AS and A level Mathematics Practice Paper – Algebra (part 2) – Mark scheme

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AS and A level Mathematics Practice Paper – Algebra (part 2) – Mark scheme Question Scheme Marks 1 Either Or y 2=4−4 x+x 2 x 2=4−4 y+ y 2 M1 4 (4−4 x+ x 2 )−x 2=11 or 2 2 4 (2 ) 11  ...  x x 4 y 2−(4−4 y+ y 2 )=11 or 2 2 4 (2 ) 11 y y    M1 3 x 2−16 x+5=0 3 y 2 +4 y−15=0 Correct 3 terms A1 (3 x−1)( x−5 )=0 , x=.. . (3 y−5 )( y+3 )=0 , y=.. . M1 x= 1 3 x=5 y= 5 3 y=−3 A1 y= 5 3 y=−3 x= 1 3 x=5 M1A1 (7 marks) 2 2 2 2 2 2 4 4 (2 4) 20 0 4 2 4 or 2 ( 4) 10( 4) 0 y x x x x y x y x y y y                  or M1 2 2 8 36 16 0 or 2 2 24 0 x x y y       M1 A1     4 (2 1)( 4) 0 ... or 2 ( 4)( 3) 0 ... x x x y y y           M1 0.5, 4 or 4, 3 x x y y     A1 cso Sub into y x   2 4 or Sub into 4 2 y x   M1 3, 4 and 4, 0.5 y y x x     A1 (7 marks) This study source was downloaded by 100000818171864 from CourseHero.com on 06-25-2021 04:25:48 GMT -05:00 1 https://www.coursehero.com/file/56333064/AS-Mathematics-Practice-Paper-Algebra-part-2-MSdocx/ This study resource was shared via CourseHero.comAS and A level Mathematics Practice Paper – Algebra (part 2) – Mark scheme Question Scheme Marks 3 2 2 4 1 ( 4 1) 5 2 0 y x x x x         M1 2 21 10 1 0 x x    A1       1 1 7 3 7 1 3 1 0 , x x x        dM1 A1 y = – 3 7 , 1 3 M1 A1 (6 marks) 4(a) 2 x k x k     4 (1 2 ) 5 ( 0) M1 So 2 x kx k    8 0 * A1cso (2) 4(b) 2 (8 ) 4 k k  M1A1 1 16 k  (oe) A1 (3) 4(c) 2 1 1 2 16 x x   0 so 1 2 4 ( ) 0 x x     M1 1 4 x  , 1 2 y 1 A1A1 (3) (8 marks) 5(a) 5x > 20 M1 x > 4 A1 (2) 5(b) 4 12 0 2 x  x    x x   2 6    [ = 0] M1 x = 6,  2 A1 x   2 , x > 6 M1A1ft (4) (6 marks) 6(a) 6x + x > 1 – 8 M1 x >  1 A1 (2) 6(b) (x + 3)(3x – 1) [= 0]  x = – 3 and 1 3 M1A1 1 3 < < 3  x M1A1ft This study source was downloaded by 100000818171864 from CourseHero.com on 06-25-2021 04:25:48 GMT -05:00 2 https://www.coursehero.com/file/56333064/AS-Mathematics-Practice-Paper-Algebra-part-2-MSdocx/ This study resource was shared via CourseHero.comAS and A level Mathematics Practice Paper – Algebra (part 2) – Mark scheme Question Scheme Marks (4) (6 marks) 7(a) 3 7 3 x x    4 10 x  M1 5 5 2.5, , o.e. 2 2 x x x    A1 (2) 7(b) Obtain 2 x x   9 36 and attempt to solve 2 x x    9 36 0 e.g.( 12)( 3) x x   = 0 so x = , or 9 81 144 2 x    M1 12, –3 A1    3 12 x M1A1 (4) 7(c) 2.5 12  x A1cso (1) (7 marks) 8(a) b 2 – 4ac = (k − 3)2 – 4(3 − 2k) M1 k 2 – 6k + 9 – 4(3 − 2k) > 0 or (k − 3)2 – 12 + 8k > 0 or better M1 k 2 + 2k − 3 > 0 A1 cso (3) 8(b) (k + 3)(k – 1) [= 0] M1 Critical values are k = 1 or k = – 3 A1 (choosing “outside” region) M1 k > 1 or k < – 3 A1 cao (4) (7 marks) 9(a) Method 1: Attempts 2 b ac  4 for a k   ( 3), b 6 and their c c k  M1     2 2 b a c k k      4 6 4 3 5 A1 2 ( 4 ) b ac   2    4 8 96 k k or 2    ( 4 ) b ac 2 4 8 96 k k   (with no prior algebraic errors) B1 As 2 b a c   4 0 , then 2     4 8 96 0 k k and so, 2 k k    2 24 0 A1 Method 2: M1 This study source was downloaded by 100000818171864 from CourseHero.com on 06-25-2021 04:25:48 GMT -05:00 3 https://www.coursehero.com/file/56333064/AS-Mathematics-Practice-Paper-Algebra-part-2-MSdocx/ This study resource was shared via CourseHero.comAS and A level Mathematics Practice Paper – Algebra (part 2) – Mark scheme Question Scheme Marks Considers 2 b ac 4 for a k   ( 3), b 6 and their c c k      2 6 4 3 5    k k A1 2 4 8 96 0 k k    or 2     4 8 96 0 k k or 