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Question

If ¯¯¯a,¯¯b,¯¯c are non coplanar vectors and λ is a real number, then the vectors ¯¯¯a+2¯¯b+3¯¯c,λ¯¯b+4¯¯c and (2λ1)¯¯c are non-coplanar for

A
All values of λ
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B
No value of λ
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C
All except two values of λ
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D
All except three values of λ
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Solution

The correct option is C All except two values of λ
For three vectors to be non-coplanar, their scalar triple product should not be equal to 0
Let A=a+2b+3c,B=λb+4c,C=(2λ1)c
for these to be non-coplanar
A.B×C0
(a+2b+3c).((λb+4c)×(2λ1)c)0
(a+2b+3c).(λ(2λ1)b×c)0 (c×c=0)
λ(2λ1)a.b×c0
Now as a,b,c are non-coplanar,
a.b×c0
The above equation will satisfy for all λ except for λ=0,λ=12

Hence, answer is option C

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