KC Sinha Solution Class 12 Chapter 30 3D Geometry : Plane (त्रिविमीय ज्यामिति : समतल ) Exercise 30.1 (Q11-Q20)

 Exercise 30.1

Question 11

In each of the following cases, determine the direction cosines of the normal to the plane and its distance from the origin

(i) $2 x-3 y+4 z-6=0$

(ii) $2 x+3 y-z=5$

(iii) $x+y+z=1$

(iv) $5 y+8=0$

(v) $z=2$

Sol :

Question 12

Find the direction cosines of the unit vector perpendicular to the plane $\vec{r} \cdot(6 \hat{i}-3 \hat{j}-2 \hat{k})+1=0$ passing through the origin.

Sol :

TYPE-III

Question 13

Find the angle between the planes whose vector equations are $\vec{r} \cdot(2 \hat{i}+2 \hat{j}-3 \hat{k})=5$ and $\vec{r} \cdot(3 \hat{i}-3 \hat{j}+5 \hat{k})=3$

Sol :

Question 14

Find the angle between the planes

(i) $2 x-y+z=6$ and $x+y+2 z=7$.

(ii) $7 x+5 y+6 z+30=0$ and $3 x-y-10 z+4=0$

(iii) $3 x-6 y+2 z=7$ and $2 x+2 y-2 z=5$

(iv) $2 x+y-2 z=5$ and $3 x-6 y-2 z=7$

Sol :

Question 15

Determine whether the following pair of planes are parallel or perpendicular, and in case they are neither, find the angle between them.

(i) $2 x-y+3 z-1=0$ and $2 x-y+3 z+3=0$

(ii) $2 x-2 y+4 z+5=0$ and $3 x-3 y+6 z-1=0$

(iii) $2 x+y+3 z-2=0$ and $x-2 y+5=0$

(iv) $4 x+8 y+z-8=0$ and $y+z-4=0$

(v) $3 x-4 y+5 z=0$ and $2 x-y-2 z=5$

Sol :

Question 16

Find the angle between the line $\frac{x+1}{2}=\frac{y}{3}=\frac{z-3}{6}$ and the plane $10 x+2 y-11 z=3$

Sol :

TYPE-IV

Question 17

(i) Find the equation of the plane containing point (1,-1,2) and perpendicular to each of the planes 2x+3y-2z=5 and x+2y-3z=8

Sol :

(ii) Find the equation of the plane passing through the point (-1,-1,2) and perpendicular to each of the planes : 2x+3y-3z=2 and 5x-4y+z=6

Sol :

Question 18

Find the equation of the plane passing through the point (-1,3,2) and perpendicular to each of the planes x+2y+3z=5 and 3x+3y+z=0.

Sol :

Question 19

Find the vector and cartesian equations of the planes

(i) that passes through the point (1, 4, 6) and the normal vector to the plane is $\hat{i}-2 \hat{j}+\hat{k}$

(ii) that passes through the point (1,0,-2) and normal vector to the plane is $\hat{i}+\hat{j}-\hat{k}$

Sol :

Question 20

If O be the origin and the coordinates of P be (1,2,-3) then find the equation of the plane passing through P and perpendicular to OP

Sol :