Equation of Coordinate Planes

Q. 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 $P?$

1. $(1,0,1)$
2. $(2,-1,1)$
3. $(0,1,1)$
4. $(-1,1,2)$

Q. Let the equation of the plane, that passes through the point $(1,4,-3)$ and contains the line of intersection of the planes $3x-2y+4z-7=0$ and $x+5y-2z+9=0,$ be $\alpha x+\beta y+\gamma z+3=0,$ then $\alpha +\beta +\gamma$ is equal to

1. $-23$
2. $-15$
3. $23$
4. $15$

Q. Let $P$ be the plane passing through the point $(1,2,3)$ and the line of intersection of the planes $\overrightarrow{r}.(\hat{i}+\hat{j}+4\hat{k})$ and $\overrightarrow{r}.(-\hat{i}+\hat{j}+\hat{k})=6.$
Then which of the following points does NOT lie on $P?$

1. $(-8,8,6)$
2. $(4,2,2)$
3. $(3,3,2)$
4. $(6,-6,2)$

Q. Equation of a plane at a distance $\sqrt{\frac{2}{21}}$ from the origin, which contains the line of intersection of the planes $x-y-z-1=0$ and $2x+y-3z+4=0$, is :

1. $4x-y-5z+2=0$
2. $3x-4z+3=0$
3. $-x+2y+2z-3=0$
4. $3x-y-5z+2=0$

Q. The equation of the plane passing through the point $(1,2,–3)$ and perpendicular to the planes $3x+y-2z=5$ and $2x-5y-z=7$ is:

1. $6x-5y+2z+10=0$
2. $11x+y+17z+38=0$
3. $6x-5y-2z-2=0$
4. $3x-10y-2z+11=0$

Q. The equation of the plane passing through the point $(1,2,–3)$ and perpendicular to the planes $3x+y-2z=5$ and $2x-5y-z=7$ is:

1. $3x-10y-2z+11=0$
2. $6x-5y+2z+10=0$
3. $6x-5y-2z-2=0$
4. $11x+y+17z+38=0$

Q. The equation of the plane passing through the point $(1,2,–3)$ and perpendicular to the planes $3x+y-2z=5$ and $2x-5y-z=7$ is:

1. $3x-10y-2z+11=0$
2. $6x-5y-2z-2=0$
3. $11x+y+17z+38=0$
4. $6x-5y+2z+10=0$