Version 1.0. Level 2 Certificate in Further Mathematics. Practice Paper Set 4. Paper /2. Mark Scheme

Transcription

1 Version 1.0 Level Certificate in Further Mathematics Practice Paper Set 4 Paper 8360/ Mark Scheme

2 Mark Schemes Principal Examiners have prepared these mark schemes for practice papers. These mark schemes have not, therefore, been through the normal process of standardising that would take place for live papers. It is not possible to indicate all the possible approaches to questions that would gain credit in a live examination. The principles we work to are given in the glossary on page 3 of this mark scheme. Evidence of any method that would lead to a correct answer, if applied accurately, is generally worthy of credit. Accuracy marks are awarded for correct answers following on from a correct method. The correct method may be implied, but in this qualification there is a greater expectation that method will be appropriate and clearly shown. Further copies of this Mark Scheme are available to download from the AQA Website: Copyright 013 AQA and its licensors. All rights reserved. AQA Education (AQA) is a registered charity (number ) and a company limited by guarantee registered in England and Wales (number ). Our registered address is AQA, Devas Street, Manchester 5 6EX.

3 AQA Level Certificate in Further Mathematics / - Set 4 - Paper - Practice Paper Glossary for Mark Schemes These examinations are marked in such a way as to award positive achievement wherever possible. Thus, for these papers, marks are awarded under various categories. M A B ft SC M dep B dep oe Method marks are awarded for a correct method which could lead to a correct answer. Accuracy marks are awarded when following on from a correct method. It is not necessary to always see the method. This can be implied. Marks awarded independent of method. Follow through marks. Marks awarded for correct working following a mistake in an earlier step. Special case. Marks awarded within the scheme for a common misinterpretation which has some mathematical worth. A method mark dependent on a previous method mark being awarded. A mark that can only be awarded if a previous independent mark has been awarded. Or equivalent. Accept answers that are equivalent. eg, accept 0.5 as well as 1 [a, b] Accept values between a and b inclusive Allow answers which begin 3.14 eg 31.4, 314, Use of brackets It is not necessary to see the bracketed work to award the marks. 3

4 Practice Paper - Paper - Set / - AQA Level Certificate in Further Mathematics Paper - Calculator 1 positive odd B1 Exactly two boxes selected in each negative even B1 positive odd B1 (a) P = QR 1 P + 1 = QR Correct step after their first step P 1 R = Q (b) = 5Q x + 8x = 180 oe eg 7x = 180 8x x = 1 7x = 84 and 8x = 96 May be seen on diagram their if correct May be seen on diagram x + 60 = 84 and angle PQR = 84 and Yes ft ft from A0 oe eg x + 60 = 84 and angle SRQ = 96 and angle PQC = 96 and Yes Angles may be seen on diagram 4

5 AQA Level Certificate in Further Mathematics /- Set 4 - Paper - Practice Paper 4(a) (0, 0) and (6, 8) B1 Seen or implied (6) + 8 oe eg or 100 or (b) Do not overlap and complete reason Bft ft their distance in (a) Example 1: B1ft Do not overlap and partial reason Do not overlap and < 10 Example : Do not overlap and diagram drawn showing 4, 5 and 10 in appropriate places Example 3: Do not overlap and there will be a gap of 1 Example 1 Do not overlap and 9 Example Do not overlap and diagram drawn showing two of 4, 5 and 10 B1 4 and 5 seen 5(a) n 4 B B1 ) 5 (n = n 10 seen or implied eg 1 4 n 5(b) c + 1 B B1 for each correct term Do not accept c + c 0 for B Accept c 0 as an alternative to 1 for B1 5(c) ( 1 ) 4 oe 1 or (a) x x oe eg x(30 x) + x(0 x) + (30 x)(0 x) 600 4x oe eg 4(150 x ) 6(b) (30 x) or (0 x) B1 Seen or implied x(30 x)(0 x) ft their three dimensions but all must be in terms of x x(600 60x 40x + 4x ) dep ft their three dimensions but must involve product of two linear expressions 600x 100x + 4x 3 5

6 Practice Paper - Paper - Set / - AQA Level Certificate in Further Mathematics 7(a) ST sin54 40 sin( ) ST sin54 40 sin101 (ST =) 40 sin 101 sin 54 dep Must use 101 Allow M if this is the first line of working 3.966(..) Only award if correct working for both M marks are seen 7(b) sin 5 = w 3.97 cos 65 = w sin 5 dep Allow 3.97 cos 65 [13.9, ] Allow 14 if correct method seen SC [1.6, 1.6] SC1 [4.4, ] 8 (AX =) (= 18) oe eg (AX =) 18 or [4., 4.43] 10 their 18 (= 8) 10 their 18 dep [9.055, 9.1] or 8 9(a) x 4 B B1 < x 4 or x < 4 or < x < 4 SC1 Fully correct response in words 9(b) 3 g(x) 5 B B1 3 < g(x) 5 or 3 g(x) < 5 or 3 < g(x) < 5 SC1 Fully correct response in words 9(c) 1. B1 3 B1 9(d) 4 B B1 4 or 8 oe 6

