2010 H2 Mathematics (9740)
JC 2Preliminary Examination Paper 2
Mark Scheme
1 /since as
2 /
(i)maximum value of
(ii)
or
3 / (a) ;
Where tangents // x-axis:
Therefore, the exact x-coordinates are and
(b)
Using GC, (correct to 3 sig fig)
4 / (i)
(ii)
Height of X above the ground = 5 units
Let be the required angle.
Hence,
(iii)
Therefore, equation of plane
(iv)Normal vector to plane OBDCis
Let be the required angle.
5 /
Differentiate w.r.t x :
(i) Differentiate w.r.t x, we have
When ,
(ii)
(iii)When,
6 / (i) [or ]
(ii) P( “3” is taken) = 1 – P(“3’ is not taken) =
P(exactly 2 numbered discs | “3” is taken)
=
or
P(exactly 2 numbered discs | “3” is taken)
=
7 / Since n is large (n>50), by Central Limit Theorem,
Sample sum S approx.
We want
From G.C., . Thus least n is 97.
8 / (i)
(ii) Ignoring Charles first and tie Alex/Benjamin
=> arrange 10 distinct objects around round table: ways.
Swap between Alex/Benjamin: 2! Ways
Insert Charles, not next to Alex: 9 available slots
Hence total no of arrangements = = 6531840
(iii) Case 1: A+A+A+G ways
Case 2: A+A+G+G ways
Total no of ways = ways
9 / X ~ N(200, 302)
Y ~ N(350, 602)
(i)
P(pays more than $5) =
P(pays less than $4) =
Required prob. =
(ii)
Required prob. =
Let C= mass of an empty regular cup
I = mass of ice-cream content of a regular cup
Then
Since C and I are indept r.v.,
Var (X) = Var (C) + Var (I)
Hence Var (I) = Var (X) – Var (C) =
10 / (i) Let X be number of scratches per plastic sheet.
X~ P(λ).
=
(ii) With = 0.12,
Since is large and ,
approx.
approx.
Alternatively:
[B1]
Since n=500 is large and np=3.3246 <5, [M1]
approx. [B1]
approx. [A1]
(iii)
Let A and B be the no. of scratches in the 25 and 20 plastic sheets from manufacturer A and B respectively.
i.e.
i.e.
Since A and B are indept Poisson R.V.,
Since
approx.
11 / (a)(i) The list generated is likely to introduce cyclical pattern/periodicity in the students’ marks (sampling interval of ). The students selected may be of the same standard and hence sample is biased. For example, if we start sampling very near the top of the list, then students with very good results is likely to be over-represented.
(a)(ii)
Select number of students from each yearproportional to the total number of students in each year:
1st Year =
2nd Year =
3rd Year =
4th Year =
Within each year, randomly select the required number of students.
Advantage: likely to obtain a more representative sample.
(b)(i)
an unbiased estimate of population mean
an unbiased estimates of population variance
(b)(ii)
To test:
against
at 5% level of significance.
Under
Test statistic: i.e. T~ t(11)
We reject Ho if
From GC, , we reject H0 and conclude at 5% significant level that the mean mark has increased (after the implementation of the new portal).
Assumption: The marks follow a normal distribution.
Since
And
As such, the conclusion would be unchanged if a Z-test were performed instead.
12 / (i)
Equation of the regression line of y on x is
(correct to 3 s.f.)
(ii) Withdrawal 5 (120, 94)
Spend a lot of cash in a relatively short time
(iii)
Remove the exceptional case:
Corresponding equation of the regression line of y on x is
(correct to 3 s.f.)
At
The number of hours until Alice’s next visit is 103 hours if she withdraws $50.
Since r = 0.949 is close to 1 (when the exceptional case was removed) and the scatter diagram suggest that a linear model is a good fit for the remaining data. The amount of withdrawal of $50 is within the range of the data collected, hence the estimate is reliable.
(iv)
Corresponding equation of the regression line of y on ln x is
At
Linear model not appropriate as y (number of hours till next withdrawal) is negative.
(v) Since r = 0.426 is much smaller than r = 0.949 obtained in (iii), this suggests that the data come from two different population, eg. Alice may have a very different spending pattern in November and December due to the pending festive season.