Gcse Math Question Paper
Sp N ec e ifi w ca tio n
General Certificate of Secondary Education January 2012
Mathematics Unit T4 Higher Tier
[GMT41]
WEDNESDAY 11 JANUARY
9.15 am–11.15 am
MARK SCHEME
7280.01
GCSE MATHEMATICS Introduction The mark scheme normally provides the most popular solution to each question. Other solutions given by candidates are evaluated and credit given as appropriate; these alternative methods are not usually illustrated in the published mark scheme. The solution to a question gains marks for correct method and marks for accurate working based on this method. The marks awarded for each question are shown in the right hand column and they are prefixed by the letters M, A and MA as appropriate. The key to the mark scheme is …show more content…
given below: M A indicates marks for correct method. indicates marks for accurate working, whether in calculation, reading from tables, graphs or answers. Accuracy marks may depend on preceding M (method) marks, hence M0 A1 cannot be awarded i.e. where the method is not correct no marks can be given. indicates marks for combined method and accurate working.
MA
A later part of a question may require a candidate to use an answer obtained from an earlier part of the same question. A candidate who gets the wrong answer to the earlier part and goes on to the later part is naturally unaware that the wrong data is being used and is actually undertaking the solution of a parallel problem from the point at which the error occurred. If such a candidate continues to apply correct method, then the candidate’s individual working must be followed through from the error. If no further errors are made, then the candidate is penalised only for the initial error. Solutions containing two or more working of transcription errors are treated in the same way. This process is usually referred to as “follow-through marking” and allows a candidate to gain credit for that part of a solution which follows a working or transcription error. It should be noted that where an error trivialises a question, or changes the nature of the skills being tested, then as a general rule, it would be the case that not more than half the marks for that question or part of that question would be awarded; in some cases the error may be such that no marks would be awarded. Positive marking: It is our intention to reward candidates for any demonstration of relevant knowledge, skills or understanding. For this reason we adopt a policy of following through their answers, that is, having penalised a candidate for an error, we mark the succeeding parts of the question using the candidate’s value or answers and award marks accordingly. Some common examples of this occur in the following cases: (a) a numerical error in one entry in a table of values might lead to several answers being incorrect, but these might not be essentially separate errors; (b) readings taken from candidates’ inaccurate graphs may not agree with the answers expected but might be consistent with the graphs drawn. When the candidate misreads a question in such a way as to make the question easier, only a proportion of the marks will be available (based on the professional judgement of the examiner).
7280.01
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General Marking Advice (i) If the correct answer is seen in the body of the script and the answer given in the answer line is clearly a transcription error, full marks should be awarded.
(ii) If the answer is missing, but the correct answer is seen in the body of the script, full marks should be awarded. (iii) If the correct answer is seen in working but a completely different answer is seen in the answer space, then some marks will be awarded depending on the severity of the error. (iv) Work crossed out but not replaced should be marked. (v) In general, if two or more methods are offered, mark only the method that leads to the answer on the answer line. If two (or more) answers are offered (with no solution offered on the answer line), mark the poorest answer. (vi) For methods not provided for in the mark scheme, give as far as possible equivalent marks for equivalent work. (vii) Where a follow through mark is indicated on the mark scheme for a particular part question, the marker must ensure that you refer back to the answer of the previous part of the question. (viii)Unless the question asks for an answer to a specific degree of accuracy, always mark at the greatest number of significant figures seen. E.g. the answer in the mark scheme is 4.65 and the candidate then correctly rounds to 4.7 or 5 on the answer line. Allow full marks for 4.65 seen in the working. (ix) Anything in the mark scheme which is in brackets (...) is not required for the mark to be earned, but if …show more content…
present it must be correct. (x) For any question, the range of answers given in the mark scheme is inclusive. Quality of written communication In GCSE Mathematics, particular questions are identified where candidates must demonstrate the quality of their written communication. In particular, candidates must: (i) ensure that text is legible and that spelling, punctuation and grammar are accurate so that meaning is clear (i.e. comprehension and meaning is clear by using the correct notation and labelling conventions); (ii) select and use a form and style of writing appropriate to their purpose and to complex subject matter (i.e. reasoning, explanation or argument is correct and appropriately structured to convey mathematical reasoning); and (iii) organise information clearly and coherently, using specialist vocabulary where appropriate (i.e. the mathematical methods and processes used are coherently and clearly organised and appropriate mathematical vocabulary used). This assessment may be through, for example, an explanation of the geometrical properties of a given shape or, for example, through concise mathematical argument in a multi-step problem.
