3 x 2 / 3 2. PhysicsAndMathsTutor.com. Question Answer Marks Guidance 1 5x(x + 1) 3(2x + 1) = (2x + 1)(x + 1) M1*

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1 Question Answer Marks Guidance 5( + ) 3( + ) ( + )( + ) * dep* Multiplying throughout by ( + )( + ) or combining fractions and multiplying up oe (eg can retain denominator throughout) Condone a single numerical error, sign error or slip provided that there is no conceptual error in the process involved Do not condone omission of brackets unless it is clear from subsequent work that they were assumed eg 5( + ) 3( + ) ( + )( ) gets 5( + ) 3( + ) gets M0 5( + )( + ) 3( + )( + ) ( + )( + ) gets M0 5( + ) 3( + ) ( + ) gets, just, for slip in omission of ( + ) Multiplying out, collecting like terms and forming quadratic ( 0). Follow through from their equation provided the algebra is not significantly eased and it is a quadratic. Condone a further sign or numerical error or a minor slip when rearranging oe (not fortuitously obtained check for double errors) (3 + )( ) 0 Solving their three term quadratic ( 0) provided b 4ac 0. Use of correct quadratic equation formula (if formula is quoted correctly then only one sign slip is permitted, if the formula is quoted incorrectly M0, if not quoted at all substitution must be completely correct to earn the ) or factorising (giving their term and one other term when factors multiplied out) or comp. the square (must get to the square root stage involving and arithmetical errors may be condoned provided their 3 / 3 seen or implied) /3 or cao for both obtained (condone or better) (If no factorisation (oe) seen B for each answer stated following correct quadratic)

2 Question Answer Marks Guidance 3 A B C ( )(4 ) 4 correct form of partial fractions ( condone additional coeffs eg A B BUT 4 ** is M0 ) A B C D 4 * or 3 A(4 + ) + (B + C)( ) Multiplying through oe and substituting values or equating coeffs at LEAST AS FAR AS FINDING A VALUE for one of their unknowns (even if incorrect) Can award in cases * and ** above Condone a sign error or single computational error for but not a conceptual error Eg 3 A( ) + (B + C)(4 + ²) is M0 3( )(4 + ²) A(4 + ²) + (B + C)( ) is M0 Do not condone missing brackets unless it is clear from subsequent work that they were implied. Eg 3 A(4 + ²) + B + C( ) 4A + A² + B + C C is M0 4A + A² + B B² + C C is 6 8A, A ¾ coeffs: 0 A B B ¾ constants: 0 4A + C C ½ oe [SC B A 3/4 from cover up rule can be applied, then the applies to the other coefficients] A B NB A 3/4 is A0 ww (wrong working) 4 oe oe [In the case of * above, all 4 constants are needed for the final ] Ignore subsequent errors when recompiling the final solution provided that the coeffs were all correct

3 Question Answer Marks Guidance () 3() ()() Multiplying throughout by ( + )( + ) or combining fractions and multiplying up oe (eg can retain denominator throughout) Condone a single numerical error, sign error or slip provided that there is no conceptual error in the process involved Do not condone omission of brackets unless it is clear from subsequent work that they were assumed eg 4( + ) 3( + ) ( + )( ) gets 4( + ) 3( + ) gets M0 4( + )( + ) 3( + )( + ) ( + )( + ) gets M0 4( + ) 3( + ) ( + ) gets, just, for slip in omission of ( + ) D (3 + )( ) 0 Multiplying out, collecting like terms and forming quadratic 0. Follow through from their equation provided the algebra is not significantly eased and it is a quadratic. Condone a further sign or numerical error or minor slip when rearranging. or 6² 4 0 oe, (not fortuitously obtained - check for double errors) Solving their three term quadratic provided b² 4ac 0. Use of correct quadratic equation formula (can be an error when substituting into correct formula) or factorising (giving their correct ² and constant terms when factors multiplied out) or comp the square oe. soi /3 or cao for both obtained (accept 4/6 oe, or eact decimal equivalent (condone 0.667or better)) SC B with or without any working

4 A B C 4 ()( ) A( + ) + (B + C) ( + ) ½ : ¼ A A 4/5 coeff of : 0 A + B B /5 constants: A + C C /5 B B B correct form of partial fractions mult up and equating or substituting oe soi for omission of B or C on numerator, M0,, then ( -/, A 4/5) B, B0, B0 is possible. A D B C for,, then B for both A4/5 and D0, B, B is possible. isw for incorrect assembly of final partial fractions following correct A,B & C. condone omission of brackets for second only if the brackets are implied by subsequent working.

5 ( )(+) + B ( + )( ) +( ( )( +) (3 ) ( + ) )( ( correct method for addition of fractions (3 ) or do not isw for incorrect subsequent cancelling or ( ( ++ + ( ) + )+)( ( 3 + ( )(+) correct method for addition of fractions (3 )( + ) (3 )( + ) ( + ( ) ) B accept denominator as ²- or(-)(+) do not isw for incorrect subsequent cancelling (3 ) ( )

6 6 4 ( ) 4. (. ( )...) 4 4 (...) dealing with 4 (or terms in 4, 4 correct binomial coefficients correct unsimplified epression for (+/4) or (4+) cao, etc) Valid for < /4 < 4 < < 4 B 7(i) 3 A B ( y)( y) y y Ay ( ) B(y) ( y)( y) 3 A(y + ) + B(y ) y 3 3A A y 3 3B B substituting, equating coeffs or cover up (ii) dy ( y)(y) d 3d y 3 ( y)( y) d ( ) dy 3 d ( y) y ln(y ) ln(y + ) 3 + c y 3 ln c y y e 3 c e 3 c.e Ae 3 * y Bft B E separating variables ln(y ) ln(y + ) ft their A,B 3 + c anti-logging including c

7 ( )(+) ( + ( + + )( ) ) 3 4 ( + )( ) combining fractions correctly factorising and cancelling (may be 3²+-8) 9(i) ( ). ( 3 ) (+4 ).4 +! (4 ) r < 4 < - ½ < < ½ Va lid f o binomial epansion with p -/ + 4 (ii) ( + 4 )( ) substitituting their and epanding ft their epansion (of three terms) cao 0 (i) ( A 0.5[ +.060]. (3 s f.). (ii) / ( + e ) + e + ( e ) +...! + e e * 8 (iii) I (+ e e ) d 8 e + e 6 ( e + e 4 ) ( e + e ) (3 s.f.) cao [] E Correct epression for trapezium rule Binomial epansion with p ½ Correct coeffs integration substituting limits into correct epression

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PhysicsAndMathsTutor.com Question Answer Marks Guidance x x x mult throughout by (x + )(x ) or combining fractions and mult up oe (can retain denominator throughout). Condone a single computational error provided that there xx)