F.4 Summer Holiday Assignment – Conventional Questions
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F.4 Summer Holiday Assignment
Conventional Questions
Name : ____________________ Class :_________ (
) Mark :__________
Instruction:
1. All questions should be done either in test pad / new book.
2. All working steps SHOULD BE SHOWN clearly.
3. The answer is either exact or correct to 3 significant figures.
Factorization and miscellaneous
Section A1
1.
(HKCEE 07) Make p the subject of the formula 5 p  7  3( p  q) .
2.
(HKCEE 07) Factorize
(a) r 2  10r  25 ,
(b) r 2  10r  25  s 2 .
3.
(3 marks)
(HKCEE 06) Factorize
(a) 3b  ab ,
(b) 9  a 2 ,
© 9  a 2  3b  ab .
4.
(HKCEE 04) Make x the subject of the formula y 
5.
(HKCEE 04) Factorize
(3 marks)
2
.
ax
(a) a 2  ab  2a  2b ,
(b) 169 y 2  25 .
6.
(3 marks)
(4 marks)
(HKCEE 03) Factorize
(a) x 2  ( y  z ) 2 ,
(b) ab  ad  bc  cd .
7.
(3 marks)
Solve each of the following equations by the factor method.
(a) 2x2 + 5x + 2 = 0
(b) (2x – 1)(x + 3) = x + 3
I (x – 2)2 – (3x + 1)2 = 0
(3 marks)
F.4 Summer Holiday Assignment – Conventional Questions
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2
2 1 
8. Solve 4  x   9  0 using the factor method.
3 5 
9. Show that the quadratic equation (x  2h)(x  2k)  1 = 0 has two distinct real roots for any real
numbers h and k.
10. Solve each of the following equations using any algebraic method.
(Give the answers correct to 3 significant figures if necessary.)
2
(a)
(b)
(c)
2
a 1
a 1
     
5 2
 3 4
2
x  1  (2  x)(1  x)
( x  0.25)(0.6  0.2 x)  1.9  0
Section A2
11. A ladder with length 10 m leans against a wall. The bottom of the ladder is 6 m from the wall.
(a) Find the height of the top end above the ground.
(b) If the bottom of the ladder is pulled away from the wall further for x m so that the top
slides down by the same distance, find x.
2
52
 0 , where k is a constant, is
12. Given that one root of the equation x 2  (k  4) x  k 2 
3
9
twice the other root.
4k
(a) Prove that one root is
.
3
(b) Find the two possible values of k.
I If k is a positive integer, and given that a, b are the roots of the equation, where a ＞ b,
(i) find the value of a2  b2,
(ii) write down the values of a and b.
F.4 Summer Holiday Assignment – Conventional Questions
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13.
The figure shows the graph of the equation y = ax2 + bx + c.
(a) Find the values of a, b, c.
(b) Consider ax2 + bx + c = 2. Given that ,  are the roots of the equation, when  ＞ ,
(i) find the value of α  β ,
(ii) find the value of α 3  β 3 ,
(iii) solve the equation by the quadratic formula.
14. (a)
(b)
If f(x) = 5x2 + 14x
g(x) = x2
the following equations.
(i) f(x
g(x) = 0
(ii) f(x) + 29 = 4g(x) + 32x
x
x satisfying each of
A function is called an odd function if f x
f(x).
(i) Show that f(x) = x3
x is an odd function.
(ii) Using the result in (b)(i), find the value of f(55) + f
A function is called an even function if g(x) = g x).
(i) Show that g(x) = x4 x2 is an even function.
(ii) Using the result in I(i), find the value of g(55) + g
Section B
15. Joyce plans to launch a cattle farm. She builds a fence of length 55 m around three sides of a
rectangular coastal land ABCD. Since BC is the coastline, it is not bounded by the fence.
Suppose AB = x m.
(a) Express the area of rectangle ABCD in terms of x.
(b) Find the area of the rectangle when x = 10.
Later, Joyce finds that an exit is missing. Therefore she changes the design so that AE,
F.4 Summer Holiday Assignment – Conventional Questions
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which is 5 m long, is not bounded by the fence.
(c) Express the area of rectangle ABCDE in terms of x.
(d) Find the area of the rectangle when x = 10.
