accurately drawn diagram - · pdf fileiv 11. the time period, t seconds, of a pendulum is...
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IV
1.
A B
C Diagram accurately drawn
NOT
The diagram shows a triangle ABC.
Angle ABC is exactly 90°.
AB = 83 mm correct to 2 significant figures.
BC = 90 mm correct to 1 significant figures.
(a) Calculate the upper bound for the area of triangle ABC.
……………….. mm2
(2)
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Angle CAB = x°.
(b) Calculate the lower bound for the value of tan x°.
…………………….. (2)
ABC is the cross section of a triangular prism.
The upper bound for the volume of the prism is 51561.25 mm3.
The lower bound for the volume of the prism is 45581.25 mm3.
(c) Write down the volume of the prism, in cm3, to an appropriate degree of accuracy.
……………… cm3
(2)
(Total 6 marks)
2.
L m
x°
Elliot did an experiment to find the value of g m/s2, the acceleration due to gravity.
He measured the time, T seconds, that a block took to slide L m down a smooth slope
of angle x°.
He then used the formula g = xT
L
sin
22
to calculate an estimate for g.
T = 1.3 correct to 1 decimal place.
L = 4.50 correct to 2 decimal places.
x = 30 correct to the nearest integer.
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(a) Calculate the lower bound and the upper bound for the value of g.
Give your answers correct to 3 decimal places.
Lower bound .........................................
Upper bound .......................................... (4)
(b) Use your answers to part (a) to write down the value of g to a suitable degree of accuracy.
Explain your reasoning.
..............................................................................................................................
..............................................................................................................................
.................................... (1)
(Total 5 marks)
3. x = 2p, y = 2
q
(a) Express in terms of x and/or y,
(i) 2p + q
....................................
(ii) 22q
.....................................
(iii) 2p – 1
.................................... (3)
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xy = 32
and 2xy2 = 32
(b) Find the value of p and the value of q.
p = ..............................
q = .............................. (2)
(Total 5 marks)
4. Use your calculator to work out the value of
491.123.6
32.4)8191.7( 3
Give your answer correct to 3 significant figures.
………………………… (Total 3 marks)
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5. Prove that,
(n + 1)2 – (n – 1)
2
is a multiple of 4, for all positive integer values of n.
(Total 3 marks)
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6.
r cm R cm
Diagram accurately drawn
NOT
The diagram represents two metal spheres of different sizes.
The radius of the smaller sphere is r cm.
The radius of the larger sphere is R cm.
r = 1.7 correct to 1 decimal place.
R = 31.0 correct to 3 significant figures.
(a) Write down the upper and lower bounds of r and R.
Upper bound of r = ………………………………
Lower bound of r = ………………………………
Upper bound of R = ………………………………
Lower bound of R = ……………………………… (2)
(b) Find the smallest possible value of R – r.
…………………………… (1)
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The larger sphere of radius R cm was melted down and used to make smaller spheres of radius r
cm.
(c) Calculate the smallest possible number of spheres that could be made.
…………………………… (4)
(Total 7 marks)
7. The shutter speed, S, of a camera varies inversely as the square of the aperture setting, f.
When f = 8, S = 125
(a) Find a formula for S in terms of f.
.......................... (3)
(b) Hence, or otherwise, calculate the value of S when f = 4
S = .................... (1)
(Total 4 marks)
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8. Martin won the 400 metre race in the school sports with a time of 1 minute.
The distance was correct to the nearest centimetre.
The time was correct to the nearest tenth of a second.
(a) Work out the upper bound and the lower bound of Martin’s speed in km/h.
Give your answers correct to 5 significant figures.
Upper bound ......................................... km/h
Lower bound ........................................ km/h (5)
(b) Write down an appropriate value for Martin’s speed in km/h.
Explain your answer.
.......................................................................................................................................
....................................................................................................................................... (1)
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The table shows the number of people in each age group who watched the school sports.
Age group 0 – 16 17 – 29 30 – 44 45 – 59 60 +
Number of people 177 111 86 82 21
Martin did a survey of these people.
