Your Practice Set – Applications and Interpretation for IBDP Mathematics
( g ) On the following diagram , sketch the following trajectory for this particular solution .
[ 2 ]
2 . Let x and y be the populations , in thousands , of horses and zebras in a national park respectively . The changes in the populations can be modelled by the coupled differential
equations
�dx
��3x
� dt �� . dy
�4y�x
�� dt
( a ) When x � 2 , state the range of values of y such that d y
10 dt � . [ 1 ]
The system can be expressed by a matrix equation
and
�dx �
� dt �
X � � � and dy � � � dt
�
M , where �1 � �2.
X � MX , where M is a 2� 2 matrix ,
�x �
X � � � are two 2� 1 matrices . Let �
1
and �
2 be the eigenvalues of
�y �
( b ) Find det ( M��I ) , giving the answer in terms of � .
( c ) Hence , write down the values of �
1
and �
2
.
Let v
1
and v
2 be the eigenvectors of M corresponding to �
1
and �
2 respectively .
[ 2 ]
[ 2 ]
( d ) Write down v
1
and v
2
.
[ 2 ]
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