9 3 5     k k    (with no prior algebraic errors) B1 and so, 2 k k    2 24 0 following correct work A1 (4) 9(b) Attempts to solve 2 k k    2 24 0 to give k = ( Critical values, k   6, 4.) M1 2 k k    2 24 0 gives - 4 6   k M1A1 (3) (7 marks) 10(a) 2 b ac    4 0 e.g. 2 4 4( 1)( 5) 0     p p or 2 0 4 4( 1)( 5)     p p or 2 4 4( 1)( 5)    p p or 2 4( 1)( 5) 4 p p    M1 A1 2 4 6 5    p p 2 p p    6 1 0 A1* (3) 10(b) 2 p p p      6 1 0 ... M1 p  3 8 A1 p p     3 8 or 3 8 M1A1 (4) (7 marks) 11(a) 2 2 2 6 4 " "3 7 or 7 7 2 6 4 3 3 px px p x y y y p p p                      M1 Examples dM1 This study source was downloaded by 100000818171864 from CourseHero.com on 06-25-2021 04:25:48 GMT -05:00 4 https://www.coursehero.com/file/56333064/AS-Mathematics-Practice-Paper-Algebra-part-2-MSdocx/ This study resource was shared via CourseHero.comAS and A level Mathematics Practice Paper – Algebra (part 2) – Mark scheme Question Scheme Marks           2 2 2 2 2 2 6 4 3 7 0 , 2 6 4 3 7 0 7 7 2 6 4 0 , 2 10 9 8 0 3 3 2 6 4 3 7 px px p x px px p x y y p p p y py p y p y px px p x                                       E.g.       2 2 b ac p p p       4 6 3 4 2 4 7 ,       2 2 b ac p p p     4 10 9 4 2 8 ddM1 4p2  20p + 9 < 0 * A1* (4) 11(b)  2 9 2 1 0 ... p p p         to obtain p = M1 9 1 , 2 2 p  A1 1 1 2 2   p 4 M1 A1 (4) (8 marks) 12(a) P x   20 6 o.e B1 20 6 40 x x     M1 x 1.7 A1* (3) 12(b) Mark parts (b) and (c) together 2 A x x x x x x       2 (2 1) 2 (6 3) 16 8 B1 2 16 8 120 0 x x    M1 Try to solve their 2 2 15 0 x x    e.g. (2 5)( 3) 0 x x    so x = M1 Choose inside region M1 5 3 2   x or 5 0 2  x ( as x is a length ) A1 (5) 12(c) 5 1.7 2  x B1cao (1) (9 marks) This study source was downloaded by 100000818171864 from CourseHero.com on 06-25-2021 04:25:48 GMT -05:00 5 https://www.coursehero.com/file/56333064/AS-Mathematics-Practice-Paper-Algebra-part-2-MSdocx/ This study resource was shared via CourseHero.comAS and A level Mathematics Practice Paper – Algebra (part 2) – Mark scheme Source paper Question number New spec references Question description New AOs 1 C1 2011 4 2.4 Simultaneous equations 1.1b 2 C1 2016 5 2.3 and 2.4 Simultaneous equations - one linear, one quadratic 1.1b 3 C1 2015 2 2.3 and 2.4 Solution of simultaneous equations 1.1b 4 C1 2013 10 2.4 Simultaneous equations, one linear one quadratic 1.1b, 3.1a, 3.2 5 C1 Jan 2012 3 2.5 Inequalities 1.1b 6 C1 2013 5 2.5 Inequalities 1.1b, 2.1, 2.2a, 2.4, 2.5, 3.1b, 3.3 7 C1 2014 3 2.5 Solution of linear and quadratic inequalities 1.1b, 2.2a 8 C1 Jan 2011 8 2.3 and 2.5 Quadratics, Inequalities, Polynomials, Factor theorem 1.1b 9 C1 Jan 2013 9 2.3 and 2.5 Quadratics, Inequalities 1.1b, 2.1 and 2.2a 10 C1 2015 5 2.3 and 2.5 Discriminant, solution of inequality by formula 1.1b, 2.1 and 2.4 11 C1 2016 8 2.3, 2.4, 2.5, 2.6 and 2.7 Inequalities and discriminant 1.1b, 2.1 and 3.1a 12 C1 June 2014R 6 2.5 Solution of linear and quadratic inequalities 2.1, 2.2a, 2.4, 3.1b, 3.4 1 C1 2011 4 2.4 Simultaneous equations 1.1b 2 C1 2016 5 2.3 and 2.4 Simultaneous equations - one linear, one quadratic 1.1b 6 This study source was downloaded by 100000818171864 from CourseHero.com on 06-25-2021 04:25:48 GMT -05:00 https://www.coursehero.com/file/56333064/AS-Mathematics-Practice-Paper-Algebra-part-2-MSdocx/ This study resource was shared via CourseHero.com Powered by TCPDF (www.tcpdf.org [Show More]

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