Exam 1
April 13, 2023 | Author: Anonymous | Category: N/A
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Time:
91 minutes
Marks:
84 marks
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Page 1 of 36
1.
y = cos x°
Here is a sketch of
for values of x from 0 to 360 Not drawn accurately
One solution of
cos x° = k
is
x = α
where
0 < α < 90
Circle the other solution for values of x from 0 to 360
x = 90 + α
x = 360 – α
x = 270 + α
x = 180 – α
(Total 1 mark)
2.
Circle the order of rotational r otational symmetry of this parallelogram.
0
1
2
4 (Total 1 mark)
3.
b is five more than a. Circle the correct equation.
b = 5a
a = 5b
a=b+5
b=a+5 (Total 1 mark)
4.
Circle the equation that has the solution x = 2
4 x = 2
2 – x = 2
x – 2 = 0 (Total 1 mark)
Page 2 of 36
5.
Circle the prime number. 21
23
25
27 (Total 1 mark)
6.
200 grams of grapes cost $1.06 Work out the cost of 500 grams of grapes.
Answer $ (Total 2 marks)
7.
Here is an isosceles triangle. Not drawn accurately
Work out the size of angle x.
Answer
° (Total 2 marks)
Page 3 of 36
8.
Work out the value of
3a + 5b
when
a=9
and
b = –2
Answer (Total 2 marks)
9.
Here is some information about the cost of theatre tickets. Tickets
Adult $35.50 Junior $27.00 Senior $30.75 Special Offer
1 junior ticket free with every 2 adult tickets
Sara wants 6 adult tickets 5 junior tickets and 1 senior ticket. Work out the total cost of her tickets.
Answer $ (Total 4 marks)
Page 4 of 36
10.
On the grid, draw a square with the same area as the given shape.
(Total 2 marks)
11.
At 09:00 the temperature in a new freezer iis s 22°C It decreases at 4°C per hour. At what time will it reach –18°C?
Answer (Total 2 marks)
12.
Zak puts these letter tiles in a bag.
He takes one tile at random.
Page 5 of 36
(a)
Circle the chance that he takes a letter A. Impossible
Unlikely
Evens
Likely
Certain (1)
(b)
Write down the probability that he takes a letter T or a letter C. Give your answer as a fraction.
Answer (1)
(c)
Zak adds some more letter tiles to the bag. He takes one tile at random. The probability that he takes a letter M is Work out the probability that he does not take a letter M.
Answer (1) (Total 3 marks)
13.
A game has two two spinners, A and B.
A player spins both spinners. The score is number on Spinner A – number on Spinner B
Page 6 of 36
(a)
Complete the table to show the possible scores. Spinner B 1
1
2
2
3
3
–1
1 2 Spinner A
0
3
3
4
(2)
(b)
Work out the fraction of the scores that are less than 0
Answer (1) (Total 3 marks)
14.
The lengths of fish in two groups are shown in the stem-and-leaf diagrams. Key: 1 | 3 represents 13 cm Group A
0 1 2 3 4
7 0 0 3
5
6
8 2 3 3
Group B
4 8 4
9 4
0 1 2 3 4
4
9 3 5 0 5
4 5 0
4 7 0
8 1
9 2
9
5
Complete these statements. In total, the number of fish longer than 30 cm was In Group A, the longest fish was
cm
In Group B, the median length was
cm (Total 3 marks)
15.
A tiling pattern is made from 48 octagons and 54 squares. One quarter of the octagons are blue. One third of all the shapes are blue.
Page 7 of 36
How many blue squares are there?
Answer (Total 3 marks)
16.
(a)
25 beads are red or yellow. There are three more red beads than yellow beads. Work out the number of red beads.
Answer (1)
(b)
36 beads are black or white. There are twice as many black beads as white beads. Work out the number of black beads.
Answer (2) (Total 3 marks)
17.
Here are six number cards. One number is missing.
Page 8 of 36
The mean of the six numbers is 24 Work out the missing number.
Answer (Total 2 marks)
18.
(a)
The diagram shows shapes P and Q.
Describe fully the single transformation that maps shape P to shape Q.
(2)
Page 9 of 36
(b)
The diagram shows shapes P and R.
Describe fully the single transformation that maps shape P to shape R.
(3) (Total 5 marks)
19.
Tea bags are sold in boxes of 80 and boxes bo xes of 60
Page 10 of 36
Which box is better value? You must show your working.
Answer (Total 3 marks)
The equation of a line is y = 2 – 5 x
20.
