JEE Main 2022 · 26 Jul · Shift 2 · Mathematics Q27

← JEE Main 2022 · 26 Jul · Shift 2

MathematicsInteger+4 / -1Question 27 of 90

The plane passing through the line $L:lx-y+3(1-l)z=1$ , $x+2y-z=2$ and perpendicular to the plane $3x+2y+z=6$ is $3x-8y+7z=4$ . If $\theta$ is the acute angle between the line L and the $y$ -axis, then $415\cos^2\theta$ is equal to ______.

Your answer

Answer and worked solution

Official answer: 125

Pencil. $(lx-y+3(1-l)z-1)+\lambda(x+2y-z-2)=0$ must be $3x-8y+7z=4$ . Matching coefficients gives $\lambda=-\frac16$ , $l=\frac23$ .

Direction of L $=(2,-3,3)\times(1,2,-1)=(-3,5,7)$ .

Angle with the $y$ -axis. $\cos^2\theta=\frac{25}{83}$ , so $415\cos^2\theta=125$ .

Answer: 125

Topics:Equation of a plane Direction cosines and ratios

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JEE Main 2022 · 26 Jul · Shift 2, Mathematics question 27