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1. The graph of y = x2 – 2x – 3 is shown on the axes below.
x
y
–4 –3 –2 –1 0 1 2 3 4 5 6
20
10
(a) Draw the graph of y = 5 on the same axes.
(b) Use your graph to find:
(i) the values of x when x2 – 2x – 3 = 5
(ii) the value of x that gives the minimum value of x2 – 2x – 3
Working:
Answers:
(b) (i) ……………………………………...
(ii) ……………………………………..
(Total 4 marks)
2
2. The costs charged by two taxi services are represented by the two parallel lines on the following
graph. The Speedy Taxi Service charges $1.80, plus 10 cents for each kilometre.
2.60
2.40
2.20
2.00
1.80
1.60
1.40
1.20
1.00
c
k0 2 4 6 8
distance (km)
cost ($)
Speedy Taxi Service
Economic Taxi Service
(a) Write an equation for the cost, c, in $, of using the Economic Taxi Service for any number
of kilometres, k.
3
(b) Bruce uses the Economic Taxi Service.
(i) How much will he pay for travelling 7 km?
(ii) How far can he travel for $2.40?
Working:
Answers:
(a) …………………………………………..
(b) (i) ……………………………………..
(ii) ……………………………………..
(Total 4 marks)
3. The line L1 shown on the set of axes below has equation 3x + 4y = 24. L1 cuts the x-axis at A
and cuts the y-axis at B. Diagram not drawn to scale
y
x
B
C
AO
M
L
L
1
2
(a) Write down the coordinates of A and B. (2)
4
M is the midpoint of the line segment [AB].
(b) Write down the coordinates of M. (2)
The line L2 passes through the point M and the point C (0, –2).
(c) Write down the equation of L2.
(2)
(d) Find the length of
(i) MC; (2)
(ii) AC. (2)
(e) The length of AM is 5. Find
(i) the size of angle CMA; (3)
(ii) the area of the triangle with vertices C, M and A. (2)
(Total 15 marks)
4. A rectangle has dimensions (5 + 2x) metres and (7 – 2x) metres.
(a) Show that the area, A, of the rectangle can be written as A = 35 + 4x – 4x2.
(1)
(b) The following is the table of values for the function A = 35 + 4x – 4x2.
x –3 –2 –1 0 1 2 3 4
A –13 p 27 35 q r 11 s
(i) Calculate the values of p, q, r and s.
5
(ii) On graph paper, using a scale of 1 cm for 1 unit on the x-axis and 1 cm for 5 units
on the A-axis, plot the points from your table and join them up to form a smooth
curve. (6)
(c) Answer the following, using your graph or otherwise.
(i) Write down the equation of the axis of symmetry of the curve,
(ii) Find one value of x for a rectangle whose area is 27 m2.
(iii) Using this value of x, write down the dimensions of the rectangle. (4)
(d) (i) On the same graph, draw the line with equation A = 5x + 30.
(ii) Hence or otherwise, solve the equation 4x2 + x – 5 = 0.
(3)
(Total 14 marks)
5. The following diagram shows the lines l1 and l2, which are perpendicular to each other.
Diagram not to scale
y
x(5, 0)
(0, 7)
(0, –2)
l
l
1
2
6
(a) Calculate the gradient of line l1.
(b) Write the equation of line l1 in the form ax + by + d = 0 where a, b and d are integers, and
a > 0.
Working:
Answers:
(a) ..............................................
(b) ..............................................
(Total 8 marks)
6. The conversion formula for temperature from the Fahrenheit (F) to the Celsius (C) scale is given
by C = 9
)32–(5 F.
(a) What is the temperature in degrees Celsius when it is 50° Fahrenheit?
7
There is another temperature scale called the Kelvin (K) scale.
The temperature in degrees Kelvin is given by K = C + 273.
(b) What is the temperature in Fahrenheit when it is zero degrees on the Kelvin scale?
Working:
Answers:
(a) ..................................................................
(b) ..................................................................
(Total 8 marks)
7. The four diagrams below show the graphs of four different straight lines, all drawn to the same
scale. Each diagram is numbered and c is a positive constant.
x
x
x
x
y
y
y
y
c
c
c
c
Number 1
Number 2
Number 3
Number 4
0
0
0
0
8
In the table below, write the number of the diagram whose straight line corresponds to the
equation in the table.
Equation Diagram number
y = c
y = – x + c
y = 3 x + c
y = 3
1 x + c
(Total 8 marks)
8. The diagrams below are sketches of some of the following functions.
(i) y = ax (ii) y = x
2 – a (iii) y = a – x
2
(iv) y = a – x (v) y = x – a
x
x
x
x
y
y
y
y
(a)
(c)
(b)
(d)DIAGRAMS NOTTO SCALE
9
Complete the table to match each sketch to the correct function.
Sketch Function
(a)
(b)
(c)
(d)
Working:
(Total 8 marks)
9. A student has drawn the two straight line graphs L1 and L2 and marked in the angle between
them as a right angle, as shown below. The student has drawn one of the lines incorrectly.
3
2
1
–1
–4 –3 –2 –1 0 1 2 3 4 x
y
90°
L
L
2
1
Consider L1 with equation y = 2x + 2 and L2 with equation y = –4
1x + 1.