(a)
A point on the line has a y-coordinate of –9 Work out the x-coordinate of the point.
Answer (3)
(b)
Is (1, 3) a point on the line? Tick a box.
Yes
No
Give a reason for your answer.
(1) (Total 4 marks)
Page 11 of 36
21.
Here is a right-angled triangle. Not drawn accurately
Four of these triangles are arranged to make the shaded shape. Not drawn accurately
(a)
Work out the area of the shaded shape.
Answer
cm2 (2)
(b)
Work out the perimeter of the shaded shape.
Answer
cm (2) (Total 4 marks)
Page 12 of 36
22.
A car travels 56 kilometres in in 1 hour 45 minutes. Circle the average speed of the car in kilometres per hour. 32
38.6
81.2
98 (Total 1 mark)
23.
Simplify Circle your answer.
t –0.4
t –3
t 7
t 0.4 (Total 1 mark)
of the students in a primary school are boys.
24.
Write the ratio
number of boys : number of girls
Answer
in its simplest form.
: (Total 1 mark)
25.
In the Venn diagram, ξ represents the students in a college
B represents the percentage of students who study Biology C represents the percentage of students who study Chemistry.
One student is chosen at random. (a)
Work out the probability that the student studies exactly one of Biology and Chemistry.
Answer (1)
Page 13 of 36
(b)
Work out the probability that the student does not study Biology or Chemistry.
Answer (2) (Total 3 marks)
26.
Use approximations to 1 significant figure to estimate the value of
You must show your working.
Answer (Total 3 marks)
27.
Factorise fully
8 y3 – 12 y
Answer (Total 2 marks)
Page 14 of 36
28.
(a)
Write
as a single fraction.
Give your answer in its simplest form.
Answer (2)
(b)
Simplify Circle your answer.
(1) (Total 3 marks)
29.
Adam decides to have a bath. At 07:40 he turns the taps on. At 07:50 he turns the taps off and gets into into the bath. He stays in the bath for 15 minutes.
Page 15 of 36
The bath then takes 5 minutes to empty at a steady rate.
(a)
Complete the graph. (2)
(b)
1 litre = 1000 cm3 Work out the rate at which the water pours into the bath. Give your answer in cubic centimetres per second.
Answer
cm3/s (3) (Total 5 marks)
30.
Use a ruler and compasses for this question. The map shows airports J, K and L. Rescue planes patrol the sea. Planes from J patrol the part of the sea closer to J than to K. Planes from L patrol the part of the sea less than 400 km from L. Planes from K patrol the rest of the sea.
Page 16 of 36
On the map, show the part of the sea patrolled by planes from K. Scale: 1 cm represents 100 km
(Total 4 marks)
31.
2 coffees and 2 juices cost $9.60 4 coffees and 1 juice cost $15.00
Page 17 of 36
Work out the cost of 1 coffee and the cost of 1 juice.
Coffee $ Juice $ (Total 4 marks)
32.
Use trigonometry to work out the size of angle x. Not drawn accurately
Page 18 of 36
Answer
° (Total 2 marks)
33.
Work out 0.9 as a fraction of 0.24 Circle your answer.
(Total 1 mark)
34.
The vector
translates A to B.
Circle the vector that translates B to A.
(Total 1 mark)
35.
Simplify Circle your answer. 56
58
510
516 (Total 1 mark)
Page 19 of 36
Mark schemes 1.
x = 360 − a B1
[1]
2.
2 B1
[1]
3.
b=a+5 B1
[1]
4.
x − 2 = 0 B1
[1]
5.
23 B1
[1]
6.
Correct method for cost of any factor or multiple of 500 e.g. 1.06 ÷ 200 or (1 g =) 0.0053 or 1.06 ÷ 2 or (100 g =) 0.53 or 1.06 ÷ 4 or (50 g =) 0.265 or 1.06 ÷ 10 or (20 g =) 0.106 or 1.06 ÷ 20 or (10 g =) 0.053 or 1.06 ÷ 100 or (2 g =) 0.0106 or 1.06 × 5 or (1000 g =) 5.3(0) or 1.06 × 2.5 oe M1
2.65 A1
Additional Guidance
1.06 × 200 ÷ 500 Zero
[2]
Page 20 of 36
7.
(180 − 30) ÷ 2 or 90 − 15 or 75 may be seen on diagram M1
285 A1
[2]
8.
