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SSI MATHEMATICS MODULE SPM (Paper 2) 2009
SIMULTANEOUS EQUATIONS
1. Calculate the value of x and of y that satisfy the following simultaneous linear
equations : 5 m - n = 2
2 m + 3
1 n = 3
2. Calculate the value of x and of y that satisfy the following simultaneous linear
equations :
2 x + 3 y = - 4 3 x - 5 y = 13
3. Calculate the value of x and of y that satisfy the following simultaneous linear
equations : 2 d + 3 e = 9
3
d - e = 2
4. Calculate the value of x and of y that satisfy the following simultaneous linear
equations :
v + 2
1 w = 1
3 v - w = 13
5. Calculate the value of x and of y that satisfy the following simultaneous linear
equations : 6 p - 3 q = 8
2 p - 3
2 q = 2
MATHEMATICAL REASONING
1. (a) Complete each of the following statements with the quantifier “all” or
“some” so
that it will become a false statement.
(i) ………. of the perfect squares are odd numbers.
(ii) ……….. hexagons have six sides. .
(b) Complete the premise in the following argument:
Premise 1 : If set M is a subset of set N, then NNM =∪ .
Premise 2 : ………………………………………………
Conclusion : Set M is not a subset of set N.
(c) State the converse of the following statement and hence determine
whether its
Converse is true or false.
If y < -2, then y < -1 (5 marks)
Answer:
(a) (i) ………. of the perfect squares are odd numbers.
(ii) ……….hexagons have six sides.
(b) Premise 2 :……………………………………………………………………...
(c) ………………………………………………………………………
2. (a) Determine whether the following sentence is a statement or a non-statement.
“ x + y = 2 ”
(b) Write down the converse of the following implication, hence state
whether the converse is true or false.
If x is an odd number, then 2x is an even number.
(c) Make a conclusion by induction for a list of numbers 3, 17, 55, 129,
… which follows the following pattern:
Conclusion: ……………………………………………………………………
(5 marks)
Answer:
(a) …………………………………………………………………………………
………….
(b) …………………………………………………………………………………
………….
(c) Conclusion:
……………………………………………………………………………….
3. (a) Complete the premise in the argument below.
Premise 1 : If n = 10, then 2 x n = 20.
Premise 2 : …………………………..
Conclusion: 10≠n
(b) Using the induction method, make a conclusion for the numerical
sequence 2, 14, 34, 62, … with the pattern shown below:
2 = 4(1)2 – 2
14 = 4(2)2 – 2
34 = 4(3)2 – 2
62 = 4(4)2 – 2
(c) Determine whether the following are true of false statements.
(i) 42 = 16 or 5.22
3=
(ii) 3 – 2 = 6 and -6 < -8
Answer: (a) Premise 2:………………………………………………………………………...
3 = 2 (1) 3 + 1
17 = 2 (2) 3 + 1
55 = 2 (3) 3 + 1
129 = 2 (4) 3 + 1
(b) Conclusion:……………………………………………………………
………….
(c) (i) ………………………
(d) (ii)……………………… 4. a) Combine the statements by using “and’ or ‘or’ to form a false statement
(i) 3 < 4 _____ - 4 > 2
(ii) 4
1 = 0.25 _____
5
1 = 0.30
b) Complete the argument below.
Premise 1 : If a < 0 , then -1 × a > 0
Premise 2 : -1 × a > 0
Conclusion : ______________________________________________
c) Write two implications for the following compound statements.