(a) Write down the gradients of L1 and L2 using the given equations.
(b) Which of the two lines has the student drawn incorrectly?
10
(c) How can you tell from the answer to part (a) that the angle between L1 and L2 should not
be 90°?
(d) Draw the correct version of the incorrectly drawn line on the diagram.
Working:
Answers:
(a) ..................................................................
(b) ..................................................................
(c) ..................................................................
(Total 8 marks)
10. The following table gives the postage rates for sending letters from the Netherlands. All prices
are given in Euros (€).
Destination Weight not more than 20 g Each additional 20 g or part of 20 g
Within the Netherlands
(zone 1) €0.40 €0.35
Other destinations
within Europe (zone 2) €0.55 €0.50
Outside Europe
(zone 3) €0.80 €0.70
(a) Write down the cost of sending a letter weighing 15 g from the Netherlands to a
destination within the Netherlands (zone 1).
(b) Find the cost of sending a letter weighing 35 g from the Netherlands to a destination in
France (zone 2).
11
(c) Find the cost of sending a letter weighing 50 g from the Netherlands to a destination in
the USA (zone 3).
Working:
Answers:
(a) ..................................................................
(b) ..................................................................
(c) ..................................................................
(Total 8 marks)
11. Two functions are defined as follows
f (x) =
6for6–
60for–6
xx
xx
g (x) = 2
1x
(a) Draw the graphs of the functions f and g in the interval 0 ≤ x ≤ 14, 0 ≤ y ≤ 8 using a scale
of 1 cm to represent 1 unit on both axes.
(5)
(b) (i) Mark the intersection points A and B of f (x) and g (x) on the graph.
(ii) Write down the coordinates of A and B. (3)
(c) Calculate the midpoint M of the line AB. (2)
(d) Find the equation of the straight line which joins the points M and N. (4)
(Total 14 marks)
12
13
12. The diagrams below include sketches of the graphs of the following equations where a and b are
positive integers.
0
y
x
2.y
x0
3. y
x0
4.y
x0
1.
Complete the table to match each equation to the correct sketch.
Equation Sketch
(i) y = ax + b
(ii) y = –ax + b
(iii) y = x2 + ax + b
(iv) y = x2 – ax – b
14
Working:
(Total 8 marks)
13. The profit (P) in Swiss Francs made by three students selling homemade lemonade is modelled
by the function
P = – 2
20
1x + 5x – 30
where x is the number of glasses of lemonade sold.
(a) Copy and complete the table below
x 0 10 20 30 40 50 60 70 80 90
P 15 90 75 50
(3)
(b) On graph paper draw axes for x and P, placing x on the horizontal axis and P on the
vertical axis. Use suitable scales. Draw the graph of P against x by plotting the points.
Label your graph. (5)
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(c) Use your graph to find
(i) the maximum possible profit; (1)
(ii) the number of glasses that need to be sold to make the maximum profit; (1)
(iii) the number of glasses that need to be sold to make a profit of 80 Swiss Francs; (2)
(iv) the amount of money initially invested by the three students. (1)
(d) The three students Baljeet, Jane and Fiona share the profits in the ratio of 1:2:3
respectively. If they sold 40 glasses of lemonade, calculate Fiona’s share of the profits. (2)
(Total 15 marks)
16
14. The perimeter of this rectangular field is 220 m. One side is x m as shown.
x m
W m
(a) Express the width (W) in terms of x.
(b) Write an expression, in terms of x only, for the area of the field.
(c) If the length (x) is 70 m, find the area.
Working:
Answers:
(a) …………………………………………..
(b) …………………………………………..
(c) …………………………………………..
(Total 4 marks)
17
15. The diagram shows the graph of y = x2 – 2x – 8. The graph crosses the x-axis at the point A, and
has a vertex at B.
y
x
B
A O
(a) Factorize x2 – 2x – 8.
(b) Write down the coordinates of each of these points
(i) A;
(ii) B.
Working:
Answers:
(a) …………………………………………..
(b) (i) ……………………………………..
(ii) ……………………………………..
(Total 4 marks)
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16. Diagram 1 shows a part of the graph of y = x2.
y
x0
Diagram 1
Diagrams 2, 3 and 4 show a part of the graph of y = x2 after it has been moved parallel to the
x-axis, or parallel to the y-axis, or parallel to one axis, then the other.
y y y
x x x0 0 0
3
2 2
3
Diagram 2 Diagram 3 Diagram 4
Write down the equation of the graph shown in
(a) Diagram 2;
(b) Diagram 3;
(c) Diagram 4. (Total 4 marks)
19
17. The perimeter of a rectangle is 24 metres.
(a) The table shows some of the possible dimensions of the rectangle.
Find the values of a, b, c, d and e.
Length (m) Width (m) Area (m2)
1 11 11
a 10 b
3 c 27
4 d e
(2)
(b) If the length of the rectangle is x m, and the area is A m2, express A in terms of x only.
(1)
(c) What are the length and width of the rectangle if the area is to be a maximum? (3)
(Total 6 marks)
18. (a) Solve the equation x2 – 5x + 6 = 0.
(b) Find the coordinates of the points where the graph of y = x2 – 5x + 6 intersects the x-axis.
