Section B ( 55 marks )
10 . The two functions f and g are given as x � 0 .
1 x
2
��
� f ( x) � e sin� x�
� 3 � and
1 x
2 g( x) � e ,
( a ) Find the three smallest values of x such that g( x) � f ( x) � 0.
[ 6 ] Consider the set of solutions of the equation g( x) � f ( x) � 0, where x � 0 . Let x n be the n th smallest value of this set of solutions .
( b ) ( i ) Show that x1 , x2, x
3, ... is an arithmetic sequence .
( ii ) Find the general term of the sequence , giving the answer in the form xn �an � b .
[ 4 ]
Consider the region R bounded by y � f ( x)
, y � g( x)
, x� x2
, x� x3 and the x -axis .
( c ) Find the expression for the area of R , giving the answer in the sum of multiple integrals in its simplest form . The calculation of the value of the area is not required .
[ 4 ]
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