JPJC 2021 J1 H2 Maths promo Exam Questions

Name:____________________________________ Class:_____________ JURONG PIONEER JUNIOR COLLEGE JC1 End-of-Year Examination 2021 MATHEMATICS 9758/01 Higher 2 29 September 2021 Paper 1 3 hours Additional materials: Answer Paper Cover Page List of Formulae (MF 26) READ THESE INSTRUCTIONS FIRST Write your name and civics class on all the work you hand in. Write in dark blue or black pen on both sides of the paper. You may use an HB pencil for any diagrams or graphs. Do not use staples, paper clips, glue or correction fluid. This document consists of 5 printed pages and 1 blank page. [Turn over 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 an approved graphing calculator. Unsupported answers from a graphing calculator are allowed unless a question specifically states otherwise. Where unsupported ans wers from a graphing 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 by [ ] at the end of each question or part question.

2 1 The parametric equations of a curve C are 2e , 2 e , 2.t tx t y t t      (i) Sketch C. [2] (ii) Find the equation of the normal to the curve at the point P where t = 0. [4] 2 A curve C has equation 3 ay bx cx   , where a, b and c are constants. It is given that C passes through the point with coordinates (1, 3) and has a stationary point (5, 21). (i) Find the values of a, b and c. [4] (ii) Hence find the equations of the asymptotes of C. [2] 3 It is given that 2 1 3tan 4x y y   . Find d d y x in terms of x and y. [4] Hence, find the exact value of d d y x when y = 1 , given that x > 0. [3] 4 (i) Sketch the curve with equation ln( 1), 1y x x   , stating the equation of asymptote and intercepts with the axes. On the same diagram, sketch the curve with equation 29 , 3 3,y x x     stating the intercepts with the axes. [4] (ii) Use your answer in part (i), solve the inequality 2ln( 1) 9x x   . [2] (iii) Hence solve the inequality 2 4ln( 1) 9x x   . [3] 5 The function f is defined by   21f : e 1 , 23 xx x    . (i) Find f1 and write down the its domain. [3] (ii) Explain why the solution of f(x) = f1(x) satisfies the equation 2e 3 1x x   and find the value of this solution. [3] The function g is defined by 2g : 1x x  , x  . (iii) Show that gf exists . Hence find the composite function gf , stating its domain and the corresponding range. [4]

3 6 (a) Given that u v = 0 , what can be deduced about the vectors u and v? [2] (b) Referred to an origin O, the position vectors of two points A and B are a and b respectively. A line l has vector equation given by r =  1 3  a 2b a , where  . The point N is the foot of perpendicular from A to l. It is given that 2, 1 a b and a is perpendicular to b. (i) Find the position vector of N in terms of a and b. [5] (ii) Find the exact area of triangle OAN. [3] 7 (a) The graph of y = f(x) has asymptotes x = 1, y = 2 and a minimum point at (–1, –1) as shown in the diagram. It cuts the x-axis at the origin and at (–2, 0). Sketch the following graphs on separate diagrams, labeling clearly the asymptotes, turning points and intercepts on the axes where applicable. (i) y = f 2x + 1, [3] (ii) y = )f( 1 x . [3] (b) The curve whose equation is y = 1 1x undergoes, in succession, the following transformations: A: A reflection in the y  axis. B: A translation of 5 units in the negative x  direction. Give the equation of the resulting curve. [3] y = 2 x = 1 0 x (–1, –1) –2 y [Turn over

4 8 (a) A closed container is constructed using a sheet of metal with area 100  cm2. The container comprises 2 shapes, a cone and a cylinder. The slant height, l, of the cone is 10 cm. Given that the cone and the cylinder share the same height h, and radius r, (i) show that r rrh 2 10100 2  , [1] (ii) use differentiation to find, the exact maximum volume of the container, proving that it is a maximum. [6] [Volume of Cone = hr 2 3 1  , Curved Surface Area of Cone = rl ] (b) The height of an upright cone is twice the radius, r, of its circular base. It is known that the volume of the cone is increasing at the rate of 15 cm 3 min 1 when the radius is 3 cm. Find the rate of increase of the base area of the cone at this instant. [4]

5 9 A plane 1 has equation x + y + z = 3. A line passes through the points P and Q with position vectors i + j + 2k and 4i + 2j − k respectively. (i) Find the exact length of projection of PQ   onto 1 . [3] (ii) Find the position vector of the point of intersection of line PQ and 1 . [3] A plane 2 is parallel to the y-z plane and contains the point (− 2, 1, 4). (iii) Find the cartesian equation of 2 . [2] (iv) A point S(a , 7, b) lies on both 1 and 2 .Write down the values of a and b . [2] (v) Hence or otherwise, find a vector equation of the line of intersection of 1 and 2 . [2] 10 (a) Find 1sin 2 e d . 1 x x x     [2] (b) Express    2 2 6 1f ( ) 3 1 3 x xx x x     in the form 23 1 3 A Bx C x x   , where the values of A ,B and C to be determined. [3] Hence find  f dx x . [2] (c) Find the exact value of 2 2sin d 8 x x        . [3] - END OF PAPER -

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