For $a, b \in \mathbb{Z}$ and $|a - b| \leq 10$, let the angle between the plane $P: ax + y - z = b$ and the line $l: x - 1 = a - y = z + 1$ be $\cos^{-1}\left(\frac{1}{3}\right)$. If the distance of the point $(6, -6, 4)$ from the plane $P$ is $3\sqrt{6}$, then $a^4 + b^2$ is equal to

JEE Main 2023 · MathsMultiple choiceSingle correctHardMulti-step

JEE Main 8 April 2023, Shift 2, Maths Q14

Question 14 of 90 in this shift, Maths question 14 of 30, Section A.

For

a, b \in \mathbb{Z}

and

|a - b| \leq 10

, let the angle between the plane

P: ax + y - z = b

and the line

l: x - 1 = a - y = z + 1

\cos^{-1}\left(\frac{1}{3}\right)

. If the distance of the point

(6, -6, 4)

from the plane

P

3\sqrt{6}

, then

a^4 + b^2

is equal to

(1)25
(2)32Official answer
(3)48
(4)85

Official answer

Option 2

NTA final key.

Chapter: Three Dimensional Geometry
Topic: Planes in 3D
Idea tested: Angle Between a Line and a Plane

Same topic in other shifts

All Planes in 3D questions

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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
8 Apr 2023, Shift 1 · Q15If the equation of the plane containing the line $x+2y+3z-4=0=2x+y-z+5$ and perpendicular to the plane…MediumSingle correct
10 Apr 2023, Shift 2 · Q14Let the image of the point $P(1, 2, 6)$ in the plane passing through the points $A(1, 2, 0)$ , $B(1, 4, 1)$ and $C(0, 5, 1)$ be…MediumSingle correct
11 Apr 2023, Shift 1 · Q2Let $(\alpha, \beta, \gamma)$ be the image of the point $P(2, 3, 5)$ in the plane $2x + y - 3z = 6$ . Then $\alpha + \beta + \gamma$ is…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.