Important Questions for CBSE Class 11 Maths Chapter 12 - Introduction to Three Dimensional Geometry
CBSE Class 11 Maths Chapter-12 Important Questions - Free PDF Download
1 Marks Questions
1. Name the octants in which the following lie. (5,2,3)
Ans. I
2. Name the octants in which the following lie. (-5,4,3)
Ans. II
3. Find the image of (-2,3,4) in the y z plane
Ans. (2, 3, 4)
4. Find the image of (5,2,-7) in the 
plane
Ans. (5, 2, 7)
5. A point lie on X –axis what are co ordinate of the point
Ans.
6. Write the name of plane in which 
axis and 
- axis taken together.
Ans.
Plane
7. The point
lie in which octants
Ans.
8. The point
lie in which plane
Ans.
9. A point is in the XZ plane. What is the value of y co-ordinates?
Ans. Zero
10. What is the coordinates of XY plane
Ans.
11. The point 
lie in which octants.
Ans. II
12. The distance from origin to point 
is:
Ans.
4 Marks Questions
1.Given that P(3,2,-4), Q(5,4,-6) and R(9,8,-10) are collinear. Find the ratio in which Q divides PR
Ans. Suppose Q divides PR in the ratio 
:1. Then coordinator of Q are
But, coordinates of Q are (5,4,-6). Therefore
These three equations give
.
So Q divides PR in the ratio 
or 1:2
2. Determine the points in 
plane which is equidistant from these point A (2,0,3) B(0,3,2) and C(0,0,1)
Ans. We know that Z- coordinate of every point on -plane is zero. So, let 
be a point in -plane such that PA=PB=PC
Now, PA=PB
PA2=PB2
Putting 
in (i) we obtain 
Hence the required points (3,2,0).
3. Find the locus of the point which is equidistant from the point A(0,2,3) and B(2,-2,1)
Ans. Let 
be any point which is equidistant from A(0,2,3) and B(2,-2,1). Then
PA=PB
PA2=PB2
4. Show that the points A(0,1,2) B(2,-1,3) and C(1,-3,1) are vertices of an isosceles right angled triangle.
Ans. We have
And 
Clearly AB=BC and AB2+BC2=AC2
Hence, triangle ABC is an isosceles right angled triangle.
5. Using section formula, prove that the three points A(-2,3,5), B(1,2,3), and C(7,0,-1) are collinear.
Ans.Suppose the given points are collinear and C divides AB in the ratio 
Then coordinates of C are
But, coordinates of C are (3,0,-1) from each of there equations, we get 
Since each of there equation give the same value of V. therefore, the given points are collinear and C divides AB externally in the ratio 3:2.
6. Show that coordinator of the centroid of triangle with vertices A(
), B(
), and C(
) is 
Ans. Let D be the mid point of AC. Then
Coordinates of D are 
Let G be the centroid of 
Then G, divides AD in the ratio 2:1. So coordinates of D are
i.e. 
7. Prove by distance formula that the points 
and 
are collinear.
Ans.Distance
Distance
Distance
The paints A.B.C. are collinear.
8. Find the co ordinate of the point which divides the join of 
and 
in the ratio 
internally 
externally
Ans.Let paint 
be the required paint.
(i)For internal division
Required paint 
(ii)For external division.
Required point 
9. Find the co ordinate of a point equidistant from the four points 
and 
Ans.Let be the required point
According to condition
Now 
Similarly 
and 
and 
are mid points of side 
respectively,
Then 
Adding eq (1),(4) and (7) we get
Adding eq. (2),(5) and (8)
And 
Hence co-ordinate of 
10. Find the ratio in which the join the 
and
is divided by the plane 
Also find the co-ordinate of the point of division
Ans. Suppose plane divides and 
in the ratio 
at pain 
Then co-ordinate of paint
Point lies on the plane
Points must satisfy the equation of plane
Required ratio 5:7
11. Find the centroid of a triangle, mid points of whose sides are 
Ans. Suppose co-ordinate of vertices of 
are
Adding eq. (3), (6) and (9)
Co-ordinate of centroid
12. The mid points of the sides of a 
are given by 
find the co ordinate of A, B and C
Ans. Suppose co-ordinate of point 
are
and 
respectively let
and are mid points of side
and
respectively
Adding eq. (1), (4) and (7)
Similarly 
Subtracting eq. (1), (4) and (7) from (10)
Now subtracting eq. (2), (5) and (8) from (11)
Similarly 
co-ordinate of point 
and are
and 
13. Find the co-ordinates of the points which trisects the line segment PQ formed by joining the point 
and 
Ans. Let R and S be the points of trisection of the segment PO. Then
R divides PQ in the ratio 1:2
Co-ordinates of point
=
Similarly 
S divider PQ in the ratio 2:1
co-ordinates of point S
14. Show that the point 
taken in order form the vertices of a parallelogram. Do these form a rectangle?
