2007 AJC Prelims Paper 2

Anderson Junior College Preliminary Examination 2007 H2 Mathematics Paper 2 Section A: Pure Mathematics (40 marks) 1a) Differentiate 13tan (ln )x− with respect to x. [2] b) The curve C has parametric equations 222 ,1x tyt t t= += − + , where t is a non-zero parameter. (i) Show that the gradient of the curve at any point ( x,y) satisfies the equation 2 3 d( 2 1 ) d2 2 y tt xt −= − . [2] (ii) The line x = p is a tangent to the curve C. By using the result in (i), find the exact value of p. [2] 2. Express ( in the form )23r+ 2( 1) ( 1)rA r B r+ ++ − , where A and B are constants. Using the method of difference, find in terms of n. [5] () 1 23 2 n r r r = +∑ Hence, or otherwise, find an expression for in terms of n. [2] 1 2 2 ) 3 2 (− = ∑ + r n n r r 3. In an Argand diagram, the point A represents the fixed complex number a , where 0a r g ( ) 2a π<< . The complex numbers z and w are such that 2iza a−= and ww i a=+ . Sketch, in a single diagram, the loci of the points representing z and w [3] Find a) the minimum value of zw− in terms of a , [1] b) the range of values of 1arg z ⎛⎞ ⎜⎟⎝⎠ in terms of arg(a). [3] Page 2 This study source was downloaded by 100000851604598 from CourseHero.com on 11-29-2022 21:05:55 GMT -06:00 https://www.coursehero.com/file/5568601/2007-AJC-Prelims-Paper-2/

4a) By completing the square, or otherwise, describe the geometrical transformation by which the curve can be obtained from the curve . [2] 22 45xy y−−− = 0 22 1xy−= b) The diagram below shows the graph of ( )fy= x ⎟ ⎟ with asymptotes y = 2 and x = 0. The curve has turning points at (-2, 2) and (3, -2). y )2, 2− 5 2 On separate diagrams, sketch the graphs of (i) () x=− [ 3 ] 2 fy (ii) ()f' x= [ 3 ] y Show all intercepts, asymptotes and turning points clearly on your diagrams if they can be found. 5. Relative to an origin O, the point A has position vector 6 2 6 ⎛⎞ ⎜ ⎜⎜⎟−⎝⎠ , the line has equation , 1l 51 22 81 ⎛⎞ ⎛ ⎜⎟ ⎜=+ λ⎜⎟ ⎜⎜⎟ ⎜−−⎝⎠ ⎝ r 0 ⎞ ⎟ ⎟⎟⎠ λ∈ and the plane 1Π has Cartesian equation 54 . 31 5xyz−+= (i) Find the shortest distance from A to the plane 1Π and determine whether A and the origin O are on the same side or on opposite sides of 1Π . [3] (ii) 2Π is the plane that passes through point A and contains the line 1l . Find the acute angle between the planes 1Π and 2Π . [4] (iii) The plane 3Π has Cartesian equation 8x ya zb+ += . Find the values of a and b if the planes 1Π , 2Π and 3Π intersect along a common line. [5] x ( ( )3, 2− 2O Page 3 This study source was downloaded by 100000851604598 from CourseHero.com on 11-29-2022 21:05:55 GMT -06:00 https://www.coursehero.com/file/5568601/2007-AJC-Prelims-Paper-2/

Section B: Statistics (60 marks) 6. From the 10 letters of the word “LEGITIMISE”, find (i) the number of possible arrangements if no two vowels are next to each other, [2] (ii) the number of 3-letter code-words that can be formed. [3] 7. In a certain country, 16% of the total popula tion are aged 60 years or more, and 18% of the total population have myopia. Furthermore, 65% of those aged 60 years or more have myopia. Given that a person is chosen at random from the population, find the probability that the chosen person is (i) less than 60 years old and has myopia, [2] (ii) aged 60 years or more, given that the person does not have myopia, [3] (iii) either aged 60 years or more, or has myopia, or both. [2] 8. A manufacturer produces pens in two colours, red and blue. On average, 35% of the pens produced are red pens. (a) Find the probability that in a sample of 10 pens, more than half of them are red pens. [ 2 ] (b) The manufacturer offers a special price on red pens by selling them in packs of six. Pens are selected at random to fill each pack. Given that five red pens have been selected, find the least value of n such that the probability that at most n more pens need to be selected to complete a pack of 6 red pens is greater than 0.98. [4] 9. An IT services company has two help-lines which receive calls independently and at random times. The help-line manned by Alice receives calls at a rate of 1 call per hour and the help-line manned by Brenda receives calls at a rate of 3.6 calls per hour. (a) Find the probability that both help-lines receive a total of less than 5 calls in each of 3 successive 30-minute periods. [2] (b) Find the probability that Alice receives at most 1 call in a randomly chosen 30- minute period, given that both help-lines receive a total of less than 5 calls. [3] (c) The help-lines are open for 8 hours a day. By using a suitable approximation, find the probability that, in a random sample of 60 da ys, Alice receives less than 6 calls on at least ten but less than twenty days. [4] Page 4 This study source was downloaded by 100000851604598 from CourseHero.com on 11-29-2022 21:05:55 GMT -06:00 https://www.coursehero.com/file/5568601/2007-AJC-Prelims-Paper-2/

10. A manufacturer claims that his slimming diet helps people lose weight. A random sample of 20 people whose original weights were 80kg took the diet for a month and their new weights, x kg, were summarized as follows: ( 80) 18.7x−= −∑ , 2( 80) 102.5x−=∑ (a) Assume that the populat ion variance is not known. (i) Test whether or not the manufacturer’s claim is justified at the 5% level of significance. You should state any assumptions that you need to make. [6] (ii) Explain what is meant by the expression “at the 5% level of significance” in the context of this question. [1] (b) Suppose the population va riance is now known to be 10 kg. Determine the set of values of the mean weight, x kg of a random sample of 20 people which will lead to the rejection of the null hypothesis at the 5% level of significance. [2] If the test is now conducted at the 8% level of significan ce, explain briefly what can be concluded about the manufacturer’s claim if a random sample of 20 people yielded a sample mean weight of 78 kg. [1] 11. Large beer cans contain a volume of beer which is normally distributed with mean 500 ml and standard deviation 3.3 ml, while small beer cans contain a volume of beer which is also normally distributed with mean 340 ml and standard deviation 2.4. The volume of beer in any can is independent of the volume of beer in any other can. (a) If 5% of the small cans produced contain more than k ml of beer each, find the value of k. [2] (b) Find the probability that the volume of beer in two large cans differ by not more than 1 0 m l . [ 3 ] (c) Beer is also sold in bottles, each of whic h contains four times the volume of a large can. Find the probability that a crate of six bottles contain in total more than 12040 ml o f b e e r . [ 3 ] A sample of n large cans and a sample of n small cans are selected. If there is a probability of at most 0.01 that the average volume of beer in a small can exceeds half the average volume of beer in a large can by more than 92 ml, find the least value of n. [4] Page 5 This study source was downloaded by 100000851604598 from CourseHero.com on 11-29-2022 21:05:55 GMT -06:00 https://www.coursehero.com/file/5568601/2007-AJC-Prelims-Paper-2/

12(a) The marks of two short quizzes in Mathematics, x and y, of 10 students are shown in the table: Student 1 2 3 4 5 6