CJC 2012 JC2 H2 Prelim Paper 1 Questions

9740/01/PRELIMS/2012 [Turn over CATHOLIC JUNIOR COLLEGE General Certificate of Education Advanced Level Higher 2 JC2 Preliminary Examination MATHEMATICS 9740/01 Paper 1 22 AUGUST 2012 3 hours Additional Materials: List of Formulae (MF15) READ THESE INSTRUCTIONS FIRST Write your name and class 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. Do not use staples, paper clips, highlighters, glue or correction fluid. Answer all the questions. 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 states otherwise. Where unsupported answers from a graphic calculator are not allowed in a question, you are required to present the mathematical steps using mathematical notations and not calculator commands. You are reminded of the need for clear presentation in your answers. At the end of the examination, arrange your answers in NUMERICAL ORDER. Place this cover sheet in front and fasten all your work securely together. The number of marks is given in brackets [ ] at the end of each question or part question. Name: ___________________________ Class: ______ __________ Question 1 2 3 4 5 6 7 8 9 10 11 Total Marks Total 6 9 10 11 10 7 12 8 9 10 8 100 This document consists of 6 printed pages. Catholic Junior College

2 9740/01/PRELIMS/2012 [Turn over 1 It is known that w satisfies the equation 112 +≤− ww . (i) If ∈w ℝ, find the range of values of w. [2] Hence, state the range of values of w for which . 11 12 ≥+ − w w [1] (ii) If ∈w ℂ, find a cartesian equation for the locus of w and hence sketch it on an Argand diagram. [3] 2 (a) Mr Thrift makes use of a special offer from a bank to obtain an interest-free loan of $2000. He decides to pay $50 in the first month. On the first day of each subsequent month, he pays $10 more than in the previous month. How many complete months would it take for him to fully repay the debt? [3] (b) On 1 January 2012, Mr. Spendalot uses a credit card to borrow $2000 from a bank, at an interest rate of 2% a month. He repays the ba nk $50 on the 10 th of each month. Interest is charged on the balance at the end of ea ch month. (i) Calculate the outstanding amount at 1 January 2013. (ii) How many months does he take to repay the entire lo an? [4] [2] 3 A sequence K,,, 321 uuu is such that 01 =u and ( )( )[ ] 2 2 1 21 4 ++ −++=+ nn nnuu nn , for all +∈ Zn . (i) Prove by mathematical induction that ( ) 2 1 1 + −= n nun for all positive integers n. [4] (ii) Hence, find ( )( )[ ]∑ = ++ −+N n nn nn 2 2 2 21 4 and state the sum to infinity. [3] (iii) Hence, or otherwise, evaluate ∑ ∞ = + −+ 2 2 2 )] 1([ 32 n nn nn . [3] 4 Two lines, l1 and l2, have equations 2, 12 1 =−−=− yzx and 243 +=−=+ zyx respectively. (i) Determine whether l1 and l2 intersect, and state the coordinates of the intersection point, if any. (ii) Find the acute angle between l1 and l2. (iii) Find an equation for the line l3, where l3 is the reflection of l1 about l2. [3] [2] [6]

3 9740/01/PRELIMS/2012 [Turn over 5 (a) The graph of a function f( x) has undergone the following transformations: Step 1: Translation in the negative x-direction by 1 unit Step 2: Translation in the positive y-direction by 5 units Step 3: Scaling parallel to the x-axis by a factor of 2 1 The function of the resulting graph is 944)h( 2 ++= xxx . Express h(x) in terms of f(x). Hence, obtain an expression for function f(x). [5] (b) The ceiling function,  x , is a function which gives the smallest integer greater than or equal to x, e.g.     27 . 2, 32 . 2 −=−= . The graph of  xy = is shown below: (i) Show that     10 09 . 35 . 22 =+ . A function g(x) is defined by       + 2 1: g xx a , where /g1876 ∈ ℝ/g2878. (ii) Evaluate ) 3g( )12 . 0g( + . (iii) Sketch the graph of )g( xy = for 40 ≤< x . (iv) State the range of g for all positive real values of x. [1] [2] [1] [1] 1 3 – 1 – 3 0

4 9740/01/PRELIMS/2012 [Turn over 6 The sketch below shows the graph of )f( xy = . The curve intersects the y-axis at the point      − 2 11 , 0 , has a maximum point at ( )ba, and a minimum point at ( )dc, . The equation of the asymptotes are 3=y , 1−=x and 2=x . On separate diagrams, sketch the following graphs indicating the points corresponding to the stationary points, axial intercepts and asymptotes where necessary. (i) |) f(| xy = [2] (ii) )f( 1 xy = [3] (iii) )( ' fxy = [2] 7 A curve C is defined by the parametric equations πθπθθ ≤≤−=+= where ,sin , 1cos 2 yx . (i) Sketch the curve C, giving the coordinates of any points of intersection with the x- and y- axes. [2] (ii) Find the area enclosed by the curve C for πθπ ≤≤− . [4] (iii) The normal to the curve at the point ( )θθ sin , 1cos 2 + , where 20 πθ << , meets the x- and y- axes at Q and R respectively. The origin is denoted by O. Find the area of triangle OQR , leaving your answer in terms of .θ [6]

5 9740/01/PRELIMS/2012 [Turn over 8 Candy is being stored in a closed container in the form of a regular hexagonal prism, with sides x cm and height h cm (as shown in the figure below). Given that the container has a volume of 972 cm 3, (i) show that its base area given by 2 33 2x cm 2. [1] (ii) Using differentiation, find the minimum area of the material, A cm 2, that is used to make the container, leaving your answer to 2 decimal places. [6] (iii) Given that the cost of the material for the packaging is $ 0.05 per 100 cm 2, find the minimum cost required for the container. [1] 9 Let x and y be variables such that xy 1tan 1) 1ln( −+=+ . (i) Show that x yxx yx d d)21 (d d)1 ( 2 2 2 −=+ . [2] (ii) Find the Maclaurin series for y, up to and including the term in x3, giving the coefficients in terms of e. [4] (iii) Given that the first two non-zero terms in the Maclaurin series for y are equal to the first two non-zero terms in the series expansion of bx a + 1 , where a and b are non- zero constants, find a and b in terms of e. [3] 10 (a) Find the roots of the equation iz 3223 −= in exponential form. [3] (b) Given that iz 3= is a root of the equation 018 11 234 =++++ bz zzaz (i) find the values of the real numbers a and b. [3] (ii) Hence solve the equation exactly. [4] x

6 9740/01/PRELIMS/2012 [Turn over 11 On a single Argand diagram, sketch the following loci: (i) 53 =−z [2] (ii) ( )      −=− − 2 1tan 8arg 1πz [2] (iii) izz 532 +−=+ [2] State the complex number p that represents the point of intersection of the loci in (i) and (ii) , in the form a + i b, where a and b are exact integers to be determined. The complex number w satisfies the inequalities 53 ≤−w ( )      −≤− − 2 1tan 8arg 1πw iww 532 +−≥+ . Illustrate the locus of w on the same Argand diagram. [1] [1]   THE END  