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2 1 Express 12 71 xx as a single simplified fraction. [1] Without using a calculator, solve 12 71 xx d . [3] 2 (i) Find 12d tand xx

. [1] (ii) Hence, or otherwise, evaluate 1 12 0 tan dxx x ´µ¶ 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 771

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 OMu JJJ G JJJJG in terms of a and b.[ 3 ] (iii) State the geometrical meaning of PQ OM PQ u JJJ G JJJJG JJJ G .[ 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 z z

6\ . (i) Find algebraically the range of f. [3] (ii) Sketch C, indicating all essential features. [4] (iii) Describe a pair of transformations 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 ´µ¶ . Hence find an approximate value for 0.5 0 sin(ln(1 )) dx xx ´µ¶ .[ 3 ] www.KiasuExamPaper.com 772

4 8( a ) A sequence of numbers 123 6 4,,,,aaa a ! is such that 1 ,nnaa d where 16 3ndd and is a constantd . The 64 numbers fill the 64 squares in the 88u 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 T , where 30, 4r SST! , find 2 66aw w and

2arg 66aw w in terms of r and T .[ 4] 10 The point M has position vector relative to the origin O, given by 651 1ij k . The line 1l has equation 27 32 y zx , and the plane S has equation 42 3 0xy z . (i) Show that 1l lies inS .[ 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 S , giving your answer correct to the nearest 0.1q . [3] 1l 1st row www.KiasuExamPaper.com 773

5 [Turn over 11 [It is given that the volume of a cylinder with base radius r and height h is 2rhS 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 TT , where 0 2 ST , 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 ST T , where k is a constant in terms of a and b.[ 3 ] (iii) Given that 1TT is the value of T which gives the maximum value of V, show that 1T satisfies the equation 23cos cos 1 0TT . Hence, find the value of 1T .[ 4] At 6 ST , 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 §· ¨¸ ¨¸¨¸ ©¹ [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 774