Working:
Answers:
(a) …………………………………………..
(b) ..................................................................
(Total 4 marks)
20
19. The diagram below shows part of the graph of y = ax2 + 4x – 3. The line x = 2 is the axis
of symmetry. M and N are points on the curve, as shown.
y
x0
N
M
x = 2
21
(a) Find the value of a.
22
(b) Find the coordinates of
(i) M;
(ii) N.
Working:
Answers:
(a) ..................................................................
(b) (i) ……………………………………...
(ii) ……………………………………...
(Total 4 marks)
23
20. Consider the graphs of the following functions.
(i) y = 7x + x2;
(ii) y = (x – 2)(x + 3);
(iii) y = 3x2 – 2x + 5;
(iv) y = 5 – 3x – 2x2.
24
Which of these graphs
(a) has a y-intercept below the x-axis?
(b) passes through the origin?
(c) does not cross the x-axis?
(d) could be represented by the following diagram?
y
xO
Working:
Answers:
(a) ..............................................
(b) ..............................................
(c) ..............................................
(d) ................................................
(Total 8 marks)
25
21. The figure below shows part of the graph of a quadratic function y = ax2 + 4x + c.
8
6
4
2
–2 –1 1 2 3 4 x
y
(a) Write down the value of c.
(b) Find the value of a.
(c) Write the quadratic function in its factorized form.
Working:
Answers:
(a) ..................................................................
(b) ..................................................................
(c) ..................................................................
(Total 8 marks)
26
22. The graph of the function f : x 30x – 5x2 is given in the diagram below.
Diagramnot to scale
f x( )
xO A
(a) Factorize fully 30x – 5x2.
(b) Find the coordinates of the point A.
(c) Write down the equation of the axis of symmetry.
Working:
Answers:
(a) ..................................................................
(b) ..................................................................
(c) ..................................................................
(Total 8 marks)
27
23. The graph of the function y = x2 – x – 2 is drawn below.
x
y
A B
C
0
(a) Write down the coordinates of the point C.
(b) Calculate the coordinates of the points A and B.
Working:
Answers:
(a) ..................................................................
(b) ..................................................................
(Total 8 marks)
28
24. The graph of the function f (x) = x2 – 2x – 3 is shown in the diagram below.
A B
C
0
y
x
Diagram not to scale
(a) Factorize the expression x2 – 2x – 3.
(b) Write down the coordinates of the points A and B.
(c) Write down the equation of the axis of symmetry.
29
(d) Write down the coordinates of the point C, the vertex of the parabola.
Working:
Answers:
(a) ..................................................................
(b) ..................................................................
(c) ..................................................................
(d) ..................................................................
(Total 8 marks)
30
25. The diagram shows a point P, 12.3 m from the base of a building of height h m. The angle
measured to the top of the building from point P is 63°.
12.3
63°P
h m
(a) Calculate the height h of the building.
Consider the formula h = 4.9t2, where h is the height of the building and t is the time in seconds
to fall to the ground from the top of the building.
(b) Calculate how long it would take for a stone to fall from the top of the building to the
ground.
Working:
Answers:
(a) .................................................
(b) .................................................
(Total 6 marks)
31
26. (a) Represent the function y = 2x2 – 5, where x {–2, –1, 0, 1, 2, 3} by a mapping diagram.
x y
(b) List the elements of the domain of this function.
(c) List the elements of the range of this function.
Working:
Answers:
(c) ...................................................
(b) ...................................................
(Total 6 marks)
27. (a) Sketch the graph of the function y = 2x2 – 6x + 5.
(b) Write down the coordinates of the local maximum or minimum of the function.
(c) Find the equation of the axis of symmetry of the function. (Total 6 marks)
32
28. The figure below shows the graphs of the functions y = x2 and y = 2
x for values of x between –2
and 5.
The points of intersection of the two curves are labelled as B, C and D.
y
x
C
A
B
D
–2 –1 1 2 3 4 5
20
15
10
5
(a) Write down the coordinates of the point A. (2)
(b) Write down the coordinates of the points B and C. (2)
(c) Find the x-coordinate of the point D. (1)
(d) Write down, using interval notation, all values of x for which 2x ≤ x
2.
(3)
(Total 8 marks)
33
29. The diagram below shows the graph of y = c + kx – x2, where k and c are constants.
Q
y
xO P(5, 0)
(a) Find the values of k and c.
(b) Find the coordinates of Q, the highest point on the graph.
Working:
Answers:
(a) .................................................
(b) .................................................
(Total 8 marks)
30. A small manufacturing company makes and sells x machines each month. The monthly cost C,
in dollars, of making x machines is given by
C(x) = 2600 + 0.4x2.
The monthly income I, in dollars, obtained by selling x machines is given by
I(x) = 150x – 0.6x2.
34
(a) Show that the company’s monthly profit can be calculated using the quadratic function
P(x) = – x2
+ 150x – 2600. (2)
(b) The maximum profit occurs at the vertex of the function P(x). How many machines
should be made and sold each month for a maximum profit? (2)
(c) If the company does maximize profit, what is the selling price of each machine? (4)
(d) Given that P(x) = (x – 20) (130 – x), find the smallest number of machines the company
must make and sell each month in order to make positive profit. (4)
(Total 12 marks)
31. The function Q (t) = 0.003t2 – 0.625t + 25 represents the amount of energy in a battery after t
minutes of use.
