2019 J2 VJC H2 Math Prelim (with ans)

2 1 Express 12 71 xx  as a single simplified fraction. [1] Without using a calculator, solve 12 71 xx  . [3] 2 (i) Find  12d tand xx  . [1] (ii) Hence, or otherwise, evaluate  1 12 0 tan dxx x    exactly. [3] 3( i) Find  2d 32d xxx .[ 2] (ii) Find the equation of the tangent to the curve 232 xyx at the point where 1x  , giving your answer in exact form. [3] 4 The graph for f( )yx is given below, where 10yx , 6y  and 4x  are asymptotes. The turning points are ( 3, 5) and (6, 0) , and the graph intersects the y-axis at 0,6 . On separate diagrams, sketch the graphs of (i) f( )yx ,[ 3] (ii) 1 f( )y x . [3] y O x 0,6 +0DWK3UHOLP9-& www.KiasuExamPaper.com 860

3 [Turn over 5 Referred to the origin O, points P and Q have position vectors 3a and a + b respectively. Point M is a point on QP extended such that PM:QM is 2:3. (i) Find the position vector of point M in terms of a and b.[ 2] (ii) Find PQ OM    in terms of a and b.[ 3] (iii) State the geometrical meaning of PQ OM PQ       .[ 1] 6 A curve C has equation fyx , where the function f is defined by 2 12 3f: , , 5 , 1 45 xxx x x xx      6 . (i) Find algebraically the range of f. [3] (ii) Sketch C, indicating all essential features. [4] (iii) Describe a pair of transformatio ns which transforms the graph o f C on to the graph of 2 9 6 xy xx   .[ 2] 7 Given that 1sin ln(1 )yx  , where 01 ,x show that 2d(1 ) 1d yxy x  .[ 2] (i) By further differentiation, find the Maclaurin expansion of y in ascending powers of x, up to and including the term in 2x .[ 4] (ii) Use your expansion from (i) and integration to find an approximate expression for sin(ln(1 )) dx xx    . Hence find an approximate value for 0.5 0 sin(ln(1 )) dx xx    .[ 3] www.KiasuExamPaper.com 861

4 8( a) A sequence of numbers 123 6 4,,,,aaa a  is such that 1 ,nnaa d  where 16 3n and is a constantd . The 64 numbers fill the 64 squares in the 88 grid in such a way that 18 to aa fills the first row of boxes from left to right in that order. Similarly, 91 6 to aa fills the second row of boxes from left to right in that order. 1st column 1 2 34567 8 91 0 1 6 aaaaaaaa aa a  # Given that the sums of the numbers in the first rowand in the third columnare 58 and 376 respectively, find the values of 1a and .d [4] (b) A geometric series has first term a and common ratio r, where a and r are non-zero. The sum to infinity of the series is 2. The sum of the six terms of this series from the 4 th term to the 9th term is 63 .256 Show that 93512 512 63 0.rr   Find the two possible values of r, justifying the choice of your answers. [5] 9 One of the roots of the equation 3 66 0za z , where a is real, is w. (i) Given that 2iwb — , where b is real, find the exact values of a and * w w .[ 6] (ii) Given instead that iewr  , where 30, 4r     , find 2 66aw w and  2arg 66aw w in terms of r and  .[ 4] 10 The point M has position vector relative to the origin O, given by 651 1ij k . The line 1l has equation 27 32 y zx    , and the plane  has equation 42 3 0xy z  . (i) Show that 1l lies in  .[ 2] (ii) Find a cartesian equation of the plane containing and M.[ 3] The point N is the foot of perpendicular from M to 1l . The line 2l is the line passing through M and N. (iii) Find the position vector ofN and the area of triangle OMN .[ 5] (iv) Find the acute angle between 2l and  , giving your answer correct to the nearest 0.1 . [3] 1l 1st row www.KiasuExamPaper.com 862