7 AQA Level Certificate in Further Mathematics /- Set 4 - Paper - Practice Paper 10 Enlargement B1 Scale factor Centre (0,0) B1 B1 11(a) (n + 1) + (n + 1) ( (n + n)) n + n n + 1 n n (= n + ) oe Example 1: n + 3n + + n + 1 n n (= n + ) Example : n + n + n n + 1 n n (= n + ) Must be clearly shown and have brackets removed = n + is not needed but if simplification shown it must be correct 11(b) n + = 3 n = 15 their 15 + their 15 or (their ) + their and 7 7

8 Practice Paper - Paper - Set / - AQA Level Certificate in Further Mathematics 1(a) x = 36 B1 4x = 1 or Simplifies their quadratic to ax = b or (x + 1)(x 1) or correctly factorises their quadratic or substitutes correctly for their quadratic 1 1 ft ft from B0 and Must have solutions 1(b) 7 = 8x 3 oe eg 7 3 = x 8 Must have x 3 3 = x dep oe Solutions 3 and 3 is A0 8

9 AQA Level Certificate in Further Mathematics /- Set 4 - Paper - Practice Paper 13 SQR = x Working may be seen on the diagram throughout QRS or QSR = 180 their x 90 x 7x + their (90 x) = 180 or oe eg 6x = 90 x + their (90 x) = 7x 15 Alternative method 1 QRS = 180 7x Working may be seen on the diagram throughout QSR = their (180 7x) x + their(180 7x) = 7x oe eg 1x = Alternative method QRS = 180 7x Working may be seen on the diagram throughout SQR = 180 their (180 7x) oe eg SQR = 14x 180 x = their (14x 180) oe eg 1x = Alternative method 3 QSR = 5x Working may be seen on the diagram throughout SRQ = their 5x 7x + their 5x = 180 oe eg 1x = Alternative method 4 QSR = 5x Working may be seen on the diagram throughout SRQ = their 5x SQR = x and oe eg 1x = 180 their 5x + their 5x + x =

10 Practice Paper - Paper - Set / - AQA Level Certificate in Further Mathematics 14 sin (= [ 4.844, 4.8]) oe eg sin [.8,.844] [ , 317.] ft ft 540 their [.8,.844] Alternative method sin (= [ 4.844, 4.8]) oe eg sin [ , 317.] [.8,.844] ft ft 540 their [ , 317.] x x = 5 (0 =) x 8x x x (= 0) 8 ( 8) 4 3 oe eg ft their 3-term quadratic [3.58, 3.6] and [0.4, 0.4] oe eg and [3.16, 3.] ft oe eg 40 or 10 ft their 3-term quadratic solutions Alternative method + 8x x = 5 x 3 oe 4x = (x ) 4 = 3 oe eg (x ) = 5 ft their 3-term quadratic [3.58, 3.6] and [0.4, 0.4] oe eg and + [3.16, 3.] ft oe eg 5 or 10 ft their 3-term quadratic solutions 10

11 AQA Level Certificate in Further Mathematics /- Set 4 - Paper - Practice Paper 16 sin x + cos x or cosxsinx + sinxcosx oe eg cosx cosx + sinx sinx One correct element Does not have to be seen in a matrix sin x cos x cos x sin x sin x cos x oe cos x sin x sin x cos x sin x cos x All 4 correct elements in correct positions in a matrix 1 0 oe eg I or identity (matrix) Any two of: a + 5b = 5 or a + 8b = 6 or 3a + 1b = 9 or 4a + 17b = 14 Makes their pair of equations have a common coefficient and subtraction attempt eg a + 10b = 10 a + 8b = 6 and b = 4 b = a = 5 B B1 Any one of: a + 5b = 5 or a + 8b = 6 or 3a + 1b = 9 or 4a + 17b = 14 Allow up to two errors Alternative method (First differences) a + 3b a + 4b a + 5b (Second differences) b b b = Any one of: a + 5b = 5 or a + 8b = 6 or 3a + 1b = 9 or 4a + 17b = 14 B1 At least correct a = 5 B1ft ft their b 11

12 Practice Paper - Paper - Set / - AQA Level Certificate in Further Mathematics 18 3m(8 3m) or m(4 9m) (8 + 3m) (8 3m) 3m 8 3m Further incorrect work is A0 19 cos (x) = 1 8 cos 1 8 dep 1 [ , 48.] Allow 48 if correct method seen SC [53.5, ] SC1 [0.84, ] 0(a) (1, 0) B1 0(b) ( y =) + x x x + x x d y ( =) 1 x dep oe dx ft their expansion (At x = 0, d y =) 1 dx ft their dy dx grad normal = 1 ft dy ft their dx Must have gained M3 y = (their grad normal) x + y = x + if correct y = x + and x = y = 0 oe eg y + x = and + 0 = 1 a b(3a ab 5b ) or ab(3a 3 a b 5ab ) or a (3a b ab 5b 3 ) a b(3a 5b)(a + b) 1

13 AQA Level Certificate in Further Mathematics /- Set 4 - Paper - Practice Paper (a) y = 10 B1 (b) x = B1 (c) Increasing B1 (d) d y ( =) 3px 6x + 8 dx At least terms correct 3p () = 0 oe dy Substitutes x = in their which must be dx a quadratic and equates to zero 1 (p =) () 3 () r = 10 3 oe Substitutes x = in their px 3 3x + 8x + r and equates to ft (r =) ft their p if M3 gained 3 13