Question 8:
(a) Diagram 3 shows the standard normal distribution graph.
On Diagram 3, shade the region(s) to represent P(|Z| ≥ a). [1 mark]
(b) Diagram 4 shows the standard normal distribution graph which is divided into three equal parts.
State
(i) the vakue of k,
(ii) the mode for part A.
[2 marks]
Answer:
(a)
(b)(i) k = 0.431
(b)(ii) Mode for part A
k = -0.431
(a) Diagram 3 shows the standard normal distribution graph.
On Diagram 3, shade the region(s) to represent P(|Z| ≥ a). [1 mark](b) Diagram 4 shows the standard normal distribution graph which is divided into three equal parts.
State(i) the vakue of k,
(ii) the mode for part A.
[2 marks]
Answer:
(a)
(b)(i) k = 0.431(b)(ii) Mode for part A
k = -0.431
Question 9:
(a) Diagram 5 shows the conversation between Nash and Cikgu Wong.
Is Nash’s statement true? Give your justification. [1 mark]
(b) In a shooting competition, Dev fires n shots and p is the chance that his shots hit the target whereas q is the otherwise chance. The chance of his shots hit the target at most one is 11 times of the chance of not hitting the target.
(i) Show that n (1 – q) = 10q.
(ii) Hence, or otherwise, find the value of p and of q if the mean is 6.
[4 marks]
Answer:
(a) True. Shoe sizes can be counted.
(b)(i) $$ \begin{aligned} P(X \leqslant 1) & =11 P(X=0) \\ { }^n C_0\left(p^0\right)\left(q^n\right)+{ }^n C_1\left(p^1\right)\left(q^{n-1}\right) & =11^n C_0\left(p^0\right)\left(q^n\right) \\ q^n+n p q^{n-1} & =11 q^n \\ n p q^{n-1} & =10 q^n \\ \frac{n p q^n}{q} & =10 q^n \end{aligned} $$
(b)(ii)
$$ \begin{aligned} &\text { Compare, }\\ &\begin{aligned} \frac{n p}{q} & =10 \\ n(1-q) & =10 q \\ n(1-q) & =10 q \\ n p & =10 q \\ 6 & =10 q \\ q=\frac{3}{5}, p & =\frac{2}{5} \end{aligned} \end{aligned} $$
(a) Diagram 5 shows the conversation between Nash and Cikgu Wong.
Is Nash’s statement true? Give your justification. [1 mark](b) In a shooting competition, Dev fires n shots and p is the chance that his shots hit the target whereas q is the otherwise chance. The chance of his shots hit the target at most one is 11 times of the chance of not hitting the target.
(i) Show that n (1 – q) = 10q.
(ii) Hence, or otherwise, find the value of p and of q if the mean is 6.
[4 marks]
Answer:
(a) True. Shoe sizes can be counted.
(b)(i) $$ \begin{aligned} P(X \leqslant 1) & =11 P(X=0) \\ { }^n C_0\left(p^0\right)\left(q^n\right)+{ }^n C_1\left(p^1\right)\left(q^{n-1}\right) & =11^n C_0\left(p^0\right)\left(q^n\right) \\ q^n+n p q^{n-1} & =11 q^n \\ n p q^{n-1} & =10 q^n \\ \frac{n p q^n}{q} & =10 q^n \end{aligned} $$
(b)(ii)
$$ \begin{aligned} &\text { Compare, }\\ &\begin{aligned} \frac{n p}{q} & =10 \\ n(1-q) & =10 q \\ n(1-q) & =10 q \\ n p & =10 q \\ 6 & =10 q \\ q=\frac{3}{5}, p & =\frac{2}{5} \end{aligned} \end{aligned} $$
Question 10:
(a) A school bookstore is having a stock clearance promotion which involves 8 different pens and 10 different notebooks. Anvi will spend RM24 in the promotion. Table 1 shows the prices for a unit of each item.
Find the number of different ways Anvi can buy the item if
(i) she wants to buy notebooks only,
(ii) the number of notebooks is at most equal to the number of pens.
[4 marks]
(b) Diagram 6 shows an example of five-character password formed by using letters and digits on a computer screen.
Find the number of different passwords if
(i) the first three characters are A8a without repetition,
(ii) the first character must be a non-vowel capital letter and followed by at least two prime numbers placed adjacently.
[4 marks]
Answer:
(a)(i) $$ \begin{aligned} \frac{\mathrm{RM} 24}{4} & =6 \\ { }^{10} C_6 & =210 \text { ways } \end{aligned} $$
(a)(ii) $$ \begin{aligned} & \left({ }^8 C_4 \times{ }^{10} C_4\right)+\left({ }^8 C_6 \times{ }^{10} C_3\right)+\left({ }^8 C_8 \times{ }^{10} C_2\right) \\ & =18105 \text { ways } \end{aligned} $$
(b)(i) $$ 1 \times 1 \times 1 \times 59 \times 58=3422 $$
(b)(ii) $$ 21 \times 4 \times 4 \times 58 \times 62 $$ (non-vowel capital letter followed by 2 prime numbers) + $$ 21 \times 4 \times 4 \times 4 \times 58 $$ (non-vowel capital letter followed by 3 prime numbers) + $$ 21 \times 4 \times 4 \times 4 \times 4 $$ (non-vowel capital letter followed by 4 prime numbers) $$ \text { = } 1291584 \text { ways } $$
(a) A school bookstore is having a stock clearance promotion which involves 8 different pens and 10 different notebooks. Anvi will spend RM24 in the promotion. Table 1 shows the prices for a unit of each item.
Find the number of different ways Anvi can buy the item if(i) she wants to buy notebooks only,
(ii) the number of notebooks is at most equal to the number of pens.
[4 marks]
(b) Diagram 6 shows an example of five-character password formed by using letters and digits on a computer screen.
Find the number of different passwords if(i) the first three characters are A8a without repetition,
(ii) the first character must be a non-vowel capital letter and followed by at least two prime numbers placed adjacently.
[4 marks]
Answer:
(a)(i) $$ \begin{aligned} \frac{\mathrm{RM} 24}{4} & =6 \\ { }^{10} C_6 & =210 \text { ways } \end{aligned} $$
(a)(ii) $$ \begin{aligned} & \left({ }^8 C_4 \times{ }^{10} C_4\right)+\left({ }^8 C_6 \times{ }^{10} C_3\right)+\left({ }^8 C_8 \times{ }^{10} C_2\right) \\ & =18105 \text { ways } \end{aligned} $$
(b)(i) $$ 1 \times 1 \times 1 \times 59 \times 58=3422 $$
(b)(ii) $$ 21 \times 4 \times 4 \times 58 \times 62 $$ (non-vowel capital letter followed by 2 prime numbers) + $$ 21 \times 4 \times 4 \times 4 \times 58 $$ (non-vowel capital letter followed by 3 prime numbers) + $$ 21 \times 4 \times 4 \times 4 \times 4 $$ (non-vowel capital letter followed by 4 prime numbers) $$ \text { = } 1291584 \text { ways } $$