“ Three points A , B , and C are on a straight line if and only if lines AB
and BC have common point and they are parallel “
Implication 1 :
______________________________________________________
______________________________________________________________
______
Implication 2 :
______________________________________________________
____________________________________________________________________
5. a) Complete the following by using symbol “ > “ or “ < “ to form a
i) false statement : 2
1 ________
3
1 ,
ii) true statement : - 4 _________ 6
b) Combine the two statements below to form a true statement
Statement 1 : 3 + ( -2 ) = 5
Statement 2 : 16 is a perfect square
Answer :
____________________________________________________________
c) Complete the following arguments
i) Premise 1 : All cows eat grass
Premise 2 : ______________________________________________
Conclusion : Si Hitam eat grass
ii) Premise 1 : All octagons have eight sides of equal length
Premise 2 : LMNOPGRS is an octagons
Conclusion : ______________________________________________
VOLUME AND SURFACE AREA
Question 1
Question 2
Question 3
Question 4
Question 5
STRAIGHT LINE
10.1 Diagram below shows a straight line PQ and a straight line RS drawn on a
Cartesian plane. PQ is parallel to RS.
y
P (–3,5)
R (0, 4)
S ( –2, 0) O x
Q
Find
(a) the equation of the straight line PQ.
(b) the x-intercept of the straight line PQ. [5 marks]
10.2 In diagram below, O is the origin, point R lies on the y-axis and point P lies on
the x-axis. Straight line RU is parallel to the x-axis and straight line PR is
parallel
to straight line ST. The equation of the straight line PR is x – 2y = 14.
y
S
P
O x
T (–12, –5)
R U
(a) State the equation of the straight line RU
(b) Find the equation of the straight line ST and hence, state its x-intercept.
[5 marks]
10.3 In diagram below, OPQR is a parallelogram and O is the origin.
y
R (–4, 12)
Q
x O
P (–3, –6)
Find
(a) the equation of the straight line PQ
(b) the y-intercept of the straight line QR.
[5 marks]
10.4 In diagram below, the graph shows that PQ, QR, and RS are straight lines.
P is on the y-axis and Q is on the x-axis. OP is parallel to QR
and PQ is parallel to RS.
y
S(–8, 7)
Q
O x
R
P
The equation of PQ is 2x + y + 5 = 0
(a) State the equation of the straight line QR
(b) Find the equation of the straight line RS and hence, state its y-intercept.
[5 marks]
10.5 In diagram below, O is the origin. The point R is on the x-axis while the point
S is on the y-axis. The straight line QP is parallel to the x-axis. The straight
lines QR and PS are parallel. Given that the gradient of the straight line PS is
–2 .
y
P(–5, 8) Q(3, k)
O R x
S
Find
(a) the value of k,
(b) the equation of the straight line PS,
(c) the coordinates of the point R.
[5 marks]
MATRICES
11.1 (a) It is given that
412
3n
is the inverse matrix of
−
−
12
38
Find the value of n.
(b) Write the following simultaneous linear equations as matrix equation: 8u – 3v = –16
–2u + v = 3
Hence, using matrices , calculate the value of u and of v.
[6 marks]
Answer :(a)
(b)
11.2 It is given that matrix P =
−
21
34 and matrix Q = k
− 41
h2
such that PQ =
10
01.
(a) Find the value of k and of h.
(b) Using matrices, find the value of x and of y that satisfy the following
simultaneous linear equations:
4x – 3y = – 4 x + 2y = 21
[6 marks] Answer :(a) k = .......................
h = .......................
(b)
11.3 (a) The inverse matrix of
−
−
54
23 is
−
−
34
25
p
1
Find the value of p.
(b) Using matrices, calculate the value of x and of y that satisfy the following
simultaneous linear equations:
3x – 2y = – 16
4x – 5y = – 19
[6 marks]
Answer :(a)
(b)
11.4 M is a 2 × 2 matrix where M
−
−
34
56 =
10
01
(a) Find the matrix M.
(b) Write the following simultaneous linear equations as matrix equation:
6x – 5y = – 7
4x – 3y = – 4
Hence, calculate the value of x and of y using matrices ,
[6 marks]
Answer :(a)
(b)
11.5 Given that the matrix equation
=
−
−
−
−
10
01
m9
28
89
23
n
1
(a) Find the value of n and of m.
