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12 3472/2 TOPIC: CIRCULAR MEASURES 1. Diagram 6 shows a sector POQ of a circle, centre O. The point A lies on OP, the point B lies on OQ and AB is perpendicular to OQ. The length of OA = 8 cm and POQ .It is given that OA:OP = 4:7. 6 [Use π = 3.142] Calculate P (a) the length, in cm, of AP. [1 marks] A (b) the perimeter, in cm, of the shaded region. [5 marks] 8 cm (c) the area, in cm2, of the shaded region. [4 marks] 6 O B Q Diagram 6 2. Diagram 9 shows two circles. The larger circle has centre X and radius 12 cm. The smaller circle has centre Y and radius 8 cm. The circles touch at point R. The straight line PQ is a common tangent to the circles at point P and point Q. Given that PXR radiants. [Use π = 3.142] (a) show that θ = 1.37 ( to two decimal places). [2 mark] (b) calculate the length, in cm, of the minor arc QR, [3 marks] Q Q 8 cm Q (c) calculate the area, in cm2, of the shaded region. [5 marks] Y R P θ Q 12 cm X Diagram 9 3472/2 – ADDITIONAL MATHEMATICS PAPER 2 - PROGRAM LATIH TUBI SPM 2012 13 3472/2 TOPIC: INTEGRATIONS 1. Diagram 5 shows part of the curve y = k(x – 1)3, where k is a constant. dy The curve intersects the straight line x = 3 at point A. At point A, dx = 24. (a) Find the value of k. [3 marks] (b) Hence, calculate y y = k(x – 1) 3 (i) the area of the shaded region, P. (ii) the volume generated, in terms of π, when the region R which is A bounded by the curve, the x-axis and the y-axis, is resolved through 360o about the x-axis. [7 marks] P x O R x=3 Diagram 5 2. Diagram 7 shows the straight line OQ and the straight line y = k intersecting the curve x = 4y – y 2 at point Q. 9 It is given that the area of the shaded region is 2 unit2. y (a) Find the value of k. [ 6 marks] (b) The region enclosed by the curve 4 and the y-axis is revolved through Q 360o about the y-axis. y=k Find the volume of revolution, in terms of π. [4 marks] x = 4y – y2 x O Diagram 7 3472/2 – ADDITIONAL MATHEMATICS PAPER 2 - PROGRAM LATIH TUBI SPM 2012 14 3472/2 TOPIC: LINEAR PROGRAMMING 1. Use graph paper to answer this question. An institution offers two computer courses, P and Q. The number of participants for course P is x and for course Q is y. The enrolment of the participants is based on the following constarints: I : The total number of participants is not more than 100. II : The number of participants for course Q is not more than 4 times the number of participants for course P. III : The number of participants for course Q must exceed the number of participants for course P by at least 5. (a) Write three inequalities, other than x 0 and y 0, which satisfy all of the above constraints. [3 marks] (b) By using a scale of 2 cm to 10 participants on both axes, construct and shade the region R that satisfies all the above constraints. [3 marks] (c) By using your graph from (b), find (i) the range of the number of participants for course Q if the number of participants for course P is 30. (ii) the maximum total fees per month that can be collected if the fees per month for courses P and Q are RM50 and RM60 respectively. [4 marks] 3472/2 – ADDITIONAL MATHEMATICS PAPER 2 - PROGRAM LATIH TUBI SPM 2012 15 3472/2 2. Use graph paper to answer this question. A workshop produces two types of rack, P and Q. The production of each type of rack involves two processes, making and painting. Table 4 shows the time taken to make and paint a rack of type P and a rack of type Q. Time taken (minutes) Rack Making Painting P 60 30 Q 20 40 The workshop produces x racks of type P and y racks of type Q per day. The production of racks per day is based on the following constraints: I. The maximum total time for making both racks is 720 minutes. II. The total time for painting both racks is at least 360 minutes. III. The ratio of the number of racks of type P to the number of racks of type Q is at least 1 : 3. (a) Write three inequalities, other than x 0 and y 0, which satisfy all the above constraints. [3 marks] (b) Using a scale of 2 cm to 2 racks on both axes, construct and shade the region R which satisfies all the above constraints. [3 marks] (c) By using your graph in part (b), find (i) the minimum number of racks of type Q if 7 racks of type P are produced per day, (ii) the maximum total profit per day if the profit from one rack of type P is RM24 and