5 [Turn over 11 [It is given that the volume of a cylinder with base radius r and height h is 2rh and the volume of a cone with the same base radius and height is a third of a cylinder.] A manufacturer makes double-ende d coloured pencils that allow us ers to have two different colours in one pencil. The manufacturer determines that the sh ape of each coloured pencil is formed by rotating a trapezium PQRS completely about the x-axis, such that it is a solid made up of a cylinder and two cones. The volume, V 3cm , of the coloured pencil should be as large as possible. It is given that the points P, Q, R and S lie on the curve 22 22 1xy ab  , where a and b are positive constants. The points R and S are  ,0a and  ,0a respectively, and the line PQ is parallel to the x-axis. (i) Verify that  cos , sinPa b  , where 0 2  , lies on the curve 22 22 1xy ab  . Write down the coordinates of the point Q. [2] (ii) Show that V can be expressed as  2sin 2cos 1Vk   , where k is a constant in terms of a and b.[ 3] (iii) Given that 1 is the value of  which gives the maximum value of V, show that 1 satisfies the equation 23cos cos 1 0  . Hence, find the value of 1 .[ 4] At 6   , the manufacturer wants to change one end of the coloured penc il to a rounded-end eraser. The eraser is formed by rotating the arc PS completely about the x-axis. (iv) Find the volume of the eraser in terms of a and b. [3] 12 A ball-bearing is dropped from a point O and falls vertically through the atmosphere. Its speed at O is zero, and t seconds later, its velocity is msv 1 and its displacement from O is mx .The rate of change of v with respect to t is given by 10 – 0.001v2. (i) Show that 5 5 e1100 . e1 t tv      [4] (ii) Find the value of v0 , where 0v is the value approached by v for large values of t.[ 1] (iii) By using chain rule, form an equation relating dd d, and dd d xv v tt x . Given that d d xv t , form a differential equation relating v and x. Show that 500100 1 e . x v   [5] (iv) Find the distance of the ball-bearing from O after 5 seconds, giving your answer correct to 2 decimal places. [3] www.KiasuExamPaper.com 863

2 Section A: Pure Mathematics [40 Marks] 1 Express 2cos sin 2  in the form sin sinab , where a and b are constants to be found. [2] Hence, find the exact value of D , where 0 D , for which cos sin 2313cos cos e d 4 122 e   D             .[ 5] 2( i) Show that 231 2 1 2 3 2 5 (2 1)(2 3)(2 5) Ar B rr r r r r      , where A and B are constants to be found. [2] (ii) Hence find 1 29 (2 1)(2 3)(2 5) n r r rr r    . (There is no need to express your answer as a single algebraic fraction.) [4] (iii) It is given that 1 29 (2 1)(2 3)(2 5) n r r rr r    is within 0.01 of the sum to infinity. Write down an inequality in terms of n, and hence find the smallest possible value of n. [3] 3 The function f is defined by 2 51 1f: , , 2 2 xxxx xx   6 . (i) Find the equations of the asymptotes of the curve f( )yx .[ 3] (ii) Determine whether f has an inverse, justifying your answer. [2] Given that the function g is defined by g : f( ), , 2 4xx x x 6 - , find  1g x and state the domain of 1g .[ 4] Sketch the graph of  1ggyx  .[ 2] www.KiasuExamPaper.com 864

3 [Turn over 4 A curve C has parametric equations 2 4xt  , lnty t ,w h e r e 0t  . (i) Show that  2 3 ln 1 4d d tty xt  .[ 3] (ii) Find the exact coordinates of the turning point on C, and explain why it is a maximum. [4] (iii) Sketch C.[ 3] (iv) Show that the area bounded by C and the lines 13x  and 5x  is given by 3 2 1 ln d 4 t t t     . Find the area, giving your answer to 4 decimal places. [3] Section B: Probability and Statistics [60 Marks] 5 Mr and Mrs Lee participate in a game show, together with 3 othe r men and 5 other women. In the first round, the 10 participants are grouped into 5 pairs. (i) Find the number of ways the pairings can be done if there is