1.
(a)
Factorize x2 – 3x – 10.
(b)
Solve the equation x2 – 3x – 10 = 0.
Working:
Answers:
(a) ..................................................................
(b) ..................................................................
(Total 4 marks)
2.
The diagram shows part of the graph with equation y = x2 + px + q. The graph cuts the x-axis
at –2 and 3.
y
6
4
2
?
?
0
?
1
2
3
4
x
?
?
?
1
Find the value of
(a)
p;
(b)
q.
Working:
Answers:
(a) ..................................................................
(b) ..................................................................
(Total 4 marks)
3.
The diagram shows the parabola y = (7 – x)(l + x). The points A and C are the x-intercepts and
the point B is the maximum point.
y
B
A
0
C
x
2
Find the coordinates of A, B and C.
Working:
Answer:
......................................................................
(Total 4 marks)
4.
The diagram shows the graph of the function y = ax2 + bx + c.
y
x
3
Complete the table below to show whether each expression is positive, negative or zero.
Expression
positive
negative
zero
a
c
b2 – 4ac
b
Working:
(Total 4 marks)
4
5.
(a)
Express f (x) = x2 – 6x + 14 in the form f (x) = (x – h)2 + k, where h and k are to be
determined.
(b)
Hence, or otherwise, write down the coordinates of the vertex of the parabola with
equation y – x2 – 6x + 14.
Working:
Answers:
(a) ..................................................................
(b) ..................................................................
(Total 4 marks)
5
6.
(a)
Express y = 2x2 – 12x + 23 in the form y = 2(x – c)2 + d.
The graph of y = x2 is transformed into the graph of y = 2x2 – 12x + 23 by the transformations
a vertical stretch with scale factor k followed by
a horizontal translation of p units followed by
a vertical translation of q units.
(b)
Write down the value of
(i)
k;
(ii)
p;
(iii)
q.
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(Total 6 marks)
6
7.
The equation x2 – 2kx + 1 = 0 has two distinct real roots. Find the set of all possible values of k.
Working:
Answer:
…………………………………………..
(Total 6 marks)
8.
Three of the following diagrams I, II, III, IV represent the graphs of
(a)
y = 3 + cos 2x
(b)
y = 3 cos (x + 2)
(c)
y = 2 cos x + 3.
7
Identify which diagram represents which graph.
y
I
y
II
4
2
1
2
? 

–
?
x
? 

–





3

5
2
4
1
–
x


y
IV
? 


x


?
y
III


3



x


2
1
?
–
? 



Working:
Answers:
(a) ..................................................................
(b) ..................................................................
(c) ..................................................................
(Total 4 marks)
8
9.
The diagram shows the graph of y = f (x), with the x-axis as an asymptote.
y
B(5, 4)
x
A(? , ? )
(a)
On the same axes, draw the graph of y =f (x + 2) – 3, indicating the coordinates of the
images of the points A and B.
(b)
Write down the equation of the asymptote to the graph of y = f (x + 2) – 3.
Working:
Answer:
(b) ..................................................................
..
(Total 4 marks)
9
10.
 
The vectors i , j are unit vectors along the x-axis and y-axis respectively.






The vectors u = – i + 2 j and v = 3 i + 5 j are given.
(a)




Find u + 2 v in terms of i and j .



A vector w has the same direction as u + 2 v , and has a magnitude of 26.
(b)



Find w in terms of i and j .
Working:
Answers:
(a) ..................................................................
(b) ..................................................................
(Total 4 marks)
10
11.
The quadrilateral OABC has vertices with coordinates O(0, 0), A(5, 1), B(10, 5) and C(2, 7).
(a)
Find the vectors OB and AC .
(b)
Find the angle between the diagonals of the quadrilateral OABC.
Working:
Answers:
(a) ..................................................................
(b) ..................................................................
(Total 4 marks)
12.
  2 4
 and D =
Let C = 
 1 7
 5 2

 .
1 a
The 2 × 2 matrix Q is such that 3Q = 2C – D
(a)
Find Q.
(3)
(b)
Find CD.
(4)
(c)
Find D–1.
(2)
(Total 9 marks)
11
13.
 3 2
 and B =
Let A = 
 k 4
(a)
2A− B;
(b)
det (2A− B).
 2 2

