( f ) Hence , express n
T in terms of n .
The initial number of citizens choosing each manufacturers from the two countries are both 3500 .
[ 3 ]
( g ) Find , after five years , the number of citizens purchasing computers from American manufacturers .
[ 4 ]
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2 . The weather condition of a town is studied every day . Assume that the weather is either sunny or cloudy . If it is sunny on a particular day , then there will be 40 % of chance that the next day is cloudy . If it is cloudy on a particular day , then there will be 36 % of chance that the next day is sunny .
( a ) Find T , the transition matrix representing the changes in the weather conditions on a particular day .
[ 2 ]
( b ) Find the characteristic polynomial of T . [ 2 ]
( c ) Hence , write down the values of �
1
and �
2
, the eigenvalues of T , where �1 � �2. [ 2 ]
Let v
1
and v
2 be the eigenvectors of T corresponding to �
1
and �
2 respectively .
( d ) Write down v
1
and v
2
.
It is given that matrix . n n �1
T � PD P , where P is a 2� 2 matrix and D is a 2� 2 diagonal
[ 2 ]
( e ) Write down
( i ) P ;
( ii ) n
D . n
( f ) Hence , express T in terms of n .
[ 3 ]
On a particular day , the probability that it is sunny is two times the probability that it is cloudy .
[ 3 ]
( g ) Find , after a week , the probability that it is cloudy . [ 4 ] www . seprodstore . com
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