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