7/30/2019 Term I (2011-2012) Unit 2 - Module 1

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7/30/2019 Term I (2011-2012) Unit 2 - Module 1

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Show that there is only one value of t for whichd

3d

 y

 x= and state that value. [6]

5. (a) Express

 1

( 4)( 4) x x x− +

in partial fractions. [3]

(b) Hence, find the exact value of 

 

3

1

1d

( 4)( 4) x

 x x x− +∫ Giving your answer in the form ln ,

a

bλ 

     

where a and b are integers. [4]

6. It is given that21

0e d , for 0, 1, 2, . . .

n x

n I x x n

−= =∫ (a) Find 1. I  [3]

(b) Show that 212 ( 1) , for 2.en n I n I n−= − + − ≥ [5]

7. Find the general solution of the equation

22

2

d d4 4 65 65 8 73.

dd

 y y y x x

 x x+ + = + + [7]

Show that, whatever the initial conditions, 21 as .

 y x

 x→ → ∞ [1]

8. A function is defined by

212( ) e .

 x

 f x x−

=

(a) Show that21

2 2( ) (1 )e . x

 f x x−′ = − [3]

(b) Find the exact coordinates of the stationary points of the curve ( ). y f x= [5]

(c) The area of the finite region bounded by the  x-axis, the curve and the line 1 x = is  A,

where

 

21

12

0e d .

 x

 A x x

−

= ∫ Using the substitution21 ,

2u x= show that

 

12

0

e d ,u

 A u−= ∫  [3]

and hence evaluate the value of  A exactly. [2]

2

7/30/2019 Term I (2011-2012) Unit 2 - Module 1

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END OF TEST

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