Download tanjong katong girls` school mid-year examination 2010 secondary

TANJONG KATONG GIRLS’ SCHOOL
MID-YEAR EXAMINATION 2010
SECONDARY THREE
4038
ADDITIONAL MATHEMATICS
Friday
30 April 2010
Additional Materials:
Answer Paper
READ THESE INSTRUCTIONS FIRST
s
h
e
f
a
2 h 15 min
C
Write your name, class and register number on all the work you hand in.
t
a
M
Write in dark blue or black pen on both sides of the paper, and use a pencil for
drawing graphs and diagrams. Do not use staples, highlighters or correction fluid.
Answer all the questions.
Write your answers on the separate writing paper provided.
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.
e
h
The use of a scientific calculator is expected, where appropriate.
You are reminded of the need for clear presentation in your answers.
T
At the end of the examination, fasten all your work securely together.
The number of marks is given in brackets [
] at the end of each question or part
question.
The total marks for this paper is 90.
Setter :
Markers :
Miss Yeo LS
Miss Lee SN, Miss Sharifah, Mr Seah CS, Miss Yeo LS
This Question Paper consists of 4 printed pages, including this page.
2
Answer all questions.
Section A [40 marks]
1
Determine the range of values of m for which
4 x  m  1x  8  3 x  7
2
has
(i)
(ii)
2
Given that ln 2 = a and ln 3 = b,
3
(i)
(ii)
3
4
e
f
a
2 real distinct roots,
no real roots.
express ln 24e
numbers,
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C
in the form p(qa + rb + s), where p, q, r and s are real
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find x such that ln x =
1
2a  3b  .
3
e
h
T
3  2 3 
2
(a)
(b)
[3]
[6]
cm2. Given that the base BC = (6  2 3 ) cm,


find the height of the triangle in the form a  b 3 cm, where a and b are real
numbers.
5
[3]
2
Find the range of values of p for which the curve y  x  3x  p  2 intersects
the line y  9 x  30 at two distinct points.
State the value of p for which the line is a tangent to the curve.
Triangle ABC has area
[5]
[1]
[6]
Solve the inequality x2 x  3  9 , where x > 0.
[4]
Calculate the range of values of c for which x3 x  4  6 x  c for all
real values of x.
[4]
TKGS Sec 3 Mid-year Exam 2010
4038 Additional Mathematics
3
6
At the beginning of 1990, the number of a certain species of birds was estimated to
to be 70000. The population decreases so that after a period of n years, the population
 0.08 n
p was 70000 e
. Estimate
(i)
the population at the beginning of year 2000,
[3]
(ii)
the year in which the population is one-fifth of those presented at the
beginning of 1990.
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h
Section B [50 marks]
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2
7
The roots of the equation 3x  1  8 x are  and .
(i)
State the values of  + and .
(ii)
Find the quadratic equation in x whose roots are
e
f
a
[5]
C
[2]
1
1
and
,
2
2
2
leaving your answer in the form ax  bx  c  0 , where a, b and c
e
h
are integers.
8
T
3
[5]
2
It is given that f(x) = 2 x  11x  5 x  18 .
(i)
(ii)
Show that x  2 is a factor of f(x).
3
2
[1]
2
Given that 2 x  11x  5 x  18 = (x  2) (ax  bx  c) where a, b and c are
constants, find the values of a, b and c.
Hence solve the equation f(x) = 0.
[5]
2
(iii)
3
Use the solutions of f(x) = 0 to solve the equation 2  5 x  18 x  11x .
TKGS Sec 3 Mid-year Exam 2010
[4]
4038 Additional Mathematics
4
9
(a)
The cubic polynomial f(x) is such that the coefficient of x3 is 2 and the
roots of f(x) = 0 are 1, k and k, where k > 0. It is given that f(x) has a
remainder of 42 when divided by x  4.
(b)
(i)
Find the value of k.
[4]
(ii)
Show that f(x) has a remainder of 18 when divided by x .
[1]
x2 + 2x  3 .
x 1
10
11
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h
x
 8 2   21 .
 
(a)
Solve the equation 4
(b)
3
2
2 x  3x  22 x  11
Express
in partial fractions.
2
x  2 x  3
e
h
Solve the equation
4x  2
(a)
(b)
e
f
a
The polynomial f(x) leaves remainders of 2 and 10 when divided by
x  1 and x + 3 respectively. Find the remainder when f(x) is divided by
 (3)7 
2x  3
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C
[5]
[5]
[6]
,
[6]
2 log 9 9 x   log 3 4 x  8  log 3  x  5 .
[6]
8
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THE END
TKGS Sec 3 Mid-year Exam 2010
4038 Additional Mathematics