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Question

If a×(b×c)+(a.b)b=(42βsinα)b+(β21)c and (c.c)a=c, where b and c are non-collinear, then the value of scalars α and β, are

A
β=1,α=(4n+1)π2;nI
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B
β=1,α=(4n1)π2;nI
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C
β=1,α=(4n+1)π2;nI
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D
β=1,α=(4n1)π2;nI
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Solution

The correct option is A β=1,α=(4n+1)π2;nI
Given:

a×(b×c)+(a.b)b=(42βsinα)b+(β21)c

(a.c)b(a.b)c+(a.b)b=(42βsinα)b+(β21)c

Comparing LHS and RHS, we get
a.c+a.b=(42βsinα) ...(1)
a.b=1β2 ...(2)
From (1) and (2),
a.c=β22β+3sinα ...(3)

Now, it is also given that
(c.c)a=c
a=cc.c or a.c=c.cc.c=1

Therefore, from equation (3), we have,
β22β+2sinα=0
sinα=1+(12β+β2)=1+(1β)2

Since, sinα1(1β)2=0
β=1 and hence sinα=1
α=2nπ+π2, where nI

Hence, option A.

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