SAJC H2 Math P2 Question

[Turn Over ST ANDREW’S JUNIOR COLLEGE PRELIMINARY EXAMINATION MATHEMATICS Higher 2 9740/2 Monday 15 SEP 2015 3 hours READ THESE INSTRUCTIONS FIRST Write your name, civics group and index number on all the work you hand in. Write in dark blue or black pen on both sides of the paper. You may use a soft pencil for any diagrams or graphs. Answer all the questions. Total marks is 100. Give non-exact numerical answers correct to 3 signi ficant figures, or 1 decimal place in the case of angles in degrees, unless a different level of accuracy is specified in the question. You are expected to use a graphic calculator. Unsupported answers from a graphic calculator are a llowed unless a question specifically state otherwise. Where unsupported answers from a graphic calculator are not allowed in a question, you are required to present the mathematic steps using mathematical notations and not calculator commands. You are reminded of the need for clear presentation in your answers. The number of marks is given in brackets [ ] at the end of each question or part question. At the end of the examination, fasten all your work securely together. This document consists of 6 printed pages includi ng this page.

2 [Turn Over Pure Mathematics (40 marks) 1 It is given that ( ) ( ) ( ) ( ) 2 2 6 3 f 3 9 x x x x − = − + . (i) Using partial fractions, find ( ) f d x x ∫ . [5] The diagram above shows the curve with equation ( )fy x = . (ii) Find the exact area bounded by the curve ( )fy x = − and the x-axis, between the values of 0x = and 3x = . [3] 2 (a) Adrian has signed up at a driving centre to learn how to drive. His first lesson is 40 minutes long. Each subsequent lesson is 5 minutes longer than the previous lesson, so that the second lesson is 45 mi nutes long, the third lesson is 50 minutes long, and so on. The centre requires a student to have attended at least 60 hours of lessons before he is qualified to take the driving test. Find the minimum number of lessons that Adrian has to attend before he can take the test. [4] (b) A sequence of real numbers 1 2 3 , , ,... u u u , where 1 0u ≠ , is defined such that the ( )1 th n + term of the sequence is equal to the sum of the fi rst n terms, where .n +∈ ℤ Prove that the sequence 2 3 4 , , ,... u u u follows a geometric progression. [3] Hence find 1 1 N r r u + = ∑ in terms of 1 and . u N [2] y x 0

3 [Turn Over 3 The equations of lines 1l and 2l are given as follows: 1 31 : 1 4 , 20 l λ λ     = − + −    ∈          r ℝ , and 2 3 6 : 3, 5 1 y z l x − − == − . (i) Find the coordinates of N, the foot of perpendicular of the point )(4, 5, 3 A onto 1l . [3] (ii) The plane 1Π contains the line 1l and is parallel to 2l . Find the Cartesian equation of the plane 1Π . [3] Another plane 2Π is defined by 3 1 4 1     − =       ri and intersects 1Π in a line 3l . (iii) Find a vector equation of 3l . [2] A plane 3Π has an equation 2 6 tx y z d − + = . (iv) What can be said about t and d if the three planes 1 2 3 , , Π Π Π have exactly one point in common? [2] 4 (a) Solve the equation 4 1 3 0, w i + − = expressing the roots in the form ire θ , where r > 0 and π θ π − < ≤ . Show the roots on an Argand diagram, showing the relationship between them clearly. [5] (b) A complex number z = ix y + has modulus r and argument θ , where 0 2 πθ< < . The complex numbers v and w are defined by iv y x = − + and 2 2 2 i w x y xy = − + . (i) Express v and w in terms of z. [2] (ii) Hence, or otherwise, express vw in exponential form in terms of r and θ . [2] (iii) If 4 4 3i vw = − − , solve for z in exponential form, giving your answer in exact form. [4]

4 [Turn Over Statistics (60 marks) 5 A school has a total of 1000 students. A company w hich sells stationery products in a bookstore inside the school, intends to select a sa mple of 50 students within the school to perform a survey with regards to its products, r epresentative of the opinions of both male and female students. (i) In the context of the question, describe how quota sampling could be carried out to select the 50 students, and explain a disadvantage of quota sampling. [2] (ii) State the name of an appropriate sampling method t hat does not have this disadvantage and describe how it can be carried out . [2] (iii) The stationery product also has retail outlets outs ide the school catering to the general public. Explain why it is not realistic for the company to carry out the sampling method in part (ii) to obtain a representa tive sample of 50 customers at their retail outlets at any given month. [1] 6 For events A and B, it is given that ( )59 1 17 ( ) , ( ' ) , ' 100 3 41 P B P B A P A B = = = . Find (i) ( ), P A [3] (ii) ( ). P A B ∩ [2] 7 Studies have shown that 74% of patients who suffer from an allergy are relieved of its symptoms after taking a new drug. (i) A hospital tested the new drug on 15 randomly chos en patients with the allergy. Find the probability that at least half of the pati ents are relieved of the symptoms. [2] (ii) Another n patients were added to the group of 15 patients to form a new bigger group of patients. It is given that the probability of at least 2 patients from this new group not being relieved of the symptoms is at least 0.99. Ex press this information as an inequality in n, and hence find the least value of n. [4] The drug was reformulated and the success rate of the improved drug was found to be 92%. (iii) The hospital decides to test the reformulated drug on another group of patients. Using a suitable approximation, find the probabilit y that, out of 36 patients, more than 30 patients are relieved of the symptoms. [3]

5 [Turn Over 8 The table shows the number y (in millions) of cell-phone subscribers in a count ry from 2001 to 2010, where t represents number of years from 2000. t 1 2 3 4 5 6 7 8 9 10 y 1.6 2.7 4.4 6.4 8.9 13.1 19.3 28.2 38.2 48.7 The relationship between y and t is given by the formula ty ab = , where a and b are constants. (i) Using the substitution ln I y = , show that the relation between I and t is linear. [1] (ii) Find the equation of the estimated regression line of I on t and hence give estimates for a and b. [2] (iii) Find the (product moment) correlation coefficient between I and t. [1] (iv) Predict the number of cell-phone subscribers in t he year 2015. Comment on the reliability of your prediction. [3] (v) It is required to estimate the value of t for which I = 1.5. Explain which of the regression lines I on t or t on I, should be used. Use the equation of your choice to find the value of t when I = 1.5. [3] 9 Farmer Chan found that the mean mass of his previo us crop of tomatoes was 0µ grams. He decides to try a new type of fertiliser for the present crop of tomatoes. The manufacturers of the new fertiliser claim that it w ill increase the mean mass of tomatoes. The farmer intends to test their claim by taking a random sample of size 50 from the present crop. A random sample of 50 tomato es gives the following data 3500 x =∑ , 2 245220.5 x =∑ where X is the random variable representing the mass of to matoes in grams, after application of the new fertiliser. (i) Calculate the unbiased estimates of the population mean and variance. [2] (ii) If 0 69.4 µ = ,