12
05.04.09 sundayherald
MATHEMATICS
STANDARDGRADE : GENERAL LEVEL (2008 PAPER 1)
1.
To enter the castle she needs the
correct four digit code.
(b) 6.39 × 9
(c) 8.74 ÷ 200
The computer gives her some clues:
(d) 5 of 420
6
2.
Marks
Marks
5. Samantha is playing the computer
game “Castle Challenge”.
Carry out the following calculations.
(a) 12.76 – 3.18 + 4.59
Marks
In the “Fame Show”, the percentage of
telephone votes cast for each act is shown
below.
Plastik Money
Brian Martins
Starshine
Carrie Gordon
•
•
•
only digits 1 to 9 can be used
each digit is greater than the one before
the sum of all four digits is 14.
(a) The first code Samantha found was 1, 3, 4, 6.
23%
35%
30%
12%
Use the clues to list all the possible codes in the table below.
1
3
4
6
Altogether 15 000 000 votes were cast.
How many votes did Starshine receive?
3.
Marks
AB and BC are two sides of a kite ABCD.
y
6
4
B
3
C
2
(b) The computer gives Samantha another clue.
•
–6
–4
–2
O
2
4
x
6
three of the digits in the code are prime numbers
What is the four digit code Samantha needs to enter the castle?
–2
A
Marks
6.
–4
-3
–6
12
-9
-8
(a) Plot point D to complete kite ABCD.
1
5
7
-11
(b) Reflect kite ABCD in the y-axis.
4.
Marks
Europe is the world’s second smallest continent.
Find the three numbers from the circle which add up to –10.
Write this number in scientific notation.
You must show your working.
3.
ANSWERS
1.
y
6
3.
(c)0·0437
(d)350
4500000
4.
5.
5
4
C
3 B
(a)14·17
(b)57·51
2.
The circle above contains seven numbers.
Its area is approximately 10 400 000 square kilometres.
2
1
0
–6 –5 –4 -3 –2 –1
–1
–2
A
D
1
2
3
4
5
6
x
6.
7.
8.
–3
–4
–5
–6
9.
1·04 × 107
(a) 1238, 1247, 12 56, 2345
(b) 2345
–9, –8, 7
£1·22
(a) 23
(b)
220°
sundayherald 05.04.09
7.
13
Marks
The cost of sending a letter depends on the size
of the letter and the weight of the letter.
HIGHER (2008 PAPER 1, SECTION A)
SECTION A
Weight
Format
Letter
Large Letter
ALL questions should be attempted.
Cost
1st Class Mail
2nd Class Mail
0–100 g
34p
24p
0–100 g
48p
40p
101–250 g
70p
60p
251–500 g
98p
83p
501–750 g
142p
120p
1.
A sequence is defined by the recurrence relation
un+1 = 0.3un + 6 with u10 = 10.
What is the value of u12?
A
C
2.
6.6
8.7
7. 8
8
7
9. 6
B
C
D
The x-axis is a tangent to a circle with centre (–7, 6) as shown in the diagram.
y
Claire sends a letter weighing 50 g by 2nd class mail.
She also sends a large letter weighing 375 g by 1st class mail.
C(–7, 6)
Use the table above to calculate the total cost.
8.
Marks
Four girls and two boys decide to organise
a tennis tournament for themselves.
O
Each name is written on a plastic token and
put in a bag.
What is the equation of the A
circle?
(x + 7) + (y – 6) = 1
(a) What is the probability that the first token drawn from the bag has a
girl’s name on it?
A
B
(x + 7)2 + (y – 6)2 = 1 B
(x + 7) + (y – 6) 49 C
2
(x –+ 7)
7)2 +
+ (y
(y +
– 6)
49
(x
6) = 36
C
(x – 7)2 + (y + 6)2 = 36 D
(x + 7)2 + (y – 6)2 = 36
(b) The first token drawn from the bag has a girl’s name on it.
This token is not returned to the bag.
What is the probability that the next token drawn from the bag has a
boy’s name on it?
"k"
3.
Marks
9.
C
4.
O
A
70 °
T
In the diagram above:
x
"−#
The vectors u = −1
"1$
"0"
and v = "4# are perpendicular.
"k$
What is the value of k?
A
0
B
3
C
4
D
5
A sequence is generated by the recurrence relation un+1 = 0.4un – 240.
What is the limit of this sequence as n → ∞ ?
A
– 800
B
– 400
C
200
D
400
B
5.
The diagram shows a circle, centre (2, 5) and a tangent drawn at the point (7, 9).
What is the equation of this tangent?
• O is the centre of the circle
• AB is a tangent to the circle at T
• angle BTC = 70 °.
y
(7, 9)
Calculate the size of the shaded angle TOC.
O
6.
(2, 5)
x
A
y – 9 = − 5 (x – 7) B
4
C
y–7=
4
(x – 9)
5
D
4
y + 9 = − (x + 7)
5
y+9=
5
(x + 7)
4
What is the solution of the equation 2 sin x − 3 = 0 where
A
π
6
C
3π
4
2π
3
5π
D
6
B
π
≤ x ≤ π?
2
14
05.04.09 sundayherald
The adiagram
shows
a line
L; theLangle
between
L and
the 12.
positive
direction
ofRSTU,
the
7. The diagram shows
line L; the
angle
between
and the
positive
direction
of the
In the
diagram
VWXY represents a cuboid.
e direction of the x-axis is 135°, as shown.
→
→
→"
SR represents vector f, ST represents vector g and SW represents vector h.
y
→
Express VT in terms of f, g and h.
Y
L
V
135°
O
X
W
x
h
U
R
What is the gradient of line L?