(e) Compared with the result in (b), find the percentage increase of the area in (d), correct to
3 significant figures.
After Joyce’s father takes a look at the design, he finds that she forgets to build a resting room.
Therefore Joyce changes the design again and builds a rectangular room of length 5 m, width
3 m on the right of the farm.
(f)
Express the area of the room in terms of x, and find its area when x = 10.
Function and Graph
Section A(1)
1. If f x   3x 2  4 x  5 ，find the value of f 2
2.
3.
4.
5.
(2 marks)
Given f x  4x  3 ，find f  f x 。
(2 marks)
Given f x  5x  2 and g x   x  6 ，find g f x  。
If f x  6x  1 and g x   2 x 2  7 ，evaluate g f 1 。
Sketch roughly the graph of y = f(x) , where f x   5( x  3) 2  2
2
(2 marks)
(2 marks)
(3 marks)
y
6. The figure shows the graph of y = f(x), where f x    x 2  x  2 ，
(a) Find the range of x so that f x   0 ,
(1 mark)
(b) Find the range of x so that f x   3 .
(2 marks)
x
-2
7.
The figure shows the graph of
-1
y
y = x 2  4x  q
Find the coordinates of A and B.
(3 marks)
1
y = x2 – 4x + q
3
A
B
x
F.4 Summer Holiday Assignment – Conventional Questions
8.
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The figure shows the graph of y = x  ax  b
Find a and b and the coordinates of P.
2
y
(4 marks)
y = x2 +ax + b
P
9.
1
The figure shows the graph of y = ax  x  c
(a) find c,
y
y = ax 2  x  c
1 2
(b) show that a = ax.  bx  c
4
2
5
x
1
(4 marks)
10.
What straight line should be drawn to the graph of
equation x2 + 2x – 6 =0?
y = 2x2 + 4x – 1 in order to solve the
(2 marks)
Section A(2)
1. Consider f ( x)  x 2  kx  3 where k is a constant, and g ( x)  x 2  1 .
If f (2)  5 , find the value(s) of
(a) k,
(b) x when f ( x)  3g ( x).
2.
(2 marks)
(3 marks)
Suppose f (x) = ax2 + bx + 3, where a and b are constants. If f (2) = f (0) and f (– 2) = 35,
(a)
(b)
find a and b.
what is the maximum or minimum point of the graph of f (x)?
3.
(4 marks)
(3 marks)
y
y = x2 + px + q
5
R
5
x
The graph of y = x2 + px + q passes through (0 , 5), (5 , 0) and point R as shown in the diagram.
(a) Find the values of p and q
(3 marks)
F.4 Summer Holiday Assignment – Conventional Questions
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(b) What is the coordinate of point R ?
(2 marks)
(c) Find a quadratic equation in x with roots equal to two times the roots of equation x2 + px +
q = 0.
(2 marks)
4. An object is launched at 20 meters per second (m/s) from a 60-meter tall platform. The
equation for the object's height s metres (from the
ground) at time t seconds is s = –5t2 + 20t + 60.
(a) When does the object hit the ground?
(b)
(2 marks)
Can the object reach a height of 90 meters?
Explain your answer with working steps.
(3 marks)
5. The Figure shows the graph of y = ax 2  bx  c which passes through (1, 36) and touches
x-axis. Its y-intercept is 16.
(a) Find c.
(1 mark)
(b) Show that b  64a and a +b = 20.
(c) Hence, find the values of a and b.
(d) Find the vertex of the graph.
2
y = ax 2  bx  c
(2 marks)
(3 marks)
(2 marks)
y
ax  bx  c
2
16
0
x
F.4 Summer Holiday Assignment – Conventional Questions
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Section B
1. In the figure, the graph of y = ax2 + bx + c passes through the points A(1, 0), B(3, 0) and C(0,
y = ax2 + bx + c
3).
(a) Find the values of a, b and c.
[4 marks]
(b) If a horizontal line passing through C is drawn and
cuts the graph at the other point D, find
(i) the coordinates of D,
(ii) the area of the quadrilateral ABDC.
[3 marks]
[3 marks]
2. Tommy makes and sells schoolbags. He found that the cost for each schoolbag ($C) is given by
C  3x 2  18 x  k where x is the number of schoolbags being made (in thousands).