He used a stratified sample of exactly 50 people according to age group.
(c) Work out the number of people from each age group that should have been in his sample
of 50.
Complete the table.
Age group 0 – 16 17 – 29 30 – 44 45 – 59 60 + Total
Number of people
in sample
(3)
(Total 9 marks)
9.
Number of girls Number of boys
Year 10 108 132
Year 11 90 110
The table gives information about Year 10 and Year 11 at Mathstown School.
Mathstown School had an end of term party.
40% of the students in Year 10 and 70% of the students in Year 11 went to the party.
Work out the percentage of all students in Years 10 and 11 who went to the party.
................................ % (Total 3 marks)
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10. In a factory, chemical reactions are carried out in spherical containers.
The time, T minutes, the chemical reaction takes is directly proportional to the square of the
radius, R cm, of the spherical container.
When R = 120, T = 32
Find the value of T when R = 150
T = ............................... (Total 4 marks)
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11. The time period, T seconds, of a pendulum is calculated using the formula
T = 6.283 g
L
where L metres is the length of the pendulum and g m/s2 is the acceleration due to gravity.
L = 1.36 correct to 2 decimal places.
g = 9.8 correct to 1 decimal place.
Find the difference between the lower bound of T and the upper bound of T.
..................................... (Total 5 marks)
12. Bill invests £500 on 1st January 2004 at a compound interest rate of R% per annum.
The value, £V, of this investment after n years is given by the formula
V = 500 × (1.045)n
(a) Write down the value of R.
R = …………………
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(b) Use your calculator to find the value of Bill’s investment after 20 years.
£…………………….. (2)
(Total 3 marks)
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13.
Peter transports metal bars in his van.
The van has a safety notice “Maximum Load 1200 kg”.
Each metal bar has a label “Weight 60 kg”.
For safety reasons Peter assumes that
1200 is rounded correct to 2 significant figures
and 60 is rounded correct to 1 significant figure.
Calculate the greatest number of bars that Peter can safely put into the van if his assumptions are
correct.
…………………. (Total 4 marks)
14. Simplify fully
(a) ( )3 2 4xy
……………………. (2)
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(b) x x
x x
2
2
3
8 15
……………………. (3)
(Total 5 marks)
15. John weighs 88 kg and Sophie weighs 65 kg.
Both weights have been rounded to the nearest kg.
Explain why their minimum combined weight is 152 kg.
………………………………………………………………………………………………
………………………………………………………………………………………………
………………………………………………………………………………………………
……………………………………………………………………………………………… (Total 1 mark)
16. Calculate the value of
12.547.48.23
67.183.9 2
…………… (Total 3 marks)
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17.
B
D
A
C
Diagram NOT accurately drawn
ABC is a right angled triangle.
D is the point on AB such that AD = 3DB.
AC = 2DB and angle A = 90.
Show that sin C = 20
k, where k is an integer.
Write down the value of k.
k = …………………………… (Total 4 marks)
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18. r is inversely proportional to t.
r = 12 when t = 0.2
Calculate the value of r when t = 4.
…………………………… (Total 3 marks)
19. In a sale, normal prices are reduced by 12%.
The sale price of a DVD player is £242.
Work out the normal price of the DVD player.
£ ………………………… (Total 3 marks)
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20. Work out (3.4 × 1012
) ÷ (1.2 × 103
)
Give your answer in standard form, correct to 3 significant figures.
………………………………….. (Total 2 marks)
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21. A clay bowl is in the shape of a hollow hemisphere.
7.7 cm8.2 cm
Diagram NOT accurately drawn
The external radius of the bowl is 8.2 cm.
The internal radius of the bowl is 7.7 cm.
Both measurements are correct to the nearest 0.1 cm.
The upper bound for the volume of clay is k cm3.
Find the exact value of k.
k = ……………………….. (Total 4 marks)
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22. y is inversely proportional to x2.
Given that y = 2.5 when x = 24,
(i) find an expression for y in terms of x
y = ...............................
(ii) find the value of y when x = 20
y = ...............................
(iii) find a value of x when y = 1.6
x = ............................... (Total 6 marks)
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23.
