# SPM Additional Mathematics (Model Test Paper) – Question Only 7 – 13

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 = y2 + 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 360o about the y-axis. [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 = hkx + 1, where h and k are constants.

 x 1 2 3 4 5 6 y 4 5.7 8.7 13.2 20 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) 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.
[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 ml2.
(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
[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 cm2, 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 cm2, 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’= 43.2o, 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]

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]