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1

JEE Main 2021 (Online) 27th August Morning Shift

Equation of a plane at a distance $${2 \over {\sqrt {21} }}$$ from the origin, which contains the line of intersection of the planes x $$-$$ y $$-$$ z $$-$$ 1 = 0 and 2x + y $$-$$ 3z + 4 = 0, is :
A
$$3x - y - 5z + 2 = 0$$
B
$$3x - 4z + 3 = 0$$
C
$$- x + 2y + 2z - 3 = 0$$
D
$$4x - y - 5z + 2 = 0$$

Explanation

Required equation of plane

$${P_1} + \lambda {P_2} = 0$$

$$(x - y - z - 1) + \lambda (2x + y - 3z + 4) = 0$$

Given that its dist. From origin is $${2 \over {\sqrt {21} }}$$

Thus, $${{|4\lambda - 1|} \over {\sqrt {{{(2\lambda + 1)}^2} + {{(\lambda - 1)}^2} + {{( - 3\lambda - 1)}^2}} }} = {{\sqrt 2 } \over {\sqrt {21} }}$$

$$\Rightarrow 21{(4\lambda - 1)^2} = 2(14{\lambda ^2} + 8\lambda + 3)$$

$$\Rightarrow 336{\lambda ^2} - 168\lambda + 21 = 28{\lambda ^2} + 16\lambda + 6$$

$$\Rightarrow 308{\lambda ^2} - 184\lambda + 15 = 0$$

$$\Rightarrow 308{\lambda ^2} - 154\lambda - 30\lambda + 15 = 0$$

$$\Rightarrow (2\lambda - 1)(154\lambda - 15) = 0$$

$$\Rightarrow \lambda = {1 \over 2}$$ or $${{15} \over {154}}$$

for $$\lambda = {1 \over 2}$$ reqd. plane is $$4x - y - 5z + 2 = 0$$
2

JEE Main 2021 (Online) 26th August Evening Shift

Let P be the plane passing through the point (1, 2, 3) and the line of intersection of the planes $$\overrightarrow r \,.\,\left( {\widehat i + \widehat j + 4\widehat k} \right) = 16$$ and $$\overrightarrow r \,.\,\left( { - \widehat i + \widehat j + \widehat k} \right) = 6$$. Then which of the following points does NOT lie on P?
A
(3, 3, 2)
B
(6, $$-$$6, 2)
C
(4, 2, 2)
D
($$-$$8, 8, 6)

Explanation

$$(x + y + 4z - 16) + \lambda ( - x + y + z - 6) = 0$$

Passes through (1, 2, 3)

$$- 1 + \lambda ( - 2) \Rightarrow \lambda = - {1 \over 2}$$

$$2(x + y + 4z - 16) - ( - x + y + z - 6) = 0$$

$$3x + y + 7z - 26 = 0$$
3

JEE Main 2021 (Online) 26th August Evening Shift

A hall has a square floor of dimension 10 m $$\times$$ 10 m (see the figure) and vertical walls. If the angle GPH between the diagonals AG and BH is $${\cos ^{ - 1}}{1 \over 5}$$, then the height of the hall (in meters) is :

A
5
B
2$$\sqrt {10}$$
C
5$$\sqrt {3}$$
D
5$$\sqrt {2}$$

Explanation

$$A(\widehat j)\,.\,B(10\widehat i)$$

$$H(h\widehat j + 10\widehat k)$$

$$G(10\widehat i + h\widehat j + 10\widehat k)$$

$$\overrightarrow {AG} = 10\widehat i + h\widehat j + 10\widehat k$$

$$\overrightarrow {BH} = - 10\widehat i + h\widehat j + 10\widehat k$$

$$\cos \theta = {{\overrightarrow {AG} \overrightarrow {BH} } \over {\left| {\overrightarrow {AG} } \right|\left| {\overrightarrow {BH} } \right|}}$$

$${1 \over 5} = {{{h^2}} \over {{h^2} + 200}}$$

$$4{h^2} = 200 \Rightarrow h = 5\sqrt 2$$
4

JEE Main 2021 (Online) 26th August Morning Shift

A 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 lies on P?
A
($$-$$1, 1, 2)
B
(0, 1, 1)
C
(1, 0, 1)
D
(2, $$-$$1, 1)

Explanation

Equation of plane P can be assumed as

P : x + 2y + 3z + 1 + $$\lambda$$ (x $$-$$ y $$-$$ z $$-$$ 6) = 0

$$\Rightarrow$$ P : (1 + $$\lambda$$)x + (2 $$-$$ $$\lambda$$)y + (3 $$-$$ $$\lambda$$)z + 1 $$-$$ 6$$\lambda$$ = 0

$$\Rightarrow {\overrightarrow n _1} = (1 + \lambda )\widehat i + (2 - \lambda )\widehat j + (3 - \lambda )\widehat k$$

$$\therefore$$ $${\overrightarrow n _1}\,.\,{\overrightarrow n _2} = 0$$

$$\Rightarrow$$ 2(1 + $$\lambda$$) $$-$$ (2 $$-$$ $$\lambda$$) $$-$$ (3 $$-$$ $$\lambda$$) = 0

$$\Rightarrow$$ 2 + 2$$\lambda$$ $$-$$ 2 + $$\lambda$$ $$-$$ 3 + $$\lambda$$ = 0 $$\Rightarrow$$ $$\lambda$$ = $${3 \over 4}$$

$$\Rightarrow$$ $$P:{{7x} \over 4} + {5 \over 4}y + {{9z} \over 4} - {{14} \over 4} = 0$$

$$\Rightarrow$$ 7x + 5y + 9z = 14

(0, 1, 1) lies on P.

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