Your Practice Set – Applications and Interpretation for IBDP Mathematics
( c ) Hence , write down the approximate value of the displacement at
( i ) t � 0.2 ;
( ii ) t � 0.3 .
The system can be expressed by a matrix equation
and
�dv �
� dt �
X � � � and dx � � � dt
�
M , where �1 � �2.
[ 2 ] X � MX , where M is a 2� 2 matrix ,
�v �
X � � � are two 2� 1 matrices . Let �
1
and �
2 be the eigenvalues of
�x�
( d ) Find det ( M��I ) , giving the answer in terms of � .
( e ) 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 ]
( f ) Write down v
1
and v
2
.
( g ) Find
[ 2 ]
( i ) the particular solution of x ;
( ii ) the displacement at t � 0.3 .
( h ) Hence , write down the difference between the approximated displacement in ( c )( ii ) and the actual displacement in ( g )( ii ).
[ 6 ]
[ 1 ]
4 . A box connected with a spring moves with velocity v ms
� 1 and displacement m
x with respect to the starting point O . By considering the rate of change of its velocity , the relationship between the variables can be modelled by the differential equation
2 d x
�9x
� 0.
2 dt
( a ) By using dx v � , express the differential equation in a coupled system . dt
[ 1 ]
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