IJC JC2 H2 Maths 2012 Questions Paper 2

2 IJC/2012/JC2 9740/02/S/12 Section A: Pure Mathematics [40 marks] 1 The graph of ( )fy x = has a turning point at ( )0,2 and passes through the points ( )4,0 − and ( )4,0 . The graph of ( )fy x ′= has a point of inflexion at the origin. On separate diagrams, sketch the graphs of (i) ( )f 2 y x ′= − , [2] (ii) ( )fy x = , [3] stating the equations of any asymptotes, the coordi nates of any stationary points and points of intersection with the axes. 2 (i) By using the substitution 1x u= , show that 4 2 2 2 1 d 24 4 x x x π= −∫ . [4] (ii) O is the origin and A is a point on the curve 2 1 4 y x x = − where 2 2 x = . The region R is enclosed by the curve 2 1 4 y x x = − , the line OA , the line 4x = and the x-axis. Find the exact area of R. [4] 3 A sequence of positive numbers 1 2 3 , , , ... x x x satisfies the recurrence relation 1 10 3 , n n x x + = − for 1, 2, 3, ... n = . (i) Given that the sequence converges to l, find the value of l. [2] (ii) Prove that ( ) ( ) 2 2 1 3n n x l l x + − = − . [2] (iii) Use the result in part (ii) to show that if nx l < , then 1nx l + > . [2] 4− 4 y x O ( )fy x = 1y = 2x = 2x = − 2 ( )fy x ′= 2x = 2x = − O y x

3 IJC/2012/JC2 9740/02/S/12 4 A sector of angle θ radians and radius 4 cm π is used to form the curved surface of a right circular cone such that there is no overlapping. Prove that the volume of the cone is 2 4 6 3 8 4 cm 3 π π θ θ − . [3] Show that the maximum volume of the cone is 43p q π cm 3 as θ varies, where p and q are integers to be determined. [5] [The formula for the arc length of a sector of a ci rcle is s r θ= .] 5 The diagram shows a solid with a horizontal rectang ular base OABC in which the lengths of OA and OC are 12 m and 8 m respectively. M and N are the mid-points of OC and AB respectively, and D and E are 7 m and 5 m vertically above M and N respectively. The point O is taken as the origin for position vectors. Perpe ndicular vectors i, j, k are such that i and j are parallel to OA and OC respectively, and k is perpendicular to the plane OABC . (i) Find a vector equation of the line DB . Hence find the foot of perpendicular from E to the line DB . [5] (ii) Find the acute angle between line DB and the plane OBE . [4] (iii) Find the length of projection of DE onto DB . [3] (iv) State, giving a reason, whether lines DB and AC are coplanar. [1] M D O B A E C N 7 5 8 12 i j k [Turn over

4 IJC/2012/JC2 9740/02/S/12 Section B: Statistics [60 marks] 6 The town council of a particular housing estate wis hes to find out whether the residents of the housing estate want to have a wet market built. A s urvey is to be carried out on 10% of the households in the housing estate. (i) Explain how a systematic sample might be carried ou t. [2] (ii) Describe one other method of sampling that could b e used, and state one advantage that this method has over systematic sampling. [2] 7 A die is biased and the probability, p, of throwing a six is known to be less than 1 . 6 An experiment consists of recording the number of six es in 25 throws of the die. In a large number of experiments, the standard deviation of t he number of sixes is 1.5. Show that the value of p is 1 . 10 Hence find the probability that at least 6 but fe wer than 10 sixes are recorded during a particular experiment. [4] The biased die is now thrown 40 times. Find the mo st likely number of sixes obtained. [1] 8 Call-outs at fire stations are classified as either genuine or false, and occu r at random times. All call-outs are independent of one another. (a) At a fire station in town A, on average there are two genuine call-outs in a w eek, and one false call-out in a two-week period. (i) Find the probability that there are fewer than 6 g enuine call-outs in a randomly chosen two-week period. [2] (ii) Using a suitable approximation, estimate the proba bility that the total number of call-outs in a randomly six-week period exceeds 19. [4] (b) At a fire station in town B, on average there are m genuine call-outs in a week. Given that the probability of at most o ne genuine call-out in a randomly chosen week is 0. 08, write down an equation for the va lue of m and find this value numerically. [2]

5 IJC/2012/JC2 9740/02/S/12 9 The mileage, X km per litre of petrol, of a particular model of c ar is a random variable with mean µ km per litre. A petroleum company had modifications done to their car petrol with the intention of improving mileage in cars. The company conducted a pilot test of the modified petrol with 13 cars of that particular model. The r esults are summarised by ( )12 6.09 x − = ∑ , ( ) 2 12 20.853 x − = ∑ . (i) Find unbiased estimates of the population mean an d variance. [3] (ii) It is given that 0=µ µ for the unmodified petrol. A test is carried out a t the 5% significance level to determine if the modified pet rol improves mileage in cars. Find the set of values of 0µ for the company to have insignificant evidence tha t the modified petrol improves mileage in cars. [4] (iii) Based on observations over a long period, it is fo und that that 12 µ = . Find the least significance level for the company to have signific ant evidence that the modified petrol improves mileage in cars. [2] State an assumption which you need to make for the above tests to be valid. [1] 10 A class of twenty-five pupils consists of 15 girls and 10 boys. At the beginning of the year four pupils are to be chosen at random to form an i nterim class committee comprising of “Chairperson”, “Vice-Chairperson”, “Treasurer” and “Secretary”. Find (i) the probability that exactly two girls are chosen, [2] (ii) the probability that the Chairperson and Vice-Chai rperson are of opposite sex, [2] (iii) the probability that the Chairperson and Vice-Chai rperson are both boys given that the Treasurer and Secretary are of opposite sex. [3] State with a reason whether or not the events ‘Chai rperson and Vice-Chairperson are both boys’ and ‘Treasurer and Secretary are of opposite sex’ are independent. [2] [Turn over

6 IJC/2012/JC2 9740/02/S/12 11 An athletics coach believes that athletes with long er legs can run faster. He selected 10 of his athletes and recorded their leg lengths, x metres and their timings, t seconds, in a 100 m race. The results are given in the table. x 0.70 0.76 0.80 0.84 0.85 0.89 0.92 0.95 0.98 1.00 t 13.90 12.73 12.12 11.89 11.80 11.42 11.29 10.94 11.0