(b) Hence, using matrices , calculate the value of u and of w that satisfy the
following matrix equation:
−=
−
−
12
6
w
u
89
23
[6 marks]
Answer :(a)
(b)
SETS
( 1 )
The following Venn Diagrams show three sets G, K and M such that universal set
E = G UK UM
a) shade the region for G ∩ M ∩ K’
K
b) shade the region for (G ∪ M ) ∩ K’
K
(2) Each of the following Venn diagrams shows sets P, Q, and R. On the diagram,
shade the region which represents
(a) P’ U Q b) P ∩ Q∩R’
R P Q
P Q R
(3) The Venn Diagram in the answer space shows sets P,Q and R.On the diagram
provided, shade
a) K U L
J
L
K
b) L’ ∩ J U K’
G M
G M
J L
(4) Given that the universal set E = P UQ U R, the shaded region in the Venn
diagram represent.
a) P Q
R
b) P Q
R
( 5 ) The Venn diagram which represents the universal set E = A UB U C.
By using the symbols U and/or W and/or complement of set, express the
relationship between sets A, B and C.
A
a)
B C
A b)
B C
LINES AND PLANES IN 3-DIMENSIONS
( 1 ) S R
P Q
D C
10cm
A B
The figure shows a cube.
Calculate the angle between BS and planes ABCD.
(2) The figure shows a pyramid with a rectangular base,KLMN. The triangle OKN is
a vertical plane. OK = 10 cm, OM = 13 cm
O 10cm
K N
12 cm
L M
a) Find the length of ON, b) Name the angle between line OM and the plane KLMN.
c) Find the angle between the plane OKL and the plane OMN.
Answer: (a)_________________________ (b)_________________________
(c)_________________________
(3) The figure shows a right prism with rectangular base PQRS.
T
U S
5 cm
P R
5cm 12 cm
Q
TURS, PTS and QUR are vertical planes.
a) Find the length of PR
b) Name the angle between the plane PUR and the plane TURS.
c) Find the angle between the line PR and the plane TURS.
Answer: (a)_____________________________ (b)_____________________________
(c)_____________________________
(4) The figure shows a cuboid with horizontal base KLMN.
S R
8cm
N M
P Q
10 cm
K 12 cm L
a) Name the angle between the line PR and the plane SRMN.
b) Name the angle between the line PR and the plane KLQP.
c) Name the angle between the plane PLR and the plane KLMN.
d) Calculate the angle between line PR and the plane SRMN.
Answer: (a)_______________________________
(b)_______________________________
(c)_______________________________
(d)_______________________________
(5) H G
10 cm
D C
E F
12 cm
A 14cm B
The above figure shows a cuboid.Calculate
a) the length of BH
b) the angle between line BH and base ABCD
Answer: (a)____________________________
(b)____________________________
QUADRATIC EQUATIONS
(1)
Solve x2 – 5
4 = x
Answer: (a)_____________________________
(2) Solve: 4 = x
x + 11 x + 2
Answer:________________________
(3)
Solve: s + 4 + 2 = 3
3 s
Answer:_________________________
(4) Solve: 7 – 6x
3x – 2 = x
Answer:________________________
(5) Find the roots: x + 3 = 2 5 x
TRANSFORMATION (Paper 2)
1 The diagram below shows three straight lines, AB, CD and EF, which are the
images of line XY under a combined transformation.
Determine the image of XY for each of the following transformations.
(a) (i) A reflection in the y- axis followed by another reflection in the line y = x. ( 2 marks )
(ii) An anticlockwise rotation of 90o about the origin followed
by a translation
−1
4. ( 2 marks )
(iii) A translation
−
−
1
1 followed by a reflection in the x-axis.
( 2 marks )
(b) In the Figure 1, triangle PQR is the image of triangle SQT under a certain enlargement.
P
S
R 8cm T 6cm Q
(i) State the scale factor of the enlargement. ( 2 marks )
(ii) If the area of the triangle PQR is 16cm2, find the area of triangle
SQT.