8.
f
3
2
A
11
−−
22
B
−
C
−−
11
D
1 1
2 2
A
C
g
S
→
VT = f + g + h
→
→
VT = − f + g − h
B
D
T
→
VT = f − g + h
→
VT = − f − g + h
The diagram shows part of the graph of a function with equation y = f(x).
13.
y
The diagram shows part of the graph of a quadratic function y = f(x).
The graph has an equation of the form y = k(x – a)(x – b).
(0, 4)
y
O
x
y = f(x)
12
(3, –3)
Which of the following diagrams shows the graph with equation y = –f(x – 2)?
y
A
B
(2, 4)
O
y
(–2, –4)
D
10.
(5, 3)
(2, –4)
sin a =
15.
3 , find an expression for sin(x + a).
5
4
3
sin x + cos x
5
5
2
3
sin x − cos x
5
5
Here are two statements about the roots of the equation x + x + 1 = 0:
17.
Which of the following is true?
A
Neither statement is correct.
A
Neither statement is correct.
ly statement (1)
B
C
Only statement (1) is correct.
Only statement (2) is correct.
Only statement (2) is correct.
D
Both statements are correct.
E(–2, –1, 4), P(1, 5, 7) and F(7, 17, 13) are three collinear points.
P lies between E and F.
What is the ratio in which P divides EF?
B
y = 3(x + 1)(x + 4)
y = 12(x – 1)(x – 4)
D
y = 12(x + 1)(x + 4)
Find
∫ 4 sin (2x + 3) dx.
cos (
3)
A
– 4cos
3) + c
cos ((2x + 3)
B
C
–2cos
3) +
+ cc
4cos (2x
(2x +
+ 3)
C
4cos (2x + 3) + c
D
8cos (2x + 3) + c
What is the derivative of (x3 + 4)2?
1 3
A (3x2 + 4)2
B
(x + 4)3
3 ( + 4)
6x2(x3 + 4)
D
2(3x2 + 4)–1
2x2 + 4x + 7 is expressed in the form 2(x + p)2 + q.
What is the value of q?
2
C
y = 3(x – 1)(x – 4)
C
C
16.
(1) the roots are equal;
(2) the roots are real.
11.
14.
x
O
Given that 0 ≤ a ≤ π and
2
3
A sin x +
B
5
3
4
C
sin x − cos x
D
5
5
A
y
x
(0, –2)
x
4
What is the equation of the A
graph?
y 3(x – 1)(x – 4)
x
O
(3, 5)
O
9.
O 1
(1, 3)
x
(5, –3)
C
y
A
5
B
7
C
9
D 11
2
A function f is given by f(x) = 9 − x .
What is a suitable domain
A x ≥ of
3 f?
18.
A
x≥3
B
C
x ≤≤3x ≤ 3
–3
C
–3 ≤ x ≤ 3
D
–9 ≤ x ≤ 9
Vectors p and q are such that |p| = 3, |q| = 4 and p.q = 10.
Find the valueAof q.(p
0 + q).
A
1:1
B
1:2
A
B
0
14
B
C
14
26
C
1:4
D
1:6
C
26
D 28
sundayherald 05.04.09
19.
15
x
The diagram shows part of the graph whose equation is of the form y = 2m .
ANSWERS
What is the value of m?
y
SECTIONA
(3, 54)
x
O
A
3
C
8
2.D
3.C
4.B
5.A
6.B
7.C
8.D
9.B
10.A 11.B 12.C
13.A 14.B 15.C 16.A
2
B
1.C
17.C 18.C 19.B 20.D
SECTIONB
D 18
21.(a) (–1,4)maximum
20.
(1,0)minimum
The diagram shows part of the graph of y = log3(x – 4).
The point (q, 2) lies on the graph.
y
(b)
so(x–1)isafactor
(q, 2)
(5, 0)
(i)x=1,f(x)=0
y = log3(x – 4)
(c)
x
O
(ii)(x–1)(x–1)(x+2)
( ) (
)(
)(
)
y
(c)
(–1, 4)
What is the value of q?
A
(0, 2)
6
B
7
C
8
(–2, 0)
D 13
[END OF SECTION A]
SECTION B
ALL questions should be attempted.
22.
21.
A function f is defined on the set of real numbers by f(x) = x3 – 3x + 2.
(b)(1,3)
) Find
the coordinates
of stationary
the stationary
points
(a) Find
the coordinates
of the
points
on the curve y = f(x) and
) and determine their nature.
(b) (i) Show that (x – 1) is a factor of x3 – 3x + 2.
23.
(ii) Hence or otherwise factorise x3 – 3x + 2 fully.
meets
thethe
axes
and
hence
sketch
the curve.
)both
State
coordinates
of
thewhere
points
where
curve
with equation
(c) State the
coordinates
of the
points
the
curvethe
with
equation
y = f(x) y = f(x)
y = f(x) meets both the axes and hence sketch the curve.
22.
The diagram shows a sketch of the
curve with equation y = x3 – 6x2 + 8x.
(a) Find the coordinates of the points
on the curve where the gradient of
the tangent is –1.
y
y = x3 – 6x2 + 8x
O
(b) The line y = 4 – x is a tangent to
this curve at a point A. Find the
coordinates of A.
23.
Functions f, g and h are defined on suitable domains by
f(x) = x2 – x + 10, g(x) = 5 – x and h(x) = log2 x.
(a) Find expressions for h(f(x)) and h(g(x)).
(b) Hence solve h(f(x)) – h(g(x)) = 3.
[END OF SECTION B]
(a)(1,3),(3,–3)
x
(a)h(f(x))=log2(x2–x+10)
h(g(x))=log2(5–x)
(b)x=3,–10
(1, 0)
x