9 © ucles 2016 0580/23/m/j/16 [turn over 19 at the start of an experiment there are 20 000...
TRANSCRIPT
Linear programming
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0580/22/O/N/14© UCLES 2014 [Turn over
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2 4 6 8 10 121 3 5 7 9 11
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5
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–1
0
y
x
By shading the unwanted regions of the grid, fi nd and label the region R which satisfi es the following four inequalities.
y [ 0 x [ 4 2y Y x 2y + x Y 12[3]
__________________________________________________________________________________________
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110°NOT TOSCALE
A
CB
Triangle ABC is isosceles with AB = AC. Angle BAC = 110° and the area of the triangle is 85 cm2.
Calculate AC.
Answer AC = .......................................... cm [3]__________________________________________________________________________________________
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© UCLES 2012 0580/21/M/J/12 [Turn over
For
Examiner's
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y
x
7
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1
–1
0–8 –7 –6 –5 –4 –3 –2 –1 1 2 3 4 5 6 7
The region R contains points which satisfy the inequalities
y Y 2
1
x + 4, y [ 3 and x + y [ 6.
On the grid, label with the letter R the region which satisfies these inequalities.
You must shade the unwanted regions. [3]
15 The scale of a map is 1 : 500 000 .
(a) The actual distance between two towns is 172 km.
Calculate the distance, in centimetres, between the towns on the map.
Answer(a) cm [2]
(b) The area of a lake on the map is 12 cm2
.
Calculate the actual area of the lake in km2
.
Answer(b) km2
[2]
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0580/23/O/N/16© UCLES 2016
24
00
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2
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1 2 3 4
R
5 6 7 8 9 10 11x
y
Find the three inequalities that define the unshaded region, R.
.................................................
.................................................
................................................. [5]
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0580/21/M/J/15© UCLES 2015 [Turn over
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–4 –3 –2 –1 0 1x
1
2
2 3 4 5 6 7 8
y
3
4
5
6
7
8
9
Write down the 3 inequalities which define the unshaded region.
Answer .................................................
.................................................
................................................. [4]__________________________________________________________________________________________
16 Georg invests $5000 for 14 years at a rate of 2% per year compound interest.
Calculate the interest he receives. Give your answer correct to the nearest dollar.
Answer $ ................................................ [4]__________________________________________________________________________________________
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0580/23/M/J/16© UCLES 2016 [Turnover
19 At the start of an experiment there are 20 000 bacteria. The number of bacteria increases at a rate of 30% per hour.
(a) Work out the number of bacteria after 4 hours.
.................................................. [2]
(b) After how many whole hours, from the start of the experiment, will the number of bacteria be greater than one million?
......................................... hours [2]
20
10 32 54
1
2
3
4
5
y
x
R
Find four inequalities that define the region, R, on the grid.
..................................................
..................................................
..................................................
.................................................. [4]
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0580/21/O/N/16© UCLES 2016 [Turnover
20 A train travels for m minutes at a speed of x metres per second.
(a) Find the distance travelled, in kilometres, in terms of m and x. Give your answer in its simplest form.
.......................................... km [2]
(b) When m = 5, the train travels 10.5 km.
Find the value of x.
x = ................................................ [2]
21y
x
4
0 1 4
R
NOT TOSCALE
Write down the three inequalities that define the unshaded region, R.
.................................................
.................................................
................................................. [4]
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0580/22/M/J/16© UCLES 2016
23
0
1
2
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6
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y
x1 2 3 4 5
The region R satisfies these inequalities.
y G 2x 3x + 4y H 12 x G 3
On the grid, draw and label the region R that satisfies these inequalities. Shade the unwanted regions. [5]
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0580/41/O/N/13© UCLES 2013
ForExaminer′s
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9y
x1 2 3 4 5 6 7 8 9 10
5
4
3
2
1
0
L2
L3
L1
R
(a) Find the equations of the lines L1, L2 and L3.
Answer(a) L1 ...............................................
L2 ...............................................
L3 ............................................... [5]
(b) Write down the three inequalities that defi ne the shaded region, R.
Answer(b) ................................................
................................................
................................................ [3]
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0580/41/O/N/13© UCLES 2013 [Turn over
ForExaminer′s
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(c) A gardener buys x bushes and y trees. The cost of a bush is $30 and the cost of a tree is $200. The shaded region R shows the only possible numbers of bushes and trees the gardener can buy.
(i) Find the number of bushes and the number of trees when the total cost is $720.
Answer(c)(i) ...................................... bushes
...................................... trees [2]
(ii) Find the number of bushes and the number of trees which give the greatest possible total cost. Write down this greatest possible total cost.
Answer(c)(ii) .......................................... bushes
.......................................... trees
Greatest possible total cost = $ ............................................... [3]_____________________________________________________________________________________
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0580/41/O/N/15© UCLES 2015
4 Ali buys x rose bushes and y lavender bushes.
He buys: • at least 5 rose bushes • at most 8 lavender bushes • at most 15 bushes in total • more lavender bushes than rose bushes.
(a) (i) Write down four inequalities, in terms of x and/or y, to show this information.
Answer(a)(i) ................................................
................................................
................................................
