5000 n 50n = 0 (*) S n = [ 49 + ( n )( ) ] ( = n( 50 n) ) A A c.s.o () (d) ( n 00)( n 50) = 0 n = 50 or 00 A//0 () (e) u < 0 n = 00 not sensible 00 B f.t. () () B requires at least terms, must include first and last terms, an adjacent term dots and + signs. st for reversing series. Must be arithmetic with a, d (or a, l) and n. nd d for adding, must have S and be a genuine attemt. Either line is sufficient. Deendent on st (NB Allow first marks for use of l for last term but as given for final mark ) for using a = 49 and d = ± in a + ( n ) d formula. (c) for using their a, d in S n A any correct exression Acso for utting S n =5000 and simlifying to given exression. No wrong work (d) Attemt to solve leading to n =... A//0 Give AA0 for correct value and AA for both correct (e) B f.t. Must mention 00 and state u 00 < 0 (or loan aid or equivalent) If giving f.t. then must have n Core Mathematics C June 005 Advanced Subsidiary/Advanced Level in GCE Mathematics

9 0 x =, y = = 0 ( =0 is OK) B () (c) dy = x 4 x + 8 (= x 8x + 8) dx dy When x =, = m = 7 dx Equation of tangent: y 0 = 7( x ) y = 7 x + dy = m gives x 8 x + 8 = 7 dx ( x 8x + 5 = 0) ( x 5)( x ) = 0 x = () or 5 x = 5 A A c.a.o A (5) y = y = 5 or A (5) () st some correct differentiation ( x n x n for one term) st A correct unsimlified (all terms) nd substituting x (= P ) in their dx dy clear evidence rd using their m to find tangent at. st dy forming a correct equation their = gradient of their tangent dx (c) nd dy for solving a quadratic based on their leading to x = dx rd for using their x value in y to obtain y coordinate MR For misreading (0, ) for (, 0) award B0 and then A as in scheme. Then allow all M marks but no A ft. (Max 7) 666 Core Mathematics C June 005 Advanced Subsidiary/Advanced Level in GCE Mathematics

10 GENERAL PRINCIPLES FOR C MARKING Method mark for solving term quadratic:. Factorisation ( x + bx + c) = ( x + )( x + q), where q = c, leading to x = ( ax + bx + c) = ( mx + )( nx + q), where q = c and mn =. Formula Attemt to use correct formula (with values for a, b and c). a, leading to x =. Comleting the square Solving x + bx + c = 0 : ( x ± ) ± q ± c, 0, q 0, leading to x = Method marks for differentiation and integration:. Differentiation Power of at least one term decreased by. ( x n x n ). Integration Power of at least one term increased by. ( x n x n+ ) Use of a formula Where a method involves using a formula that has been learnt, the advice given in recent examiners reorts is that the formula should be quoted first. Normal marking rocedure is as follows: Method mark for quoting a correct formula and attemting to use it, even if there are mistakes in the substitution of values. Where the formula is not quoted, the method mark can be gained by imlication from correct working with values, but will be lost if there is any mistake in the working. Exact answers Examiners reorts have emhasised that where, for examle, an exact answer is asked for, or working with surds is clearly required, marks will normally be lost if the candidate resorts to using rounded decimals. Answers without working The rubric says that these may gain no credit. Individual mark schemes will give details of what haens in articular cases. General olicy is that if it could be done in your head, detailed working would not be required. Most candidates do show working, but there are occasional awkward cases and if the mark scheme does not cover this, lease contact your team leader for advice. Misreads A misread must be consistent for the whole question to be interreted as such. These are not common. In clear cases, lease deduct the first A (or B) marks which would have been lost by following the scheme. (Note that marks is the maximum misread enalty, but that misreads which alter the nature or difficulty of the question cannot be treated so generously and it will usually be necessary here to follow the scheme as written). Sometimes following the scheme as written is more generous to the candidate than alying the misread rule, so in this case use the scheme as written. 666 Core Mathematics C June 005 Advanced Subsidiary/Advanced Level in GCE Mathematics