( 4 marks )
2. The following diagrams shows two points, A and B, on a Cartesian plane. y
Transformation K is the translation
−
3
2 and transformation L is an
anticlockwise rotation of 90o about the centre (0 , -1).
(a) (i) State the coordinates of the image of point A under transformation
K.
( 2 marks )
(ii) State the coordinates of the image of point B under the following
transformations. ( 4 marks )
(a) K2 (b) KL
(b) In the following diagram, quadrilateral Y is the image of quadrilateral
Y is the image of quadrilateral X under a transformation R followed by
a transformation S.
Describe transformations R and S in full. ( 6 marks )
3. Based on the Figure 1, triangle ABC is mapped on to triangle DEF under the
transformations below.
a. (i) enlargement at the centre (-6, 1) with a scale factor of 2,
(ii) reflection on the line x = 1,
(iii) rotation of 90o clockwise about the centre (3, 2)
Draw the final image in Figure 1. ( 8 marks )
x
0
A●
B ●
2 –
•
-2 –
•
-4 –
-2
4 –
•
4 -4 2
B
C D
X
G
F
Y
A
E
Figure 1
b. R1 is a reflection on the x-axis and R2 is another reflection on the y-
axis.
Describe a single transformation equivalent to R1R2.
( 4 marks )
4. (a) In Figure 2, A is the image of B under an anticlockwise rotation.
Figure 2
State
(i) the coordinates of the centre of rotation, ( 2 marks )
(ii) the angle of rotation, ( 2 marks )
(iii) the coordinates of the image of W under the same transformation.
( 4 marks )
(b) C is the image of B under a combined transformation XY, Describe
completely X and Y. ( 4 marks )
5. Based on Figure 3, draw the object PQRS of which the vertices are P(-6, -2), Q(-3,-2), R(-4,-3) and S(-5,-3). Given F is an enlargement with a scale factor
of 2 at the centre of enlargement (-7, -4), G is a reflection on the line x = 3 and
H is a translation of
3
1.
Figure 3
(a) On the same Figure, draw the image of PQRS under the transformations (i) F and label the image as P’Q’R’S’, ( 2 marks )
(ii) GF and label the image as P”Q”R”S”. ( 2 marks )
(b) Next, P”Q”R”S” is transformed under H. State the coordinates of the four
vertices P”Q”R”S” after the transformation and label them as ABCD
respectively. ( 8 marks )
Question 14 ( Statistics)
1 The data in the diagram below is the heights, in cm of 33 students
151
125 124 147 147 145 148 152 135 150 150
127
132 122 156 133 145 140 152 151 155 147
152 148
149 150 154 154 148 146 132 158 155
a) Using the data above and a class interval of 5, complete the following table
Height (cm)
Frequency Midpoint
121-125
b) Based on the table constructed in (a)
i) state the modal class
ii) estimate the mean pages
c) For this part of the question, use a graph paper
By using the scale of 2cm to 5 cm on the horizontal axis and 2 cm to 1 student
on the vertical axis, draw a frequency polygon for the data
2 The data in the diagram below shows the marks obtained by a group of students
in a test
42 88 39 85 72 86 66 70
62 50 40 74 41 66 53 79
64 84 51 69 36 76 55 48
80 60 42 58 35 76 59 55 49 50 56 68 58 67 36 77
a) i) find the range of the data
ii) Based on the data, construct a frequency table as shown below
Mark
Frequency Midpoint Upper boundary
20-29
b) By using the table constructed in (a), estimate the mean mark obtained
c) For this part of the question, use a graph paper
By using the scale of 2cm to 10 marks on the horizontal axis and 2 cm to 1
student on the vertical axis, draw a frequency polygon to represent the data
above
3. The data in the diagram below shows the time taken to finish reading a
passage, in seconds, by a group of students
