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Q.

Let P1 denote the equation of a plane to which the vector (i^+j^)  is normal and which contains the line whose equation  r→=i^+j^+k^+λ(i^−j^−k^) and P2 denote the equation of the plane containing the line L and a point with position vector j^ Which of the following holds good?

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a

The equation of  x+y=2

b

The equation of r→⋅(i^−2j^+k^)=2

c

The acute angle between P1 and P2 is cot−1⁡(3)

d

The angle between the plane P2 and the line L is tan−1⁡3

answer is A.

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Detailed Solution

Plane P1, contains the line r→=i^+j^+k^+λ(i^−j^−k^)  hence contains the point i^+j^+k^   and is normal to vector (i^+j^)(r→−(i^+j^+k^))⋅(i^+j^)=0or  x+y=2Plane P2, contains the liner→=i^+j^+k^+λ(i^−j^−k^) and point j^Hence, equation of plane is x−0y−1z−01−01−11−01−1−1=0or  x+2y−z=2ff θ is the acute angle between P1 and P2,then cos⁡θ=n1→⋅n2→|n→|n2→(i^+j^)⋅(i^+2j^−k^)2⋅6 =32⋅6=32θ=cos−1⁡32=π6  As L is contained in P2⇒θ=0
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