It is known that when x = 4, the cost for each schoolbag is $75.
(a) Find the value of k.
(2 marks)
2
(b) The figure shows the graph of y  x  6 x  40 . By adding a suitable line, find the
number of schoolbags so that the cost for each schoolbag is $120.
(4 marks)
(c) By using completing the square method, find the minimum cost for each schoolbag and
its corresponding number of schoolbags.
(4 marks)
y
2
y = x – 6x + 40
70
65
60
55
50
45
40
35
30
25
20
15
10
5
0
1
2
3
4
5
6
7
8
9
10
x
F.4 Summer Holiday Assignment – Conventional Questions
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Equation of straight line
Section A(1)
1.
Find the equation of the straight line passing through (2, 0) and having a slope of 3.
2.
Find the equation of the straight line passing through (3, 6) and (4, –2).
3.
Find the equation of the straight line having x-intercept and y-intercept of 3 and 2 respectively.
4.
Find the slope of the straight line 5 x  8 y  3  0 .
5.
Find the x-intercept and y-intercept of the straight line 28  7 y .
6.
If x  3 y  8  0 passes through (2k, k–2), find the value of k.
7.
Find the inclination of the straight line which has a slope of
8.
Find the equation of the straight line which passes through (0, 0) and is parallel to another
straight line 3 x  2 y  9  0 .
9.
Find the equation of the straight line which passes through (3, –1) and is perpendicular to
another straight line x  y  7 .
3.
10. Find the equation of straight line which passes through (0, 5) and also the point of intersection
of 2 x  y  5  0 and 4 x  3 y  5  0 .
Section A(2)
1.
ABCD is a right-angled trapezium. If A=(7, –8), B=(0, –11), C=(x, y) and D=(1, 6). It is given
A
that AB//CD, AB  BC and BC  CD .
B
(a)
Find the slope of AB.
D
(b)
(c)
Find the slopes of BC and CD in terms of x and y.
Hence, or otherwise, find the coordinates of C.
C
F.4 Summer Holiday Assignment – Conventional Questions
2.
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Two straight lines L1 : kx  6 y  2  0 and L2 : x  2 y  3  0 are given.
(a) Find the slope of L1 in terms of k.
(b) If L1 // L2 , find the value of k.
(c) If L1  L2 ,
(i) find the value of k.
(ii) find the coordinates of the point of intersection.
3.
∆ABC has vertices A(–1, 0), B(2, 0) and C(0, 3).
(a) Find the equation of the perpendicular bisector of AB.
(b) (i) Find the slope of BC.
(ii) Find the equation of the perpendicular bisector of BC.
(c)
4.
C
A
B
If the perpendicular bisectors of AB and BC intersect at a point O. Show that the three
perpendicular bisectors of ∆ABC pass through the same point.
Two points A and B are marked in the diagram. A is
rotated 90˚ about (0, 0) in the clockwise direction and
becomes A’. B is rotated 90˚ about (0, 0) in the
anti-clockwise direction and becomes B’.
(a) Write down the coordinates of A’ and B’.
(b) Find the equation of the straight line joining A’
and B’.
(c) Is AB//A’B’? Explain your answer.
5.
In the figure, A(–10, 10) and B(18, –6) are two points. The perpendicular bisector l of the
line segment AB cuts AB at M and the x-axis at P.
(a) Find the equation of l.
(b) Find the length of BP.
(c) If N is the mid-point of AP, find the length of MN.
F.4 Summer Holiday Assignment – Conventional Questions
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Section B
1.
In the figure, L1 : 2x  y  8  0 and L2 : 5x  4 y  8  0 cut the y-axis at A and
B respectively. The two lines intersect at a point P.
(a) Write down the coordinates of A and B.
(b) Find the coordinates of P.
(c) Find the ratio of area of ∆AOP to area of ∆BOP.
(d) Suppose C and D are the mid-points of AP and BP respectively.
(i) Find the ratio of area ∆CDP to area of quadrilateral ABDC.
(ii) If E is a moving point on CD, describe the change in the area of ∆AEB.
Explain your answer.
Trigonometry
Section A(1)
1.
In the figure, ABC is a triangle right-angled at B. If AB = 6 and tan C = 3, find BC.
C
B
2.