7280.01
3
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1
(a) LCM of 8 and 10 = 40 10th October (b) 135.66 = 120% 135.66 # 20 or 113.05 120 = £22.61
MA1 A1 MA1 MA1 A1 A1 A3 A1 A1 A1 A1, A1, A1
AVAILABLE MARKS
5
2
(a) 23, 43, 52, 55 (b) Correctly drawn graph (c) Correct reading (d) Identifying 37 55 – 37 = 18 (e) box plot median, quartiles, limits of range
10
3
(a) 3x = 9 x=3 y = –2 (b) 4x + 1 + 4x - 6 = 21 12 12 4x + 1 + 4x - 6 = 21 8x = 26 x = 26 8 or 13 4 or 3.25
MA1 MA1 MA1 MA1 MA1 MA1 MA1 MA1 A1 MA1 MA1 A1 MA1 MA1, MA1 3 6 6
4
(a) tan 48° = RS 13 RS = 13 tan 48° RS = 14.4 (14.438) (b) PR = :(5 + 13 ) = :194 (13.9) RPS = tan–1 (13 tan 48/:194) RPS = 46.03 (tan (14.4/13.9) = 46.01)
-1 2 2
5
(a) 5 2 (b) y = 5 x + 2 2
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6
(a) Angle EBC = 90° (b) Angle ABE = 32° (c) Angle BAE = 116° (d) Angle BCE = 64° (e) Angle BED = 46°
A1 A1 A1 A1 A1
AVAILABLE MARKS
5
7
(a) e.g. Only sampling Year 8 pupils (b) e.g. Pupils at end of list miss out (c) (i) 57 # 50 600 4.75 → 5 (ii) No, because 82 # 50 = 6.83 . . . → 7 600 and 41 # 50 = 3.416 . . . " 3 600 or using 1 12
A1 A1 MA1 A1
MA2
6
8
(a) 3 (p - 25q ) 3 (p + 5q) (p - 5q) (b) (x - 8) (x + 3) = 0 8 or –3
2
2
MA1 MA2 MA2 MA1 6
9
(a) –1 3
1 y =- 3 x + 4
MA1 MA1
2
(b) 12p - 21pq + 8pq - 14q 12p - 13pq - 14q
2 2
2
MA2 MA1 5
7280.01
5
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10 (a) y = k2 x k 4= 9 k = 36 y = 36 2 x (b) y = 36 144 =1 4 (c) When x is doubled the denominator becomes 4 times bigger therefore y becomes 4 times smaller 11 (a) Area = 1 80 # 264 sin 44 2 = 7335.6 m2 (b) Let M = midpoint AM = 80 + 132 - 2 # 80 # 132 cos 44 AM = :8631.543417 AM = 92.9 12 (a) x + (x - 5) = 144 x + x - 10x + 25 = 144 2x - 10x - 119 = 0 (b) x = 10 !
1052 4 x = 10.6 13 (a) 70 and 20 (b) Two bars drawn to correct heights (6.8 and 9.6) (c) 7.5 + 52.5 60 (d) x # 20 or 144 = 3.6, 3.6 # 20 = 72 60 40 x h = x # 20 = x 3 60 3
6
2 2 2 2 2 2 2 2
AVAILABLE MARKS
MA1 A1 MA1
A2 MA1
5
A1, A1 units
MA1 MA1 A1 MA1 A1 MA1 MA2 MA1 A1, A1 A1, A1 MA1 A1 MA1 A1 8 6 6
7280.01
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4 -3 3 14 a 8 k = a 27 k = d3 27 n = a 3 k = 81 27 8 8 2 16 -2 2 -2 0.25 = a 1 k = 4 = 16 4 81 # 16 = 81 16
4 4
4
MA2 MA1 A1 A1 MA1 MA1 A1 MA1 MA1 MA1 MA1 MA1
AVAILABLE MARKS
4
15 (a) Since triangle ANB is isosceles (180 – 120)/2 = 30 (b) AN = 25 sin 30 sin 120 AN = 25 sin 30 sin 120 12.5 AN 12.5 AN = cos 30° AN = 14.4 (14.43376) cos 30° =
or
AN = 14.4 (14.43376)
(c) Let x = length of OA and OB 2 2 2x = 25 2 x = : (25 /2) = 17.68 16 x + 2qx + q 2q = -8
2 2 2
6
q = -4
p = (-4) = 16
3
17
2 (3x - 2) + 6 (x + 1) =1 (x + 1) (3x - 2) 6x - 4 + 6x + 6 = 3x - 2x + 3x - 2 3x - 11x - 4 = 0 (3x + 1) (x - 4) = 0 x =1 3 or 4
2 2
M1, A1 MA1 MA1 MA2 MA1 7
18 C y x y G y z x F Total
7280.01
D y
CGD = 180 – (x + y) x + y + y + z = 180 (cyclic quad)
z E
y + z = 180 – (x + y) CGD = FED A3 3 100
7
[Turn over
7280.01
8
[Turn over
7280.01
9
[Turn over
General Certificate of Secondary Education January 2012
Mathematics Unit T4 Higher Tier
[GMT41]
WEDNESDAY 11 JANUARY
9.15 am–11.15 am
MARK SCHEME
7280.01
GCSE MATHEMATICS Introduction The mark scheme normally provides the most popular solution to each question. Other solutions given by candidates are evaluated and credit given as appropriate; these alternative methods are not usually illustrated in the published mark scheme. The solution to a question gains marks for correct method and marks for accurate working based on this method. The marks awarded for each question are shown in the right hand column and they are prefixed by the letters M, A and MA as appropriate. The key to the mark scheme is …show more content…
given below: M A indicates marks for correct method. indicates marks for accurate working, whether in calculation, reading from tables, graphs or answers. Accuracy marks may depend on preceding M (method) marks, hence M0 A1 cannot be awarded i.e. where the method is not correct no marks can be given. indicates marks for combined method and accurate working.