3 × 9 (+) 5 × −2 or 27 or −10 M1
17 A1
Additional Guidance
3 × 9 = 27a and 5 × −2 = −10b M1A0
27a and −10b Zero
[2]
9.
6 × 35.5(0) or 213 M1
267 implies M2 M1
their 213 + their 54 + 30.75 dep on M1M0 or M0M1 M1dep
297.75 A1
Additional Guidance
213 + 135 + 30.75 M1M0M1
378.75 A0
[4]
Page 21 of 36
10.
5 by 5 square B1 25 seen or different shape with area 25 squares drawn B2
[2]
11.
(22 − −18) ÷ 4 or 10 (hours) oe M1
19:00 or 7 pm A1
Additional Guidance
Answer of 7(am) M1A0
Build down must be fully correct to gain marks [2]
12.
(a)
Unlikely B1
(b) oe fraction B1
(c)
or 0.86(66…) or 0.87 or 86.(66…)% or 87% B1
[3]
13.
(a) 0
0
−1
−1
−2
−2
1
1
0
0
−1
−1
2
2
1
1
0
0
3
3
2
2
1
1
B1 one correct row or column B2
(b) oe fraction ft their completed table B1ft
Page 22 of 36
Additional Guidance
Ignore incorrect cancelling once correct fraction seen [3]
14.
12 B1
56 5|6 is B0 B1
28.5 2|85 is B0 B1
[3]
15.
48 ÷ 4 or 12 M1
(48 + 54) ÷ 3 or 102 ÷ 3 or 34 oe M1
22 A1
[3]
16.
(a)
14 B1
(b)
36 ÷ 3 or 12 oe M1
24 A1
[3]
17.
24 × 6 or 144 or
M1
39 A1
[2]
Page 23 of 36
18.
(a)
Reflection B1
in y-axis oe B1
Additional Guidance
Do not accept flip for reflection B0
Condone line y for y-axis 2nd B1
Any other transformation transformation mentioned or implied such such as rotation rotation or translation loses the mark for reflection e.g. moved left 4 and reflected 1st B0
(b)
Rotation B1
90° anticlockwise or 270° clockwise
oe B1
Origin or (0, 0) or O oe
B1
Additional Guidance
Do not accept turn for rotation B0
Any other transformation transformation mentioned or implied such such as reflection reflection or translation loses the mark for rotation e.g. moved left 4 and rotated 1st B0
[5]
Page 24 of 36
19.
Alternative method 1
Cost per bag for one box e.g. 1.45 ÷ 80 or 0.018… M1
Cost per bag for both boxes e.g. 1.45 ÷ 80 or 0.018… and 1.05 ÷ 60 or 0.017(5) M1dep
0.018… and 0.17(5) and Box B oe A1
Alternative method 2
Bags per $ for one box e.g. 80 ÷ 1.45 or 55.(…) M1
Bags per $ for both boxes e.g. 80 ÷ 1.45 or 55.(…) and 60 ÷ 1.05 or 57.(…) M1
55.(…) and 57.(…) and Box A1
Alternative method 3
method for cost of 60 As or 80 Bs M1
1.08(75) or 1.4(0) A1
1.08(75) (and 1.05) and Box B or 1.4(0) (and 1.45) and Box B A1
Alternative method 4
Method for scaling up or down both prices to consistent numbers of tea bags e.g. for 20 tea bags, 1.45 ÷ 4 and 1.05 ÷ 3 M1
Page 25 of 36
Correct prices for consistent numbers of tea bags e.g. 0.36(25) and 0.35 A1
Correct prices for consistent numbers of tea bags and Box B e.g. 0.36(25) and 0.35 and Box B A1
Additional Guidance
Ignore any units stated Correct values seen alongside other incorrect working max 2 marks
Correct comparable values ($) for method 4: Bags
2
4
5
8
10
20
240
480
A
0.03625
0.0725
0.090625
0.145
0.18125
0.3625
4.35
8.7
B
0.035
0.07
0.0875
0.14
0.175
0.35
4.2
8.4 [3]
20.
(a)
Alternative method 1 −9 = 2 − 5 x M1
5 x = 2 − −9 or −9 − 2 = −5 x or 5 x = 11 or −11 = −5 x M1dep
A1
Alternative method 2 −5 x = y − 2 oe M1
M1dep
A1
Page 26 of 36
Additional Guidance
Accept answer in the form (2.2, −9) M2A1
(b)
No ticked with any valid reason given No and e.g. when x = 1, y is negative or when x = 1, y = −3 or when y = 3, x is negative or when y = 3, x = −0.2 or (1, 3) is above the line oe B1
Additional Guidance
3 ≠ −3 oe and No ticked B1
1 × −5 + 2 = −3 and No ticked B1
[4]
21.
