2012 A-level H2 Mathematics (9740) Paper 2 Question 10 Suggested Solutions

All solutions here are SUGGESTED. Mr. Teng will hold no liability for any errors. Comments are entirely personal opinions.

(i)
Firstly, gold coins are scattered randomly in a randomly chosen region of $1m^2$ .
Secondly, the mean number of gold coins found in any randomly chosen region of $1m^2$ is constant.
Lastly, the finding of a gold coin in a randomly chosen region of area $1m^2$ is independent of the finding of another gold coins.

(ii)
Let X denote the number of gold coins in a region of $1m^2$ .
$\mathrm{P}(X \ge 3)$

$= 1 - \mathrm{P}(X \le 2)$

$= 0.0474$

(iii)
Let Y denote the number of gold coins found in a region of $x~m^2$ .
$Y ~\sim~ \mathrm{Po} (0.8x)$

$\mathrm{P} (Y = 1) = 0.2$

Using the Graphing Calculator, $x = 0.324$

(iv)
Let W denote the number of gold coins found in a region of $100m^2$
$W ~\sim~ \mathrm{Po} (80)$

Since $\lambda = 80 > 10$ , $W ~\sim~ \mathrm{N} (80, 80)$ approximately

$\mathrm{P} (W \ge 90)$
$= \mathrm{P} (W \ge 89.5)$ by continuity correction
$\approx 0.144$

(v)
Let C and S denote the number of gold coins and pottery shards in in a region of $50m^2$ respectively.

$C ~\sim~ \mathrm{Po} (40)$
Since $\lambda = 40 > 10$ , $C ~\sim~ \mathrm{N} (40, 40)$ approximately.
$S ~\sim~ \mathrm{Po} (150)$
Since $\lambda = 150 > 10$ , $S ~\sim~ \mathrm{N} (150, 150)$ approximately.

$C + S ~\sim~ \mathrm{N} (190, 190)$ approximately

$\mathrm{P} (C + S \ge 200)$
$\mathrm{P} (C + S \ge 199.5)$
$\approx 0.245$

(vi)
$S + 3C ~\sim~ \mathrm{N} (30, 510)$ approximately

$\mathrm{P} (S + 3C \ge 0)$
$\mathrm{P} (S + 3C \ge -.05)$ by continuity correction
$\approx 0.912$

KS Comments:

For (iii), students can also introduce the poisson formula from MF15 and solve by hand. Please rm. to apply continuity correction for (iv). I took awhile to understand (vi) too. :/ Students should read carefully if their english is as bad as mine. For (vi), students should note that it is necessary that they approximate to normal before consider the difference of the distributions. Be reminded that we cannot take the difference of poisson distributions.

About

KS Teng

KS has been teaching H2/H1 Mathematics and IB mathematics for the more than 10 years. Having taught students from all various Junior Colleges, KS adapts to students' abilities and help them better understand the topics. As someone who loves to teach mathematics and sees it as a truly useful tool in life, KS seeks to enable students to appreciate math. Therefore, his tuition mission is to motivate and cultivate students to be independent and confident thinkers.

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