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1. 1. ADDITIONAL MATHEMATICS FORM 4 2007mozac 1 ADDITIONAL MATHEMATICS FORM 4 MODULE 4 STATISTICS CIRCULAR MEASURE
2. 2. ADDITIONAL MATHEMATICS FORM 4 2007mozac 2 7 STATISTICS  PAPER 1 1 The mean of a list of numbers x – 1, x + 3, 2x + 4, 2x – 3, x + 1 and x – 2 is 7. Find (a) the value of x, (b) the variance of the numbers. Answer: (a) x = .……………………… (b) …………………………… 2 The mean of a list of numbers 3k , 5k + 4, 3k + 4 , 7k – 2 and 6k + 6 is 12. Find (a) the value of k, (b) the median of the numbers. Answer: (a) k = .……………………… (b) …………………………… 3 Given a list of numbers 8, 9, 7, 10 and 6. Find the standard deviation of the numbers. Answer: ………………………….
3. 3. ADDITIONAL MATHEMATICS FORM 4 2007mozac 3 4 The set of positive numbers 3, 4, 7, 8,12, x, y has a mean 6 and median 7. Find the possible values of x and y. Answer: x = …………………………….. y = …………………………….. 5 The test marks of a group of students are 15, 43, 47, 53, 65, and 59. Determine (a) the range, (b) the interquartile range of the marks. Answer: (a) …………………………… (b) …………………………… 6 The mean of five numbers is q . The sum of the squares of the numbers is 120 and the standard deviation of the numbers is 4m. Express q in terms of m. Answer : ……………………………
4. 4. ADDITIONAL MATHEMATICS FORM 4 2007mozac 4 7 The sum of the 10 numbers is 170 and the sum of the squares of the numbers is 2930. Find the variance of the 10 numbers. Answer: ……………………………… 8 Score 0 1 2 3 4 Frequency 7 10 p 15 8 The table shows the scores obtained by a group of contestants in a quiz. If the median is 2, find the minimum value of p. Answer: ……………………………… 9 The numbers 3, 9, y , 15, 17 and 21 are arranged in ascending order. If the mean is equal to the median, determine the value of y. Answer : y = …………………………… 10 Number 41 – 45 46 – 50 51 – 55 56 – 60 61 – 65 Frequency 6 10 12 8 4 The table above shows the Additional Mathematics test marks of 40 candidates. Find the median of the distribution. Answer:............................................. Number of goals 1 2 3 4 5
5. 5. ADDITIONAL MATHEMATICS FORM 4 2007mozac 5 11 The table above shows the number of goals score in each match in a football tournament. Calculate the mean and the standard deviation of the data. Answer : mean = …………………………… standard deviation = ……………... 12 Given the set of positive numbers n, 5, 11. (a) Find the mean of the set of numbers in terms of n. (b) If the variance is 14, find the values of n. Answer: (a) …………………………… (b) n = ..……………………… 13 The mean and standard deviation for the numbers x1, x2, …, xn are 74 and 26 respectively. Find the (a) mean for the numbers 3x1 + 5 , 3x2 + 5, …, 3xn + 5, (b) variance for the numbers 4x1 + 2 , 4x2 + 2, …, 4xn + 2. Answer: (a) …………………………… (b) …………………………… Frequency 7 6 4 2 1
6. 6. ADDITIONAL MATHEMATICS FORM 4 2007mozac 6 14 The mean of the data 2, h, 3h, 11, 12 and 17 which has been arranged in an ascending order, is p. If each of the element of the data is reduced by 2, the new median is 8 9 p. Find the values of h and p. Answer: h = …………………………… p = …………………………… 15 The table above shows a set of numbers arranged in ascending order where p is a positive integer. (a) Express the median of the set of the of numbers in terms of p. (b) Find the possible values of p. Answer: (a) ………………………….. (b) p = …………………...….  PAPER 2 16 A set of examination marks x1, x2, x3, x4, x5, x6 has a mean of 7 and a standard deviation of 14. (a) Find (i) the sum of the marks, x. (ii) the sum of the squares of the marks, x2 . (b) Each mark is multiplied by 3 and then 4 is added to it. Find, for the new set of marks, (i) the mean, (ii) the variance. Number 2 p – 1 7 p + 4 10 12 Frequency 2 4 2 3 3 2
7. 7. ADDITIONAL MATHEMATICS FORM 4 2007mozac 7 17 Length (mm) 16 – 19 20 – 23 24 – 27 28 – 31 32 – 35 36 – 39 Frequency 2 8 18 15 6 1 The table above shows the lengths of 50 leaves collected from a tree. (a) Calculate (i) the mean, (ii) the variance length of the leaves. (b) Without drawing an ogive, find the interquartile range length of the leaves. 18 Set R consists of 40 scores, y, for a certain game with the mean of 9 and standard deviation of 5. (a) Calculate y and y2 . (b) A number of scores totaling 200 with a mean of 10 and the sum of the squares of these scores of 2700, is taken out from set R. Calculate the mean and variance of the remaining scores in set R. 19 A set of data consists of 10 number. The sum of the numbers is 150 and the sum of the squares of the numbers is 2 472. (a) Find the mean and variance of the 10 numbers. (b) Another number is added to the set of data and the mean is increased by 1. Find (i) the value of this number, (ii) standard deviation of the set of 11 numbers. 20 The table shows the frequency distribution of the scores of the scores of a group of pupils in a game. Score Number of pupils 10 – 19 1 20 – 29 2 30 – 39 8 40 – 49 12 50 – 59 m 60 – 69 1 (a) It is given that the median score of the distribution is 42. Calculate the value of m. (b) Use the graph paper provided by the invigilator to answer this question. Using a scale of 2 cm to 10 scores on the horizontal axis and 2 cm to 2 pupils on the vertical axis, draw a histogram to represent the frequency distribution of the scores. Find the mode score. (c) What is the mode score if the score of each pupil is increased by 5?