20 37 24 40 30 24
48 29 30 33 44 27
34 30 21 38 34 24
30 42 26 30 26 49 30 41 21 35 26 28
a) i) state the range of the data
ii) Based on the data, construct a frequency table as shown below
Time (s)
Frequency Midpoint Upper boundary
20-24
b) Based on the table, i) state the modal class
ii) estimate the mean of the reading time for the students
c) For this part of the question, use a graph paper
By using the scale of 2cm to 5 seconds on the horizontal axis and 2 cm to 1
student on the vertical axis, draw a histogram based on the table above
4. The data in the diagram below shows number of pages of story books on a
book rack
28 11 23 27 20 15 30 24
27 25 17 19 16 23 25 26
29 17 23 12 20 21 23 28
29 19 33 24 23 24 25 31
a) Using the data above and a class interval of 3, complete the following table below
Number of pages
Frequency Midpoint
10-12
b) By using the table constructed in (a), estimate the mean pages
c) For this part of the question, use a graph paper
By using the scale of 2cm to 3 pages on the horizontal axis and 2 cm to
1 student on the vertical axis, draw a histogram
d) State one information which can be obtained from the histogram
5 The data in the diagram below the masses in gram of 40 packages handled by
a forwarding company
370
390 370 390 350 370 310 370
320
320 260 430 380 230 290 350
240
310 340 340 310 300 260 330
200
280 350 290 342 340 360 360
370 322
333 328 241 271 201 310
a) Using the data above and a class interval of 30, complete the following
table
Mass(kg) Upper boundary Frequency Cumulative
frequency
200-229
230-259
b) For this part of the question, use a graph paper By using the scale of 2cm to 30 g on the horizontal axis and 2 cm to
5 packages on the vertical axis, draw an ogive
c) From the ogive in (b)
i) find the third quartile
ii) explain briefly the meaning of the third quartile
TOPIC : PLAN AND ELEVATION
1. (a) Diagram 1(i) shows a solid right prism with rectangular base FGPN on a horizontal table. EFGHJK is the uniform cross section of the prism . Rectangle
EKLM is an inclined plane. Rectangle JHQR is a horizontal plane. EF, KJ and HG are vertical edges.
5 cm
2 cm
2 cm
7 cm5 cm
5 cm
E
ML
J
R
F
N
K
P
G
Diagram 1(i)
Q
H
Draw a full scale plan of the solid.
(b) A solid cuboid is joined to the prism in Diagram 1(i) at the vertical plane
PQRLMN. The combined solid is as shown in Diagram 1(ii). The square base
FGSW is a horizontal.
2 cm
7 cm
7 cm
7 cm5 cm
5 cm
E
ML
J
R
F
N
K
P
G
S
X
T
V
U
W
H
Q
C
D
Diagram 1(ii)
Draw full scale
(i) the elevation of the combined solid on a vertical plane parallel to FG as viewed from C,
(ii) the elevation of the combined solid on a vertical plane parallel to GPS as viewed from D.
2. Diagram 2(i) shows a solid consisting of a cuboid and a half cylinder which are joined at the plane HJMN. The base GDEF is on a horizontal plane and HJ = 3
cm.
HN
M
L
ED
G
J
KF
3 cm
4 cm
6 cm
Diagram 2(i)
X
Draw to full scale, the elevation of the solid on a vertical plane parallel to DG as
viewed from X.
(b) A solid right prism is joined to the solid in Diagram 2(i) at the vertical plane
ELMW. The combined solid is as shown in Diagram 2(ii). Trapezium PMWU is
the uniform cross-section of the prism and PQRM is an inclined plane. The base
DEUTSWFG is on a horizontal plane and EU = 2 cm.
HN
M
L
ED
G
J
KF
3 cm
4 cm
6 cm
Y
2 cm
5 cm
QP
U T
W S
R
Diagram 2(ii)
Draw to full scale,
(i) the plan of the combined solid,
(ii) the elevation of the combined solid on a vertical plane parallel to UT as
viewed from Y.
