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

Free PDF download of Important Questions for CBSE Class 12 Maths Chapter 11 â€“ Three Dimensional Geometry prepared by expert Maths teachers from latest edition of CBSE(NCERT) books, On CoolGyan.Org to score more marks in CBSE board examination.

You can also DownloadÂ __Maths Revision Notes Class 12__Â to help you to revise complete Syllabus and score more marks in your examinations.

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**CBSE Class 12 Mathematics Important Questions Chapter 11 â€“ Three Dimensional Geometry**

**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 a_{1}Â = 3, b_{1}Â = 4, c_{1}Â = 5 and a_{2}Â = 4, b_{2}Â = -3, c_{2}Â = 5

**4. What is the direction ratios of the line segment joining P(x _{1}Â y_{1}Â z_{1}) and Q (x_{2}Â y_{2}Â z_{2})**

**Ans.**Â x

_{2}Â â€“ x

_{1}, y

_{2}Â â€“ y, and z

_{2}-z

_{1}Â 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 x_{1}=-3, y_{1}Â = 1, z_{1}Â = 5

a_{1}Â = -3, b_{1}=1, c_{1}= 5

x_{2}Â = -1, y_{2}=2, z_{2}Â = 5

a_{2}Â = -1, b_{2}Â = 2, c_{2}Â = 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

a^{2}+b^{2}+c^{2}Â = (-18)^{2}Â + (12)^{2}Â + (-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.

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**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

A_{1}Â x +B_{1}y +C_{1}Z + D = 0 , A_{2Â }x + B_{2}y + C_{2Â }Z + D_{2}Â = 0

A_{1}Â = 3, B_{1}Â = -6, C_{1}Â = 2

A_{2}Â = 2, B_{2}Â = 2, C_{2Â }= -2

**6. Find the shortest between the lÂ _{1}Â and l_{2}Â 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

a_{1}a_{2}+b_{1}b_{2}+c_{1}c_{2}=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 l _{1Â }and l_{2}Â 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:**

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**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