2012 NYJC H2 Math Prelim P1 Question

[Turn Over NANYANG JUNIOR COLLEGE JC2 PRELIMINARY EXAMINATION Higher 2 MATHEMATICS 9740/01 Paper 1 11 Sep 2012 3 hours Additional Materials: Answer Papers List of Formulae (MF15) READ THESE INSTRUCTIONS FIRST Write your name and class on every script 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 significant 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 allowed 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, fasten all your work securely together. The number of marks is given in brackets [ ] at the end of each question or part question. This document consists of 6 printed pages.

2 2012 NYJC 9740/01 [Turn Over 1 It is given that 3 2 f ( ) x ax bx cx d = + + + , where a, b, c and d are constants. The curve C with equation f ( ) y x = passes through (0, –1) and has a maximum point at ( –1, 1). The area bounded by C, the x-axis and the lines x = 2 and 3x = is 31 4 units 2. Given that f( x) > 0 for 2 3 x< < , find the values of a, b, c and d. [5] 2 The vectors a and b are given by (sin ) (cos ) a i j k θ θ = + + and (sin ) (cos ) b = i j k φ φ + + , where 0 θ φ π ≤ ≤ ≤ . Find an expression for a b × in terms of δ , where 1 2 ( ) δ φ θ = − . [5] Deduce that the angle α between a and b is given by 2sin sin 1 cos α δ δ = + . [2] 3 The equation of a curve is given by 3 2 3 4 3 2 x x y y + = − . Find d d y x in terms of x and y, simplifying your answer. [2] The curve meets the line y x = − at point P. Find (i) the coordinates of P and [2] (ii) the equation of the tangent at P. [2] The tangent to the curve at P cuts the x-axis at R and the normal to the curve at P cuts the y-axis at Q. If O denotes the origin, use the results above to give a geometrical description of the quadrilateral OQPR. [1]

3 2012 NYJC 9740/01 [Turn Over 4 (a) Solve the inequality 1 2sin 0 x− > for π0 2x≤ ≤ . Hence, evaluate the exact value of π 2 0 1 2sin d x x −∫ . [4] (b) Use the substitution x = sin θ to show that 2 cos 1 d d 2cos 2 1 1 4 x x θ θ θ = − −∫ ∫ . Given that 0 cos π d 42cos 2 1 α θ θ θ = −∫ , find the value of α , given that 0 2 πα< < . [4] 5 A curve f ( ) y x = undergoes in succession, the following transformations: A: A reflection in the y-axis B: A translation of 2 units in the direction of the x-axis C: A translation of a− units in the direction of the y-axis, where 1a > (i) The equation of the resulting curve is g( ) y x = , where 1 3 g( ) 5 ax x −= − + . Determine the equation of the curve f ( ) y x = , in terms of a and x . [4] (ii) Sketch, on the same diagram, the graphs of g( ) y x = and 1g ( ) y x −= , indicating clearly the equations of the asymptotes and the axial intercep ts. [4] 6 (a) In a triangle with vertices A, B and C, angle BAC is a right-angle and angle ABC = 3 xπ − . (i) Show that 1 3 tan 3 tan AB x AC x += − . [1] (ii) Hence, show that when x is small enough for x2 and higher powers of x to be neglected, then AB a bx AC ≈ + , where a and b are exact constants to be determined. [3]

4 2012 NYJC 9740/01 [Turn Over (b) A curve is defined by the equation ( ) 2 2 d1 1 d yx xy x x+ + = + and (0, 1) is a point on the curve. (i) Find the Maclaurin’s expansion of y up to and including the term in x2. [3] (ii) Hence, find the series expansion of e y , up to and including the term in x2. [3] 7 (a) An arithmetic progression has first term a and common difference d, where a and d are non- zero. The first, third and seventh terms of the a rithmetic progression are three consecutive positive terms of a geometric progression with com mon ratio r. (i) Show that 2r = . [3] (ii) The first term of the geometric progression is one -tenth that of the first term of the arithmetic progression. Find the smallest value of n such that the sum of the first n terms of the geometric progression exceeds the sum of t he first 2 n terms of the arithmetic progression. [3] (b) Each time that a ball falls vertically on to a hor izontal floor, it rebounds to three-fifth of the height from which it fell. It is initially droppe d from a point h m above the floor. (i) Find the distance travelled by the ball just befor e it strikes the floor for the third time in terms of h. [1] (ii) Show that the total distance travelled by the ball cannot exceed 4 h m. [3] 8 (a) Solve the equation iz 5 = − 32, giving your roots in the form reiθ , where r > 0 and − π < θ ≤ π . Sketch on an Argand diagram the points P1, P2, P3, P4 and P5 representing these roots, where P1 represents the root with the smallest argument and P1 P2 P3 P4 P5 is a polygon described in an anticlockwise sense. Find the area of P1 P2 P3 P4 P5. [6] (b) On an Argand diagram sketch clearly the locus of P where P represents the complex number z such that z satisfies both | z – 2 – 2 i| ≤ 1 and arg( z – 1) = arg(1 + 3 i). Find the range of values of arg( z – 3 – 2 i), given that |z – 2 – 2 i| ≤ 1 and arg( z – 1) = arg(1 + 3 i). [4]

5 2012 NYJC 9740/01 [Turn Over 9 (a) A sequence of negative real numbers 1 2 3 , , ,... x x x satisfies the relation 1 1 2 n n x x + = − − , for n ≥ 1. Given that the sequence converges to l, find the exact value of l. [2] (b) (i) By expressing ( ) 2 2r r + in the form 2 A B r r + + , where A and B are real constants to be determined, show that ( ) ( ) ( )1 1 3 2 3 2 4 2 1 2 n r n r r n n = += − + + + ∑ . [3] (ii) Prove the result in (i) using mathematical induction. [4] (iii) Deduce the value of ( )2 1 2r r r ∞ = +∑ . [2] 10 The points P and Q have position vectors 3 4 i j k + − and 5 7 6 i j k + + respectively. (i) The plane ∏ passes through Q and is perpendicular to PQ . The equation of ∏ is ,r a b c λ µ = + + where ,λ µ ∈ /Rbb and vectors b and c are perpendicular to each other. Write down a suitable vector a and explain why 3 2 i j − can be taken as b. Find a suitable vector c. [5] The point R has position vector 6 43 8 i j k + + . The line l passing through the points P and R intersect ∏ at point S with position vector s. (ii) Explain why the lines with equations r a b α= + and r s c β= + , where ,α β ∈ /Rbb, will intersect. [2] (iii) Find the position vector of point S and determine whether point S lies on PR produced. [5]

6 2012 NYJC 9740/01 [Turn Over 11 In a city of 5 million people, the number of customers of a new company increases at a rate proportional to the number of people who are not its custome rs. The company determines that in order for its business to be profitable, it must have at least 1 million customers at any point in time. The company has 1 million customers at the end of the third year of operation, starting from a customer base of 0. Assume that the population remains unchanged at any point in time. (i) State a differential equation involving y and t, where y is the nu