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
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1 Marks Questions
1. Name the octants in which the following lie. (5,2,3)
2. Name the octants in which the following lie. (-5,4,3)
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
6. Write the name of plane in which axis and - axis taken together.
7. The point lie in which octants
8. The point lie in which plane
9. A point is in the XZ plane. What is the value of y co-ordinates?
10. What is the coordinates of XY plane
11. The point lie in which octants.
12. The distance from origin to point is:
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
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
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
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
7. Prove by distance formula that the points and are collinear.
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
(ii)For external division.
9. Find the co ordinate of a point equidistant from the four points and
Ans.Let be the required point
According to condition
and and are mid points of side respectively,
Adding eq (1),(4) and (7) we get
Adding eq. (2),(5) and (8)
Hence co-ordinate of
10. Find the ratio in which the join the andis 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)
Subtracting eq. (1), (4) and (7) from (10)
Now subtracting eq. (2), (5) and (8) from (11)
co-ordinate of point and are
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
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
also mid point of QS is
Then PR and QS have same mid points.
PR and QS bisect each other. It is a Parallelogram.
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
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)
Subtracting eq. from we get
Similarly, adding eq.
Subtracting eq. from
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
4. Show that the plane divides the line joining the points and in the ratios
Ans. Suppose the plane divides the line joining the points and in the ratio
Plane Passing through
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
|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
3|PA| = 2|PB|
9 PA2=4 PB2