8. 8. ADDITIONAL MATHEMATICS FORM 4 2007mozac 8 8 CIRCULAR MEASURE  PAPER 1 1 Convert (a) 5420to radians. (b) 406 radians to degrees and minutes. Answer : (a) .......................................... (b) ......................................... 2 Answer : ...................................... 3 The area of a sector of a circle with radius 14 cm is 147 cm2 . Find the perimeter of the sector. Answer :....................................... The diagram on the left shows a sector OAB with centre O and radius 9 cm. Given that the perimeter of the sector OAB is 30 cm. Find the angle of AOB in radian. O A B 9 cm9 cm
9. 9. ADDITIONAL MATHEMATICS FORM 4 2007mozac 9 4 Answer :....................................... 5 Answer : ..................................... 6 Answer : ..................................... 2 rad 6 cm O A B The diagram on the left shows a circle with a sector OAB and centre O . Find the area of the major sector OAB in cm2 and state your answer in terms of π. O R Q P 2 cm 10 cm The diagram on the left shows a sector of a circle OPQ with centre O and OPR is a right angle triangle. Find the area of the shaded region. O A B The diagram on the left shows an arc of a circle AB with centre O and radius 4 cm. Given that the area of the sector AOB is 6 cm2 . Find the length of the arc AB.
10. 10. ADDITIONAL MATHEMATICS FORM 4 2007mozac 10 7 Answer : ...................................... 8 Answer :....................................... 9 Answer : ...................................... O P Q R S2 cm 0.8 rad The diagram shows two sectors OPQ and ORS of concentric circles with centre O. Given that POQ = 08 radian and OP = 3PR, find the perimeter of the shaded region. The diagram shows a semicircle of OPQR with centre O. Given that OP = 10 cm and QOR = 30. Calculate the area of the shaded region. P O R Q 30 10 cm The diagram shows a circle with centre O. Given that the major arc AB is 16cm and the minor arc AB is 4cm. Find the radius of the circle. O A B
11. 11. ADDITIONAL MATHEMATICS FORM 4 2007mozac 11 10 Answer : ...................................... 11 Answer : (a) r = ................................... (b) θ= ................................... 12 Answer : ……………………………… O R S  The diagram on the left shows a sector ROS with centre O. Given the length of the arc RS is 724 cm and the perimeter of the sector ROS is 25 cm. Find the value in radians. O A B  r cm The diagram on the left shows a sector with centre O. Given that the perimeter and the area of the sector is 14 cm and 10 cm2 respectively. Find (a) the value of r, (b) the value of θin radians. O A B 60 8 cm The diagram on the left shows a sector OAB of a circle with centre O. Find the perimeter of the shaded segment.
12. 12. ADDITIONAL MATHEMATICS FORM 4 2007mozac 12 13 Answer : (a) .......................................... (b) .........................................  PAPER 2 14 The above diagram shows two arcs AB and DE, of two circles with centre O. OBD and OCE are straight lines. Given OB = BD, find (a) the length of arc AB, (b) the area of segment DE, (c) the area of the shaded region. 15 The diagram on the right shows the position of a simple pendulum which swings from P and Q. Given that POQ = 25° and the length of arc PQ is 12.5 cm, calculate (a) the length of OQ, (b) the area swept out by the pendulum. O P Q O P Q R S T The diagram on the left shows a circle PRTSQ with centre O and radius 3 cm.Given RS = 4 cm and POQ = 130. Calculate (a) ROS , in degrees and minutes, (b) the area of segment RST, (c) the perimeter of the shaded region. 70 OA B C D E6 cm
13. 13. ADDITIONAL MATHEMATICS FORM 4 2007mozac 13 16 The diagram above shows a semicircle ACBE with centre C and a sector of a circle OADB with O. Given BAO = 35and OA = OB = 7 cm. Calculate (a) the diameter AB, (b) the area of the triangle AOB, (c) the area of the shaded region, (d) the perimeter of the shaded region. 17 The diagram above shows two circles PAQB with centres O and A respectively. Given that the diameter of the circle PAQB = 12 cm and both of the circles have the same radius. (a) Find POA in radians. (b) Find the area of the minor sector BOP. (c) Show that the area of the shaded region is (12 – 9 3 ) cm 2 the perimeter of the shaded region is (4+ 6 3 ) cm. O A B P O A B C D E   35
14. 14. ADDITIONAL MATHEMATICS FORM 4 2007mozac 14 18 The diagram below shows the plan of a garden. PCQ is a semicircle with centre O and has radius of 8 cm. RAQ is a sector of a circle with centre A and has a radius of 14 m. Sector COQ is a lawn. The shaded region is a flower bed and has to be fenced. It is given that AC = 8 m and COQ = 1956 radians. Using π= 3142, calculate (a) the area, in m2 , of the lawn, (b) the length, in m, of the fence required for fencing the flower bed, (c) the area, in m2 , of the flower bed. R Q R C P A O