JEE Advanced 2018 · Paper 1 · Mathematics Q3

MathematicsMultiple correct+4 / -2Question 39 of 54

Let $P_1: 2x + y - z = 3$ and $P_2: x + 2y + z = 2$ be two planes. Then, which of the following statement(s) is (are) TRUE?

The line of intersection of

P_1

and

P_2

has direction ratios

1, 2, -1

The line

\dfrac{3x-4}{9} = \dfrac{1-3y}{9} = \dfrac{z}{3}

is perpendicular to the line of intersection of

P_1

and

P_2

The acute angle between

P_1

and

P_2

60^\circ

P_3

is the plane passing through the point

(4, 2, -2)

and perpendicular to the line of intersection of

P_1

and

P_2

, then the distance of the point

(2, 1, 1)

from the plane

P_3

\dfrac{2}{\sqrt{3}}

Answer and worked solution

Official answer: C, D

Normals. $\vec n_1=(2,1,-1)$ and $\vec n_2=(1,2,1)$ .

Line of intersection.
$\vec n_1\times\vec n_2=(1\cdot1-(-1)\cdot2,\ (-1)\cdot1-2\cdot1,\ 2\cdot2-1\cdot1)=(3,-3,3)\parallel(1,-1,1).$

(A) The direction is $(1,-1,1)$ , not $(1,2,-1)$ . False.
(B) Rewrite the line as $\dfrac{x-4/3}3=\dfrac{y-1/3}{-3}=\dfrac z3$ . Its direction is $(3,-3,3)$ , parallel to the line of intersection, not perpendicular. False.
(C) $\cos\theta=\dfrac{|2+2-1|}{\sqrt6\sqrt6}=\dfrac12$ , so the acute angle is $60^\circ$ . True.
(D) $P_3$ has normal $(1,-1,1)$ and passes through $(4,2,-2)$ : $x-y+z=0$ . The distance of $(2,1,1)$ is
$\frac{|2-1+1|}{\sqrt3}=\frac2{\sqrt3}.$
True.

(C), (D)

JEE Advanced 2018 · Paper 1, Mathematics question 3