(a) State the amount of energy held by the battery immediately before it was used.
(b) Calculate the amount of energy available after 20 minutes.
(c) Given that Q (10) = 19.05, find the average amount of energy produced per minute for the
interval 10 t 20.
(d) Calculate the number of minutes it takes for the energy to reach zero.
35
Working:
Answers:
(a) .....................................................
(b) .....................................................
(c) .....................................................
(d) ..................................................... (Total 6 marks)
32. (a) A function f (x) is defined by f (x) = 2x2 – 10x + 60, –5 x 8.
x –5 0 2 5 8
f (x) 160 a b 60 108
(i) Write down the values of a and b. (2)
(ii) Using the values in the above table, draw the graph of f (x) on a set of coordinate
axes. Use a scale of 1 cm to represent 1 unit on the horizontal axis and 1 cm to
represent 20 units on the vertical axis. (4)
(iii) Show that the coordinates of the vertex of the graph are (2.5, 47.5) (3)
(iv) State the values of x for which the function is increasing. (2)
36
(b) A second function h (x) is defined by:
h (x),= 80, 0 x 8.
(i) On the same axes used for part (a), draw the graph of h (x). (2)
(ii) Find the coordinates of the point at which f (x) = h (x). (2)
(iii) Find the vertical distance from the vertex of the graph
of f (x) to the line h (x). (2)
(Total 17 marks)
33. The graph of a quadratic function f (x) intersects the horizontal axis at (1, 0) and the equation of
the axis of symmetry is x = −1.
(a) Write down the x-coordinate of the other point where the graph of y = f (x) intersects the
horizontal axis.
(b) y = f (x) reaches its maximum value at y = 5.
(i) Write down the value of f (−1).
(ii) Find the range of the function y = f (x).
37
Working:
Answers:
(a) ...................................................
(b) (i)............................................
(ii)........................................... (Total 6 marks)
34. The figure below shows the graphs of the functions f (x) = 2x + 0.5 and g (x) = 4 − x
2 for values
of x between –3 and 3.
6
3
–3
–3 –2 –1 1 2 3
g x( )
f x( )
x
y
A
B
0
(a) Write down the coordinates of the points A and B.
(b) Write down the set of values of x for which f (x) g (x).
38
Working:
Answers:
(a) .................................................
(b) ................................................. (Total 6 marks)
39
35. A football is kicked from a point A (a, 0), 0 a 10 on the ground towards a goal to the right
of A.
The ball follows a path that can be modelled by part of the graph
y = − 0.021x2 +1.245x −6.01, x , y 0.
x is the horizontal distance of the ball from the origin
y is the height above the ground
Both x and y are measured in metres.
(a) Using your graphic display calculator or otherwise, find the value of a. (1)
(b) Find dx
dy.
(2)
(c) (i) Use your answer to part (b) to calculate the horizontal distance the ball has
travelled from A when its height is a maximum.
(ii) Find the maximum vertical height reached by the football. (4)
(d) Draw a graph showing the path of the football from the point where it is kicked to the
point where it hits the ground again. Use 1 cm to represent 5 m on the horizontal axis and
1 cm to represent 2 m on the vertical scale. (4)
The goal posts are 35 m from the point where the ball is kicked.
(e) At what height does the ball pass over the goal posts? (2)
(Total 13 marks)
40
36. The following curves are sketches of the graphs of the functions given below, but in a different
order. Using your graphic display calculator, match the equations to the curves, writing your
answers in the table below.
(the diagrams are not to scale)
A B C
D E F
0
0
0
00
0
y
x
y
x
y
x
y
x
y
x
y
x
Function Graph label
(i) y = x3 + 1
(ii) y = x2 + 3
(iii) y = 4 − x2
(iv) y = 2x + 1
(v) y = 3−x
+ 1
(vi) y = 8x − 2x2 − x
3
(Total 6 marks)
41
37. The diagram below shows the graph of a line L passing through (1, 1) and (2, 3) and the graph P
of the function f (x) = x2 − 3x − 4
y
x
L P
0
(a) Find the gradient of the line L. (2)
(b) Differentiate f (x). (2)
(c) Find the coordinates of the point where the tangent to P is parallel to the line L. (3)
(d) Find the coordinates of the point where the tangent to P is perpendicular to
the line L. (4)
(e) Find
(i) the gradient of the tangent to P at the point with coordinates (2, −6);
(ii) the equation of the tangent to P at this point. (3)
(f) State the equation of the axis of symmetry of P. (1)
(e) Find the coordinates of the vertex of P and state the gradient of the curve
at this point. (3)
(Total 18 marks)
42
38. In an experiment researchers found that a specific culture of bacteria increases in number
according to the formula
N = 150 × 2t,
where N is the number of bacteria present and t is the number of hours since the experiment
began.
Use this formula to calculate
(a) the number of bacteria present at the start of the experiment;
(b) the number of bacteria present after 3 hours;
(c) the number of hours it would take for the number of bacteria to reach 19 200.
Working:
Answers:
(a) …………………………………………..
(b) …………………………………………..
(c) …………………………………………..
