IJC H2 MATH P1 (9740)

INNOVA JUNIOR COLLEGE JC 2 PRELIMINARY EXAMINATION in preparation for General Certificate of Education Advanced Level Higher 2 CANDIDATE NAME CLASS INDEX NUMBER MATHEMATICS Paper 1 Additional Materials: Answer Paper Cover Page List of Formulae (MF 15) 9740/01 28 August 2017 3 hours READ THESE INSTRUCTIONS FIRST Do not open this booklet until you are told to do so. Write your name, class and index number 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 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.      Innova Junior College  [Turn over

2    IJC/2017/JC2 9740/01/August/17   1 Prove by the method of mathematical induction that   0 !1 ! 1 n r rr n    . [5] 2 (i) Find ec o s d x n nx x , where n is a positive constant. [3] (ii) Hence find the exact value of  2 ec o s d x n nx x   . [2] 3 The vectors p and q are given by p  2i + j +ak and q  bi + j, where a and b are non-zero constants. (i) Find (2 5 ) (2 5 )pq pq in terms of a and b. [2] Given that the i- and j- components of the answer to part (i) are equal, find the value of b. [1] Use the value of b you have found to solve parts (ii) and (iii). (ii) Given that the magnitude of (2 5 ) (2 5 ) pq pq is 80, find the possible exact values of a. [2] (iii) Given instead that 25pq and 25pq are perpendicular, find the exact value of p . [3] 4 A graphic calculator is not to be used in answering this question. (a) The equation 32 30 0wp wq w   , where p and q are real constants, has a root 2iw . Find the values of p and q, showing your working. [3] (b) The equation 2 (5 2 i ) ( 2 1 i ) 0zz     has a root 3izu  , where u is real constant. Find the value of u and hence find the second root of the equation in cartesian form, iab , showing your working. [5]

3    IJC/2017/JC2 9740/01/August/17   5 A sequence 123, , ,. . .uu u is such that  22 1 21 nu nn   and   1 22 2 11 nnuu nn n    , for all 2n . (i) Find   22 2 2 11 N n nn n  . [3] (ii) Explain why   22 2 2 11n nn n    is a convergent series, and state the value of the sum to infinity. [2] (iii) Using your answer in part (i), find  22 1 2 (1 ) 2 N n N nn n  . [2] 6 (i) The variables x and y are related by d() 2 d yxy k y x  and 1y  at 0x , where k is a constant. Show that 22 2 dd d() ( 1 ) 0 ddd yy yxy k xxx      . [1] By further differentiation of this result, find the Maclaurin series for y, up to and including the term in 3x , giving the coefficients in terms of k. [4] (ii) Given that x is small, find the series expansion of 2 1g( ) sin 2 2 x x    in ascending powers of x, up to and including the term in 2x . If the coefficient of 2x in the expansion of g( )x is equal to twice the coefficient of 2x in the Maclaurin series for y found in part (i), find the value of k. [4]    [Turn over

4    IJC/2017/JC2 9740/01/August/17   7 A population of a certain organism grows from an initial size of 5. After 5 days, the size of the population is 20, and after t days, the size of the population is M. The rate of growth of the population is modelled as being proportional to  22100 M . (i) Write down a differential equation modelling the population growth and find M in terms of t. [6] (ii) Find the size of the population after 15 days, giving your answer correct to the nearest whole number. [2] (iii) Find the least number of days required for the population to exceed 80. [2] 8 It is given that  3 21 0 2 , f( ) 2 3 2 4 , 1o t h e r w i s e . xx xx x        Sketch, on separate diagrams, for 08 x  , the graphs of (i) f( )yx and state the range of f, [5] (ii) 1 f( )y x . [4] In each graph, indicate clearly the coordinates of the end points, points of intersection with the axes and stationary point, if any. State clearly the equation of any asymptote. (iii) Deduce the value of 4 6 f( )dx x    . [1]

5    IJC/2017/JC2 9740/01/August/17   9 Given that f( ) s i n2 c o s2x xx , express f( )x as  sin 2Rx  , where 0R , 0 2  and R and are constants to be found. [2] (i) Describe a sequence of transfor mations involved that transformed sinyx to f( )yx . [3] (ii) Sketch the graph of f( )yx for 30 8x  , indicating clearly the exact coordinates of the maximum point and the end points of the graph. [3] (iii) The region bounded by the curve f( )yx , the line 8x  and both axes is rotated about the y-axis through 2 radians. Find the volume of the solid of revolution correct to 4 decimal places. [4]   10 The plane 1p and 2p have equations 28x yz  and 332 4xyz  respectively. It is given that the point A has coordinates  6, 0 , 2c , where c is a constant. (i) Find the coordinates of the foot of perpendicular from A to 1p . Express your answer in terms of c. [4] (ii) The point B is the mirror image of A in 1p . If B lies in 2p , find the value of c. [3] (iii) 1p and 2p intersect in a line l. Find a vector equation of l. [1] Another plane 3p has equation mx z n  , where m and n are constants. (iv) Given that all three planes meet in the line l, find m and n. [2] (v) Given instead that the three planes have no point in common, what can be said about the values of m and n? [2]    [Turn over

6    IJC/2017/JC2 9740/01/August/17   11 [It is given that the volume of a circular cone with base radius r and height h is 21 3 rh and the volume and surface area of a sphere of radius r are 34 3 r and 24 r respectively.] In a distant Northern kingdo m of Drivenbell, Elsanna bu ilds a spherical snowball with radius 3 m. The snowball is inscribed in a right conical container of base radius r m and height h m. The container is specially designed to allow the snowball to remain intact with fixed radius 3 m (see diagram). (i) By considering the slant height of the cone, show that  2 3 6h hr h   . [3] (ii) Use differentiation to find the values of h and r that give a minimum volume for the container. Find the value of the minimum volume. [6] The snowball is being removed from the c ontainer and it starts to melt under room temperature. (iii) Assuming that the snowball remains spherical as it melts, find the rate of decrease of its volume at the instant when the radius of the sphere is 2.5m, given that the surface area is decreasing at 0.75 2m per minute at this instant. [5]

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