Jacob answered 80% of the questions in a test correctly.
He answered 32 of the questions correctly.
Work out the total number of questions in the test.
………………….. (Total 3 marks)
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24. Simplify fully 5x3y
4 × 7xy
2
…………………………… (Total 2 marks)
25. The weight of a piece of wire is directly proportional to its length.
A piece of wire is 25 cm long and has a weight of 6 grams.
Another piece of the same wire is 30 cm long.
Calculate the weight of the 30 cm piece of wire.
………………….. grams (Total 2 marks)
26. Write the number 620 000 000 in standard form.
…………………………… (Total 1 mark)
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27.
C
E F
D
B
20º
x
50º
Diagram NOT accurately drawn
BCEF is a trapezium.
EC is parallel to FDB.
CD is parallel to EF.
Angle CBD = 50°. Angle DEF = 20°. Angle EFD = 90°.
EF = x.
(a) Express, in terms of x,
(i) the length of DF,
…………….
(ii) the area of triangle DEF.
……………. (3)
(b) Work out the percentage of the trapezium BCEF that is not shaded.
……………. % (4)
(Total 7 marks)
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28. Correct to 2 significant figures, the area of a rectangle is 470 cm2.
Correct to 2 significant figures, the length of the rectangle is 23 cm.
Calculate the upper bound for the width of the rectangle.
………………………. cm (Total 3 marks)
29. (a) Write 5 720 000 in standard form.
............................. (1)
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p = 5 720 000
q = 4.5 × 105
(b) Find the value of 2)( qp
qp
Give your answer in standard form, correct to 2 significant figures.
................................ (2)
(Total 3 marks)
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30.
A
B
C
12 cm7 cm
65º
D
Diagram NOT accurately drawn
ABC is a triangle.
ADC is a straight line with BD perpendicular to AC.
AB = 7 cm.
BC = 12 cm.
Angle BAD = 65°.
Calculate the length of AC.
Give your answer correct to 3 significant figures.
......................... cm (Total 6 marks)
31. Calculate the reciprocal of 0.8
................................................ (Total 1 mark)
32. (a) Write 0.000 000 000 054 in standard form.
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....................................... (1)
S = 12.6 R2
R = 0.000 000 000 054
(b) Use the formula to calculate the value of S.
Give your answer in standard form, correct to 3 significant figures.
S = .......................... (2)
(Total 3 marks)
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33. The equation
x2 + 2x = 410
has a solution between 7 and 8.
Use a trial and improvement to find this solution.
Give your solution correct to 1 decimal place.
You must show ALL your working.
x = ........................... (Total 4 marks)
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34.
R Q
P
5.8 m
43º
Diagram NOT accurately drawn
PQR is a triangle.
Angle Q = 90°.
Angle R = 43°.
PR = 5.8 m.
Calculate the length of QR.
Give your answer correct to 3 significant figures.
............................... m (Total 3 marks)
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35.
Diagram NOT accurately drawn
This 12-sided window is made up of squares and equilateral triangles.
The perimeter of the window is 15.6 m.
Calculate the area of the window.
Give your answer correct to 3 significant figures.
......................... m2
(Total 6 marks)
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36.
x = pq
qp
p = 4 × 108
q = 3 × 106
Find the value of x.
Give your answer in standard form correct to 2 significant figures.
x = ....................................... (Total 3 marks)
37. Use your calculator to work out the value of 86.3–73.2
12
(a) Write down all the figures on your calculator display.
……………………………………. (2)
(b) Give your answer to an appropriate degree of accuracy.
……………………. (1)
(Total 3 marks)
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38.
For k > 0 each graph matches with one of the equations,
y = kx y = k x y = x
k y = kx
2
Match each graph to its equation,
Equation Graph
y = kx
y = k x
y = x
k
y = kx2
(Total 3 marks)
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39. Simplify fully (3xy2)4
…………………. (Total 2 marks)
40. Write 2
10 18 in the form p + q 2 , where p and q are integers.
p = ..….…….
q = …………. (Total 2 marks)