(Total 4 marks)
43
39. The following diagram shows part of the graph of an exponential function f(x) = a–x
, where
x .
f x( )y
x
P
0
(a) What is the range of f ?
(b) Write down the coordinates of the point P.
(c) What happens to the values of f(x) as elements in its domain increase in value?
Working:
Answers:
(a) …………………………………………..
(b) ..................................................................
(b) …………………………………………..
(Total 4 marks)
44
40. Under certain conditions the number of bacteria in a particular culture doubles every 10 seconds
as shown by the graph below.
8
7
6
5
4
3
2
1
0 10 20 30
Time(seconds)
Numberof
bacteria
(a) Complete the table below.
Time (seconds) 0 10 20 30
Number of bacteria 1
(b) Calculate the number of bacteria in the culture after 1 minute.
Working:
Answer:
(b) ..................................................................
(Total 4 marks)
45
41. The number (n) of bacteria in a colony after h hours is given by the formula n = 1200(30.25h
).
Initially, there are 1200 bacteria in the colony.
(a) Copy and complete the table below, which gives values of n and h.
Give your answers to the nearest hundred.
time in hours (h) 0 1 2 3 4
no. of bacteria (n) 1200 2100 2700
(2)
(b) On graph paper, draw the graph of the above function. Use a scale of 3 cm to represent 1
hour on the horizontal axis and 4 cm to represent 1000 bacteria on the vertical axis. Label
the graph clearly. (5)
(c) Use your graph to answer each of the following, showing your method clearly.
(i) How many bacteria would there be after 2 hours and 40 minutes?
Give your answer to the nearest hundred bacteria.
(ii) After how long will there be approximately 3000 bacteria? Give your answer to the
nearest 10 minutes. (4)
(Total 11 marks)
46
42. The graph below shows the curve y = k(2x) + c, where k and c are constants.
0
y
x–6 –4 –2 2 4
10
–10
Find the values of c and k.
Working:
Answers:
.......…………………………………………..
..........................................................................
(Total 4 marks)
47
43. The following diagram shows the graph of y = 3–x
+ 2. The curve passes through the points (0,
a) and (1, b).
Diagram not to scale
y
x
(0, ) a(1, ) b
O
(a) Find the value of
(i) a;
(ii) b.
48
(b) Write down the equation of the asymptote to this curve.
Working:
Answers:
(a) (i) ........................................
(ii) ........................................
(b) ...............................................
(Total 8 marks)
44. The diagram below shows a part of the graph of y = ax. The graph crosses the y-axis at the point
P. The point Q (4, 16) is on the graph.
y
xO
P
Q (4, 16)
Diagram not to scale
49
Find
(a) the coordinates of the point P;
(b) the value of a.
Working:
Answers:
(a) ..................................................................
(b) ……………………………………..........
(Total 8 marks)
45. The diagram shows a chain hanging between two hooks A and B.
The points A and B are at equal heights above the ground. P is the lowest point on the chain.
The ground is represented by the x-axis. The x-coordinate of A is –2 and the x-coordinate of
B is 2. Point P is on the y-axis. The shape of the chain is given by y = 2x + 2
–x where –2 ≤ x ≤ 2.
Chain
Ground
–2 0 2 x
yA B
P
(a) Calculate the height of the point P.
50
(b) Find the range of y. Write your answer as an interval or using inequality symbols.
Working:
Answers:
(a) ..................................................................
(b) ..................................................................
(Total 8 marks)
46. The number of bacteria (y) present at any time is given by the formula:
y = 15 000e–025t
, where t is the time in seconds and e = 2.72 correct to 3 s.f.
(a) Calculate the values of a, b and c to the nearest hundred in the table below:
Time in seconds (t) 0 1 2 3 4 5 6 7 8
Amount of bacteria (y)
(nearest hundred)
a 11700 9100 7100 b 4300 3300 2600 c
(3)
(b) On graph paper using 1 cm for each second on the horizontal axis and 1 cm for each
thousand on the vertical axis, draw and label the graph representing this information. (5)
(c) Using your graph, answer the following questions:
(i) After how many seconds will there be 5000 bacteria? Give your answer correct to
the nearest tenth of a second.
51
(ii) How many bacteria will there be after 6.8 seconds? Give your answer correct to the
nearest hundred bacteria.
(iii) Will there be a time when there are no bacteria left? Explain your answer. (6)
(Total 14 marks)
47. The area, A m2, of a fast growing plant is measured at noon (12:00) each day. On 7 July the area
was 100 m2. Every day the plant grew by 7.5%. The formula for A is given by
A = 100 (1.075)t
where t is the number of days after 7 July. (on 7 July, t = 0)
The graph of A = 100(1.075)t is shown below.
400
300
200
100
–6 –4 –2 0 2 4 6 8 10 12 14 16 18
A
t
7 July
(a) What does the graph represent when t is negative? (2)
(b) Use the graph to find the value of t when A = 178. (1)
(c) Calculate the area covered by the plant at noon on 28 July. (3)
(Total 6 marks)
52
48. The graph below shows the temperature of a liquid as it is cooling.
100
90
80
70
60
50
40
30
20
10
05 10 15 20 25 30 35 40 45 50 55 60 x
y
Time in minutes
Tem
per
atu
re (
°C)
(a) Write down the temperature after 5 minutes.