MA
A later part of a question may require a candidate to use an answer obtained from an earlier part of the same question. A candidate who gets the wrong answer to the earlier part and goes on to the later part is naturally unaware that the wrong data is being used and is actually undertaking the solution of a parallel problem from the point at which the error occurred. If such a candidate continues to apply correct method, then the candidate’s individual working must be followed through from the error. If no further errors are made, then the candidate is penalised only for the initial error. Solutions containing two or more working of transcription errors are treated in the same way. This process is usually referred to as “follow-through marking” and allows a candidate to gain credit for that part of a solution which follows a working or transcription error. It should be noted that where an error trivialises a question, or changes the nature of the skills being tested, then as a general rule, it would be the case that not more than half the marks for that question or part of that question would be awarded; in some cases the error may be such that no marks would be awarded. Positive marking: It is our intention to reward candidates for any demonstration of relevant knowledge, skills or understanding. For this reason we adopt a policy of following through their answers, that is, having penalised a candidate for an error, we mark the succeeding parts of the question using the candidate’s value or answers and award marks accordingly. Some common examples of this occur in the following cases: (a) a numerical error in one entry in a table of values might lead to several answers being incorrect, but these might not be essentially separate errors; (b) readings taken from candidates’ inaccurate graphs may not agree with the answers expected but might be consistent with the graphs drawn. When the candidate misreads a question in such a way as to make the question easier, only a proportion of the marks will be available (based on the professional judgement of the examiner).
7280.01
2
[Turn over
General Marking Advice (i) If the correct answer is seen in the body of the script and the answer given in the answer line is clearly a transcription error, full marks should be awarded.
(ii) If the answer is missing, but the correct answer is seen in the body of the script, full marks should be awarded. (iii) If the correct answer is seen in working but a completely different answer is seen in the answer space, then some marks will be awarded depending on the severity of the error. (iv) Work crossed out but not replaced should be marked. (v) In general, if two or more methods are offered, mark only the method that leads to the answer on the answer line. If two (or more) answers are offered (with no solution offered on the answer line), mark the poorest answer. (vi) For methods not provided for in the mark scheme, give as far as possible equivalent marks for equivalent work. (vii) Where a follow through mark is indicated on the mark scheme for a particular part question, the marker must ensure that you refer back to the answer of the previous part of the question. (viii)Unless the question asks for an answer to a specific degree of accuracy, always mark at the greatest number of significant figures seen. E.g. the answer in the mark scheme is 4.65 and the candidate then correctly rounds to 4.7 or 5 on the answer line. Allow full marks for 4.65 seen in the working. (ix) Anything in the mark scheme which is in brackets (...) is not required for the mark to be earned, but if …show more content…
present it must be correct. (x) For any question, the range of answers given in the mark scheme is inclusive. Quality of written communication In GCSE Mathematics, particular questions are identified where candidates must demonstrate the quality of their written communication. In particular, candidates must: (i) ensure that text is legible and that spelling, punctuation and grammar are accurate so that meaning is clear (i.e. comprehension and meaning is clear by using the correct notation and labelling conventions); (ii) select and use a form and style of writing appropriate to their purpose and to complex subject matter (i.e. reasoning, explanation or argument is correct and appropriately structured to convey mathematical reasoning); and (iii) organise information clearly and coherently, using specialist vocabulary where appropriate (i.e. the mathematical methods and processes used are coherently and clearly organised and appropriate mathematical vocabulary used). This assessment may be through, for example, an explanation of the geometrical properties of a given shape or, for example, through concise mathematical argument in a multi-step problem.
