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SULIT 3472/1 3472/1 Form Four Name : ………………..…………… Additional Mathematics Paper 1 Form : ………………………..…… Oct 2006 2 hours SEKTOR SEKOLAH BERASRAMA PENUH KEMENTERIAN PELAJARAN MALAYSIA PEPERIKSAAN AKHIR TAHUN TINGKATAN 4 2006 ADDITIONAL MATHEMATICS For examiner’s use only Form Four Total Marks Question...
SULIT
3472/1
3472/1 Form Four Additional Mathematics Paper 1 Oct 2006 2 hours
Name : ………………..…………… Form : ………………………..……
SEKTOR SEKOLAH BERASRAMA PENUH KEMENTERIAN PELAJARAN MALAYSIA
PEPERIKSAAN AKHIR TAHUN TINGKATAN 4 2006 ADDITIONAL MATHEMATICS Form Four Paper 1 2 hours
DO NOT OPEN THIS QUESTION PAPER UNTIL INSTRUCTED TO DO SO
1
Write your name and class clearly in the space provided on this page.
2
Read the information on page 2 carefully.
3
This question paper must be handed in at the end of the examination.
For examiner’s use only Question 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25
Total Marks 2 3 4 2 3 3 3 4 3 3 4 4 3 4 3 3 3 3 4 3 4 3 3 3 3
Marks Obtained
TOTAL
This question paper consists of 13 printed pages
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INFORMATION FOR CANDIDATES 1.
This question paper consists of 25 questions.
2.
Answer all questions in this question paper.
3.
Give only one answer/solution to each question.
4.
Show your working. It may help you to get marks.
5.
The diagrams in the questions provided are not drawn to scale unless stated.
6.
The marks allocated for each question are shown in brackets.
7.
A list of formulae is provided on pages 2 to 3.
9.
You may use a non-programmable scientific calculator.
The following formulae may be helpful in answering the questions. The symbols given are the ones commonly used.
ALGEBRA
b b 2 4ac 2a
5
log a mn log a m log a n
am x an = a m + n
6
log a
3
am an = a m – n
7
log a mn = n log a m
4
( am )n = a m n
8
1
x
2
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m log a m log a n n
log a b
log c b log c a
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GEOMETRY
1
Distance =
x1 x 2 2 y1 y 2 2
3
A point dividing a segment of a line
x , y nx1 mx2 , ny1 my2
2 Mid point
x, y x1 x2 , y1 y 2
2
2
mn
mn
4 Area of a triangle =
1 x1 y 2 x 2 y 3 x3 y1 x 2 y1 x3 y 2 x1 y 3 2 STATISTICS
1
x
x N
2
x
fx f
4.
x x 3 N 2
x 2 x2 N
f x x f 2
fx 2 x2 f
1 N F C 5. m L 2 fm
TRIGONOMETRY 1
Arc length, s =r
2
Area of a sector, A
1 2 r 2
CALCULUS
1
dy dv du y = uv , u v dx dx dx
3
dy dy du dx du dx
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2
y
u dy , v dx
v
du dv u dx dx 2 v
[See overleaf SULIT
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4
For examiner’s use only
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Answer all questions.
1
Diagram 1 represents the relation between set R and set S. S 10 8 6 4 2 1
2
R
3
DIAGRAM 1 State (a)
the range,
(b)
the type of relation. [2 marks]
1 Answer : (a) …………………….. 2
(b) ……………………...
2
Diagram 2 illustrates the mapping of the function f and g – 1. y
x f (x)
2
z g–1 (y) 3
5 DIAGRAM 2 Given that f 1 ( x) 2 x k , find (a) the value of k, (b) the image of f 1 g (3) .
[3 marks] 2
Answer : (a) …………………….. 3
(b) ……………………...
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5
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Given the function f ( x) 1 3 x and the composite function fg ( x) 4 3 x 2 , find (a) g (x) , (b) the value of x when f
1
For examiner’s use only
( x) 5 . [4 marks]
Answer : (a) ...………………………......
3
(b)...............................................
4
Express the quadratic equation
4
x3 x in general form. 6 2x 4
[2 marks]
4 Answer : .........…………………
5
2
One of the roots of the quadratic equation 2 x 2 nx 24 0 is three times the value of the other root. Find the value of n , if n > 0 . [3 marks]
5 Answer : ……………………..
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For examiner’s use only
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6
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The quadratic equation x 2 px q 0 has roots 3 and
1 where p and q are 2
constants . Find the value of p and q. [3 marks]
Answer : p = ……........................
