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6 Sketch, on separate diagrams, the graph of k ykx 422 =+ , where

(i) k =4, [2]

(ii) k 0,

x x x ,4

tan:g , 2

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[Turn Over8 The sequence of real numbers u1, u2, u3, satisfies the recurrence relation

40

)6)(10(1

+=+

nnnn

uuu

u

for n 1.

(i) If u1 = 1 and un as n , find the exact value of . [3]

(ii) Determine the behaviour of the sequence when u1 = 11. [1]

(iii) If un < , show that un+1 > un. [3]

9 Shade on an Argand diagram the set of points representing all complex numbers z satisfying both of the following inequalities:

0i23

arg4

3

z and 53 z . [5]

Hence find the greatest and least possible values of 3i z , given that

0i23

arg4

3

z and 53 z . [3]

10 A curve is defined by the parametric equations

22 x at = , 3 y at = ,

where a is a nonzero constant. Show that the tangent to the curve at any point with

parameter t has equation 22

3

4

3at x yt += . [3]

Find the range of values of p such that the line l with equation 2 0 x py a + = intersectsthe curve at two distinct points. [3]

Given that l is a tangent to the curve, show that 014 2 =++ st t , where the values of sare to be determined. [2]

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The diagram shows a sketch of the curve )(f x y = . The lines with equations x = 2 and y = 3 are asymptotes to the curve. The intersections of the curve with the x and y axes have coordinates (1, 0), (3, 0) and (0, 2). On separate diagrams, sketch thegraphs of

(i) )(f 2 x y = , [2]

(ii))(f

1

x y = , given that 0

)2(f

1= , [4]

(iii) )(f x y = . [3]

Whenever appropriate, your sketch should indicate clearly the equations of anyasymptotes, intercept(s) and the coordinates of turning point(s).

12 Given that

44 121321 =+++++ + nnn uuuuu ,

prove that un = )4(2

4

15 n . [2]

(i) Find 1232531 +++++ nn uuuuu , giving your answers in terms of n. [3]

5

)(f x y =

x0

y

2

1 3

y = 3

x = 2

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(ii) Show that ln( u1) + ln( u2) + ln( u3) + + ln( un) = an 2 + bn, where the exact values of a and b are to be determined. [4]

[Turn over13 The planes 1 and 2 have vector equations

r. (2i + j k ) = 4, r = 2 i + j 2k + 1 (3 i + 2 j 7k ) + 1 (i j + 4k ),

respectively, where 1 and 1 are real parameters. The point A has position vector k ji 229 + .

(i) Find the coordinates of the foot of the perpendicular from A to 1 .[3]

(ii) 1 and 2 meet in a line l . Show that the vector equation of l may beexpressed in the form r = (a i + b j + ck ) + (2i j + 3k ), where is a real parameter and a, b,c are constants to be determined. [4]

(iii) The plane 3 has vector equation

r = i + j k + 2 ( pi + j + k ) + 2 (2 i j + k ),

where 2 and 2 are real parameters and p is a constant. Given that 1 , 2 and 3 meet in a line, find the value of p. [3]

14 A curve C 1 has equation )(f x y = , x > 4. The tangent to C 1 at the point (2, ln16) is parallel to the x axis. Given that for all points on C 1,

41

dd

2

2

+= x x y

,

show that the equation of C 1 is 2)4ln()4( +++= mx x x y where the exact value of m is to be determined. [6]

(i) Sketch C 1, making clear its main relevant features. [2]

(ii) A particle moves along C 1 such that x increases at a constant rate of 2 units s -1.

Find the rate of increase of y when x = 4, giving your answer in an exact form.[2]

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(iii) C 2is obtained by reflecting C 1 in the line y = x . Write down the equation of C 2. [1].

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