# Important Questions for CBSE Class 12 Maths Chapter 11 – Three Dimensional Geometry

## CBSE Class 12 Maths Chapter-11 Important Questions â€“ Free PDF Download

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## 1 Mark Questions

1. Find the directions cosines of x, y and z axis.
Ans.Â 1,0,0, 0,1,0 0,0,1

2.Find the vector equation for the line passing through the points (-1,0,2) and (3,4,6)
Ans.Â LetÂ be the p.v of the points A (-1,0,2) and B ( 3, 4 6)

3.Find the angle between the vector having direction ratios 3,4,5 and 4, -3, 5.
Ans.Â Let a1Â = 3, b1Â = 4, c1Â = 5 and a2Â = 4, b2Â = -3, c2Â = 5

4. What is the direction ratios of the line segment joining P(x1Â y1Â z1) and Q (x2Â y2Â z2)
Ans.Â x2Â â€“ x1, y2Â â€“ y, and z2-z1Â are the direction ratio of the line segment PQ.

5. The Cartesian equation of a line isÂ  Find the vector equation for the line.
Ans.Â Comparing the given equation with the standard equation form

6.Show that the lines
are coplanar.
Ans.Â x1=-3, y1Â = 1, z1Â = 5
a1Â = -3, b1=1, c1= 5
x2Â = -1, y2=2, z2Â = 5
a2Â = -1, b2Â = 2, c2Â = 5

Therefore lines are coplanar.

7. If a line has the direction ratios -18, 12, -4 then what are its direction cosines
Ans.Â a = -18, b=12, c= -4
a2+b2+c2Â = (-18)2Â + (12)2Â + (-4)2
= 484

8. Find the angle between the pair of line given by

Ans.

9. Prove that the points A(2,1,3) B(5, 0,5)and C(-4, 3,-1) are collinear
Ans.Â The equations of the line AB are

If A, B, C are collinear, C lies in equation (1)

Hence A,B,C are collinear

10. Find the direction cosines of the line passing through the two points
(2,4,-5) and (1,2,3).
Ans.Â Let P(-2,4,-5) Q (1,2,3)

11. Find the equation of the plane with intercepts 2,3 and 4 on the x, y and z axis respectively.
Ans.Â Let the equation of the plane be

12.If the equations of a line AB isÂ  find the directions ratio of line parallel to AB.
Ans.Â the direction ratios of a line parallel to AB are 1, -2, 4

13. If the line has direction ratios 2,-1,-2 determine its direction Cosines.
Ans.Â

14. The Cartesian equation of a line isÂ  . Write its vector form
Ans.Â

15. Cartesian equation of a line AB isÂ  write the direction ratios of a line parallel to AB.
Ans.Â Given equation of a line can be written is

The direction ratios of a line parallel to AB are 1, -7, 2.

## 4 Mark Questions

1. Find the vector and Cartesian equation of the line through the point (5, 2,-4) and which is parallel to the vectorÂ
Ans:
Vector equation of line is

Cartesian equation is

2. Find the angle between the lines

Ans:
LetÂ is the angle between the given lines

3. Find the shortest distance between the lines

Ans:

4. Find the direction cosines of the unit vectorÂ  to the planeÂ  passing through the origin.
Ans:

Dividing equation 1 by 7

Hence direction cosines ofÂ isÂ

5. Find the angle between the two planes 3x â€“ 6y + 2z = 7 and 2x + 2y â€“ 2z = 5
Ans:Â Comparing the giving eq of the planes with the equations
A1Â x +B1y +C1Z + D = 0 , A2Â x + B2y + C2Â Z + D2Â = 0
A1Â = 3, B1Â = -6, C1Â = 2
A2Â = 2, B2Â = 2, C2Â = -2

6. Find the shortest between the lÂ 1Â and l2Â whose vectors equations are

Ans:

7. Find the angel between lines

Ans:

The angleÂ between them is given by

8. Show that the linesÂ  Are perpendicular to each others
Ans:

ForÂ
a1a2+b1b2+c1c2=0
L.H. S

9.Find the vector equations of the plane passing through the points R(2,5,-3), Q(-2,-3,5) and T (5,3,-3)
Ans:Let

Vector equation is

10. Find the Cartesian equation of the planeÂ

Ans:Let

Which is the required equation of plane.

