Question
If the vectors →a=(clog2x)^i−6^j+2^k and →b=(log2x)^i+2^j+3(clog2x)^k make an obtuse angle for any x∈(0,∞) then c belongs to
Answer: Option C
:
C
For the vectors →a and →bto be inclined at an obtuse angle, we must have
→a.→b<0 for all x∈(0,∞)
⇒c(log2x)2−12+6c(log2x)<0 for all x∈(0,∞)
⇒cy2+6cy−12<0 for all y∈R, where y=log2x
⇒c<0 and ⇒36c2+48c<0⇒c<0and c(3c+4)<0
⇒c<0 and −43<c<0
⇒c∈(−43,0)
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:
C
For the vectors →a and →bto be inclined at an obtuse angle, we must have
→a.→b<0 for all x∈(0,∞)
⇒c(log2x)2−12+6c(log2x)<0 for all x∈(0,∞)
⇒cy2+6cy−12<0 for all y∈R, where y=log2x
⇒c<0 and ⇒36c2+48c<0⇒c<0and c(3c+4)<0
⇒c<0 and −43<c<0
⇒c∈(−43,0)
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