6
q =....……………….... 3
7
Find the range of values of x if y x 2 4 and 2 y x 11 . [3 marks]
7 Answer :
3
8
………………………...
Diagram 3 shows the graph for the function f ( x) p( x q) 2 r , where p, q and r f(x) are constants. f ( x) p( x q)2 r
4 2 0
1
5
x
DIAGRAM 3 Find the values of (a) q and r, (b) p
[4 marks]
8 Answer : (a) q = …….. r = ........... 4
(b) p =....………………..
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7
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Diagram 4 shows the graph of the function f ( x) 4 x 2 (4k 12) x 15 5k in relation to the x–axis . Find the range of values of k. [3 marks] f(x)
x DIAGRAM 4
9 Answer : k =……..……...……….....
10
Solve the equation
125
x 1
3
5 x 3 . 25 [3 marks]
10 Answer : ……………...……….....
11
3
Given log 5 2 m and log 5 7 p , express log 5 3.43 in terms of m and p. [4 marks]
11 Answer : …...…………..………... 4
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SULIT 12
8
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Solve the equation log 2 ( x 2) 1 2 log 4 (4 x) [4 marks]
12 Answer : ……………...………..... 4
13
Given points P(5, 6), Q(10, 3), R(7, n) and S(6, 1 ) are the vertices of a kite. Find the value of n. [3 marks]
13 Answer : n =…………...……….....
3
14
Diagram 5 shows points A, B and C on a Cartesian plane. Given that point A lies on the y-axis and the area of the triangle formed by the three points is 10 unit2. y A k
B (4, 1) x
O
C (2, 3) DIAGRAM 5 Calculate the value of k.
[4 marks]
14 Answer : k = ………..……….......... 4
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9
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Find the equation of the locus of the moving point M such that its distances from the
For examiner’s use only
points A (2 , 3) and B (5 , 2) are in the ratio of 1 : 2 . [3 marks]
15 Answer : ...........................................
16
3
A set of five numbers has a mean of 9. When q is added to the data the new mean is 10. Find the value of q. [3 marks]
16 Answer : q =...................................... 3
17
Table 1 shows the frequency distribution of the masses of 40 students. Mass (kg) Frequency
46-50 7
51-55 10
56-60 12
61-65 8
66-70 3
TABLE 1
Without drawing an ogive, find the median.
[3 marks]
17 Answer : …...…………..……..…... 3
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For examiner’s use only
SULIT 18
10
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For a set of four numbers, it is given that the sum of all the numbers, x , is 28 and the standard deviation is 2 3 . Find the value of x 2 . [3 marks]
18 Answer : …...…………..……..…... 3
19
In Diagram 6 , PQR is a semicircle with centre O.
Q 4 cm P
O DIAGRAM 6
R
Given that the radius is 5 cm, find the length of the arc PQ. (Use = 3.142) [4 marks]
19 4
Answer : …...…………..……..…...
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11
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Diagram 7 shows sector OAB with centre O and radius of 5p. A
0.5 rad O
C DIAGRAM 7
B
Point C is on OB such that OC : CB = 3 : 2. Given that the perimeter of the shaded region is 34 cm, calculate the value of p. (Use = 3.142) [3 marks]
20 Answer : …...…………..……..…... 3
21
Diagram 8 shows sector OPQ with centre O. O
20 cm
P
50
Q
DIAGRAM 8 Given that the radius is 20 cm and OPQ 50 . Find the area of the shaded region. (use 3.142) [4 marks] 21 Answer : …...…………..……..…...
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For examiner’s use only
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22
12
Determine the value of
lim n 1
n 2 n2 . n 1
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[3 marks]
22 Answer : …...…………..……..…... 3
23
Given that f ( x)
2x2 1 , find f ' (1) . 3x 4 [3 marks]
23 Answer : …...…………..……..…... 3
24
Find the equation of the tangent to the curve y = 3x 2 – 2x + 1 that passes through point (1, 2) . [3 marks]
24 3
Answer : …...…………..……..…...
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13
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Given that y (3 x 2) 4 , and x decreases at the rate of 4 unit per second, find the rate at which y changes when x =
For examiner’s use only
1 . 3
[3 marks]
Answer : …...…………..……..…...
