Download 1. The function Q (t) = 0.003t 2 – 0.625t + 25 represents the amount

The function Q (t) = 0.003t2 – 0.625t + 25 represents the amount of energy in a battery after t
minutes of use.
1.
(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.
(Total 6 marks)
The following figures show the graphs of y = f (x) with f (x) chosen to be various cubic functions.
The value of a is positive.
2.
y
A
y
y
B
C
a
a
a
a
x
a
x
a
y
y
E
D
a
a
x
a
(a)
x
a
x
In the table below, write the letter corresponding to the graph of y = f (x) in the space next to
the cubic function.
(Note: one of the graphs is not represented in this table)
cubic function f (x)
graph label
f (x) = x3 + a
f (x) = (x – a)3 + a
f (x) = x3
f (x) = (x – a)3
(b)
State which one of the graphs represents a function that has a positive gradient for every
value of x.
(c)
State how many of the graphs have the x-axis as a tangent at some point.
(Total 6 marks)
1
3.
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)
5
4
3
2
1
t (hours)
0
–1
(a)
Determine how many times the temperature is exactly 0°C 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 −1°C to 2°C.
(d)
Calculate the value of b.
(Total 6 marks)
4.
Two functions f (x) and g (x) are given by
f (x) =
g (x) =
1
,
x +1
2
x , x ≥ 0.
(a)
Sketch the graphs of f (x) and g (x) together on the same diagram using values of x between
–3 and 3, and values of y between 0 and 2. You must label each curve.
(b)
State how many solutions exist for the equation
(c)
Find a solution of the equation given in part (b).
1
–
x +1
2
x = 0.
(Total 6 marks)
5.
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).
(Total 6 marks)
2
6.
6
The figgure below shhows the graphhs of the funcctions f (x) = 2x + 0.5 and g (x) = 4 − x2 for
f values of
x between –3 and 3..
y
f(x)
6
B
3
A
–3
–2
–1
0
1
–
–3
(a)
W
Write
down thhe coordinatees of the pointts A and B.
(b)
W
Write
down thhe set of valuues of x for whhich f (x) < g (x).
2
3 x
g(x)
(Total 6 marks)
m
7.
7
(a)
S
Sketch
the graph of y = 3 +
(b)
W
Write
down thhe equations of
3
forr −10 ≤ x ≤10.
x +1
(
(i)
the horrizontal asympptote;
(
(ii)
the verrtical asymptoote.
(Total 6 marks)
m
8.
8
The vaalue of a car decreases
d
eachh year. This vvalue can be calculated
c
usinng the functio
on
v = 32
3 000rt, t ≥ 00, 0 < r < 1,
where v is the valuee of the car in USD, t is thee number of years
y
after it w
was first bougght and r is a
constaant.
(a)
(b)
(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.
Find how many years it will take for the value of the car to be less than 8000 USD.
(Total 6 marks)
9.
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.
(Total 6 marks)
10.
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.
4
(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).
(Total 6 marks)
11.
The mapping below is of the form f : x a a × 2x + b and maps the elements of x to elements of y.
(a)
0
–1
–2
–1.75
1
0
–1
–1.5
g
30
(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.
(Total 6 marks)
12.
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)
5
A
y
C
B
y
y
x
0
x
0
x
0
D
y
E
y
F
y
0
x
0
Function
(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 − x3
0
x
x
Graph label
(Total 6 marks)
13.
(a)
Sketch the graph of the function y =1+
sin ( 2 x)
for 0° ≤ x ≤ 360° on the
2
axes below.
(4)
6
y
2
1
180
90
270
x
360
–1
(b)
Write down the period of the function.
(1)
(c)
Write down the amplitude of the function.
(1)
(Total 6 marks)
14.
(a)
f : x → 3x − 5 is a mapping from the set S to the set T as shown below.
T
S
1
p
q
7
5
10
Find the values of p and q.
(2)
(b)
A function g is such that g (x) =
(i)
2
.
( x − 2) 2
State the domain of the function g(x).
(2)
(ii)
State the range of the function g(x).
(1)
(iii)
Write down the equation of the vertical asymptote.
(1)
(Total 6 marks)
7
15.
(a)
Consider the numbers 2,
3, −
2 and the sets
,
,
, and
.
Complete the table below by placing a tick in the appropriate box if the number is an
element of the set, and a cross if it is not.
(i)
2
(ii)
3
−
(iii)
2
3
(3)
(b)
A function f is given by f : x → 2x2 – 3x, x ∈ {–2, 2, 3}.
(i)
Draw a mapping diagram to illustrate this function.
(ii)
Write down the range of function f.
(3)
(Total 6 marks)
16.
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)
(Total 6 marks)
17.
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)
Total 6 marks)
8
18.
(a)
Factorize the expression x2 – 3x – 10.
(2)
(b)
A function is defined as f (x) = 1 + x3 for x ∈
(i)
List the elements of the domain of f (x).
(ii)
Write down the range of f (x).
, –3 ≤ x ≤ 3.
(4)
(Total 6 marks)
9