**SPM Additional Mathematics (Model Test Paper)**

**Section B**

[40 marks]

Answer any

**four**questions from this section.**Question 9**

Use graph paper to answer this question.

Table 1 shows the values of two variables,

*x*and*y*, obtained from an experiment. The variables*x*and*y*are related by the equation*y*=*hk*^{x}^{ + 1}, where*h*and*k*are constants.x |
1 |
2 |
3 |
4 |
5 |
6 |

y |
4.0 |
5.7 |
8.7 |
13.2 |
20.0 |
28.8 |

Table 1

(a) Based on table 1, construct a table for the values (

*x*+ 1) and log*y.*[2 marks](b) Plot log

*y*against (*x*+ 1), using a scale of 2 cm to 1 uint on the (*x*+ 1) –axis and 2 cm to 0.2 unit on the log*y*-axis.Hence, draw the line of best fit. [3 marks]

(c) Use your graph in 9(b) to find the value of

(i)

(ii)

(i)

*h*.(ii)

*k*. [5 marks]

*Answer and Solution:***(a)**

x + 1 |
2 |
3 |
4 |
5 |
6 |
7 |

log y |
0.60 |
0.76 |
0.94 |
1.12 |
1.30 |
1.46 |

**(b)**

**(c)**

*y*=

*hk*

^{x}^{ + 1}

log

*y =*log*h*+ (*x*+ 1) log*k*log

*y =*log*k*(*x*+ 1) + log*h*Y = log

*y**m = log k*

*X =*(*x*+ 1)

c = log hc = log h

**(i)**

log

*h*=*y*-interceptlog

*h*= 0.26

*h***= 1.82**

**(ii)**

$\begin{array}{l}\mathrm{log}k=\text{Gradient of the graph}\\ \text{}=\frac{1.46-0.60}{7-2}\\ \text{}=0.172\\ k=1.486\end{array}$

**Question 10**

(a) 20% of the students in SMK Bukit Bintang are cycling to school. If 9 pupils from the school are chosen at random, calculate the probability that

(i) exactly 3 of them are cycling to school,

(ii) at least a student is cycling to school. [4 marks]

(i) exactly 3 of them are cycling to school,

(ii) at least a student is cycling to school. [4 marks]

(b) The volume of 800 bottles of fresh milk produced by a factory follows a normal distribution with a mean of 520

(i) Find the probability that a bottle of fresh milk chosen in random has a volume of less than 515 ml.

(ii) If 480 bottles out of 800 bottles of the fresh milk have volume greater that

*ml*per bottle and variance of 1600*ml*^{2}.(i) Find the probability that a bottle of fresh milk chosen in random has a volume of less than 515 ml.

(ii) If 480 bottles out of 800 bottles of the fresh milk have volume greater that

*k ml*, find the value of*k*. [6 marks]

Answer and Solution:Answer and Solution:

**(a)(i)**

*X*~ Students in SMK Bukit Bintang who are cycling to school

*X*~

*B*(

*n*,

*p*)

*X*~

*B*(9, 0.2)

*P*(

*X*=

*r*) =

^{n}C_{r}. p^{r}. q^{n-r}Probability, exactly 3 students are cycling to school

*P*(

*X*= 3) =

^{9}C

_{3}(0.2)

^{3}(0.8)

^{6}

= 0.1761

**(a)(ii)**

At least a student is cycling to school

= 1 –

*P*(*X*= 0)= 1 –

^{9}C_{0}(0.2)^{0}(0.8)^{9}= 0.8658

**(b)(i)**

*m*

*=*520

*ml*

σ

^{2 }= 1600*ml*^{2}σ = 40

Let

*X*represents volume of a bottle of fresh milk.*X*~

*N*(520, 1600)

*P*(

*X*< 515)

$=P\left(Z<\frac{515-520}{40}\right)$

=

*P*(*Z*< – 0.125)=

*P*(*Z*> 0.125)**= 0.4502**

**(b)(ii)**

$\begin{array}{l}P\left(X>k\right)=\frac{480}{800}\\ P\left(Z>\frac{k-520}{40}\right)=0.6\\ \frac{k-520}{40}=-0.253\\ k-520=-10.12\\ k=509.88\end{array}$

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