11. find the distance between the lines l1Â and l2Â given by

Ans:

Hence line are parallel

12. Find the angle between lines

Ans:

13. Find the shortest distance between the lines

Ans:

14. Find the vector and Cartesian equations of the plane which passes through the point (5,2,-4) andÂ  to the line with direction ratios (2,3,-1)
Ans:

Vector equation is

Cartesian equation is

15. Find the Cartesian equation of the plane

Ans:

16. Find the distance of a point (2,5,-3) from the planeÂ
Ans:

17. Find the shortest distance

Ans:

18. Find the vector equation of a plane which is at a distance of 7 units from the origin and normal to the vectorÂ
Ans:

19. Find the Cartesian equation of planeÂ
Ans:Â

20. Find the angle between the lineÂ  and the plane 10x +2y-11z=3
Ans:Â

21. Find the value of P so that the linesÂ  are at right angles.
Ans:

22. Find the shortest distance between the lines whose vector equation are

Ans:Â

23. Find x such that four points A(3,2,1) B(4,x,5)(4,2,-2) and D (6,5,-1)are coplanar.
Ans:Â The equation of plane through
A(3,2,1), C(4,2,-2) and D (6,5,-1) is

The point A,B,C,D are coplanar

24. Find the angle between the two planes 2x +y-2z=5 and 3x -6y -2z = 7using vector method.
Ans.Â

25. Find the angle b/w the line

Ans:

## 6 Marks Questions

1.Find the vector equation of the plane passing through the intersection of plane And the point (1,1,1)
Ans.

Using the relation

2. Find the coordinate where the line thorough (3,-4,-5) and ((2,-3,1) crosses the plane 2x + y + z = 7
Ans.Â Given points are A(3,-4,-5)
B(2,-3,1)
Direction ration of AB are 3-2, -4+3, -5-1
1,-1,-6
Eq. of line AB

are the required point

3. Find the equation of the plane through the intersection of the planes
3x â€“ y + 2z -4 = 0 and x + y + z â€“ 2 = 0 and the point (2,2,1)
Ans.Â Equation of any plane through the
intersection of given planes can be taken as

The point (2,2,1) lies in this plane
put in eq â€¦.(i)

4. If the points (1,1p) and (-3,0,1)be equidistant from the planeÂ  , then find the value of p.
Ans.The given plane is

This plane is equidistant from the points (1,1,P) and (-3,0,1)

5. Find the equation of the plane through the line of intersection of the planes
x +y +z = 1 and 2x + 3y + 4z = 5 which isÂ  of the plane x-y + z = 0
Ans.Â Equations of any plane through the intersection of given planes are be written is

This plane is it right angle to the plane x-y+z

6. Find the distance of the point (-1,-5,-10) from the point of intersection of the lineÂ  and the planeÂ
Ans.

Are the coordinate of the point of intersection of the given line and the planeÂ

7. Find the equation of the plane that contains the point (1,-1,2) and isÂ  to each of the plane 2x+3y-2z=5 and x+2y-3z = 8
Ans.Â The equation of the plane containing the given point is
A(x-1)+B(y-2)+C(Z-3)= 0â€¦.[i]
Condition ofÂ to the plane given in (i) with the plane
2x+3y-2z=5, x+2y-3z=8
2A+3B-2C=0
A+2B-3C=0
On solving
A=-5c, B=4C
5x-4y-Z=7

8. Find the vector equation of the line passing through (1,2,3) andÂ  to the planesÂ
Ans.Â

9. Find the equation of the s point where the line through the points A(3,4,1) and B(5,1,6) crosses the XY plane.
Ans.Â The vector equation of the line through the point A and B is

Let P be the point where the line AB crosses the XY plane. Then the position vectorÂ of the point P is the form

10. Prove that if a plane has the intercepts a,b,c is at a distance of p units from the origin then

Ans.Â The equation of the plane in the
intercepts from isÂ distance of
this plane from the origin is given to be p