25 END OF THE QUESTION PAPER 3
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[See overleaf SULIT
SULIT 3472/1 Additional Mathematics Paper 1 October 2006
SEKOLAH BERASRAMA PENUH KEMENTERIAN PELAJARAN MALAYSIA
PEPERIKSAAN AKHIR TAHUN TINGKATAN 4 2006
ADDITIONAL MATHEMATICS Paper 1 MARKING SCHEME
This marking scheme consists of 7 printed pages
3472/1
© 2006 Copyright SBP Sector, Ministry of Education, Malaysia
SULIT
Sub Marks
Full Marks
(a) { 2, 4, 6, 10 }
1
2
(b) many to many
1
(a) k = 1
2
Number 1
2
Solution and marking scheme
2(2) + k = 5
3
B1
(b) 5
1
(a) g(x) = x 2 + 1
3
( 4 3 x 2 ) 1 3 x 1 f – 1 (x) = 3
4
4
B2 B1
(b) x = 16
1
x2 – 8x + 6 = 0
2
(x-3)(2x -4) = 6x
5
3
2
B1
n = 16
3
SOR = 4 =
n 2
and
POR = 3
2
12
B2
SOR = 4 =
n 2
or
POR = 3
2
12
B1
2
3
Number 6
7
8
Solution and marking scheme 3 2 5 p= 2
q=
SOR = 3
1 2
x 1 , x
3 2
1
3
POR = 3
or
1 2
B1
3
(2 x 3)( x 1) 0
B2
2(x2 + 4) x + 11
B1
(a) q = - 3
(b) p =
1
1 2
4
2 or
4 = p(5 – 3) 2 + 2
-2 < k < 3
B1
3
(k + 2)(k – 3) < 0
B2
(12 – 4k) 2 – 4(4)(15 – 5k) < 0
B1
x = 1
3
2 + (3x – 3) =
x3 2
B2
x 3
5 3( x 1)
3
1
4 = p(1 – 3) 2 + 2
10
Full Marks
2
r=2
9
Sub Marks
5 2 2 5
or
5 3( x 1) ( 5 2 ) 5
3
x 3 2
B1
3
3
Number 11
12
Solution and marking scheme 3p – 2 – 2m
4
4
B3
log 5 343 – log 5 100
B2
log 5
343 100
x =
10 3
log 2
B1
4
4
B3
x2 = 1 4 x
B2
log 2(4 x) log 2 ( x – 2 ) = 1 + 2 log 2 4 n = -2
B1
3
3 1 n 6 1 10 6 7 5 3 1 m QS = 10 6
14
Full Marks
3 log 5 7 - 2 log 5 5 – 2log 5 2
x2 2 4x
13
Sub Marks
or
3
B2
n6 m PR = 7 5
k=3
B1
4
14 + 2k = - 20
and
14 + 2k = 20
B3
14 + 2k = - 20
or
14 + 2k = 20
B2
1 14 2k 10 2
B1
4
4
Number 15
Solution and marking scheme 3x2 + 3y2 – 6x + 28y + 23 = 0 2 ( x 2) 2 ( y 3) 2 ( x 2) 2 ( y 3) 2
16
17
( x 5) 2 ( y 2) 2 ( x 5) 2 ( y 2) 2
or
q = 15
3
3
B2 B1
45 q 10 6
B2
x 45
B1
Median = 56.75
3
3
B1
x 2 244
3
x2 (7) 2 2 3 4
3
B2
B1
x = 7 S PQ = 11.71 cm
4
S PQ = 5 ( 2.342 )
B3
POQ 3.142 0.8 2.342 rad
B2
QOR
3
B2
L = 55.5
19
Full Marks
3
1 2 (40) 17 55.5 5 12
18
Sub Marks
B1
4 radian 5
5
4
Number 20
21
22
Solution and marking scheme p = 4
Full Marks
3
3
2.5p + 2p + 4p = 34
B2
S AB = 5p (0.5)
B1
82.24
to
82.33 cm2
4
1 20 2 2
3.142 80 180
1 20 2 2
3.142 80 180
and
1 20 2 2
3.142 80 180
or
-
1 20 2 sin 80 2 1 20 2 sin 80 2 1 20 2 sin 80 2
3
4
B3
B2
B1
3
lim ( n 2 )
3
B2
n 1
n 2 n 1 23
Sub Marks
B1
7
3
4 1 3 1 4 3 2 1 2 1 [ 3 1 4 ] 2
B2
4 x 3x 4 3 2 x 2 1 [3x 4 ] 2
B1
6
3
Number 24
Solution and marking scheme y = 4x – 2 2 = 4 (1) + c dy = 6x – 2 dx
25
or
or
y2 4 x 1
Sub Marks
Full Marks
3
3
B2 B1
dy 4 dx
3
dy 48 dt dy 1 12 3 2 3 4 dt 3
B2
dy = 4 (3x – 2) 3 (3) dx
B1
7
3...