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Question

Given that a and b are two unit vectors such that the angle between a and b is cos1(14). Let c be a vector in the plane of a and b, such that |c|=4 and c×b=2a×b. If c=λa+μb, where λ and μ are constants, then which of the following is/are CORRECT?

A
λ=2.
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B
Sum of all possible values of μ is 1.
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C
Product of all possible values of μ is 12.
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D
Number of distinct values of μ is 3.
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Solution

The correct option is C Product of all possible values of μ is 12.
Given c×b=2a×b
(c2a)×b=0

So, (c2a) is parallel to b
c2a=μbc=2a+μb (1)
Comparing withc=λa+μb, we get
λ=2

Now, |a|=|b|=1, ab=14
Taking dot product with a,b and c simultaneously with equation (1), we get
ac=2+μ4 (2)bc=12+μ (3)16=2ac+μbc
Using equation (2) and (3), we get
16=4+μ2+μ2+μ2μ2+μ12=0(μ+4)(μ3)=0μ=4,3

Hence, the sum of all possible values of μ is 1.
Product of all possible values of μ is 12.
Number of distinct values of μ is 2.

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