SPM Additional Mathematics (Model Test Paper)
Section B
[40 marks]
Answer any four questions from this section.
Question 7
(a)$\text{Prove that}{\left(\frac{\mathrm{cos}ec\text{}x\mathrm{sec}x}{\mathrm{sec}x\text{}\mathrm{cos}ec\text{}x}\right)}^{2}=1\mathrm{sin}2x$
[3 marks]
(b)(i) Sketch the graph of y = 1 – sin2x for 0 ≤ x ≤ 2π.
(b)(ii) Hence, using the same axes, sketch a suitable straight line to find the number of solutions to the equation $2{\left(\frac{\mathrm{cos}ec\text{}x\mathrm{sec}x}{\mathrm{sec}x\text{}\mathrm{cos}ec\text{}x}\right)}^{2}=\frac{x}{\pi}$ for 0 ≤ x ≤ 2π.
State the number of soultions.
[7 marks]
Question 8
Diagram 4 shows part of a curve x = y^{2} + 2. The gradient of a straight line QR is –1.
Find
(a) the equation of PQ, [2 marks]
(b) the area of shaded region, [4 marks]
(c) the volume of revolution, in terms of π, when the shaded region is rotated through 360^{o} about the yaxis. [4 marks]
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 yaxis.
Hence, draw the line of best fit. [3 marks]
(c) Use your graph in 9(b) to find the value of
(i) h.
(ii) k. [5 marks]
(i) h.
(ii) k. [5 marks]
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.
(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 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.
(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]
Question 11
In diagram 5, AOBDE, is a semicircle with centre O and has radius of 5 cm. ABC is a right angle triangle.
It is given that $\frac{AD}{DC}=3.85\text{and}DC=2.31\text{cm}.$
[Use π = 3.142]
Calculate
(a) the value of θ, in radians, [2 marks]
(b) the perimeter, in cm, of the segment ADE, [3 marks]
(c) the area, in cm^{2}, of the shaded region BCDF. [5 marks]SPM 2016 Additional Mathematics (Forecast Paper)
Section C
[20 marks]
Answer any two questions from this section.
Question 12
Diagram 6 shows a quadrilateral KLMN.
Calculate
(a) ∠KML, [2 marks]
(b) the length, in cm, of KM, [3 marks]
(c) area, in cm^{2}, of triangle KMN, [3 marks]
(d) a triangle K’L’M’ has the same measurements as those given for triangle KLM, that is K’L’= 12.4 cm, L’M’= 9.5 cm and ∠L’K’M’= 43.2^{o}, which is different in shape to triangle KLM.
(i) Sketch the triangle K’L’M’,
(ii) State the size of ∠K’M’L’. [2 marks]
(i) Sketch the triangle K’L’M’,
(ii) State the size of ∠K’M’L’. [2 marks]
Question 13
Table 2 shows the prices, the price indices and the proportion of four materials, A, B, Cand D used in the production of a type of bag.
Material 
Price (RM) for the year 
Price index in the year 2014 based on the year 2011 
Proportion 

2011 
2014 

A 
x 
7.20 
120 
7 
B 
8.00 
9.20 
115 
3 
C 
5.00 
5.50 
y 
6 
D 
3.00 
3.75 
125 
8 
Table 2
(a) Calculate the value of x and of y. [2 Marks]
(b) Find the composite index for the bag in the year 2014 based on the year 2011. [3 Marks]
(c) Given the cost for the production of the bag in the year 2014 is RM 115, find the corresponding cost in the year 2011. [2 Marks]
(d) From the year 2014 to year 2015, the price indices of material B and C increase by 5%, material A decrease by 10% and material D remains unchanged.
Calculate the composite index in the year 2015 based on the year 2011. [3 Marks]