buy
Content
Class
Register Number
Name
南洋女子中学校
NANYANG GIRLS' HIGH SCHOOL
BLOCK TEST 1 2013
Secondary Four
INTEGRATED MATHEMATICS 1
Monday
1 hour 30 minutes
29 April 2013
0845 – 1015
READ THESE INSTRUCTIONS FIRST
INSTRUCTIONS TO CANDIDATES
1.
Write your name, register number and class in the spaces at the top of this page.
2.
Answer questions 1 – 9 before attempting the Bonus Question.
3.
Write your answers and working on the separate writing paper provided.
4.
Omission of essential working will result in loss of marks.
5.
Give non-exact numerical answers correct to 3 significant figures, or 1 decimal place in
the case of angles in degrees, unless a different level of accuracy is specified in the
question.
INFORMATION FOR CANDIDATES
1.
The number of marks is given in brackets [ ] at the end of each question or part
question.
2.
The total number of marks for this paper is 60.
3.
You are reminded of the need for clear presentation in your answers.
This document consists of 7 printed pages.
Mathematical Formulae
1.
ALGEBRA
Quadratic Equation
For the equation ax2 + bx + c = 0 ,
x=
2.
− b ± b 2 − 4ac
.
2a
TRIGONOMETRY
Identities
sin2 A + cos2 A = 1 .
sec2 A = 1 + tan2 A .
cosec2 A = 1 + cot2 A .
sin (A ± B) = sin A cos B ± cos A sin B .
cos (A ± B) = cos A cos B m sin A sin B .
tan (A ± B) =
tan A ± tan B
.
1 m tan A tan B
sin 2A = 2 sin A cos A .
cos 2A = cos A − sin2 A = 2cos2 A − 1 = 1 − 2sin2 A .
2
tan 2A =
2 tan A
1 − tan 2 A
.
sin A + sin B = 2 sin
1
1
(A + B) cos (A – B)
2
2
sin A – sin B = 2 cos
1
1
(A + B) sin (A – B)
2
2
cos A + cos B = 2 cos
1
1
(A + B) cos (A – B)
2
2
cos A – cos B = –2 sin
1
1
(A + B) sin (A – B)
2
2
Formulae for ∆ABC
a
b
c
=
=
.
sin A sin B sin C
a2 = b2 + c2 − 2bc cos A .
∆=
1
ab sin C .
2
3
1
(a)
Copy each of the Venn diagrams below and shade the region that is indicated
by the set notation.
(i)
(ii)
A′ ∩ B′
(b)
(A ∪ C)∩ B
Write the set notation for the sets shaded in the following Venn diagrams.
(i)
2
[2]
(ii)
[2]
You are given the following sets.
ε = {Students in School X }
A = {Students studying Art}
B = {Students studying Biology}
C = {Students in the Chess Club}
D = {Students in the Drama Club}
Express in set notation the following truths about the students in School X.
3
(i)
All students in the Chess Club study Art or Biology or both .
[1]
(ii)
30 students in the Chess Club do not study Art.
[1]
(iii)
None of the students studying Art are in both the Chess Club and Drama Club.
[1]
2 a
1 2 0
It is given that AB = I, where A =
, B = b c and I is an identity
0 2 1
d 2
matrix of order n × n .
(i)
State the value of n
[1]
(ii)
Find the value of a, b, c and d.
[4]
4
4
“Delicious Cake Shop” supplies two different types of cakes to three major hotels
every day. The number of each type of cake supplied to each hotel is shown in the
table below.
Hotel Amazing
Hotel Beautiful
Hotel Charming
Apple Cake
70
20
40
Banana Cake
50
30
30
The price of each Apple Cake and Banana Cake sold to the hotels is $50 and $30
respectively.
5
(i)
Write down two matrices A and B whose product AB shows the amount (in
dollar terms) that Delicious Cake Shop will make in sales daily from each of
the three hotels. Evaluate the product AB.
[3]
(ii)
Write down a matrix C which, when multiplied to AB, will give the total sales
(in dollar terms) that Delicious Cake Shop will make daily from all the three
hotels combined.