If cos 
6
A
2
and 0    90 , find the values of sin  and tan  . (Leave your answers
5
in surd form)
3.
Solve the equation 2 sin   cos 38 for 0    90 .
4.
Simplify 2 cos(90o   ) 
5.
Prove that (cos + 1)(cos – 1)  –tan2 cos2.
cos
.
tan( 90o   )
F.4 Summer Holiday Assignment – Conventional Questions
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tan( 270  )  sin 2 (180  )
 1.
cos 2 (90  )  tan( 90  )
6.
Prove that
7.
The figure below shows the graph of y = cos x for 0º  x  360º.
Use the graph to solve
(a) cos x = – 0.5 for 0º  x  180º,
(b) 5 cos x – 3 = 0 for 0º  x  360º.
8.
Solve the following equations for 0º   < 360º.
(a) 1 – cos = sin2
(b)
9.
7 sin   cos
2
4 sin   cos
Find the value of x in the following figures.
(a)
(b)
10. In quadrilateral ABCD, DAC = 40, BAC = 35, AB = 4 cm, AC = 8 cm and AD = 5 cm.
Find the perimeter of ABCD.
F.4 Summer Holiday Assignment – Conventional Questions
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11. In trapezium ABCD, AB // DC. If AB = 10 cm, CD = 6 cm, DAB = 58 and ABC = 70.
Find the length of AD and the height of the trapezium.
Section A(2)
1.
In the figure, ABC is a triangle right-angled at C. BC = a and cos  a .
A

B
a
C
(a) Find the lengths of AB and AC in terms of a.
(b) Hence find sin  in terms of a.
(c) Using the result of (b), or otherwise, find the value of sin  if cos 
2.
The figure below shows the graph of y = sin x for 0º  x  360º.
(a) Use the given graph to sketch the graph of y = sin3x for 0º  x  360º.
(b) Find the period of y = sin3x.
1
.
3
F.4 Summer Holiday Assignment – Conventional Questions
3.
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The figure shows the graph of y = a cosx + b sinx for 0º  x  360º.
(a) Find the values of a and b.
(b) Using the graph, find the maximum and minimum values of 6 cos x – 4 sin x + 3.
(c) Using the graph, solve the equation 100 sin x – 150 cos x – 150 = 0.
4.
In the figure, PQR = 80. Find
(a) the lengths of PS and PR,
(b) the value of ,
(c) the area of PRS.
5.
In the figure, A, B and C are the centres of three circles touching each other. The radii of the
circles are a, b and c respectively.
(a) Find the lengths of AB, BC and CA in terms of a, b and c.
(b) Prove that the area of ABC is
abca  b  c  .
(c) Find the area when a = 5, b = 6 and c = 4.
(Leave the answer in surd form.)
F.4 Summer Holiday Assignment – Conventional Questions
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Section B
1.
In the figure, a triangular plane ABC leans against a vertical wall CD at C with A on the
horizontal ground AD. It is given that A, B, C and D all lie on the same plane. E is a point on
AD such that BE  AD. CAD = 40, BAD = 25, BCD = 30 and BC = 50 cm. Find
(a) ACB,
(b) AB,
(c) the height of B above the ground.
(Give the answers correct to 3 significant figures if necessary.)
2.
AB and CD are two buildings. E and F are two points on AB. The angles of depression of E and
F from the top of building CD are 30 and 45 respectively. It is given that BE = 27 m and BF
= 15 m.
(a) Find EF and EC.
(b) How far are the two buildings apart?
(c) Find the height of CD.
(Give the answers correct to 2 decimal places if necessary.)
F.4 Summer Holiday Assignment – Conventional Questions
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Application of trigonometry
Section A1
1.
If the compass bearing of X from Y is S40W, find the true bearing of X from Y.
2.
3.
4.
A ship sails for 10 km in the direction N70E from port A to port B. Then it sails for 20 km
towards south to port C. C is due south of B. Find the distance between port A and port C.
(Give the answer correct to the nearest km.)
F.4 Summer Holiday Assignment – Conventional Questions
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5.
6.
In the figure, ABCD is a rectangular wall of length 4 m. EFCD is
the shadow of the wall in which CDF  50 and
Find CF correct to 3 significant figures.
