CBSE Class 12 Maths Chapter-11 Important Questions – Free PDF Download
Free PDF download of Important Questions for CBSE Class 12 Maths Chapter 11 – 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.
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CBSE Class 12 Mathematics Important Questions Chapter 11 – Three Dimensional Geometry
1 Mark Questions
1. Find the directions cosines of x, y and z axis.
Ans. 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. Let 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. Let a1 = 3, b1 = 4, c1 = 5 and a2 = 4, b2 = -3, c2 = 5
4. What is the direction ratios of the line segment joining P(x1 y1 z1) and Q (x2 y2 z2)
Ans. x2 – x1, y2 – y, and z2-z1 are the direction ratio of the line segment PQ.
5. The Cartesian equation of a line is
Find the vector equation for the line.
Ans. Comparing the given equation with the standard equation form
6.Show that the lines
Ans. x1=-3, y1 = 1, z1 = 5
a1 = -3, b1=1, c1= 5
x2 = -1, y2=2, z2 = 5
a2 = -1, b2 = 2, c2 = 5
Therefore lines are coplanar.
7. If a line has the direction ratios -18, 12, -4 then what are its direction cosines
Ans. a = -18, b=12, c= -4
a2+b2+c2 = (-18)2 + (12)2 + (-4)2
8. Find the angle between the pair of line given by
9. Prove that the points A(2,1,3) B(5, 0,5)and C(-4, 3,-1) are collinear
Ans. 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. 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. Let the equation of the plane be
12.If the equations of a line AB is
find the directions ratio of line parallel to AB.
Ans. 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.
14. The Cartesian equation of a line is
. Write its vector form
15. Cartesian equation of a line AB is
write the direction ratios of a line parallel to AB.
Ans. 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 vector
Vector equation of line is
Cartesian equation is
2. Find the angle between the lines
Let is the angle between the given lines
3. Find the shortest distance between the lines
4. Find the direction cosines of the unit vector
to the plane
passing through the origin.
Dividing equation 1 by 7
Hence direction cosines of is
5. Find the angle between the two planes 3x – 6y + 2z = 7 and 2x + 2y – 2z = 5
Ans: Comparing the giving eq of the planes with the equations
A1 x +B1y +C1Z + D = 0 , A2 x + B2y + C2 Z + D2 = 0
A1 = 3, B1 = -6, C1 = 2
A2 = 2, B2 = 2, C2 = -2
6. Find the shortest between the l 1 and l2 whose vectors equations are
7. Find the angel between lines
The angle between them is given by
8. Show that the lines
Are perpendicular to each others
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)
Vector equation is
10. Find the Cartesian equation of the plane
Which is the required equation of plane.
11. find the distance between the lines l1 and l2 given by
Hence line are parallel
12. Find the angle between lines
13. Find the shortest distance between the lines
14. Find the vector and Cartesian equations of the plane which passes through the point (5,2,-4) and
to the line with direction ratios (2,3,-1)
Vector equation is
Cartesian equation is
15. Find the Cartesian equation of the plane
16. Find the distance of a point (2,5,-3) from the plane
17. Find the shortest distance
18. Find the vector equation of a plane which is at a distance of 7 units from the origin and normal to the vector
19. Find the Cartesian equation of plane
20. Find the angle between the line
and the plane 10x +2y-11z=3
21. Find the value of P so that the lines
are at right angles.
22. Find the shortest distance between the lines whose vector equation are
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: 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.
25. Find the angle b/w the line
6 Marks Questions
1.Find the vector equation of the plane passing through the intersection of plane
And the point (1,1,1)
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. Given points are A(3,-4,-5)
Direction ration of AB are 3-2, -4+3, -5-1
Eq. of line AB
are the required point
3. Find the equation of the plane through the intersection of the planes
3x – y + 2z -4 = 0 and x + y + z – 2 = 0 and the point (2,2,1)
Ans. 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 ….(i)
4. If the points (1,1p) and (-3,0,1)be equidistant from the plane
, 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 is of the plane x-y + z = 0
Ans. 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 line
and the plane
Are the coordinate of the point of intersection of the given line and the plane
7. Find the equation of the plane that contains the point (1,-1,2) and is
to each of the plane 2x+3y-2z=5 and x+2y-3z = 8
Ans. The equation of the plane containing the given point is
Condition of to the plane given in (i) with the plane
8. Find the vector equation of the line passing through (1,2,3) and
to the planes
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. 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 vector 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. The equation of the plane in the
intercepts from is distance of
this plane from the origin is given to be p