[1]
The graph of y = f ( x ) is shown below. The points A(−3, 0) , B(−2, 3) and C(0, 0) are
shown on the graph.
Sketch, on separate axes, the following graphs, indicating the coordinates of the
points A, B and C clearly.
(a)
y = f (2 x) − 2 ,
[3]
(b)
y = 5f (− x ) ,
[3]
(c)
y = 2 − f ( x) .
[3]
5
6
In the Venn diagram, x and y represent the number of elements in the respective
regions.
Given that n (ξ ) = 360 , n ( P ) = 135 , n (Q ) = 180 , n ( R ) = 170 ,
7
(a)
state the maximum value of y,
[1]
(b)
state the minimum value of y,
[1]
(c)
find the value of x, if y = n (Q ∩ R ) = 100 .
[3]
(a)
Given that A = 2 7 , B = 1 0 and 2A + C = B2, find C.
−1 3
3 −1
[2]
(b)
1
1 −
-1
-1
Given that P =
4 , find P . Use P to solve the simultaneous
−5 1
equations
1
x − y = −2 ,
4
−5x + y = 6 .
[4]
Hence, solve the following simultaneous equations
4 x − y = −2 ,
− 20 x + 4 y = 6 .
[2]
6
8
9
The points A(2, 4) , B(8,10) and C(6, 0) are the vertices of a triangle ABC.
(i)
Show that triangle ABC is a right angled triangle.
[2]
(ii)
Circle C1 is such that the points A, B and C all lie on its circumference.
Find the coordinates of the center of circle C1 .
[2]
(iii)
Hence, show that the equation of circle C1 is (x − 7)2 + (y − 5)2 = 26 .
[2]
(iv)
Does the point D(5, 3) lie inside C1 , outside C1 , or on the circumference of
[2]
C1 ? Explain your answer clearly.
(v)
Another circle C2 passes through the points A and B, and has the same radius
as circle C1 . Find the equation of C2 .
[3]
Answer the whole of this question on the INSERT provided.
Tie the INSERT together with the rest of your answer scripts.
1
The graph of y = − (x 2 − 4x) is partially shown in the INSERT.
5
Using the graph,
(a)
state the coordinates of the point on the curve where gradient is 0,
[1]
(b)
state the range of values of x for which the curve is decreasing,
[1]
(c)
find the range of values of x for which x 2 − 4x < 0 ,
[2]
(d)
(e)
1
find the value of k for which the line y = − x + k is a tangent to the curve,
5
[2]
by finding and inserting a suitable straight line graph, solve the equation
x2 − x − 2 = 0.
[4]
7
Bonus Question
10
The diagram below shows the arch THINK of a stone bridge where TXOYK is
horizontal. The arch forms an arc of a circle. The bridge is supported by vertical
pillars HX, IO and NY, with IO = 1m tall. The highest point of the bridge is I. The
length of TK is 10m and TX, XO, OY and YK are equal in length. Find the height of
the pillar HX.
[3]
8
2013 Sec 4 IM1 BT1 Answers
1ai
1aii
1bi
1bii
2i
30
2ii
2iii
Or
0
2
3i
1
1
2
1
2
1
3ii
4i
70 50
50
Option 1: 20 30 and
30
40 30
5000
1900
2900
OR
Option 2: 50
70
30 and
50
5000 1900
4ii
Option 1: ! 1 1
1
20
30
2900
40
30
9
Or
1
Option 2: ! 1
1
5a
5b
5c
6a
6b
6c
7a
7bi
180
0
# 10.
3 14
2
5
# 2, % 16
!
10
1
# ,% 4
2
7bii
'( *+(,
8i
8ii
Center
7,5
8iii Equation: # 7- . % 5- 26
8iv Substitute 5,3 into LHS of the equation of the circle:
/01: # 7- . % 5- 5 7- . 3 5- 8
8v
9a
9b
9c
9d
9e
Hence, D lies inside 3 .
Equation of - : # 3- . % 9- 26
2, 0.8
#52
06#64
7 1.25 8 0.1
9
insert the line % : # :
Intersection points of the graph and the straight line are the
solutions to the equation # - # 2 0.