(b) After how many minutes is the temperature 50°C?
The equation of the graph for all positive x can be written in the form y = 100(5–0.02x
).
(c) Calculate the temperature after 80 minutes.
53
(d) Write down the equation of the asymptote to the curve.
Working:
Answers:
(a) ..................................................................
(b) ..................................................................
(c) ..................................................................
(d) ..................................................................
(Total 8 marks)
54
49. It is thought that a joke would spread in a school according to an exponential model
N = 4 × (1.356)0.4t
, t 0; where N is the number of people who have heard the joke, and t is the
time in minutes after the joke is first told.
(a) How many people heard the joke initially?
(b) How many people had heard the joke after 16 minutes?
There are 1200 people in the school.
(c) Estimate how long it would take for everybody in the school to hear this joke.
Answers:
Working:
(a) .....................................................
(b) .....................................................
(c) .....................................................
(Total 6 marks)
50. (a) Write 2
3
x in the form 3x
a where a .
(b) Hence differentiate y = 2
3
x giving your answer in the form
cx
b where c
+.
(Total 6 marks)
55
51. The equation M = 90 × 2–t/20
gives the amount, in grams, of radioactive material held in a
laboratory over t years.
(a) What was the original mass of the radioactive material?
The table below lists some values for M.
t 60 80 100
M 11.25 v 2.8125
(b) Find the value of v.
(c) Calculate the number of years it would take for the radioactive material to have a mass of
45 grams.
Answers:
Working:
(a) .....................................................
(b) .....................................................
(c) .....................................................
(Total 8 marks)
56
52. A function is represented by the equation f (x) = 3(2)x + 1.
The table of values of f (x) – 3 x 2 is given below.
x –3 –2 –1 0 1 2
f (x) 1.375 1.75 a 4 7 b
(a) Calculate the values for a and b. (2)
(b) On graph paper, draw the graph of f (x) , for – 3 x 2, taking 1 cm to represent 1 unit
on both axes. (4)
The domain of the function f (x) is the real numbers, .
(c) Write down the range of f (x). (2)
(d) Using your graph, or otherwise, find the approximate value for x when f (x) = 10. (2)
(Total 10 marks)
57
53. The value of a car decreases each year. This value can be calculated using the function
v = 32 000rt, t 0, 0 r 1,
where v is the value of the car in USD, t is the number of years after it was first bought and r is
a constant.
(a) (i) Write down the value of the car when it was first bought.
(ii) One year later the value of the car was 27 200 USD. Find the value of r.
(b) Find how many years it will take for the value of the car to be less than 8000 USD.
Working:
Answers:
(a)
(b) .................................................
(i)..........................................
(ii).........................................
(Total 6 marks)
58
54. The mapping below is of the form f : x a × 2x + b and maps the elements of x to elements
of y.
0
–2
1
–1
g
–1
–1.75
0
–1.5
30
(a) (i) List the elements in the domain of f.
(ii) List the elements in the range of f.
(b) Find a and b.
(c) Find the value of g.
Working:
Answers:
(a)
(b) ...................................................
(c) ...................................................
(i)...........................................
(ii)..........................................
(Total 6 marks)
59
55. The following graph shows the temperature in degrees Celsius of Robert’s cup of coffee, t
minutes after pouring it out. The equation of the cooling graph is f (t) =16 + 74 × 2.8−0.2t
where f
(t) is the temperature and t is the time in minutes after pouring the coffee out.
40
20
00 1 1918171615141312111098765432
Tem
per
atu
re (
°C)
Time (minutes)
100
60
80
(a) Find the initial temperature of the coffee. (1)
(b) Write down the equation of the horizontal asymptote. (1)
(c) Find the room temperature. (1)
(d) Find the temperature of the coffee after 10 minutes. (1)
If the coffee is not hot enough it is reheated in a microwave oven. The liquid increases in
temperature according to the formula
T = A × 21.5t
where T is the final temperature of the liquid, A is the initial temperature of coffee in the
microwave and t is the time in minutes after switching the microwave on.
(e) Find the temperature of Robert’s coffee after being heated in the microwave for 30
seconds after it has reached the temperature in part (d). (3)
(f) Calculate the length of time it would take a similar cup of coffee, initially at 20C, to be
heated in the microwave to reach 100C. (4)
(Total 11 marks)
60
56. In an experiment it is found that a culture of bacteria triples in number every four hours.
There are 200 bacteria at the start of the experiment.
Hours 0 4 8 12 16
No. of bacteria 200 600 a 5400 16200
(a) Find the value of a. (1)
(b) Calculate how many bacteria there will be after one day. (2)
(c) Find how long it will take for there to be two million bacteria. (3)
Working:
Answers:
(a) ...................................................
(b) ...................................................
(c) ................................................... (Total 6 marks)
61
57. (a) For y = 0.5 cos 0.5 x, find
(i) the amplitude;
(ii) the period.
(b) Let y = –3 sin x + 2, where 90° ≤ x ≤ 270°.
By drawing the graph of y or otherwise, complete the table below for the given
values of y.
x y
–1
2
Working:
Answers:
(a) (i) ……………………………………..
(ii) ……………………………………..
