2007_AJC_Prelims_Paper_2

Anderson Junior College Preliminary Examination 2007 H2 Mathematics Paper 2 Section A: Pure Mathematics (40 marks) 1a) Differentiate 13tan (ln )x− with respect to x. [2] b) The curve C has parametric equations 222 ,1x tyt t t= += − + , where t is a non-zero parameter. (i) Show that the gradient of the curve at any point ( x,y) satisfies the equation 2 3 d( 2 1 ) d2 2 y tt xt −= − . [2] (ii) The line x = p is a tangent to the curve C. By using the result in (i), find the exact value of p. [2] 2. Express ( in the form )23r+ 2( 1) ( 1)rA r B r+ ++ − , where A and B are constants. Using the method of difference, find in terms of n. [5] () 1 23 2 n r r r = +∑ Hence, or otherwise, find an expression for in terms of n. [2] 1 2 2 ) 3 2 (− = ∑ + r n n r r 3. In an Argand diagram, the point A represents the fixed complex number a , where 0a r g ( ) 2a π<< . The complex numbers z and w are such that 2iza a−= and ww i a=+ . Sketch, in a single diagram, the loci of the points representing z and w [3] Find a) the minimum value of zw− in terms of a , [1] b) the range of values of 1arg z ⎛⎞ ⎜⎟⎝⎠ in terms of arg(a). [3] Page 2 This study source was downloaded by 100000851604598 from CourseHero.com on 11-29-2022 21:05:55 GMT -06:00 https://www.coursehero.com/file/5568601/2007-AJC-Prelims-Paper-2/

4a) By completing the square, or otherwise, describe the geometrical transformation by which the curve can be obtained from the curve . [2] 22 45xy y−−− = 0 22 1xy−= b) The diagram below shows the graph of ( )fy= x ⎟ ⎟ with asymptotes y = 2 and x = 0. The curve has turning points at (-2, 2) and (3, -2). y )2, 2− 5 2 On separate diagrams, sketch the graphs of (i) () x=− [ 3 ] 2 fy (ii) ()f' x= [ 3 ] y Show all intercepts, asymptotes and turning points clearly on your diagrams if they can be found. 5. Relative to an origin O, the point A has position vector 6 2 6 ⎛⎞ ⎜ ⎜⎜⎟−⎝⎠ , the line has equation , 1l 51 22 81 ⎛⎞ ⎛ ⎜⎟ ⎜=+ λ⎜⎟ ⎜⎜⎟ ⎜−−⎝⎠ ⎝ r 0 ⎞ ⎟ ⎟⎟⎠ λ∈ and the plane 1Π has Cartesian equation 54 . 31 5xyz−+= (i) Find the shortest distance from A to the plane 1Π and determine whether A and the origin O are on the same side or on opposite sides of 1Π . [3] (ii) 2Π is the plane that passes through point A and contains the line 1l . Find the acute angle between the planes 1Π and 2Π . [4] (iii) The plane 3Π has Cartesian equation 8x ya zb+ += . Find the values of a and b if the planes 1Π , 2Π and 3Π intersect along a common line. [5] x ( ( )3, 2− 2O Page 3 This study source was downloaded by 100000851604598 from CourseHero.com on 11-29