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 _________.

JEE Main 2022 · MathsNumerical answerNumerical valueHardMulti-step

JEE Main 26 July 2022, Shift 2, Maths Q27

Question 27 of 90 in this shift, Maths question 27 of 30, Section B.

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

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 _________.

Official answer

125

NTA final key.

Chapter: Three Dimensional Geometry
Topic: Planes in 3D
Idea tested: Family of Planes

Same idea in other shifts

Asked 15× in all

6 Apr 2023, Shift 1 · Q13If the equation of the plane passing through the line of intersection of the planes $2x-y+z=3$ , $4x-3y+5z+9=0$ and parallel to the line…MediumSingle correct
6 Apr 2023, Shift 2 · Q15A plane P contains the line of intersection of the plane $\vec{r}\cdot\left(\hat{i}+\hat{j}+\hat{k}\right)=6$ and…MediumSingle correct
13 Apr 2023, Shift 1 · Q8Let the equation of plane passing through the line of intersection of the planes $x+2y+az=2$ and $x-y+z=3$ be $5x-11y+bz=6a-1$ . For…MediumSingle correct
27 Jul 2022, Shift 1 · Q17If the plane $P$ passes through the intersection of two mutually perpendicular planes $2x + ky - 5z = 1$ and $3kx - ky + z = 5$ , $k < 3$ …MediumSingle correct
28 Jun 2022, Shift 1 · Q16The acute angle between the planes P $_1$ and P $_2$ , when P $_1$ and P $_2$ are the planes passing through the intersection of the planes…MediumSingle correct
28 Jun 2022, Shift 1 · Q17Let the plane $P:\vec{r}\cdot\vec{a}=d$ contain the line of intersection of two planes…HardSingle correct
26 Aug 2021, Shift 1 · Q75A plane P contains the line $x+2y+3z+1=0=x-y-z-6$ , and is perpendicular to the plane $-2x+y+z+8=0$ . Then which of the following points…MediumSingle correct
26 Aug 2021, Shift 2 · Q79Let P be the plane passing through the point $(1,2,3)$ and the line of intersection of the planes…MediumSingle correct

Question text from the official JEE Main paper published by NTA; answer from the final answer key. Chapter, topic and idea tags, difficulty and skill are Lumi’s. Lumi analyses 108 of the 174 JEE Main shifts held from 2017 to 2026.