(Total 4 marks)
62
58. The diagram below shows the graph of y = – a sin x° + c, 0 ≤ x ≤ 360.
5
4
3
2
1
0
–1
–2
–3
x
y
90 180 270 360
Use the graph to find the values of
(a) c;
(b) a.
Working:
Answers:
(a) …………………………………………..
(b) ..................................................................
(Total 4 marks)
63
59. The graph below shows part of the function y = 2 sin x + 3.
6
5
4
3
2
1
0
0º 90º 180º 270º 360º 450ºx
y
(a) Write the domain of the part of the function shown on the graph.
(b) Write the range of the part of the function shown on the graph.
Working:
Answers:
(a) ..................................................................
(b) ..................................................................
(Total 4 marks)
64
60. The graph below shows the tide heights, h metres, at time t hours after midnight, for Tahini
island.
5
4
3
2
1
01 2 3 4 5 6 7 8
Number of hours after midnight
Tide heights for Tahini
Hei
gh
t in
met
res
h
t
(a) Use the graph to find
(i) the height of the tide at 03:15;
(ii) the times when the height of the tide is 3.5 metres. (3)
(b) The best time to catch fish is when the tide is below 3 metres. Find this best time, giving
your answer as an inequality in t. (3)
Due to the location of Tahini island, there is very little variation in the pattern of tidal heights.
The maximum tide height is 4.5 metres and the minimum tide height is 1.5 metres. The height h
can be modelled by the function
h(t) = a cos(bt°) + 3.
(c) Use the graph above to find the values of the variables a and b. (4)
(d) Hence calculate the height of the tide at 13:00. (3)
(e) At what time would the tide be at its lowest point in the second 8 hour period? (2)
(Total 15 marks)
65
61. The diagrams below show the graphs of two functions, y = f(x), and y = g(x).
(b)(a) yy
x
x
2
1
0 1 2
1
0.5
–0.5
–1
–1.5
1.5y f x = ( ) y g x = ( )
–360º 360º–180º 180º
State the domain and range of
(a) the function f;
(b) the function g.
Working:
Answers:
(a) Domain of f ...............………………......
Range of f .............................................
(b) Domain of g ...............……………….....
Range of g .............................................
(Total 8 marks)
66
62. The diagram below represents a stopwatch. This is a circle, centre O, inside a square of side
6 cm, also with centre O. The stopwatch has a minutes hand and a seconds hand. The seconds
hand, with end point T, is shown in the diagram, and has a radius of 2 cm.
A
D
T
O
B
C
6 cm
r
qp
(a) When T is at the point A, the shortest distance from T to the base of the square is p.
Calculate the value of p. (2)
(b) In 10 seconds, T moves from point A to point B. When T is at the point B, the shortest
distance from T to the base of the square is q. Calculate
(i) the size of angle AOB;
(ii) the distance OD;
(iii) the value of q. (5)
(c) In another 10 seconds, T moves from point B to point C. When T is at the point C, the
shortest distance from T to the base of the square is r. Calculate the value of r. (4)
Let d be the shortest distance from T to the base of the square, when the seconds hand has
moved through an angle . The following table gives values of d and .
Angle 0° 30° 60° 90° 120° 150° 180° 210° 240° 270° 300° 330 360°
Distance p 4.7 q 3 r 1.3 1 1.3 r 3 q 4.7 p
d
67
The graph representing this information is as follows.
d
p
q
r
0º 30º 60º 90º 120º 150º 180º 210º 240º 270º 300º 330º 360ºangle
The equation of this graph can be written in the form d = c + k cos().
(d) Find the values of c and k. (4)
(Total 15 marks)
68
63. Consider the function f (x) = 2 sin x – 1 where 0 ≤ x ≤ 720°.
(a) Write down the period of the function.
(b) Find the minimum value of the function.
(c) Solve f (x) = l.
Working:
Answers:
(a) ..................................................................
(b) ..................................................................
(c) ..................................................................
(Total 8 marks)
64. The curve shown in the figure below is part of the graph of the function, f (x) = 2 + sin (2x),
where x is measured in degrees.
4
3.5
3
2.5
2
1.5
1
0.5
–0.5
0–180–150–120 –90 –60 –30 30 60 90 120 150 180 210 240
f x( )
x
69
(a) Find the range of f (x).
(b) Find the amplitude of f (x).
(c) Find the period of f (x).
(d) If the function is changed to f (x) = 2 + sin (4x) what is the effect on the period, compared
to the period of the original function?
Working:
Answers:
(a) ..................................................................
(b) ..................................................................
(c) ..................................................................
(d) ..................................................................
(Total 8 marks)
70
65. The diagram shows the graph of y = sin ax + b.
30° 60° 90° 120° 150° 180° 210° 240° 270° 300° 330° 360°
2
1
0 x
y
(a) Using the graph, write down the following values
(i) the period;
(ii) the amplitude;
(iii) b.
(b) Calculate the value of a.
Working:
Answers:
(a) (i) ...........................................................
(ii) ...........................................................
(iii) ...........................................................
(b) ..................................................................
(Total 8 marks)
71
66. Below is a graph of the function y = a + b sin (cx) where a, b and c are positive integers and x is
measured in degrees.