4m
A
CFD  60.
B
D
50
C
60
E
7.
In the figure, ABCDEF is a triangular prism in which
50, AF  5 cm and CF  6 cm. Find AC correct to 3
significant figures.
F
AFC 
C
A
B
D
50
E
F
8.
In the figure, the plane ABE of the solid is a right-angled
triangle in which EAB  90, AEB  60 and AE  5 cm. If
EBD  30 and BD  12 cm, find
(a) BE,
(b) the area of the plane BDE of the solid.
D
E
12 cm
60
30
5 cm
C
A
B
F.4 Summer Holiday Assignment – Conventional Questions
9.
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In the figure, the three legs of a tripod of a camera are put on the
three points A, B and C. If AB  5 cm, BC  7 cm and AC  8 cm,
ABC correct to 3
significant figures.
V
C
A
B
Section A2
1. The figure shows a cube with side 8 cm. P is the mid-point of
AE.
(a) Find the length of GP.
(b) Find the angle between the lines CP and GP.
(6 marks)
2.
In the figure, a pole VP with height 30 m stands vertically on the
same ground as A and B. The angles of elevation of V from A and
B are 28 and 35 respectively. A and B are
80 m apart.
(a) Find the lengths of VA and VB.
(b) Find the angle between the lines AV and BV.
(4
mark
s)
3.
In the figure, A, B and P are three points on the ground. A is
due west of P and they are 50 m apart. The true bearing of B
from P is 310 and they are 40 m apart.
(a) How far is B away from A?
(b) Two balloons start rising vertically from A and B
respectively. After one minute, the balloon at A has risen 25
m to A . The balloon at B has risen 18 m to B  .
(i) Find the distance between the two balloons after one minute.
(ii) Find the angle of elevation of B  from P.
(8 marks)
F.4 Summer Holiday Assignment – Conventional Questions
4.
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In the figure, VT is a vertical pole with height 50 m. A, T and B are three points on the same
plane. B is due east of A and they are 200 m apart. BT = 150 m. The compass bearings of A and
B from T are S55W and S50E respectively.
(a) Find the distance between A and T.
(b) Find the angle between the line VA and the plane ABT.
(c) If P is a point on AB where AP : PB = 2 : 3, find
the angle between the line VP and the plane ABT.
(8 marks)
5.
(a) Figure 1 shows a trapezium with AD = AB = DC = 5 cm
and BC = 11 cm. Find the height of the trapezium.
(b) Figure 2 shows a prism with trapezium ABCD
as the base. The length of the prism is 20 cm.
(i) Find the length of BF.
(ii) Find the angle between the line BF and
the plane BCGH.
Figure 1
Figure 2
(8 marks)
F.4 Summer Holiday Assignment – Conventional Questions
Section B
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F.4 Summer Holiday Assignment – Conventional Questions
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Numerical Answer
Factorization and miscellaneous
Section A1
1 p
3q  7
2
2
(r + 5 + s)(r + 5 - s)
3 a) b(3-a)
c) (3-a)(3+a+b)
ay  2
4 x
y
b) (3+a)(3-a)
5 a) (a-b)(a+2)
6 a) (x+y-z)(x-y+z)
b)
(13y+5)(13y-5)
b) (b-d)(a-c)
7a) x = 2
c)=
or x = 
1
4
or
x
=
8.
10 a) a=
1
2
x =
b)x= –3
3
2
65
25
or x = 
6
6
45
15
or a= 
32
8
c) x= 5or x = 
b) x= 1or x =
12. (a) One of the roots is
(c) (i) 
(a)
4k
.
