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Question

Let a,b and c be three non-zero and non-coplanar vectors and p,q and r be three vectors given by p=a+b2c, q=3a2b+c and r=a4b+2c. If the volume of the parallelopiped determined by a,b and c is V1 and the volume of the parallelopiped determined by p,q and r is V2 then V2:V1=

A
3:1
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B
7:1
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C
11:1
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D
15:1
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Solution

The correct option is D 15:1
Volume of parallelopiped is given by V1=[¯¯¯a¯¯b¯¯c]
V2=[¯¯¯p¯¯¯q¯¯¯r]
V2=(¯¯¯a+¯¯b2¯¯c).(3¯¯¯a2¯¯b+¯¯c)×(¯¯¯a4¯¯b+2¯¯c)
=(¯¯¯a+¯¯b2¯¯c).(10¯¯¯aׯ¯b+5¯¯¯aׯ¯c)
=5[¯¯b¯¯¯a¯¯c]+20[¯¯c¯¯¯a¯¯b]
=15[¯¯¯a¯¯b¯¯c]
Thus, the ratio of volumes is 15:1

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