 . Find, in terms of k,
 1 3
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(Total 6 marks)
14.
 5 – 2
 .
Consider the matrix A = 
7 1 
(a)
Write down the inverse, A–l.
(2)
12
(b)
B, C and X are also 2 × 2 matrices.
(i)
Given that XA + B = C, express X in terms of A–1, B and C.
(ii)
6 7 
 , and
Given that B = 
 5 – 2
 – 5 0
, find X.
C = 
 – 8 7
(4)
(Total 6 marks)
15.
 3
 
 1 x  1
 and B =  x  .
Let A = 
3 1 4 
 2
 
(a)
Find AB.
(b)
 20 
The matrix C =   and 2AB = C. Find the value of x.
 28 
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(Total 6 marks)
13
16.
Each year for the past five years the population of a certain country has increased at a steady
rate of 2.7% per annum. The present population is 15.2 million.
(a)
What was the population one year ago?
(b)
What was the population five years ago?
Working:
Answers:
(a) ..................................................................
(b) ..................................................................
(Total 4 marks)
17.
Find the exact value of x in each of the following equations.
(a)
5x+1 = 625
14
(b)
loga (3x + 5) = 2
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(Total 6 marks)
2x
18.
Solve the equation 9x–1 =  1  .
3
Working:
Answer:
......................................................................
(Total 4 marks)
15
19.
(a)
Given that log3 x – log3 (x – 5) = log3 A, express A in terms of x.
(b)
Hence or otherwise, solve the equation log3 x – log3 (x – 5) = 1.
Working:
Answers:
(a) ..................................................................
(b) ..................................................................
(Total 6 marks)
16
20.
Given that log5 x = y, express each of the following in terms of y.
(a)
log5 x2
(b)
log5  1 
 x
(c)
log25 x
Working:
Answers:
(a) ..................................................................
(b) ..................................................................
(c) ..................................................................
(Total 6 marks)
17
21.
(a)
(b)
Let logc 3 = p and logc 5 = q. Find an expression in terms of p and q for
(i)
log c 15;
(ii)
log c 25.
Find the value of d if log d 6 =
1
.
2
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(Total 6 marks)
18
22.
Find the sum of the arithmetic series
17 + 27 + 37 +...+ 417.
Working:
Answer:
.........................................................................
(Total 4 marks)
23.
In an arithmetic sequence, the first term is 5 and the fourth term is 40. Find the second term.
Working:
Answer:
......................................................................
(Total 4 marks)
19
24.
In an arithmetic sequence, the first term is –2, the fourth term is 16, and the nth term is 11 998.
(a)
Find the common difference d.
(b)
Find the value of n.
Working:
Answers:
(a) ..................................................................
(b) ..................................................................
(Total 6 marks)
25.
Arturo goes swimming every week. He swims 200 metres in the first week. Each week he
swims 30 metres more than the previous week. He continues for one year (52 weeks).
(a)
How far does Arturo swim in the final week?
(b)
How far does he swim altogether?
Working:
Answers:
(a) ..................................................................
(b) ..................................................................
(Total 6 marks)
20
26.
Gwendolyn added the multiples of 3, from 3 to 3750 and found that
3 + 6 + 9 + … + 3750 = s.
Calculate s.
Working:
Answer:
..................................................................
(Total 6 marks)
27.
The first four terms of a sequence are 18, 54, 162, 486.
(a)
Use all four terms to show that this is a geometric sequence.
(2)
21
(b)
(i)
Find an expression for the nth term of this geometric sequence.
(ii)
If the nth term of the sequence is 1062 882, find the value of n.
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(4)
(Total 6 marks)
22
28.
The first term of an infinite geometric sequence is 18, while the third term is 8. There are two
possible sequences. Find the sum of each sequence.
Working:
Answers:
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(Total 6 marks)
29.
The diagrams below show two triangles both satisfying the conditions
AB = 20 cm, AC = 17 cm, AB̂C = 50°.
Diagrams not
to scale
B
Triangle 1
Triangle 2
A
A
C
B
C
23
(a)
Calculate the size of AĈB in Triangle 2.
(b)
Calculate the area of Triangle 1.
Working:
Answers:
(a) ..................................................................
(b) ..................................................................
(Total 4 marks)
30.
A triangle has sides of length 4, 5, 7 units. Find, to the nearest tenth of a degree, the size of the