3.Diagram 3(i) shows a solid right prism with rectangular base PEFN on a horizontal
table. The surface EFGHJ is the uniform cross-section of the prism. Rectangle
KJHL is an inclined plane and rectangle LHGM is a horizontal plane. JE and GP are
vertical edges.
F
G
M
L
K
P
E
J
H
6 cm
8 cm
4 cm
N
4 cm
7 cm
X
Diagram 3(i)
Draw full scale, the elevation of the solid on a vertical plane parallel to EF as viewed
from X.
(b) A solid right prism with uniform cross section UVWX is removed from the solid
in Diagram 3(i). The remaining solid is as shown in Diagram 3(ii). Rectangle
VSTW is a horizontal plane. UV and XW are vertical edges. VS = 3 cm and SE =
2 cm.
F
G
M
L
K
P
E
J
H
6 cm
8 cm
4 cm
N
4 cm
7 cm
Y
T
W
V
S
U
X
Diagram 3(ii)
Draw to full scale,
(i) the plan of the remaining solid,
(ii) the elevation of the remaining solid on a vertical plane parallel to PE as
viewed from Y.
4. Diagram 4 shows a solid consisting of a prism and a half cylinder joined at the
plane PADS. The horizontal base of the solid consists of an equilateral triangle and a semicircle of diameter 3 cm.
A B
C
P Q
R
4 cm
6 cm
6 cm
3 cm
T
U
S
D
X
Y
Diagram 4
Draw full scale,
(i) the plan of the solid,
(ii) the elevation of the solid on a vertical plane parallel to AB as viewed from X,
(iii)the elevation of the solid on a vertical plane parallel to ADC as viewed from
Y.
5. Diagram 5 shows a solid consisting of a prism and a half cylinder which joined at
the plane CDEFH. The surface ABMQP is the uniform cross section of the prism with a rectangular base ABCD on a horizontal table. Rectangle PQRS is a
horizontal plane, rectangle QMHR is a vertical plane and rectangle BCHM is an inclined plane. AP and MQ are vertical edges.
A
B
P
S
C
Q
R
H
F
M
E
G
D
6 cm
7 cm4 cm
3 cm
4 cm
4 cm
X
Y
Diagram 5
Draw full scale
(i) the plan of the solid,
(ii) the elevation of the solid on a vertical plane parallel to AB viewed from X,
(iii) the elevation of the solid on a vertical plane parallel to BC as viewed from
Y
GRAPH OF FUNCTIONS
13.1 (a) Complete Table 1 in the answer space for the equation y = x
24− by writing
down the values of y when x = −3 and when x = 1.5. [2 marks] (b) For this part of the question, use the graph paper provided on next page.
You
may use a flexible curve rule.
By using a scale of 2 cm to 1 unit on the x-axis and 2 cm to 5 units
on the y-axis, draw the graph of y = x
24− for − 4 4x ≤≤ . [5 marks]
(c) From your graph, find
(i) the value of y when x = − 2.9.
(ii) the value of x when y = −13 [2 marks]
(d) Draw a suitable straight line on your graph to find a value of x which
satisfies
the equation 2x2
− 5x −24 = 0
State this value of x. [3 marks]
Answer :
(a)
x −4 −3 −2 −1 1 1.5 2 3 4
y 6 12 24 −24 −12 −8 −6
Table 1 (b) Refer graph on next page
(c) (i) y = ......................
(ii) x = ......................
(d) x = ......................
Graph for Question 13
13.2(a) Table 1 shows values of x and y which satisfy the equation y = 2x2
− 4 x −
6
x −2 −1 0 1 2 3 4 4.5 5
y k 0 −6 −8 −6 m 10 16.5 24
Table 1
Calculate the value of k and of m. [2 marks]
(b) For this part of the question, use the graph paper provided on next page.
You
may use a flexible curve rule.
