If a- b- c be three unit vectors such that a - b - c - 12 b- b and c being non-parallel- If - 1 is the angle between a c and -2 is the angle between a b- then

If $t a n θ_{1}, t a n θ_{2}, t a n θ_{3}$ are the real roots of $x^{3} - (a + 1) x^{2} + (b - a) x - b = 0 w h e r e θ_{1}, θ_{2}, θ_{3}$ are acute then $θ_{1} + θ_{2} + θ_{3} =$

Q. If

tan θ_{1}, tan θ_{2}, tan θ_{3}

are the real roots of the

x^{3} - (a + 1) x^{2} + (b - a) x - b = 0

, where

θ_{1} + θ_{2} + θ_{3} \in (0, π)

, then

θ_{1} + θ_{2} + θ_{3}

is equal to

Q. If

θ_{1}

and

θ_{2}

be the angles which the lines

(x^{2} + y^{2}) ({cos}^{2} θ {sin}^{2} α + {sin}^{2} θ) = (x tan α - y sin θ)^{2}

make with the axis of

x

, then if

θ = \frac{π}{6}, tan θ_{1} + tan θ_{2}

is equal to

Q. If a,b,c are non-coplanar unit vectors such that

a \times (b \times c) = \frac{b + c}{\sqrt{2}}

, then the angle between a and b is

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If a,b,c be three unit vectors such that a×(b×c)=12b, b and c being non-parallel. If θ1 is the angle between a & c and θ2 is the angle between a & b, then

The correct option is A θ1=π/3 (1/2)b=a×(b×c)=(a⋅c)b−(a⋅b)c⇒(a⋅c−(1/2))b=(a⋅b)cThe last relation is possible only if a⋅c=1/2 and a⋅b=0 as b and c are non-parallel. Hence θ2=π/2 and θ1=π/3.

If $a, b, c$ be three unit vectors such that $a \times (b \times c) = \frac{1}{2} b, b$ and $c$ being non-parallel. If $θ_{1}$ is the angle between $a$ & $c$ and $θ_{2}$ is the angle between $a$ & $b$ , then