find the value of x2 and the value of x6, giving your answers to 3 decimal places

INSTRUCTIONS TO CANDIDATES

ANSWER ALL QUESTIONS

1. The curve C has equation

y = x 2 cos  1 x 0 < x  π 2 

The curve has a stationary point at the point P.

(a) Show, using calculus, that the x coordinate of P is a solution of the equation

x = 2 arctan 4 x

Using the iteration formula

xn+1 = 2 arctan x1 = 2n

(b) find the value of x2 and the value of x6, giving your answers to 3 decimal places. (3)

2. (a) Show that

1  cos 2x º k tan x x ¹ (90n)° Z

2sin 2x

where k is a constant to be found.

(b) Hence solve, for 0 < θ < 90°

91  cos 2θ  = 2 sec2 θ

2sin 2θ

giving your answers to one decimal place.

(Solutions based entirely on graphical or numerical methods are not acceptable.)

(6)

3. (i)   Find

12 d x

2x  12

giving your answer in simplest form.

(2)

4x  3

(ii) (a) Write in the form

x   2

A +    B      where A and B are constants to be found

x   2

(b) Hence find, using algebraic integration, the exact value of

 5 4x 

d x 8    x   2

giving your answer in simplest form.

(6)

4. The functions f and g are defined by

4x 6

f (x) =   x   5 x   R, x ¹ 5

g (x) = 5 − 2x2 x   R, x  0

(a) Solve the equation

fg(x) = 3

(b) Find f −1

(c) Sketch and label, on the same axes, the curve with equation y = g(x) and the curve with equation y = g−1(x) . Show on your sketch the coordinates of the points where each curve meets or cuts the coordinate axes.

 

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