By using a scale of 2 cm to 1 unit on the x-axis and 2 cm to 5 units
on the y-axis, draw the graph of y = 2x2
− 4 x − 6 for − 2 5x ≤≤ .
[4 marks]
(c) From your graph, find
(i) the value of y when x = − 1.5,
(ii) the values of x when y = − 5 [3 marks]
(d) Draw a suitable straight line on your graph to find a value of x which
satisfies the equation 2x2 + x − 23 = 0 for − 2 5x ≤≤ .
State this value of x. [3 marks]
Answer :
(a) (i) k = ......................
(ii) m = ......................
(b) Refer graph on next page
(c) (i) y = ......................
(ii) x = ......................
(d) x = ......................
Graph for Question 13
13.3(a) Complete Table 1 in the answer space for the equation y = −2x2 + x + 23
[2 marks]
Answer :
(a)
x −2 −1 −0.5 1 2 3 4 4.5 5
y 13 22 22 17 8 −13 −22
Table 1
(b) For this part of the question, use the graph paper provided on next page.
You
may use a flexible curve rule.
By using a scale of 2 cm to 1 unit on the x-axis and 2 cm to 5 units
on the y-axis, draw the graph of y = −2x2 + x + 23 for − 2 ≤ x ≤ 5.
[4 marks]
Answer :
(b) Refer graph on next page
(c) From your graph, find (i) the value of y when x = 3.5..
(ii) the value of x when y = −10 [2 marks]
Answer :
(c) (i) y = ......................
(ii) x = ......................
(d) Draw a suitable straight line on your graph to find all values of x which
satisfy
the equation 2x2 − 3x = 8 for − 2 ≤ x ≤ 5.
State these values of x. [4 marks]
Answer :
(d) x = ......................
Graph for Question 13
13.4(a) Complete Table 1 in the answer space for the equation y = x3
−3 x + 10
[2 marks]
Answer :
(a)
x −3 −2 −1 0 1 1.5 2 2.5 3
y −8 12 10 8 8.9 18.1 28
Table 1
(b) For this part of the question, use the graph paper provided on next page.
You
may use a flexible curve rule.
By using a scale of 2 cm to 1 unit on the x-axis and 2 cm to 5 units
on the y-axis, draw the graph of y = x3 −3 x + 10 for − 3 3x ≤≤ .
[4 marks]
Answer :
(b) Refer graph on next page
(c) From your graph, find
(i) the value of y when x = −1.5. (ii) the value of x when y = 20. [2 marks]
Answer :
(c) (i) y = ......................
(ii) x = ......................
(d) Draw a suitable straight line on your graph to find all values of x which satisfy
the equation x3
− 6 x + 3 = 0 for − 3 3x ≤≤
State these values of x. . [4 marks]
Answer :
(d) x = ......................
Graph for Question 13
13.5(a) Complete Table 1 in the answer space for the equation y = −x 3 − 3x + 5
[2 marks]
Answer :
(a)
X −3 −2 −1 0 1 2 3 3.5
Y 41 9 5 1 −9 −48.4
Table 1
(b) For this part of the question, use the graph paper provided on next page.
You
may use a flexible curve rule.
By using a scale of 2 cm to 1 unit on the x-axis and 2 cm to 10 units
on the y-axis, draw the graph of y = −x 3 − 3x + 5 for –3 ≤ x ≤ 3.5. .
[4 marks]
Answer :
(b) Refer graph on next page
(c) Draw a suitable straight line on your graph to find all values of x which
satisfy the equation x 3 = x + 5 for –3 ≤ x ≤ 3.5.
State this value of x. [3 marks]
Answer :
(c) x = ......................
(d) On the same graph, shade and label "R", the region which satisfies
the three inequalities x ≥ 0 , y ≥ 20 and y ≤ –10x + 40 .
[3 marks]
Answer :
(d) Refer graph on next page.
Graph for Question 13