RI H2 MATHS P2 Qns

RI H2 Mathematics 2017 Prelim Exam Paper 2 Question 1 Referred to the origin O, the points A, B and C have position vectors a, b and c respectively such that 23aij k , 523 bij k and 42 ci j k . (i) Given that M is the mid-point of AC, use a vector product to find the exact area of triangle ABM. [4] (ii) Find the position vector of the point N on the line AB such that MN  is perpendicular to AB  . [4] 2 (a) (i) Show that   121 2 .11 1 1rr rr r r    [1] (ii) Hence find  3 4 .11 n r rr r  (There is no need to express your answer as a single algebraic fraction). [4] (b) Amy and her brother Ben are saving money togeth er for their family trip. In the first week of 2017, Amy saves $25 and Ben saves $2. In each subsequent week, Amy saves $4 more than the amount she saved in the previous week, and Ben saves 22% more than the amount he saved in the previous week. (i) Which is the first week in which Ben saves more than Amy in that week? [2] (ii) They need a combined total of $2400 for the trip. How many complete weeks do Amy and Ben need to save before they can achieve their targeted amount? [2] 3 The function f is defined as follows. f: 3 s i n c o s , , 0 .xx x x x    (i) Write  f x as sin( )Rx  , where R and  are constants with exact values to be found. [2] (ii) Sketch the graph of =f ,y x stating the axial intercepts, and find the range of f. [3] (iii) Hence, solve  f1 x  exactly. [2] The function g is defined as follows: g: 2c o s , , . 66x xx x b      (iv) Write down the largest exact value of b, for 1g to exist. [1] (v) Taking the value of b found in part (iv), show that the composite function 1gf exists and solve  1gf x x  exactly. [3]

4 The line 1l has equation 1 31 2 4 x yz  and the line 2l has equation 11 434 x zy . (i) Show that 1l and 2l are skew lines. [3] (ii) Find a cartesian equation of the plane p which is parallel to 1l and contains 2l . [3] (iii) The point (0, ,1)A a is equidistant from p and 1l . Calculate the possible values of a exactly. [ 6 ] 5 For events X and Y, it is given that  1P| 2XY  ,  2P| 3YX  and  5P 6XY . Find (i) P, X [3] (ii) P' .XY [2] 6 The power consumption of a randomly chosen Effixion laptop has a normal distribution. The salesman at Elf Superstore claims that the av erage power consumption of an Effixion laptop is 100 watts. The power consumption, w watts, is measured for a random sample of 50 Effixion laptops. The results are summarised as follows. ( 100) 26w 2( 100) 273w Test whether this data provides evidence at the 3% level of significance, that the salesman has made an understatement. [6] The power consumption of another random sample of 50 Effixion laptops is measured. It is found that the sample variance is 6.25. Using th is sample only, find the set of values of w , correct to 2 decimal places, for which the test w ould result in the rejection of the null hypothesis in favour of the alternative hypothesis at the 1% level of significance. [4] 7 An unbiased cubical die has the number 1 on one face, the number 2 on two faces and the number 3 on three faces. Adrian invites Benny to play a game. In each round, Benny rolls the die twice. Adrian pays Benny $ a if the total score is 2 and $3 if th e total score is 3. However, if the total score is 4, Benny pays Adrian $2. No payment is made otherwise. (i) Find the probability that Adrian pays Benny at least 5 times in 20 rounds. [4] The random variable X represents Benny’s winnings in each round. (ii) Given that 6a  , find the probability distribution of X . Hence, help Benny decide if he should accept Adrian’s invitation to play the game. Justify your answer. [5] (iii) Determine the value of a for the game to be fair. [1] 8 (a) In Country S, each household’s monthly income per capita is calculated by taking the gross household income divided by the total number of members in the household. It is assumed

that this amount for a randomly chosen hous ehold consisting of 3 members follows a normal distribution with mean $2601 and standard deviation $768. (i) The Ministry of Education offers financia l aid to students from households consisting of 3 members each and with a household monthly income per capita lower than $1800. Find the probability that a randomly chosen household with 3 members does not qualify for financial aid. [1] (ii) It is found that there is a 50% chance that a randomly chosen household with 3 members has a gross househol d income between $5000 and $ a, where 5000.a Find the value of a, correct to the nearest dollar. [3] (b) Mr Tan is self-employed and his monthly income follows a normal distribution with mean $6000 and standard deviation $1000 whereas Mr s Tan works part-time and earns a fixed amount of $1500 a month. Their family’s monthly expenditure follows a normal distribution with mean  dollars and standard deviation 650 dollars. (i) It was found that 10% of the time they spend more than $5900 in a month. Find the value of , correct to the nearest dollar. [2] (ii) Mr and Mrs Tan save the remaining amount of their income after deducting their expenditure every month. Find the probability that their monthly savings in August and in September differ by more than $1000. [4] (iii) State an assumption needed for your calculation in part (b)(ii). [1] 9 (i) Sketch a scatter diagram that might be expected when x and y are related approximately as given in each of the cases (A ) and (B) below. In each case, your diagram should include 6 points, approximately equally spaced with respect to x, and with all x- and y-values positive. The letters ,, a n d ab c d represent constants. (A) 2 ,y ab x where a is positive and b is negative, (B) ln ,ycd x where c is positive and d is negative. [2] The following table shows the Gross Domestic Product (GDP) per capita, $ x, and infant mortality rate, y, for a sample of 9 countries. x ($) 1375 2502 10569 2966 11539 2036 4260 1433 7427 y 115 69 18 65 17 83 44 112 27 (ii) Draw a scatter diagram for these valu es, labelling the axes clearly. [2] (iii) Calculate the product moment correlation coeffi cient, and explain why its value does not necessarily mean that a linear model is th e best model for the relationship between x and y. [2]

(iv) State which of the two cases in part (i) is more appropriate for modelling the relationship between x and y. Calculate the product moment correla tion coefficient and the equation of the appropriate regression line for this case. [3] (v) Use the regression line in part (iv) to find an estimate of the infant mortality rate for a country with GDP per capita of $723. Comment on the reliability of your estimate. [3] 10 (a) It is given that the probability that 21 ra ndomly chosen people were all born on different days of the year is 0.55631, correct to 5 decimal places. Find the probability that in a random sample of 22 people, there are at least 2 people with the same date of birth. [3] [You may assume there are 365 days in a year and the probability that a person is born on any of the 365 days is the same.] (b) A soccer team consists of 1 goalkeeper, 4 defenders, 4 midfielders and 2 forwards. Country N has a squad of 3 goalkeepers, 6 defenders, 9 midfielders and 4 forwards. (i) How many diffe