JPJC 9758 2023 Prelim P1

Jurong Pioneer Junior College H2 Mathematics JC2-2023 Prelim Paper 1 Question Paper 1 A sequence 1u , 2u , 3u , … is such that 2 12 15 60nnu u n nA−= − ++ , where A is a constant, and 2.n≥ It is given that 1 2u = and 2 4u = . (i) Find A and 3u . [2] It is known that the nth term of this sequence is given by 12(2 )n nu p qn r += −+ . (ii) Find p, q and r. [3] 2 It is known that the nth term of a sequence is given by ln 1 n nu n =  + , where 1n≥ . (i) Determine if the sequence is convergent, stating your reason clearly. [1] (ii) By using the method of differences or otherwise, find 1 N n n u = ∑ in terms of N . [3] 3 A curve C has equation 11y x= − , where 0x≠ . The line l is the tangent to C at xa= , where a is positive integer. Show that the equation of l may be expressed in the form 22 2.ay x a a−= − [3] Find the value of a for which l is parallel to the line 91yx−= . Hence find the equation of l in this case. [3] 4 The curve C has equation 2 3 2 xxy x ++= − . (i) Without using a calculator, find the set of values that y can take. [4] (ii) Sketch the graph of C , stating clearly the equations of any asymptotes and the coordinates of any turning points and axial intercepts. [3] (iii) By drawing a suitable graph on the same diagram in (ii), find the range of values of k, where 0k > , such that the equation ( ) ( ) 2 22 22 2( 1) ( 2) 3 2x x x x kx+ − + ++ = − has at least one positive real root, giving your answer correct to 3 decimal places. [3]

2 5 The function f, with domain the set of all real values, is given by ( ) 2 for 0 2,4f for 2 4,24 xxx xx <≤ −= <≤− and that ( ) ( )f f4xx= + . (i) Find ( )f 45 . [1] (ii) Sketch the graph of ( )fyx= for 47 x−≤≤ and state the range of f. [3] The function g is defined by ( ) 2 g: 1 2xx −+ for , 0 1.xx∈ << (iii) Explain why the composite function gf does not exist. [1] (iv) Find an expression for ( )fg x and hence, or otherwise, find ( ) 1 1fg ( ) 2 − . [4] 6 The plane p has equation 2 2 12xyz−+= . With reference to the origin O , the point A has position vector 2 +−i j 3k . (i) Find the coordinates of the foot of perpendicular, N, from A to p. Hence determine the coordinates of the reflection of A in p. [5] (ii) Find the cartesian equations of the planes such that the perpendicular distance from each plane to p is 15. [5] 7 (a) Find 3 4 2 d1 x xx+∫ . [2] (b) Find cos 4 sin10 dx xx∫ . [2] (c) Show, by means of the substitution tanx θ= , that ( ) 31 34 300 2 d cos sin d 1 x x x π θ θθ= +∫∫ . Hence find the exact value of ( ) 31 30 2 d 1 x x x +∫ . [5]

3 8 A marathon runner and a sprinter are running around in laps on a track. Each lap is 400 m. The sprinter runs his first lap in 56 seconds, and the time he takes for each subsequent lap is 6 seconds more than the previous lap. The marathon runner runs his first lap in 1 minute, and the time he takes for each subsequent lap is 4% more than the time taken for the previous lap. (i) Show that the time the sprinter takes to complete n laps is 23 53nn+ . [2] (ii) Find the time the marathon runner takes to complete n laps, giving your answer in terms of n . [2] (iii) How many complete laps does it take for the marathon runner to first exceed a total time of 40 minutes? [3] The 2 runners decide to do a 12km race with the same starting point. (iv) Determine which runner will be the first to complete the race. [3] (v) Determine which lap the winner is on when he first overtakes the other runner. [3] 9 (a) The function f is defined as ( ) 432f 4 16x x x ax x b= −+−+ , where a and b are real and non-zero. (i) Given that two of the roots of ( )f0 x = are of the form ik , where k is real and non-zero, find these two roots and show that 16 4 0ab− += . [4] (ii) The graph of f meets the y-axis at (0, 20). Express ( )f x as a product of two quadratic factors. [4] (b) Find the modulus of the complex number ( ) ( ) 2 *zi 3i i z zz+ − , where z is a complex number. [3] [Turn over

4 10 (i) On the same axes, sketch the graphs of sin 2yx= and cosyx= for 0 2x π≤≤ , stating the exact coordinates of the points where the curves cross the axes. [2] (ii) Solve exactly the inequality sin 2x > cos x, where 0 2x π≤≤ . [2] (iii) Hence, find 2 0 sin 2 cos dx xx π −∫ without using a calculator. [4] (iv) The region bounded by the curves sin 2yx= , cosyx= and the y -axis, where 0x≥ is rotated through 2π radians about the x -axis. Find the exact volume of the solid obtained. [3] 11 [ The volume of a cone of base radius r and height h is given by 21 3V rh π= .] A disposable cup for water dispenser, in the form of a cone , is made from a circular piece of paper of radius 8 cm with negligible thickness. A sector is cut off from the piece of paper as shown in Fig. 1. The rest of the paper is then folded to form a cone with slant height 8 cm, base radius r cm and height h cm as shown in Fig. 2. Fig. 1 Fig. 2 (i) Find the maximum volume of the cone, giving your answer in exact form. [ 6] (ii) Show that 2rh= when the volume of the cone is a maximum. [2] 8 r h 8 [Turn over

5 (iii) The cone with the volume found in (i) is then inverted and is held with its axis vertical and vertex downwards . Water from the dispenser flows into the cone at a rate of 3π cm3 per second. At time t seconds after the start, the radius of the water surface is x cm as shown in Fig. 3. Find the rate of change of the height of the water in the cone after 6 seconds. Fig. 3 [4] x cm3 per second