AJC H2Maths 2012Prelim P1 Question

Page 1 of 4 AJC / 2012 Preliminary Examination / 9740 / P1 Anderson Junior College Preliminary Examination 2012 H2 Mathematics Paper 1 (9740/01) 1 The sum of the first 100 terms of an arithmetic progression is 10,000. The first, second and fifth terms of this progression are three consecutive terms of a geometric progression. Find the first term and the non -zero common difference of the arithmetic progression. [5] 2 The equation of a curve is 22 8Ax By Cy   , where A, B and C are constants. The curve has a tangent parallel to the y-axis at the point (2,1). Given that d3 d2 y x  and 2 2 d9 d4 y x  at the point where the curve cut s the positive x-axis, find the values of A, B and C. [6] 3 Without using a calculator, solve the inequality 64 123 x xx   , 3x . [4] Hence find the exact range of values of  for which 64 1213 cosec sincosec     where 02  . [3] 4 A curve with equation y = f(x) is transformed by a reflection in the y-axis, followed by a translation of 1 unit in the negative x-axis, followed by a scaling with factor 2 parallel to the y-axis. The equation of the resulting curve is given by y = g(x), where 2 1g( ) , 2 0 4 xx x      . (i) Find f(x). [3] (ii) On the same diagram, sketch the graph of y = g(x) and y = g-1(x). [3] (iii) Without evaluating g -1(x), find the exact area of the region bounded by the curve y = g-1(x), the x-axis and the line x = 1. [4] 5 Consider 2 2 2 2 .....1 2 3 2 3 4 3 4 5 ( 1)( 2) nS n n n            (i) Calculate the values of 1 2 3, and S S S . [2] (ii) Make a conjecture for nS in the form 11 2 ( )( )n a n b  , where a and b are integers. [1] (iii) Prove your conjecture by the method of induction. [5]

Page 2 of 4 AJC / 2012 Preliminary Examination / 9740 / P1 R 6 The diagram shows a right circular cone of height of 2 units, radius r and slant height l inscribed in a sphere of radius R. (i) Show that 24A R R , where A is the curved surface area of the cone. [3] (ii) If the volume of the sphere is increasing at a rate of 8 unit s3/sec, find the exact rate of change of A at the instant when R = 2 units. [4] [Curved surface area of a right circular cone with base radius r and slant height l is πrl . Volume of sphere = 34 3 R where R is the radius of the sphere.] 7 A sequence of numbers, nx , satisfy the relation  10 2 11 2 , 0 , 0 13 nn n x xx nx          . (i) If the sequence converges to a number L, find L. [2] (ii) Given that nxL for all integers 1n , show that 1nnxx  . [3] (iii) Describe, with reference to (i) and (ii), the behaviour of the sequence. [1] The difference nd between nx and L is given by nnd x L . Show that 2 1nndd  and state the range of values of nd for which this approximation is valid. [4] l 2 r

Page 3 of 4 AJC / 2012 Preliminary Examination / 9740 / P1 8 (a) The diagram below shows a region R bounded by the curve 2( 5) 3yx   and the line 10yx . Find the volume of solid formed when R is rotated through four right angles about the x-axis. [4] (b) (i) Show that 2 2 2 21cos d cos sin 55 x x xe x x e x e x C      , where C is an arbitrary constant. [4] (ii) A curve  is defined by the parametric equations 2sin , , where 22 tx t y e t      . Find the exact area of the region bounded by the curve , the tangent to the curve  at the y-axis and the line 1x . [5] 9 The curve C has equation 22xay xk   where ,0ak  . (i) Given that 22ka , find the range of values of k such that the curve C has at least one tangent parallel to the x -axis. [4] (ii) What can be said about the graph of 22xay xk   when 22ka ? [2] For the range of values of k in (i), (iii) sketch the curve of C, clearly indicating the equations of the asymptotes and any intercepts with the axis. [3] (iv) The curve 22xay xk   does not intersect the curve 2 2 4( ) 1 ykxk b    where b > 0. By drawing the 2 curves on the same diagram, find the range of values of b . [2] 2( 5) 3yx   10yx y 0 x R

Page 4 of 4 AJC / 2012 Preliminary Examination / 9740 / P1 10 The two planes p1 and p2, given by the equations 2 x + y = 1 and 8x +ay + z =4 respectively, meet at a line L which contains the point A(0,1,0). (i) Show that a = 4 and hence find the vector equation of the line L. [3] (ii) Another point B lies in the plane p1 such that AB is perpendicular to the line L. Show that AB is parallel to 0 0 1    . If the distance of B from p2 is 5 units, find the possible position vectors of point B. [5] (iii) Find the acute angle between line AB and p2. [2] (iv) A third distinct plane p3 is given by the equation 2 x+ y +z = 6. Show that p1, p2 and p3 do not meet at a common point for all values of . Comment on the geometrical relationship of the three planes when  = 0. [3] 11 The point A represents the complex number a with |a| = 1 and arg( )a  , 0 4  . The point B represents the complex number ia. The complex number z satisfies the relations i2za and iz a z a   . Sketch clearly, on a single Argand diagram, the locus of the point representing the complex number z , with reference to the points A and B. Find the angle that the locus of z makes with the positive real axis in terms of  . [4] By writing 2iaa as  iia a a , or otherwise, indicate the point C representing the complex number 2iaa on your diagram and state the geometrical relationship between the points A, B and C. [2] Hence find (i) the least value of 2iz a a , and [1] (ii) the largest value of arg( 2i )z a a . [3] END OF PAPER