3
1
3
a  1
b7
(ii)  37
14. (a)
(i) x=
4
3
3
2
7
4
Section A2
11(a)
8
13
or x = 1
or
x=–8
(b) (ii) 0
(b)
x=
0 (rejected) or
(b)
k   7 or k  3
(ii)
1
2
b , a
3
3
(b)
(i)     1
(iii)
 4 or 3
(ii) x= 3 (repeated)
(c)
18 295 200
(a) A(x) = (55 x  2 x 2 ) m 2
(b)
A(10)= 350 m2
(c) A2(x) = (60 x  2 x 2 ) m2
(d)
400 m2
Section B:
(    )
x=2
F.4 Summer Holiday Assignment – Conventional Questions
(e)
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A3(x) = (60 x  2 x 2  15) m2
(f)
14.3%
A3(10) = 385 m2
Function and graph
Section A1
(1) 9, (2) 16x-15
(3) 25x2 – 20x – 2
(4) 43
(6) (a) -2 < x< 1 (b) no solution
7. A= (1,0) and B=(3,0)
(8) a= – 6and b = 5 P=(0,5)
(9) c=1
(10) the line
is y –
11=0
Section A2
(1) (a) k=3 (b) 0 0r 1.5 (2) (a) a = 4 and b = -8, (b) (1, -1) (3)(a) p=-6, q=5
3(b)R(1,0) (c) x2 – 12x +20=0
(4) (a) 6s
(b) no
(5)(a) c=16
(c) 4,16
(d)(-2,0)
Section B
(1) (a) a=1, b=-4, c=3
(b) (i) D(4,3) (ii) 9
(2) (a) 99
(b) 7000 (c) 72/3000
Equation of Straight line
Section A1
1.
3x- y- 6= 0
3.
2x + 3y – 6 = 0
2.
4.
8x + y – 30 = 0
-5/8
5.
no x-int, y-int= 4
6.
14
7.
9.
60
x – y – 4= 0
8.
10.
3x – 2y = 0
8x + y – 5 = 0
2.
(a) –k/6 (b) 3 (c) (i)-12 (ii) (11/15, 17/15)
Section A2
3/7 (b)
y  11 y  6
,
(c) (-6, 3)
x
x 1
1.
(a)
3.
4.
5.
(a) x = 0.5 (b) (i) -3/2, (ii) 4x – 6y + 5 = 0 (c) –
(a)(5, 1), (-3, 4) (b) 3x+ 8y – 23 = 0 (c) –
(a) 7x – 4y – 20 = 0 (b) 16.3 (c) 8.14
Section B
1.
(a) (0, 8), (0, -2) (b) (40/13, 24/13) (c) 4 : 1 (d) (i) 1: 3 (ii) -Trigonometry
Section A1
1.
2
2.
cos 
3.
23.2
4.
sin
5
; tan   2
5
F.4 Summer Holiday Assignment – Conventional Questions
5.
---
7.
(a)
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6.
---
120
(b)
55 or 305 (correct to the
nearest 5)
8. (a) 0, 90 or 270
9. (a) 121
10. 19.5 cm
cm
(b)
(b)
11.
135 or 315
4.02
AD = 4.77 cm ; height = 4.08
Section A2
1.
(a)
(c)
2.
(a)
(b)
3. (a)
= –4.2
4.
5.
1  a2
(b)
1  a2
2 2
3
The required graph is:
120
a = 3; b = –2
(c)
(a)
(c)
(a)
111 or 180 (correct to the nearest 3)
PS = 9.51; PR = 11.0
13.3 sq. units
AB = a + b; BC = b + c; CA = c + a
(c)
30 2 sq. units
(b)
Maximum = 10.2; Minimum
(b)
14.8
(b)
---
(b)
66.1 cm
(b)
28.39 m
Section B
1.
2.
(a)
(c)
(a)
(c)
20
27.9 cm
32.78 m
43.39 m
Application of Trigonometry
F.4 Summer Holiday Assignment – Conventional Questions
Page 23 / 24
Section A1
1. 220°
2.
3.
4.
5.
6.
7.
8.
9.
AB = 11.9 m
(a) 56.3° (b) 2.22 cm
19 km
43°
3.54 m
4.74 cm
(a) 10 cm
(b) 30 cm2
17.3 cm2
Section A2
1. (a) 12 cm
(b) 38.9
2. (a) VA = 63.9 m; VB = 52.3 m
(b) 86.4
3. (a) 32.2 m
(b) (i) 32.9 m
4.
(ii) 24.2
(a) 99.0 m
(b) 26.8
(c) 34.5
5. (a) 4 cm
(b)(i) 21.3 cm
(ii) 10.8
Section B
(a) 12 cm
(b) (i) A’E = 20/3 cm; A’B = 25/3 cm
(b) (ii) 50.1
(iii) 56.3
(iv) No
F.4 Summer Holiday Assignment – Conventional Questions
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