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2007 AJC Prelims Paper 2

2007 AJC Prelims Paper 2 - Anderson Junior College...

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Anderson Junior College Preliminary Examination 2007 H2 Mathematics Paper 2 Section A: Pure Mathematics (40 marks) 1 a) Differentiate 13 tan (ln ) x with respect to x. [2] b) The curve C has parametric equations 22 2 ,1 x ty t 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 ) 23 r + 2( 1) ( rA r B r + ++ , where A and B are constants. Using the method of difference, find in terms of n . [5] () 1 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 ( ) 2 a π << . The complex numbers z and w are such that 2i za 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 1 arg z ⎛⎞ ⎜⎟ ⎝⎠ in terms of arg( a ). [3] Page 2

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4 a) By completing the square, or otherwise, describe the geometrical transformation by which the curve can be obtained from the curve . [2] 22 45 xy y −−− = 0 1 −= b) The diagram below shows the graph of ( ) f y = 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 f y (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 , 1 l 51 81 =+ λ −− r 0 λ ± and the plane 1 Π has Cartesian equation 54 .
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2007 AJC Prelims Paper 2 - Anderson Junior College...

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