from one rack of type Q is RM32. [4 marks] 3472/2 – ADDITIONAL MATHEMATICS PAPER 2 - PROGRAM LATIH TUBI SPM 2012 16 3472/2 TOPIC: INDEX NUMBER 1. Table 2 shows the price indices and percentage of usage of four items, P, Q, R and S, which are the main ingredients in the production of a type of biscuit. Price index for the year 1995 Percentage of usage Item based on the year 1993 (%) P 135 40 Q x 30 R 105 10 S 130 20 Table 2 (a) Calculate (i) the price of S in the year 1993 if its price in the year 1995 is RM37.70, (ii) the price index of P in the year 1995 based on the year 1991 if its price index in the year 1993 based on the year 1991 is 120. [5 marks] (b) The composite index number of the cost of biscuit production for the year 1995 based on the year 1993 is 128. Calculate (i) the value of x, (ii) the price of a box of biscuit in the year 1993 if the corresponding price in the year 1995 is RM32. [5 marks] 3472/2 – ADDITIONAL MATHEMATICS PAPER 2 - PROGRAM LATIH TUBI SPM 2012 17 3472/2 2. Table 2 shows the price indices and the price indices for the four ingredients, P, Q, R and S, used in making biscuits of a particular kind. Diagram 8 is a pie chart which represents the relative amount of the ingredients P, Q, R and S, used in the making of these biscuits. Price per kg (RM) Price index in the Ingredient year 2004 based on Year 2001 Year 2004 Q P the year 2001 P 0.80 1.00 x 120o Q 2.00 y 140 60o 100o S R 0.40 0.60 150 R S z 0.40 80 Table 2 Diagram 8 (a) Find the value of x, y and z. [3 marks] (b) (i) Calculate the composite index for the cost of making these biscuits in the year 2004 based on the year 2001. (ii) Hence, calculate the corresponding cost of making these biscuits in the year 2001 if the cost in the year 2004 was RM2 985. [5 marks] (c) The cost of making these biscuits is expected to increase by 50% from the year 2004 to the year 2007. Find the expected composite index for the year 2007 based on the year 2001. [2 marks] 3472/2 – ADDITIONAL MATHEMATICS PAPER 2 - PROGRAM LATIH TUBI SPM 2012 18 3472/2 SOLUTION OF TRIANGLES 1. Diagram 7 shows quadrilateral ABCD. A (a) Calculate 5.6 cm (i) the length, in cm, of AC (ii) ACB [4 marks] B 105 o 16.4 cm 50o C 6 cm D Diagram 7 (b) Point A’ lies on AC such that A’B = AB. (i) Sketch Δ A’BC. (ii) Calculate the area, in cm2, of Δ A’BC. [6 marks] 3472/2 – ADDITIONAL MATHEMATICS PAPER 2 - PROGRAM LATIH TUBI SPM 2012 19 3472/2 2. Diagram 12 shows a trapezium KLMN. KN is parallel to LM and LMN is obtuse. Find N (a) the length, in cm, of LN. [2 marks] 12.5 cm 80 o K M 32 o 5.6 cm L Diagram 12 (b) the length, in cm, of MN. [3 marks] (c) LMN. [3 marks] (d) the area, in cm2, of triangle LMN. [2 marks] 3472/2 – ADDITIONAL MATHEMATICS PAPER 2 - PROGRAM LATIH TUBI SPM 2012 20 3472/2 PROBABILITY DISTRIBUTIONS 1. (a) In a survey carried out in a school, it is found that 2 out 5 students have handphones. If 8 students from that school are chosen at random, calculate the probability that (i) exactly 2 students have handphones, (ii) more than 2 students have handphones. [5 marks] (b) A group of workers are given medical check up. The blood pressure of a worker has a normal distribution with a man of 130 mmHg and a standard deviation of 16 mmHg. Blood pressure that is more than 150 mmHg is classified as ‘high blood pressure’. (i) A worker is chosen at random from the group. Find the probability that the worker has a blood pressure between 114 mmHg and 150 mmHg.more than 40 kg. (ii) It is found that 132 workers have ‘high blood pressure’. Find the total number of workers in the group. [5 marks] 3472/2 – ADDITIONAL MATHEMATICS PAPER 2 - PROGRAM LATIH TUBI SPM 2012 21 3472/2 2. (a) The result of a study shows that 20% of the pupils in a city cycle to school. If 8 pupils from the city are chosen at random, calculate the probability that (i) exactly 2 of them cycle to school. (ii) less than 3 of them cycle to school. [4 marks] (b) The masses of water melons produced from an orchard follow a normal distribution with a mean of 3.2 kg and a standard deviation of 0.5 kg. Find (i) the probability that the watermelon chosen randomly from the orchard has a mass of not more than 40 kg. (ii) the value of m if 60 % of the watermelons from the orchard have a mass of more than m kg. [6 marks] 3472/2 – ADDITIONAL MATHEMATICS PAPER 2 - PROGRAM LATIH TUBI SPM 2012