2008 JJC H2 Maths 9740/02 Solution:
Section A: Pure Maths
1. Let the first term of the AP be a and common difference be d.
(reject , since a is positive)
Hence,
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2.
when
when
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3.
, k=positive constant
When ,
When ,
When :
, a parabola
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4. (i) AB:AD =2:3
AB:BD =2:1
By the Ratio Theorem,
(ii)
(iii)
Direction vector of L is
=
put x=2,
from (1),
from (2),
(iv) normal vector of the plane =
Equation of the required plane is
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5. (a)
For to be real ,
(b)(i)
(ii)
=
(iii)
,
k = 0, 1, 2
, where =
1 +iz = − iz
iz (1 + ) = − 1
iz =
iz = i tan
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Section B: Statistics
6 (a) (i) Number of ways = 5!x 6! = 86400
(ii) Number of ways = x 2! x 2!x 7! = 564480
(b) (i) Number of ways = 4!x 5! = 2880
(ii) Number of ways = 4!x 3 x 4! =1728
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7 (i) P(game ends in a draw)
= 0.3x0.2 + 0.4x0.3 + 0.3x0.5 = 0.330
(ii) P(A wins the first game)
= P(A cat, B mouse) +P( A mouse, B elephant)+ P(A elephant, B cat)
= 0.3x0.3 + 0.4x0.2 + 0.3x05 = 0.320
(iii) P(B wins 1st game, 1st game not a draw)= 1 – P(A wins) – P(draw)
= 1- 0.320 – 0.330= 0.350
P(1st game not a draw) = 1 – P(1st game ends in a draw)
= 1 – 0.330= 0.670
P( B wins 1st game, given that 1st game is not a draw)
=
(iv) P(B will win the contest)
= 0.350 + (0.330)(0.350) +
=
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8 Let X be the no. of washing machines sold per day
(i) P = 0.135(3sf)
(ii) Let Y be the no. of washing machines sold in a period of 3 days
P
=
= 0.938 (3sf)
(iii) Let T = no. of days in which no washing machine sold
n =100 is large ,
P
= 0.95942 = 0.959 (3sf)
Let n = no. of washing machine per day
P
P
P
P
Least n = 3
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9 let X be the wt. of a bundle of spinach and Y be the wt. of a bundle of kangkong.
&
(i) P= 0.159 (3sf)
(ii)
Let W =
P(3sf)
(iii) Let T =
P(3sf)
(iv) Let the required wt. be m
P
Maximum m = 137.18
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10(i)
(ii) H0:
H1:
Using G.C,
Since , we do not reject H0 and conclude at 5%
level of significance that the population mean score do not
exceed 50.
No assumption is needed since sample size is large, by Central
Limit Theorem, sample mean follows a normal
(iii) For alternate hypothesis to be accepted,
(iv) The probability of concluding that the mean Mathematics test
score to be more than 50 when it is actually 50 is .
(v) Since number of sample is small and the population variance is
unknown, use a T-test.
We assume that the population follows a normal distribution.
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11(i) In stratified sampling, every member in the strata has an equal
chance of being selected. In a quota sampling, we do not have
access to every member of the population. Members are being
selected to meet the quota set by the suveyor. Hence if members
do not fit into the criteria, they would be rejected.
(ii)To survey the popularity of a cosmetics product among women
visiting a particular shopping centre. Stratified sampling would
be difficult because of the lack of access to the sampling frame,
i.e. all the women visiting a shopping centre. Hence, it is difficult
to form strata. However, it is easy to conduct a quota sampling
by interviewing women based upon quota drawn up by age group
or income level.
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12(i) Key in data into GC.
The equation of the regression line of on is .
(ii) When , .
(iii) Since , hence the regression line of on is
almost identical to the regression line of on . Hence it is reliable
to use the regression line of on to estimate when .
(iv)
Based on the scatter diagram, the linear model is not suitable.
(v) Choose Model D: because as increases, decreases.
We calculate the new product moment correlation coefficient,
. Hence Model D is appropriate.
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