13 Maths-Mark Scheme Dulwich College
1. (a) Write down the first 10 digits shown on your calculator.
4.885216166
4.885
(b) Write your answer to (a) rounded to 3 decimal places.
4.9
(c) Write your answer to (a) rounded to 2 significant figures
2. (a) A car was bought on January 1st 2010 for £16,000. By January 1st 2011, its value had fallen by 20%.
(i) Calculate the value of the car on January pt 2011.
0.8 x 16000
= 12800
(ii) In each subsequent year, the value of the car fell by 15%. Calculate its value on January 1st 2013.
12800 x 0.85
2
= 9248
(b) A rare postage stamp was bought for £300 in 1980. Its value at that time was just 12% of its current value. Calculate the current value of the stamp.
0.12x = 300
x = 300
0.12
= 2500
1 mark
1 mark
1 mark
1 mark
2 marks
2 marks
3. Below is a sequence of numbers:
5, 8, 11, 14,....
(a) Calculate the 10th term.
32
(b) Calculate the 75th term.
3n + 2 (nth term)
3(75) + 2
= 227
( c) David saves 5p on Sunday 1st December, 8p on Monday 2nd December, 11p on Tuesday 3rd December, and so on according to the sequence above. Calculate the day and date on which David will save 77p.
3n + 2 = 77
n = 25
Day: Wednesday
Date: 25th Dec
1 mark
2 marks
3 marks
4 (a) Adam and Brian share £544 in the ratio 7 : 9. How much does Brian receive?
9
16
x 544
A1
M1
Answer:
306
2 marks
(b) Last month a local shop found that its total revenue from selling calculators and pens could be expressed in the ratio c : p. That total revenue was £150, with the greater portion coming from calculator sales. In terms of c and p, what is the difference between the revenues for calculators and pens?
c . 150 - p .150 = (c - p) x 150
c+p c+p c+p
Answer:
150 (c - p)
c+p
A1, or equivalent answer
2 marks
M1 For sensible attempt at finding appropriate fractions of the total
(c) Given that a : b = 5 : 32 and b : c = 24 : 31, find the ratio a : b : c, giving
a : b = 5: 32 = 5 : 1
32
b : c = 24: 31 = 1: 31
24
M1 For sensible attempt at finding appropriate fractions of the total
a: b : c = 5 : 1 : 31 = 15 : 96 : 124
32 24
Answer:
15: 96 :124
A1
3 marks
A1
5. Simplify the following:
(a) 8ab - 11a + 14b + 12a - 5ba - b
Answer:
3ab + a + 13b
A2
A1 if exactly two terms correct
(b) 7(2 - 4x)
Answer: 14 - 28x
A1
(c) 6t - 4 (2t - 5) + 8
Answer: 28 - 2t
A1
6t - 8t + 20 + 8
A1
(d) (y - 3) (2y + 9)
Answer:
A1
M1 at least three terms correct
2y + 9y - 6y - 27
2
2y + 3y - 27
2
(e) 36a b
3 2
24a bc
4
Answer:
3b
29c
A1 correct 3/2
A1 correct b/ac
2 marks
1 mark
2 marks
2 marks
2 marks
6. The diagram, which is not drawn to scale, shows a shape with one line of symmetry.
Calculate the area of this shape.
(12 x 15) - (1/2 x 12 x 4) + ( 1/2 x 12 x 10)
Answer: 216
A1
3 marks
M1 for correct splitting of the figure
M1 for at least two correct areas calculated
7. Calculate the length of the side marked x, giving your answer correct to 2 decimal places.
M1
23 = x + 14
x = 23 - 14 = 333
x = \333
2
2
2
2
2
2
Answer: 18.25
A1
8. (a) On the axes below, draw the line given by the equation y=2 - x
(b) Reflect shape A in the line y = 2 -x and label that reflection B.
(c) Rotate shape A through 90 degrees in the clockwise direction about the point (4,1) and label the resulting shape C.