# 1, # 2
Bonus question:
0.757 m
11
Register Number
Name
南洋女子中学校
NANYANG GIRLS' HIGH SCHOOL
BLOCK TEST 1 2013
Secondary Four
INTEGRATED MATHEMATICS 1
Monday
1 hour 30 minutes
29 April 2013
0845 – 1015
READ THESE INSTRUCTIONS FIRST
INSTRUCTIONS TO CANDIDATES
1.
Write your name, register number and class in the spaces at the top of this page.
2.
Answer questions 1 – 9 before attempting the Bonus Question.
3.
Write your answers and working on the separate writing paper provided.
4.
Omission of essential working will result in loss of marks.
5.
Give non-exact numerical answers correct to 3 significant figures, or 1 decimal place in
the case of angles in degrees, unless a different level of accuracy is specified in the
question.
INFORMATION FOR CANDIDATES
1.
The number of marks is given in brackets [ ] at the end of each question or part
question.
2.
The total number of marks for this paper is 60.
3.
You are reminded of the need for clear presentation in your answers.
This document consists of 7 printed pages.
Mathematical Formulae
1.
ALGEBRA
Quadratic Equation
For the equation ax2 + bx + c = 0 ,
x=
2.
− b ± b 2 − 4ac
.
2a
TRIGONOMETRY
Identities
sin2 A + cos2 A = 1 .
sec2 A = 1 + tan2 A .
cosec2 A = 1 + cot2 A .
sin (A ± B) = sin A cos B ± cos A sin B .
cos (A ± B) = cos A cos B m sin A sin B .
tan (A ± B) =
tan A ± tan B
.
1 m tan A tan B
sin 2A = 2 sin A cos A .
cos 2A = cos A − sin2 A = 2cos2 A − 1 = 1 − 2sin2 A .
2
tan 2A =
2 tan A
1 − tan 2 A
.
sin A + sin B = 2 sin
1
1
(A + B) cos (A – B)
2
2
sin A – sin B = 2 cos
1
1
(A + B) sin (A – B)
2
2
cos A + cos B = 2 cos
1
1
(A + B) cos (A – B)
2
2
cos A – cos B = –2 sin
1
1
(A + B) sin (A – B)
2
2
Formulae for ∆ABC
a
b
c
=
=
.
sin A sin B sin C
a2 = b2 + c2 − 2bc cos A .
∆=
1
ab sin C .
2
3
1
(a)
Copy each of the Venn diagrams below and shade the region that is indicated
by the set notation.
(i)
(ii)
A′ ∩ B′
(b)
(A ∪ C)∩ B
Write the set notation for the sets shaded in the following Venn diagrams.
(i)
2
[2]
(ii)
[2]
You are given the following sets.
ε = {Students in School X }
A = {Students studying Art}
B = {Students studying Biology}
C = {Students in the Chess Club}
D = {Students in the Drama Club}
Express in set notation the following truths about the students in School X.
3
(i)
All students in the Chess Club study Art or Biology or both .
[1]
(ii)
30 students in the Chess Club do not study Art.
[1]
(iii)
None of the students studying Art are in both the Chess Club and Drama Club.
[1]
2 a
1 2 0
It is given that AB = I, where A =
, B = b c and I is an identity
0 2 1
d 2
matrix of order n × n .
(i)
State the value of n
[1]
(ii)
Find the value of a, b, c and d.
[4]
4
4
“Delicious Cake Shop” supplies two different types of cakes to three major hotels
every day. The number of each type of cake supplied to each hotel is shown in the
table below.
Hotel Amazing
Hotel Beautiful
Hotel Charming
Apple Cake
70
20
40
Banana Cake
50
30
30
The price of each Apple Cake and Banana Cake sold to the hotels is $50 and $30
respectively.
5
(i)
Write down two matrices A and B whose product AB shows the amount (in
dollar terms) that Delicious Cake Shop will make in sales daily from each of
the three hotels. Evaluate the product AB.
[3]
(ii)
Write down a matrix C which, when multiplied to AB, will give the total sales
(in dollar terms) that Delicious Cake Shop will make daily from all the three
hotels combined.