0 50 100 150 200 250 300 350
7
6
5
4
3
2
x
y
Find the values of a, b and c. (Total 6 marks)
67. (a) Sketch the graph of the function f : x 1 + 2 sin x, where x , –360° ≤ x ≤ 360°. (4)
(b) Write down the range of this function for the given domain. (2)
(c) Write down the amplitude of this function. (1)
(d) On the same diagram sketch the graph of the function g : x sin 2x, where x ,
–360° ≤ x ≤ 360°. (4)
(e) Write down the period of this function. (1)
72
(f) Use the sketch graphs drawn to find the number of solutions to the equation
f (x) = g (x) in the given domain. (1)
(g) Hence solve the equation 1 + 2 sin x = sin 2x for 0° ≤ x ≤ 360°. (4)
(Total 17 marks)
68. The graphs of three trigonometric functions are drawn below. The x variable is measured in
degrees, with 0 ≤ x ≤ 360°. The amplitude 'a' is a positive constant with 0 < a ≤ 1.
2
90
90
90
180
180
180
270
270
270
360
360
360
x
x
y
Graph A Graph B
Graph C
0
0
a
–a
a
–a
(a) Write the letter of the graph next to the function representing that graph in the box below.
FUNCTION GRAPH
y = a cos (x)
y = a sin (2x)
y = 2 + a sin (x)
73
(b) State the period of the function shown in graph B.
74
(c) State the range of the function 2 + a sin (x) in terms of the constant a.
Working:
Answers:
(c) ...................................................
(b) ...................................................
(Total 8 marks)
75
69.
x
y
4
3
2
1
0
–1
–2
–3
–4
90° 180° 270° 360°
The graph represents the function y = 4 sin (3x).
(a) (i) Write down the period of the function.
(ii) Write down the amplitude of the function.
(b) Draw the line y = 2 on the diagram.
(c) Using the graph, or otherwise, solve the equation 4 sin (3x) = 2 for 0 x 90.
76
Working:
Answers:
(a) (i)..............................................
(ii).............................................
(c) ....................................................
(Total 8 marks)
70. The temperature (C) during a 24 hour period in a certain city can be modelled by the function T
(t) = – 3 sin (bt) +2, where b is a constant, t is the time in hours and bt is measured in degrees.
The graph of this function is illustrated below.
T ( )C
t (hours)
–1
5
4
3
2
1
0
(a) Determine how many times the temperature is exactly 0C during this 24 hour period.
(b) Write down the time at which the temperature reaches its maximum value.
(c) Write down the interval of time in which the temperature changes from −1C to 2C.
(d) Calculate the value of b.
77
Working:
Answers:
(a) .....................................................
(b) .....................................................
(c) .....................................................
(d) ..................................................... (Total 6 marks)
71. The graph of y = a sin 2x + c is shown below, −180 x 360, x is measured in degrees.
3
2
1
–180 –120 –60 60 120 180 240 300 360
–1
(a) State:
(i) the period of the function;
(ii) the amplitude of the function.
(b) Determine the values of a and c.
(c) Calculate the value of the first negative x-intercept.
78
Working:
Answers:
(a)
(b) ...................................................
(c) ...................................................
(i)...........................................
(ii)..........................................
(Total 6 marks)
79
72. The figure below shows the set U of all functions of x.
A is the set of all sine and cosine functions of x (measured in degrees).
B is the set of all functions of x with period 120.
C is the subset of A, containing functions whose amplitude is 3.
U
A
C
B
Write the following functions in the correct place in the Venn diagram.
(You may use the numbers (i) to (vi) if you wish, to achieve a better fit.)
(i) sin (x);
(ii) x2;
(iii) 3 cos (x);
(iv) 2 sin (3x);
(v) 3 cos (3x);
(vi) −3 sin (2x).
80
Working:
(Total 6 marks)
73. (a) Sketch the graph of the function y =1+ 2
)2(sin x for 0 x 360 on the
axes below. (4)
90 180 270 360
2
1
–1
–2
x
y
(b) Write down the period of the function. (1)
(c) Write down the amplitude of the function. (1)
81
Working:
Answers:
(c) ...................................................
(b) ...................................................
(Total 6 marks)
82
74. The depth, in metres, of water in a harbour is given by the function d = 4 sin (0.5t) + 7, where t
is in minutes, 0 t 1440.
(a) Write down the amplitude of d. (1)
(b) Find the maximum value of d. (1)
(c) Find the period of d. Give your answer in hours. (2)
On Tuesday, the minimum value of d occurs at 14:00.
(d) Find when the next maximum value of d occurs. (2)
Working:
Answers:
(a) .....................................................
(b) .....................................................
(c) .....................................................
(d) ..................................................... (Total 6 marks)
83
75. The graph below represents the temperature (T Celsius) in Washington measured at midday
during a period of thirteen consecutive days starting at Day 0.
These points also lie on the graph of the function
T(x) = a + b cos (cx), 0 x 12,
where a, b and c .
23
22
21
20
19
18
17
16
2 4 6 8 10 120
0
T
x
(a) Find the value of
(i) a;
(ii) b. (2)
(b) Show that c = 30. (1)
(c) Using the graph, or otherwise, write down the part of the domain for which the midday
temperature is less than 18.5. (2)
(Total 5 marks)