2 marks
2 marks
2 marks
2 marks
2 marks
9. Solve the following equations for x:
(a) 4(x + 5) = 2x + 10
2 marks
4x + 20 = 2x + 10
2x = -10
M1
M1
Answer: x = - 5
A1
(b) 2x(3x + 7) - 12 = x(6x + 10)
3 marks
6x + 14x - 12 = 6x + 10x
14x - 12 = 10x
4x = 12
M1
M1
2
Answer: x = 3
A1
(C) 8 - 5x = 2
8
2 marks
6 = 5x
3
5x = 18
M1 Either equation
Answer: x = 18
5
A1
(d) 16 + 7 = 11
2x -1
16 = 4 => 16 = 4(2x - 1)
2x -1
M1
=> 4 = 2x - 1
M1
=> 2x = 5
Answer: x = 5
2
A1
3 marks
10. Find the values of a, band c in the diagrams below.
(a)
M1 for identifying either 40 degrees or 106 degrees in diagram
Answer a: 106
A1
(b) The diagram below shows two identical squares meeting at one of their comers.
360 - 138 - 90 - 90 = 42
180 - 42
2
M1
Answer b: 69
A1
(c)
2 marks
2 marks
3 marks
11. Factorise the following expression fully: 9c + 3c d - 12c d
3
2
[ AO for 3(3c + 3c d - 4c d) ]
3
2
Answer: 3c (3 + 3c d - 4cd)
2
A1
A1
12. While taking part in a 10 km race, a runner completed the first 6500 min 26 minutes.
(a) Calculate the average speed of the runner, in km per hour, over this section of the course.
S = D = 6.5
T 26/60
M1
Answer: 15 km/hr
A1
The runner's target was to complete the entire race is under 40 minutes. For the remaining 3500 m his average speed was 16 km per hour.
(b) Show your working and conclusion clearly, determine whether the runner was successful in achieving his target.
2 marks
2 marks
4 marks
T = D = 3.5 = 7 hours
S 32 32
M1
7 x 60 = 13.125 minutes
32
M1
26 + 13.125 = 39.125 minutes
39.125 < 40
A1
Answer: Successful Not Successful
A1 Only if follows from clear (sensible) working.
13. The volume of the prism below is 2450 cm3
Calculate the length marked x in the diagram.
M1
A1
Area of cross section = 2450 = 98cm2
25
31 + 12x = 98
2x = 66
M1
x = 66
12
Answer: 5.5cm
A1
4 marks
14. 1 gallon = 3.785 litres
1 cubic inch= 0.0164 litres
Convert 2.5 gallons to cubic inches, giving your answer to 2 decimal places.
3 marks
M1 2.5 gallons = (2.5 x 3.785 = ) 9.4625 litres
M1 9.4625 litres = (9.4625 = ) 576.9817 cubic inches
0.0164
Answer: 576.98 cubic inches
A1
15. The diagram below, which is not drawn to scale, shows three sides of a regular polygon with n sides. Work out the value of n.
9x + 34 + 2x - 8 = 180
M1
11x + 26 = 180
11x = 154
x = 4
so 2x - 8 = 2(24) - 8
= 20 (ext angle) A1
no. sides = 360
20
Answer: 18
A1
16. The mean of 8 numbers ism. When one of these numbers is discarded, the mean of the remaining 7 numbers falls tom - 4. In terms of m, what was the value of the discarded number?
x, + x2 + ...... + x8 = 8
x, + x2 + ...... + x 7 = 7(m - 4)
M1 for attempt to find the two relevant sums
8m - 7(m - 4)
8m - 7 + 28
Answer: m + 28
A1
17. A palindromic number is a number that remains the same when its digits are reversed. Examples are 27572 and 5826285.
If S is the set of all whole numbers greater than 100 and less than 301, and one whole number is randomly selected from S:
(a) what is the probability that the selected number is palindromic?
3 marks
3 marks
2 marks
number of elements is S = 200
number of palindromic numbers = 20
M1 for either
Answer: 20/200 or 1/10
A1
(b) what is the probability that the selected number is palindromic or even?
Answer: 11/20
A1
(c) If this process of selection was repeated 500 times, how many times would
you expect to select a palindromic number?