(a)
× 9 × 12 or 54 M1
216 A1
(b)
12 – 9 or 3 may be implied by 18 M1
72 A1
Additional Guidance
3 seen on diagram M1
[4]
22.
32 B1
[1]
23.
t −3 B1
[1]
Page 27 of 36
24.
3:2
B1
[1]
25.
(a)
51% or 0.51 or B1
Additional Guidance
Answer 51 B0
(b)
100 − (36 + 28 + 15) or 21 or 36% + 28% + 15% or 79% oe M1
A1
Additional Guidance
21 seen in correct region on diagram M1
M1A0
[3]
26.
3 or 400 or 0.5 M1
or (33 =) 27 and ( or (33 −
=) 20
=) 7 M1
14 with correct working 3 or 27 and 400 or 20 and 0.5 must be seen A1
Page 28 of 36
Additional Guidance
Condone e.g. 3.00 or 0.500 Correct working and 14 seen but with answer 10 M1M1A1
Do not allow misread of cubed as squared [3]
27.
4 y(2 y2 − 3) or −4 y(3 − 2 y2) y(4 y2 − 6) y3 − 3 y ) or 2 B1 4(2
y2 − 12) or 2(4 y3 − 6 y ) or y(8 y(6 − 4 y2 ) y3 ) or −2 or −4(3 y − 2 y − 4 y3 ) y2 ) or −2(6 or − y(12 − 8 B2
Additional Guidance
B2 or B1 answer followed by further work
B1
B1
[2]
28.
(a) common denominator with one numerator correct M1
A1
(b) B1
[3]
29.
(a)
Horizontal line from (07:50, 70) to (08:05, 70) B1
Straight line joining (08:05, 70) to (08:10, 0) ft their (08:05, 70) joined to horizontal axis 5 minutes later with a straight line B1ft
Page 29 of 36
(b)
Alternative method 1
70 × 1000 or 70 000 or (50 − 40) × 60 or 600 M1
70 000 ÷ 600 condone 7000 ÷ 600 or 70 000 ÷ 60
M1dep
A1
Alternative method 2
70 ÷ (50 − 40) or 70 ÷ 10 or 7 (litres per minute) M1
their 7 × 1000 ÷ 60 or 7000 ÷ 60 M1dep
A1
[5]
Pair of intersecting arcs, equal radii > half JK, above and below JK
30.
M1
Perpendicular bisector of JK drawn with correct construction A1
Arc, centre L, L, radius [3.8, 4.2] cm B1
Correct area of sea identified ft their perpendicular bisector and their arc B1ft
[4]
Page 30 of 36
31.
Alternative method 1
4c + 4 j = 19.2 and 4c + j = 15 2c + 2 j = 9.6 and 8c + 2 j = 30 or c + j = 4.8 and 4c + j = 15 oe e.g. works in cents multiplies one or both equation(s) to equate coefficients allow an error in 19.2 or 30 or 4.8 M1
3 j = 4.2 6c = 20.4 or 3c = 10.2 oe correctly subtracts their equations to eliminate one variable M1dep
(coffee =) 3.4(0) and (juice =) 1.4(0) A1 one one correct correct from correct method A2
Alternative method 2
oe makes c or j the subject M1
oe e.g. 2 c + 2(15 – 4c ) = 9.6 correctly substitutes the expression to eliminate one variable M1dep
(coffee =) 3.4(0) and (juice =) 1.4(0) A1 one one correct correct from correct method A2
Additional Guidance
Accept any letter, or coffee and juice as variables Answer only (coffee =) 3.4(0) 3.4(0) and (juice =) 1.4(0) 4 marks
Page 31 of 36
4c + 4 j = 38.4 and 4c + j = 15 M1
3 j = 23.4 M1dep
juice = $7.80, coffee = $1.80 A0
[4]
32.
tan x =
or tan−1
or 90 − 34.(…) oe
any letter M1
55.6… or 56 A1
Additional Guidance
tan =
or tan
or tan−1 =
(unless recovered) M0
Answer from scale scale drawing Zero
Page 32 of 36
If using sine rule must rearrange to sin x = for M1 If using cosine rule must rearrange to cos x = for M1 Allow [1.46, 1.462] for
Allow [0.68, 0.685] for
[2]
33. B1
[1]
34. B1
[1]
35.
