Your Practice Set – Applications and Interpretation for IBDP Mathematics
61
Paper 2 – Real Life Problems
Example
X and Y are the only two cities on an island . Each month 30 % of the residents in X move to Y , and 55 % of the residents in Y choose to stay in Y . Assume that there is no net gain or net loss of the entire population on the island .
( a ) Find T , the transition matrix representing the changes in population between X and Y in a particular month .
[ 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 .
( f ) Hence , express n
T in terms of n .
The initial population of X and Y are 550 and 450 respectively .
( g ) Find , after a year , the population of X and Y .
[ 3 ]
[ 3 ]
[ 4 ]
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