[1]
The graph of y = f ( x ) is shown below. The points A(−3, 0) , B(−2, 3) and C(0, 0) are
shown on the graph.
Sketch, on separate axes, the following graphs, indicating the coordinates of the
points A, B and C clearly.
(a)
y = f (2 x) − 2 ,
[3]
(b)
y = 5f (− x ) ,
[3]
(c)
y = 2 − f ( x) .
[3]
5
6
In the Venn diagram, x and y represent the number of elements in the respective
regions.
Given that n (ξ ) = 360 , n ( P ) = 135 , n (Q ) = 180 , n ( R ) = 170 ,
7
(a)
state the maximum value of y,
[1]
(b)
state the minimum value of y,
[1]
(c)
find the value of x, if y = n (Q ∩ R ) = 100 .
[3]
(a)
Given that A = 2 7 , B = 1 0 and 2A + C = B2, find C.
−1 3
3 −1
[2]
(b)
1
1 −
-1
-1
Given that P =
4 , find P . Use P to solve the simultaneous
−5 1
equations
1
x − y = −2 ,
4
−5x + y = 6 .
[4]
Hence, solve the following simultaneous equations
4 x − y = −2 ,
− 20 x + 4 y = 6 .
[2]
6
8
9
The points A(2, 4) , B(8,10) and C(6, 0) are the vertices of a triangle ABC.
(i)
Show that triangle ABC is a right angled triangle.
[2]
(ii)
Circle C1 is such that the points A, B and C all lie on its circumference.
Find the coordinates of the center of circle C1 .
[2]
(iii)
Hence, show that the equation of circle C1 is (x − 7)2 + (y − 5)2 = 26 .
[2]
(iv)
Does the point D(5, 3) lie inside C1 , outside C1 , or on the circumference of
[2]
C1 ? Explain your answer clearly.
(v)
Another circle C2 passes through the points A and B, and has the same radius
as circle C1 . Find the equation of C2 .
[3]
Answer the whole of this question on the INSERT provided.
Tie the INSERT together with the rest of your answer scripts.
1
The graph of y = − (x 2 − 4x) is partially shown in the INSERT.
5
Using the graph,
(a)
state the coordinates of the point on the curve where gradient is 0,
[1]
(b)
state the range of values of x for which the curve is decreasing,
[1]
(c)
find the range of values of x for which x 2 − 4x < 0 ,
[2]
(d)
(e)
1
find the value of k for which the line y = − x + k is a tangent to the curve,
5
[2]
by finding and inserting a suitable straight line graph, solve the equation
x2 − x − 2 = 0.
[4]
7
Bonus Question
10
The diagram below shows the arch THINK of a stone bridge where TXOYK is
horizontal. The arch forms an arc of a circle. The bridge is supported by vertical
pillars HX, IO and NY, with IO = 1m tall. The highest point of the bridge is I. The
length of TK is 10m and TX, XO, OY and YK are equal in length. Find the height of
the pillar HX.
[3]
8
2013 Sec 4 IM1 BT1 Answers
1ai
1aii
1bi
1bii
2i
30
2ii
2iii
Or
0
2
3i
1
1
2
1
2
1
3ii
4i
70 50
50
Option 1: 20 30 and
30
40 30
5000
1900
2900
OR
Option 2: 50
70
30 and
50
5000 1900
4ii
Option 1: ! 1 1
1
20
30
2900
40
30
9
Or
1
Option 2: ! 1
1
5a
5b
5c
6a
6b
6c
7a
7bi
180
0
# 10.
3 14
2
5
# 2, % 16
!
10
1
# ,% 4
2
7bii
'( *+(,
8i
8ii
Center
7,5
8iii Equation: # 7- . % 5- 26
8iv Substitute 5,3 into LHS of the equation of the circle:
/01: # 7- . % 5- 5 7- . 3 5- 8
8v
9a
9b
9c
9d
9e
Hence, D lies inside 3 .
Equation of - : # 3- . % 9- 26
2, 0.8
#52
06#64
7 1.25 8 0.1
9
insert the line % : # :
Intersection points of the graph and the straight line are the
solutions to the equation # - # 2 0.
# 1, # 2
Bonus question:
0.757 m
11