1/10 x 50
Answer: 50
A1
2 marks
1 mark
18. Given that a, b, c and dare points on the number line such that
• a> b
• b is halfway between a and c
• the distance between a and d is three times the distance between a and c
• d < c,
calculate the value of c - b / b - d
Answer: -1/5
A1
19. Jack spends half of his money and gives one fifth of what remains to his friend. Jack is then left with £24. How much money did he start with?
3 marks
2 marks
4 x 1 x X = 24
5 2
M1
2 x = 24
5
Answer: £60
A1
4
7
3
8
*
13
21
4
5
+
*
+
3
4
20. (a) Evaluate 9
16
*
M1
12
21
+
9
16
+
6
16
13
21
*
16
20
+
*
15
20
3
16
+ 0 +
1
20
Answer: 19/80
A1
(b) Given that t < 0, simplify 3t * 8t + 10t * 7t - 2t * 3t.
Answer: -4t
A1
-5t + 0 - ( -t )
M1
21. The diagram below shows five identical circles, each with radius r cm. A square is formed with its vertices (comers) at the centres of the four outer circles and the inner circle just touches each of the four outer circles. The unshaded area inside the square is (72 - l 81t) cm2.
Showing clear working, calculate the radius r of each circle.
Answer: 3cm
A1
22.When written out in full, how many digits are there in the number 16 x 25 ?
250
500
16 x 25
250
500
= (2 )
250
500
4
+ (5 )
2
M1 for attempt to express in terms of power of 2 and 5
= 2
1000
1000
x 5
= 10
1000
M1 for arriving at power of 10
Answer: 1001
A1
2 marks
2 marks
4 marks
3 marks
GCSE examinations are marked in such a way as to award positive achievement wherever possible. Thus, for GCSE Mathematics papers, marks are awarded under various categories.
If a student uses a method which is not explicitly covered by the mark scheme the same principles of marking should be applied. Credit should be given to any valid methods. Examiners should seek advice from their senior examiner if in any doubt.
M - Method marks are awarded for a correct method which could lead to a correct answer.
A - Accuracy marks are awarded when following on from a correct method. It is not necessary to always see the method. This can be implied.
B - Marks awarded independent of method.
ft - Follow through marks. Marks awarded for correct working following a mistake in an earlier step.
SC - Special case. Marks awarded within the scheme for a common misinterpretation which has some mathematical worth.
Mdep - A method mark dependent on a previous method mark being awarded.
Bdep - A mark that can only be awarded if a previous independent mark has been awarded.
oe - Or equivalent. Accept answers that are equivalent.
eg accept 0.5 as well as 21
[a, b] - Accept values between a and b inclusive.
3.14… - Accept answers which begin 3.14 eg 3.14, 3.142, 3.1416
Use of brackets - It is not necessary to see the bracketed work to award the marks
Examiners should consistently apply the following principles
Diagrams - Diagrams that have working on them should be treated like normal responses. If a diagram has been written on but the correct response is within the answer space, the work within the answer space should be marked. Working on diagrams that contradicts work within the answer space is not to be considered as choice but as working, and is not, therefore, penalised.
Responses which appear to come from incorrect methods
Whenever there is doubt as to whether a student has used an incorrect method to obtain an answer, as a general principle, the benefit of doubt must be given to the student. In cases where there is no doubt that the answer has come from incorrect working then the student should be penalised.
Questions which ask students to show working
Instructions on marking will be given but usually marks are not awarded to students who show no working.
Questions which do not ask students to show working
As a general principle, a correct response is awarded full marks.
Misread or miscopy
Students often copy values from a question incorrectly. If the examiner thinks that the student has made a genuine misread, then only the accuracy marks (A or B marks), up to a maximum of 2 marks are penalised. The method marks can still be awarded.
Further work
Once the correct answer has been seen, further working may be ignored unless it goes on to contradict the correct answer.
Choice
When a choice of answers and/or methods is given, mark each attempt. If both methods are valid then M marks can be awarded but any incorrect answer or method would result in marks being lost.
Work not replaced
Erased or crossed out work that is still legible should be marked.
Work replaced
Erased or crossed out work that has been replaced is not awarded marks.
Premature approximation
Rounding off too early can lead to inaccuracy in the final answer. This should be penalised by 1 mark unless instructed otherwise.