largest angle.
Working:
Answer:
......................................................................
(Total 4 marks)
24
31.
The following diagram shows a triangle ABC, where BC = 5 cm, B̂ = 60°, Ĉ = 40°.
A
B
(a)
Calculate AB.
(b)
Find the area of the triangle.
40
60
5 cm
C
Working:
Answers:
(a) …………………………………………..
(b) …………………………………………..
(Total 6 marks)
25
32.
Two boats A and B start moving from the same point P. Boat A moves in a straight line at
20 km h–1 and boat B moves in a straight line at 32 km h–1. The angle between their paths is
70°.
Find the distance between the boats after 2.5 hours.
Working:
Answer:
......................................................................
(Total 6 marks)
26
33.
(a)
Factorize the expression 3 sin2 x – 11 sin x + 6.
(b)
Consider the equation 3 sin2 x – 11 sin x + 6 = 0.
(i)
Find the two values of sin x which satisfy this equation,
(ii)
Solve the equation, for 0°  x  180°.
Working:
Answers:
(a) ..................................................................
(b) (i) ...........................................................
(ii) ...........................................................
(Total 6 marks)
27
34.
(a)
Write the expression 3 sin2 x + 4 cos x in the form a cos2 x + b cos x + c.
(b)
Hence or otherwise, solve the equation
3 sin2 x + 4 cos x – 4 = 0,
0  x  90.
Working:
Answers:
(a) ..................................................................
(b) ..................................................................
(Total 4 marks)
35.
Solve the equation 3 sin2 x = cos2 x, for 0°  x  180°.
Working:
Answer:
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(Total 4 marks)
28
A student measured the diameters of 80 snail shells. His results are shown in the following
cumulative frequency graph. The lower quartile (LQ) is 14 mm and is marked clearly on the
graph.
90
80
Cumulative frequency
36.
70
60
50
40
30
20
10
0
5
0
10
15
LQ = 14
20
25
30
35
40
45
Diameter (mm)
(a)
(b)
On the graph, mark clearly in the same way and write down the value of
(i)
the median;
(ii)
the upper quartile.
Write down the interquartile range.
Working:
Answer:
(b) ..................................................................
(Total 6 marks)
29
37.
In a school with 125 girls, each student is tested to see how many sit-up exercises (sit-ups) she
can do in one minute. The results are given in the table below.
(a)
Number of sit-ups
Number of students
Cumulative
number of students
15
11
11
16
21
32
17
33
p
18
q
99
19
18
117
20
8
125
(i)
Write down the value of p.
(ii)
Find the value of q.
(3)
(b)
Find the median number of sit-ups.
(2)
(c)
Find the mean number of sit-ups.
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(2)
(Total 7 marks)
30
A taxi company has 200 taxi cabs. The cumulative frequency curve below shows the fares in
dollars ($) taken by the cabs on a particular morning.
200
180
160
140
120
Number of cabs
38.
100
80
60
40
20
10
20
30
40
50
Fares ($)
60
70
80
31
(a)
Use the curve to estimate
(i)
the median fare;
(ii)
the number of cabs in which the fare taken is $35 or less.
(2)
The company charges 55 cents per kilometre for distance travelled. There are no other charges.
Use the curve to answer the following.
(b)
On that morning, 40% of the cabs travel less than a km. Find the value of a.
(4)
(c)
What percentage of the cabs travel more than 90 km on that morning?
(4)
(Total 10 marks)
39.
The table below represents the weights, W, in grams, of 80 packets of roasted
peanuts.
Weight (W)
80 < W  85
85 < W  90
90 < W  95
Number of
packets
5
10
15
(a)
95 < W  100 100 < W  105 105 < W  110 110 < W  115
26
13
7
4
Use the midpoint of each interval to find an estimate for the standard deviation of the
weights.
(3)
(b)
Copy and complete the following cumulative frequency table for the above data.
Weight (W)
W  85
W  90
Number of
packets
5
15
W  95
W  100
W  105
W 110
W  115
80
(1)
32
(c)
A cumulative frequency graph of the distribution is shown below, with a scale 2 cm for
10 packets on the vertical axis and 2 cm for 5 grams on the horizontal axis.
80
70
60
50
Number
of
packets
40
30
20
10
80
85
90
95
100
Weight (grams)
105
110
115
Use the graph to estimate
(i)
the median;
(ii)
the upper quartile (that is, the third quartile).
Give your answers to the nearest gram.
(4)
33
(d)
Let W1, W2, ..., W80 be the individual weights of the packets, and let W be their mean.
What is the value of the sum
(W1 – W )  (W2 – W )  (W3 – W )  . . .  (W79 – W )  (W80 – W ) ?
(2)
(e)
One of the 80 packets is selected at random. Given that its weight satisfies
85 < W  110, find the probability that its weight is greater than 100 grams.
(4)
(Total 14 marks)
34