2019 J2 Hwa Chong H2 Math Prelim (with ans)

1 The number of units, D(x) of a particular product that people are willing to purchase per week in city A at a price $ x is given by the function 40320() g( )Dx x , where g( x) is a quadratic polynomial in x. The following table shows the number of units people are willing to purchase at different prices. x 581 0 D(x) 384 224 168 Find the number of units of the p roduct that people are willing to purchase at a price of $18. [4] 2 It is given that 1 1ln 1 n n n xp x    , where 11 x  and n is a positive integer. (i) Find 1 N n n p   , giving your answer in terms of N and x . [3] (ii) Hence find the sum to infinity of the series in part (i) in terms of x. [2] 3 In the triangle ABC, 1AB  , 2AC  and angle 2ABC x   radians. Given that x is sufficiently small for 3x and higher powers of x to be ignored, show that 2BC p qx rx  , where p, q, r are constants to be determined in exact form. [5] 4 A part of a hyperbola has equation given by  2(5 )f1 36 xx  , xD , where D Ž  . (i) State the largest possible setD . [1] (ii) State the equations of the asymptotes of fyx . [1] (iii) Sketch the graph of fyx for D in part (i), showing clearly all the features of the curve. [2] (iv) On separate diagrams, sketch the graphs of  1 fy x and f 'yx , showing clearly all the features of the curves. [4] +0DWK3UHOLP+&, www.KiasuExamPaper.com 289

2 5 Referred to the origin O, points A and B have position vectors a and b respectively. Point C lies on OB produced such that OC OB     where 1.  Point D is such that OCDA is a parallelogram. Point M lies on AD, between A and D, such that :1 : 2AM MD  . Point N lies on OC, between O and C, such that :4 : 3ON NC  . (i) Find the position vectors of M and N, in terms of a, b and  .[ 2] (ii) Show that the area of triangle OMD is k ab , where k is a constant to be determined. [4] (iii) The vector p is a unit vector in the direction of OD   . Give a geometrical meaning of .pa .[ 1] 6 A curve C is defined by the parametric equations 225sinxt , 2c o syt where 0 t  . (i) Find d d y x .[ 2] (ii) The tangent to C at the point where 4t  cuts the x-axis at P and the y-axis at Q, find the exact area of the triangle OPQ, where O is the origin. [4] (iii) State the equation of the normal to C where the normal is parallel to the x-axis. [1] 7 (a) (i) Find cos 2d 2ed x x     .[ 1] (ii) Hence find cos 21 sin e d2 x xx .[ 4] (b) Using the substitution 1e xu  , find 1 d1e x x .[ 4] www.KiasuExamPaper.com 290

3 8 The function f is defined by 2 2 ()f: 4 xaxa 6 for x   , 3ax a  , where a is a positive constant. (i) Sketch the graph of fyx , giving the coordinates of any stationary points and the points where the graph meets the axes. [2] (ii) If the domain of f is further restricted to axk  , state the greatest value of k for which the function 1f  exists. [1] (iii) Using the restricted domain found in part (ii), find 1f  in similar form. [3] The function g is defined by 3g( ) f 2xx    for x   , 2 23 ax a  . (iv) Explain why the composite function gf exists and find the range of gf . [3] 9 A company produces festive decorative Light Emitting Diode (LED) string lights, where micro LEDs are placed at intervals along a thin wire. In a particular design, the first LED (LED 1) is placed on one end of a wire with the second LED (LED 2) placed at a distance of d cm from LED 1, and each subsequent LED is placed at a distance 4 5 times the preceding distance as shown. (i) If the distance between LED 8 and LED 9 is 56.2 cm, find the value of d correct to 1 decimal place. [2] (ii) Find the theoretical maximum length of the wire, giving your answer in centimetres correct to 1 decimal place. [2] The LEDs consisting of three colours red, orange and yellow, ar e arranged in that order in a repeated manner, that is, LED 1 is red, LED 2 is orange, LED 3 is yellow, LED 4 is red, LED 5 is orange, LED 6 is yellow, and so on. (iii) Find the colour of the LED nearest to a point on the wire 12.9 m from LED 1. [3] (iv) If the minimum distance between any two consecutive LEDs is 1 c m so that they can be mounted on the wire, find the colour of the last LED. [3] wire LED 1 LED 2 cm LED 3 cm www.KiasuExamPaper.com 291

4 10 (a) Solve the simultaneous equations i2vu and 23 iav u , where a i s a r e a l constant. Simplify your answers to cartesian form ixy , where x and y are in terms of a.[ 4] (b) It is given that ()xk is a factor of the equation,   32 12 i 12i 12 0bx b x b x b    , where k and b are non-zero real constants. (i) Find the value of k.[ 2] (ii) Show that the roots of the equation 2 i0bx x b  are purely imaginary. [2] (iii) Hence express     32f 12 i 12i 12xb x b x b x b     as a product of three linear factors, leaving your answers in terms of b.[ 2] www.KiasuExamPaper.com 292

5 11 A container is made up of a cylinder and an inverted right circ ular cone as shown in the diagram below. The height and the diameter of the cylinder are 20 cm and 5 cm respectively. The height of the cone is 4 cm. An external device ensures liquid flows out through a small hole at the verte x of the cone into a bowl belo w at a constant rate of 18 cm3 per minute. The depth of the liquid and the radius of the liquid surface area in the container at the time t minutes are x cm and r cm respectively. The container is full of liquid initially. (a) When 4x  , find the rate of change of the depth of the liquid in the container. [4] (b) Find the rate of change of r when 2x  .[ 5] (c) The bowl as shown in the diagram in part (a) is generated by rotating part of the curve 22 1225 100 xy  which is below the x-axis through  radians about the y-axis. Assuming the bowl has negligible thickness, find the volume of the empty space in the bowl when the liquid has completely flowed from the contain er into the bowl, giving your answer correct to 3 decimal places. [3] 20 cm 4 cm 5 cm www.KiasuExamPaper.com 293

6 12 At an airport, an air traffic control room T is located in a vertical air traffic control tower, 70 m above ground level. Let (0,0,0)O be the foot of the air traffic control tower and all points (,,)xyz are defined relative to O where the units are in kilometres. Two observation posts at the points (0.8,0.6,0)M and ( 0.4, 0.9,0)N  are located within the perimeters of the airport as shown. An air traffic controller on duty at T spots an errant drone in the vicinity of the airport. The two observation posts at M and N are alerted immediately. A laser rangefinder at M directs a laser beam in the direction 2 7 1     at the errant drone to determine D, the position of the errant drone. The position D is confirmed using another laser beam from N, which passes through the point ( 0.8,0.75, 0.3), directed at the errant drone. (i) Show that D has coordinates ( 0.56, 0.24,0.12) .[ 4] A Drone Catcher, an anti-drone drone which uses a net to trap and capture errant drones, is deployed instantly from O and flies in a straight line directly to D to intercept the errant drone. (ii) Find the acute angle between the flight path of the Drone Catcher and the horizontal ground. [2] At the same time, a Jammer Gun, which emits a signal to jam the control signals of the errant drone, is fired at the errant drone. The Jammer Gun is l ocated at a point G on the plane p containing the points T, M and N. (iii) Show that the equation of p is 10.5 2.8 6.72 96      r .[ 3] It is also known that the Jammer Gun is at the foot of the perpe ndicular from the errant drone to plane p. (iv) Find the coordinates of G.[ 3] (v) Hence, or otherwise, find the distance GD in metre