NIMCET 2011 Question Paper with solutions

Question : 39 of 120

Marks: +1, -0

Let

a=x i-3 j-k

and

b=2 x i+x j-k

. Suppose that the angle between a and b is acute and the angle between b and the positive direction of the y -axis lies between

{π}/{2}

and

π

. Then, the set of all possible values of x is

Solution:

Since, the angle between the vectors

{a}

and

{b}

is acute and the angle between

b

and

y

-axis lies between

{π}/{2}

and

π

, it means they are obtuse

∴ a ⋅ b > 0 \and b ⋅ j < 0

⇒(x i-3 j-k) ⋅(2 x i+x j-k)>0

and

(2 x ˙{i}+x j-k) ⋅ j<0

⇒ 2 x^{2}-3 x-1>0

and

x<0

⇒ (2 x-1)(x-1)>0

and

x<0

⇒ (x<{1}/{2}\or x>1)

and

x<0

⇒ x<0

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