510 B1
[1]
Examiner reports 2. 3. 4. 5. 6.
7. 8.
9.
10. 11.
12. 13. 14. 15.
16. 17. 18.
This question was well answered. Common incorrect choices were 0 and 1. This question was well answered. The common incorrect choice was a = b + 5. This question was well answered. The common incorrect choice was 4 x = 2. This question was very well answered. This question was well answered. Students who worked out the cost of 100 grams or 1 gram were usually more successful than those who tried to work out the cost of 400 grams + 100 grams. The common incorrect answer was 75°. A few students subtracted 75° from 180°. This question was very well answered. Some students made errors substituting the negative value. Occasionally students substituted but also left the variables in the expression giving an answer of the form 27 a – 10b. This question discriminated well between more able and less able students. Frequently students did not use the special offer so included the full cost of 5 junior tickets. Some students thought that the senior ticket was free. Most students did not show the area of the given shape even though thou gh there was evidence that they had counted squares. However, many went on to draw a square with the correct area. Very few students worked out that the temperature had to decrease by 40° C. Most students used a build-down method and often made an arithmetic error or became muddled as the time increased and the temperature decreased. The common incorrect answer was 07:00 or 7 am. All parts of this question were very well answered, particularly part (c). Both parts of this question were very well answered. This question was less well answered. Many wrote Group A to complete the first statement. For the second statement students appeared to be looking at row length. Those who knew how to work out the median for the third statement usually did not take account of the key. Most students were able to access this problem-solving question but fewer than half were able to give a fully correct solution. Some students worked out 12 and 34 but did not use them to reach their answer. Many students worked out 12 but then found a third of the squares rather than all the shapes. Far more students were successful in part (a) than in part (b). ( b). In part (a) the common error was to halve 25 and then add 3. In part (b) common wrong answers were 18 and 27. This question was very well answered. Many Ma ny students used a successful trial approach. In part (a) most students recognised it was a reflection but many also mentioned a translation and few gave the mirror line. In part (b) again students often referred to a second transformation, usually a translation, not realising that a single transformation was required. Some students did state that it was 90° anticlockwise but it was rare to see a reference to the centre of the rotation.
Page 33 of 36
Page 34 of 36
19.
20.
21.
22. 23. 24. 25.
This question discriminated well. Students who worked out the increase in price for 20 more tea bags usually made no progress. A few students multiplied the number of tea bags by the price. Those who did use a correct method sometimes chose the wrong box for their answer. In part (a) students who substituted and then rearranged were generally more successful than those who used the other method. Many resorted to trial and error. Part (b) was badly answered. Some students stated that all values had to be negative. There were quite a few instances of students using the perimeter per imeter in part (a) and the area in part (b). A common error of those who used area in (a) was to forget to divide by 2. In part (b) for those using the perimeter the common errors were to include the dashed internal lines or to ignore the 3cm lines. This question was well answered. The common incorrect choice was 38.6. This question was very well answered. The common incorrect choice was t 7. This was less well answered. Many answers were unsimplified, for example, given as decimals or fractions. Common wrong answers included 3:5 and 2:3. Neither part of this question was well answered. In part (a) some students added all the given values. The most common wrong answer was
. Students who knew which values were
needed did not always give their answer as a probability. In part (b) the common wrong answer was
.
26.
The vast majority of students did not use approximations. It was common to see students only have one correct value and that was often 0.5.
27.
Some responses were only partially factorised but many students did d id give a correct answer. Less able students tried to combine the two terms resulting in answers such as –4 y2.
28.
In part (a) some students added the numerators and added the denominators. Part (b) was better answered. The common wrong answer was
29.
.
Part (a) was well answered with the vast majority managing to draw at least the first part of the graph. Part (b) was not done so well with many students only able to work out 70 000 but usually progressing no further.
30.
This was not well answered at this tier and there was high proportion of non-attempts. Those who drew the correct circle centred on L often then drew another centred on J.
31.
The vast majority of students used trial and error and made no n o progress. Those who used a formal method often went on to give a fully correct response.
32.
If students used trigonometry they usually chose the correct ratio and worked it out accurately. A common error was to use Pythagoras’ theorem. Other students attempted to use the angle sum of a triangle with the given values.
33.
This question was not well answered. The common incorrect choice was
.
Page 35 of 36
34. 35.
This question was very well answered. The common incorrect choice was This question was very well answered.
.
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