Ans.Mid point of PR is 
i.e. 
also mid point of QS is 
i.e.
Then PR and QS have same mid points.
PR and QS bisect each other. It is a Parallelogram.
Now 
and
diagonals an not equal
are not rectangle.
15. A point R with 
co-ordinates 4 lies on the line segment joining the points 
and 
find the co-ordinates of the point R
Ans. Let the point. R divides the line segment joining the point P and Q in the ratio , Then co-ordinates of Point R
The co-ordinates of point R is 4
co-ordinates of point R
16. If the points 
are collinear, find the values of P and q
Ans. Given points
are collinear
Let point Q divider PR in the ratio K:1
co-ordinates of point 
the value of P and q are 6 and 2.
17. Three consecutive vertices of a parallelogram ABCD are 
and 
find forth vertex D
Ans. Given vertices of 11gm ABCD
Suppose co-or dine of forth vertex 
Mid point of 
Mid point of 
Mid point of AC = mid point of BD
Co-ordinates of point 
18. If A and B be the points
and
respectively. Find the eq. of the set points P such that 
where K is a constant
Ans. Let co-ordinates of point P be
6 Marks Questions
1. Prove that the lines joining the vertices of a tetrahedron to the centroids of the opposite faces are concurrent.
Ans. Let ABCD be tetrahedron such that the coordinates of its vertices are 
, 
, 
and 
The coordinates of the centroids of faces ABC, DAB, DBC and DCA respectively
Now, coordinates of point G dividing DG1 in the ratio 3:1 are
Similarly the point dividing CG2, AG3 and BG4 in the ratio 3:1 has the same coordinates.
Hence the point 
is common to DG1, CG2, AG3 and BG4.
Hence they are concurrent.
2. The mid points of the sides of a triangle are (1,5,-1), (0,4,-2) and (2,3,4). Find its vertices.
Ans. Suppose vertices of 
ABC are 
respectively
Given coordinates of mid point of side BC, CA, and AB respectively are D(1,5,-1), E(0,4,-2) and F(2,3,4)
Adding eq. 
Subtracting eq. from 
we get
Similarly, adding eq. 
Subtracting eq. from 
Similarly 
Coordinates of vertices of ABC are A(1,3,-1), B(2,4,6) and C(1,7,-5)
3. Let
and
be two points in space find co ordinate of point
which divides 
and
in the ratio
by geometrically
Ans. Let co-ordinate of Point be 
which divider line segment joining the point 
in the ratio
Clearly 
Similarly 
and
4. Show that the plane 
divides the line joining the points 
and 
in the ratio
s
Ans. Suppose the plane divides the line joining the points and in the ratio 
Plane Passing through
Hence Proved.
5. Prove that the points 
are the vertices of a regular tetrahedron.
Ans. To prove O, A, B, C are vertices of regular tetrahedron.
We have to show that
|OA|=|OB|=|OC|=|AB|=|BC|=|CA|
|OA|=
unit
|OB|=
unit
|OC|= 
unit
|AB|=
unit
|BC|= 
unit
|CA|= 
unit
|AB| = |BC| = |CA| = |OA| = |OB| = |OC| = 2 unit
O, A, B, C are vertices of a regular tetrahedron.
6. If A and B are the points
and
respectively, then find the locus of P such that 3|PA| = 2|PB|
Ans. Given points 
and 
Supper co-ordinates of point 
|PA|= 
|PA|= 
|PB|= 
|PB|= 
3|PA| = 2|PB|
9 PA2=4 PB2