................................................ [4]
(ii) On the grid, show the information in part(a)(i) by drawing four straight lines. Label the region R where all four inequalities are true.
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1 2 3 4 5 6 7 8 9 10 11 12 13 15140
y
x
[5]
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0580/41/O/N/15© UCLES 2015 [Turnover
(b) Rose bushes cost $6 each and lavender bushes cost $4.50 each.
What is the greatest amount of money Ali could spend?
Answer(b) $ ................................................ [2]__________________________________________________________________________________________
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O
A
B
C
D
E
37°
NOT TOSCALE
A, B, C, D and E are points on the circle, centre O. Angle BAD = 37°.
Complete the following statements.
(a) Angle BED = .................... because .................................................................................................
............................................................................................................................................................. [2]
(b) Angle BOD = .................... because ................................................................................................
............................................................................................................................................................. [2]
(c) Angle BCD = .................... because ................................................................................................
............................................................................................................................................................. [2]__________________________________________________________________________________________
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0580/42/O/N/16© UCLES 2016 [Turnover
(b) A bag of sweets contains x orange sweets and y lemon sweets. Each orange sweet costs 2 cents and each lemon sweet costs 3 cents.
The cost of a bag of sweets is less than 24 cents. There are at least 9 sweets in each bag. There are at least 2 lemon sweets in each bag.
(i) One of the inequalities that shows this information is x y2 3 241+ .
Write down the other two inequalities.
.................................................
................................................. [2]
(ii) On the grid, by shading the unwanted regions, show the region which satisfies the three inequalities.
12
y
1110987654321
0 1 2 3 4 5 6 7 8 9 10 11 12x
[4]
(iii) Find the lowest cost of a bag of sweets. Write down the value of x and the value of y that give this cost.
Lowest cost = .................................... cents
x = .............................................
y = ............................................. [3]
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0580/42/M/J/15© UCLES 2015
8 Sima sells x biscuits and y cakes.
(a) (i) She sells at least 100 biscuits.
Write down an inequality in x.
Answer(a)(i) ................................................ [1]
(ii) She sells at least 120 cakes.
Write down an inequality in y.
Answer(a)(ii) ................................................ [1]
(iii) She sells a maximum of 300 biscuits and cakes altogether.
Write down an inequality in x and y.
Answer(a)(iii) ................................................ [1]
(iv) Sima makes a profit of 40 cents on each biscuit and 80 cents on each cake. Her total profit is at least $160.
Show that x + 2y ! 400.
Answer(a)(iv)
[1]
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0580/42/M/J/15© UCLES 2015 [Turnover
(b) On the grid, draw four lines to show the four inequalities and shade the unwanted regions.
400
300
200
100
100 200 300 400 5000 x
y
[6]
(c) Calculate Sima’s maximum profit. Give your answer in dollars.
Answer(c) $ ................................................ [2]__________________________________________________________________________________________
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0580/43/M/J/13© UCLES 2013
ForExaminer′sUse
3 (a) Luk wants to buy x goats and y sheep.
(i) He wants to buy at least 5 goats.
Write down an inequality in x to represent this condition.
Answer(a)(i) ............................................... [1]
(ii) He wants to buy at least 11 sheep.
Write down an inequality in y to represent this condition.
Answer(a)(ii) ............................................... [1]
(iii) He wants to buy at least 20 animals.
Write down an inequality in x and y to represent this condition.
Answer(a)(iii) ............................................... [1]
(b) Goats cost $4 and sheep cost $8. The maximum Luk can spend is $160.
Write down an inequality in x and y and show that it simplifi es to x + 2y Y 40 .
Answer(b)
[1]
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0580/43/M/J/13© UCLES 2013 [Turn over
ForExaminer′sUse
(c) (i) On the grid below, draw four lines to show the four inequalities and shade the unwanted regions.
40
35
30
25
20
15
10
5
y
x05 10 15 20 25 30 35 40
[7]
(ii) Work out the maximum number of animals that Luk can buy.
Answer(c)(ii) ............................................... [2]_____________________________________________________________________________________
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© UCLES 2012 0580/42/M/J/12
For
Examiner's
Use
3 Pablo plants x lemon trees and y orange trees.
(a) (i) He plants at least 4 lemon trees.
Write down an inequality in x to show this information.
Answer(a)(i) [1]
(ii) Pablo plants at least 9 orange trees.
Write down an inequality in y to show this information.
Answer(a)(ii) [1]
(iii) The greatest possible number of trees he can plant is 20.
Write down an inequality in x and y to show this information.
Answer(a)(iii) [1]
(b) Lemon trees cost $5 each and orange trees cost $10 each.
The maximum Pablo can spend is $170.
Write down an inequality in x and y and show that it simplifies to x + 2y Y 34.
Answer (b)
[1]
(c) (i) On the grid opposite, draw four lines to show the four inequalities and shade the unwanted
region.
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© UCLES 2012 0580/42/M/J/12 [Turn over
For
Examiner's
Use
y
x
24
22
20
18
16
14
12
10
8
6
4
2
02 4 6 8 10 12 14 16 18 20 22 24 26 28 30 32 34
[7]
(ii) Calculate the smallest cost when Pablo buys a total of 20 trees.
Answer(c)(ii) $ [2]