Determine an equation of the plane $E$ in normal form. (for verification: $E : 4 x _ { 1 } + 4 x _ { 2 } - 10 x _ { 3 } - 43 = 0$ )

The points $A _ { 1 } ( 0 | 0 | 0 ) , A _ { 2 } ( 20 | 0 | 0 ) , A _ { 3 }$ and $A _ { 4 } ( 0 | 10 | 0 )$ represent the vertices of the base of the multipurpose hall in the model, and the points $B _ { 1 } , B _ { 2 } , B _ { 3 }$ and $B _ { 4 }$ represent the vertices of the roof surface. The side wall that lies in the $x _ { 1 } x _ { 3 }$-plane in the model is 6 m high, and the opposite wall is only 4 m high.
Give the coordinates of the points $B _ { 2 } , B _ { 3 }$ and $B _ { 4 }$ and confirm that these points lie in the plane $E : x _ { 2 } + 5 x _ { 3 } - 30 = 0$.

The points $A ( 6 | 0 | 4 ) , B ( 0 | 6 | 4 ) , C ( - 6 | 0 | 4 )$ and $D$ lie in the plane $E$ and form the vertices of the square base of a pyramid ABCDS with apex $S ( 0 | 0 | 1 )$. $A , B$ and $S$ lie in the plane $F$.
Show by calculation that the triangle ABS is isosceles. Give the coordinates of point $D$ and describe the special position of plane $E$ in the coordinate system.

The points $A ( 6 | 0 | 4 ) , B ( 0 | 6 | 4 ) , C ( - 6 | 0 | 4 )$ and $D$ lie in the plane $E$ and form the vertices of the square base of a pyramid ABCDS with apex $S ( 0 | 0 | 1 )$. $A , B$ and $S$ lie in the plane $F$.
Determine an equation of plane $F$ in coordinate form.

Determine an equation of $E$ in coordinate form and show that the line $g$ lies in $E$.
(for verification: $E : 2 x _ { 1 } - x _ { 2 } + 2 x _ { 3 } + 35 = 0$ )

Determine an equation of $L$ in coordinate form and the angle $\varphi$ that $L$ makes with the $x _ { 1 } x _ { 2 }$-plane. (for verification: $x _ { 1 } + x _ { 2 } + x _ { 3 } - 19 = 0 ; \varphi \approx 55 ^ { \circ }$ )

133- Vector $\mathbf{a}$ makes an angle of $60°$ with each of the axes $Ox$ and $Oy$, and makes a right angle with the axis $z$. This vector is perpendicular to which plane? Which equation does the plane satisfy?
(1) $x - \sqrt{2}y + z = 0$ (2) $2x + 2y + \sqrt{2}z = 0$
(4) $x + y - \sqrt{2}z = 0$ (3) $x + y + \sqrt{2}z = 0$

Let $P$ be the image of the point $( 3,1,7 )$ with respect to the plane $x - y + z = 3$. Then the equation of the plane passing through $P$ and containing the straight line $\frac { x } { 1 } = \frac { y } { 2 } = \frac { z } { 1 }$ is
(A) $x + y - 3 z = 0$
(B) $3 x + z = 0$
(C) $x - 4 y + 7 z = 0$
(D) $2 x - y = 0$

The equation of the plane passing through the point $( 1,1,1 )$ and perpendicular to the planes $2 x + y - 2 z = 5$ and $3 x - 6 y - 2 z = 7$, is
[A] $14 x + 2 y - 15 z = 1$
[B] $14 x - 2 y + 15 z = 27$
[C] $14 x + 2 y + 15 z = 31$
[D] $- 14 x + 2 y + 15 z = 3$

Let $\alpha, \beta, \gamma, \delta$ be real numbers such that $\alpha^{2} + \beta^{2} + \gamma^{2} \neq 0$ and $\alpha + \gamma = 1$. Suppose the point $(3, 2, -1)$ is the mirror image of the point $(1, 0, -1)$ with respect to the plane $\alpha x + \beta y + \gamma z = \delta$. Then which of the following statements is/are TRUE?
(A) $\alpha + \beta = 2$
(B) $\delta - \gamma = 3$
(C) $\delta + \beta = 4$
(D) $\alpha + \beta + \gamma = \delta$

A plane passes through the points $A ( 1,2,3 ) , B ( 2,3,1 )$ and $C ( 2,4,2 )$. If $O$ is the origin and $P$ is $( 2 , - 1,1 )$, then the projection of $\overrightarrow { O P }$ on this plane is of length:
(1) $\sqrt { \frac { 2 } { 5 } }$
(2) $\sqrt { \frac { 2 } { 7 } }$
(3) $\sqrt { \frac { 2 } { 3 } }$
(4) $\sqrt { \frac { 2 } { 11 } }$

Let $P$ be an arbitrary point having sum of the squares of the distance from the planes $x + y + z = 0 , l x - n z = 0$ and $x - 2 y + z = 0$ equal to 9 units. If the locus of the point $P$ is $x ^ { 2 } + y ^ { 2 } + z ^ { 2 } = 9$, then the value of $l - n$ is equal to

A plane $E$ is perpendicular to the two planes $2x - 2y + z = 0$ and $x - y + 2z = 4$, and passes through the point $P(1, -1, 1)$. If the distance of the plane $E$ from the point $Q(a, a, 2)$ is $3\sqrt{2}$, then $(PQ)^2$ is equal to
(1) 9
(2) 12
(3) 21
(4) 33

If $(2, 3, 9)$, $(5, 2, 1)$, $(1, \lambda, 8)$ and $(\lambda, 2, 3)$ are coplanar, then the product of all possible values of $\lambda$ is
(1) $\frac{21}{2}$
(2) $\frac{59}{8}$
(3) $\frac{57}{8}$
(4) $\frac{95}{8}$

A vector $\vec{v}$ in the first octant is inclined to the $x$-axis at $60^{\circ}$, to the $y$-axis at $45^{\circ}$ and to the $z$-axis at an acute angle. If a plane passing through the points $(\sqrt{2}, -1, 1)$ and $(a, b, c)$, is normal to $\vec{v}$, then
(1) $\sqrt{2}a + b + c = 1$
(2) $a + b + \sqrt{2}c = 1$
(3) $a + \sqrt{2}b + c = 1$
(4) $\sqrt{2}a - b + c = 1$

If a plane passes through the points $(-1, k, 0)$, $(2, k, -1)$, $(1, 1, 2)$ and is parallel to the line $\frac{x-1}{1} = \frac{2y+1}{2} = \frac{z+1}{-1}$, then the value of $\frac{k^{2}+1}{(k-1)(k-2)}$ is
(1) $\frac{17}{5}$
(2) $\frac{5}{17}$
(3) $\frac{6}{13}$
(4) $\frac{13}{6}$

If the equation of the plane containing the line $x + 2 y + 3 z - 4 = 0 = 2 x + y - z + 5$ and perpendicular to the plane $\vec { r } = ( \hat { i } - \hat { j } ) + \lambda ( \hat { i } + \hat { j } + \hat { k } ) + \mu ( \hat { i } - 2 \hat { j } + 3 \hat { k } )$ is $a x + b y + c z = 4$ then $( a - b + c )$ is equal to
(1) 18
(2) 22
(3) 20
(4) 24