7280.01
3
[Turn over
1
(a) LCM of 8 and 10 = 40 10th October (b) 135.66 = 120% 135.66 # 20 or 113.05 120 = £22.61
MA1 A1 MA1 MA1 A1 A1 A3 A1 A1 A1 A1, A1, A1
AVAILABLE MARKS
5
2
(a) 23, 43, 52, 55 (b) Correctly drawn graph (c) Correct reading (d) Identifying 37 55 – 37 = 18 (e) box plot median, quartiles, limits of range
10
3
(a) 3x = 9 x=3 y = –2 (b) 4x + 1 + 4x - 6 = 21 12 12 4x + 1 + 4x - 6 = 21 8x = 26 x = 26 8 or 13 4 or 3.25
MA1 MA1 MA1 MA1 MA1 MA1 MA1 MA1 A1 MA1 MA1 A1 MA1 MA1, MA1 3 6 6
4
(a) tan 48° = RS 13 RS = 13 tan 48° RS = 14.4 (14.438) (b) PR = :(5 + 13 ) = :194 (13.9) RPS = tan–1 (13 tan 48/:194) RPS = 46.03 (tan (14.4/13.9) = 46.01)
-1 2 2
5
(a) 5 2 (b) y = 5 x + 2 2
7280.01
4
[Turn over
6
(a) Angle EBC = 90° (b) Angle ABE = 32° (c) Angle BAE = 116° (d) Angle BCE = 64° (e) Angle BED = 46°
A1 A1 A1 A1 A1
AVAILABLE MARKS
5
7
(a) e.g. Only sampling Year 8 pupils (b) e.g. Pupils at end of list miss out (c) (i) 57 # 50 600 4.75 → 5 (ii) No, because 82 # 50 = 6.83 . . . → 7 600 and 41 # 50 = 3.416 . . . " 3 600 or using 1 12
A1 A1 MA1 A1
MA2
6
8
(a) 3 (p - 25q ) 3 (p + 5q) (p - 5q) (b) (x - 8) (x + 3) = 0 8 or –3
2
2
MA1 MA2 MA2 MA1 6
9
(a) –1 3
1 y =- 3 x + 4
MA1 MA1
2
(b) 12p - 21pq + 8pq - 14q 12p - 13pq - 14q
2 2
2
MA2 MA1 5
7280.01
5
[Turn over
10 (a) y = k2 x k 4= 9 k = 36 y = 36 2 x (b) y = 36 144 =1 4 (c) When x is doubled the denominator becomes 4 times bigger therefore y becomes 4 times smaller 11 (a) Area = 1 80 # 264 sin 44 2 = 7335.6 m2 (b) Let M = midpoint AM = 80 + 132 - 2 # 80 # 132 cos 44 AM = :8631.543417 AM = 92.9 12 (a) x + (x - 5) = 144 x + x - 10x + 25 = 144 2x - 10x - 119 = 0 (b) x = 10 !
1052 4 x = 10.6 13 (a) 70 and 20 (b) Two bars drawn to correct heights (6.8 and 9.6) (c) 7.5 + 52.5 60 (d) x # 20 or 144 = 3.6, 3.6 # 20 = 72 60 40 x h = x # 20 = x 3 60 3
6
2 2 2 2 2 2 2 2
AVAILABLE MARKS
MA1 A1 MA1
A2 MA1
5
A1, A1 units
MA1 MA1 A1 MA1 A1 MA1 MA2 MA1 A1, A1 A1, A1 MA1 A1 MA1 A1 8 6 6
7280.01
[Turn over
4 -3 3 14 a 8 k = a 27 k = d3 27 n = a 3 k = 81 27 8 8 2 16 -2 2 -2 0.25 = a 1 k = 4 = 16 4 81 # 16 = 81 16
4 4
4
MA2 MA1 A1 A1 MA1 MA1 A1 MA1 MA1 MA1 MA1 MA1
AVAILABLE MARKS
4
15 (a) Since triangle ANB is isosceles (180 – 120)/2 = 30 (b) AN = 25 sin 30 sin 120 AN = 25 sin 30 sin 120 12.5 AN 12.5 AN = cos 30° AN = 14.4 (14.43376) cos 30° =
or
AN = 14.4 (14.43376)
(c) Let x = length of OA and OB 2 2 2x = 25 2 x = : (25 /2) = 17.68 16 x + 2qx + q 2q = -8
2 2 2
6
q = -4
p = (-4) = 16
3
17
2 (3x - 2) + 6 (x + 1) =1 (x + 1) (3x - 2) 6x - 4 + 6x + 6 = 3x - 2x + 3x - 2 3x - 11x - 4 = 0 (3x + 1) (x - 4) = 0 x =1 3 or 4
2 2
M1, A1 MA1 MA1 MA2 MA1 7
18 C y x y G y z x F Total
7280.01
D y
CGD = 180 – (x + y) x + y + y + z = 180 (cyclic quad)
z E
y + z = 180 – (x + y) CGD = FED A3 3 100
7
[Turn over
7280.01
8
[Turn over
7280.01
9
[Turn over