Enzyme, Pharmaceutical Aids, Miscellaneous Last Part of Chapter no 5th.pdf
Β
Three dimensional geometry
1. THREE DIMENSIONAL GEOMETRY
1 Mark Question :
1. If A(1, 5, 4) and B(4, 1, -2) find the direction ratio of π΄π΅βββββ .
2. Write Direction ratios of line
π₯β2
2
=
2π¦β5
β3
= π§ β 1 .
3. Write whether the line
π₯β3
3
=
π¦β2
1
=
π§β1
0
is perpendicular to x β axis, y-axis or z- axis.
4. Write the equation of plane passing through the points (a, 0,0) (0, b, 0) and (0, 0, c).
5. For what value of πΎthe line
π₯β1
2
=
π¦β1
3
=
π§β1
πΎ
is perpendicular to normal plane π =(2 πΜ+ 3πΜ β 4πΜ) = 4 .
6. For what value of πΎ, the planes x + 2y + πΎπ§ = 18 and 2x β 4y + 3z = 7 are perpendicular to each other.
7. Write the direction cosines of line whose Cartesian equation are : 2x = 3y = -z .
8. If a line makes πΌ, π½, πΎ with x-axis , y-axis and z-axis respectively, then find the value of sin2
πΌ + sin2
π½ + sin2
πΎ .
9. Find the value of K for which the lines :
π₯β1
β3
=
π¦β2
2πΎ
=
π§β3
2
and
π₯β1
3πΎ
=
π¦β1
1
=
6βπ§
5
are perpendicular to each other.
10. If the equation of the line π΄π΅βββββ is
π₯β3
1
=
π¦+2
β2
=
π§β5
4
, find the direction ratios of a line parallel to AB .
4 Marks Questions :
11. Find the value of π so that the lines
1βπ₯
3
=
7π¦β14
2π
=
5π§β10
11
and
7β7π₯
3π
=
π¦β5
1
=
6βπ§
5
are perpendicular to each other.
12. Find the distance of the point (-1, -5, -10) from the point of intersection of line π = 2πΜ β πΜ + 2πΜ + π(3πΜ + 4πΜ + 2πΜ)
and the plane π .(πΜ β πΜ + πΜ) = 5 .
13. Find the foot of perpendicular from P(1, 2, 3) on the line
π₯β6
3
=
π¦β7
2
=
π§β7
β2
. Also obtain the equation and length
of perpendicular.
14. Find the image of point (1, 6, 3) in the line
π₯
1
=
π¦β1
2
=
π§β2
3
15. Determine whether the following pairs of lines intersect or not :
π₯β1
2
=
π¦+1
3
= z,
π₯+1
5
=
π¦β2
1
=
π§β2
1
.
16. Find the shortest distance between the lines :
π₯+1
7
=
π¦+1
β6
=
π§+1
1
and
π₯β3
1
=
π¦β5
β2
=
π§β7
1
.
17. Find the shortest distance between the lines : π = πΜ + 2πΜ + πΜ + π( πΜ β πΜ + πΜ); π = 2πΜ β πΜ β πΜ + π(2πΜ + πΜ + 2πΜ) .
18. Show that the lines :
π₯β1
2
=
π¦β2
3
=
π§β3
4
and
π₯β4
5
=
π¦β1
2
= zintersect. Also find their point of intersection
19. Find whether the lines π = ( πΜ β πΜ + πΜ) + π(2πΜ + πΜ) and π = (2πΜ β πΜ) + π( πΜ + πΜ β πΜ) intersect or not. If
intersecting, find their point of intersection.
20. Find the coordinates of foot of perpendicular draw from the origin to the plane 2x β 3y + 4z = 6.
21. Find the coordinates of the point where the line through the points A (3, 4, 1) and B (5, 1, 6) crosses the XY β
plane.
22. Prove that if a plane has intercepts a, b, c and is at a distance of p units from origin, then
1
π2
+
1
π2
+
1
π2
=
1
π2
23. Find the length and foot of perpendicular from the point (1, 1, 2) to the plane 2x β 2y + 4z + 5 = 0.
2. 24. Show that the lines
π₯β1
2
=
π¦β3
4
=
π§
β1
and
π₯β4
3
=
π¦β1
β2
=
π§β1
1
are coplanar. Also find the equation of the plane containing these
lines.
25. Find the equation of plane passing through the points A(0, 0, 0) & B (3, -1, 2) and parallel to line
π₯β4
1
=
π¦+3
β4
=
π§+1
7
.
26. Find the equation of the plane passing through the point (-1, 2, 1) and perpendicular to the line joining
the points (-3, 1, 2) and (2, 3, 4). Also , find the perpendicular distance of the origin from this plane.
27. Find the perpendicular distance of the point (1, 0, 0) from the line
π₯β1
2
=
π¦+1
β3
=
π§+10
8
.
28. Find the angle between the line
π₯+1
2
=
3π¦+5
9
=
3βπ§
β6
and the plane 10x + 2y β 11z = 3.
29. Find the equation of the perpendicular drawn from the point (1, -2, 3) to the plane 2x β 3y + 4z + 9 = 0 Also,
find the co-ordinates of the foot of the perpendicular.
30. Find the Vector and Cartesian equations of the line passing through the point (1, 2, -4) and perpendicular to the
two lines
π₯β8
3
=
π¦+19
β16
=
π§β10
7
and
π₯β15
3
=
π¦β29
8
=
π§β5
β5
.
6 Marks Questions :
1. Find the equation of the plane containing the line of intersection of the planes 2x β 3y + 5z = 7 and
3x +4y- z = 11 and passing through the point (1, 0, -2) .
2. Find the foot of perpendicular distance of the point (1, 3, 4) from the plane 2x β y + z + 3 = 0. Find also, the image
of the point in plane (-3, 5, 2) .
3. If the lines
π₯β1
2
=
π¦+1
3
=
π§β1
4
and
π₯β3
1
=
π¦βπ
2
=
π§
1
intersect, then find the value of k and hence find the equation of
plane containing these lines.
4. Find the equation of plane passing through the point (-1, 3, 2) and perpendicular to each of the planes
x + 2y + 3z = 5 and 3x + 3y + z = 0.
5. Find the equation of plane that contains the point (1, -1, 2) and is perpendicular to each of the planes 2x+3y-2z=5 &
x+2y-3z = 8 .
6. Find the vector equation of the plane passing through the intersection of planes π . ( πΜ + πΜ + πΜ) = 6 and
π .(2 πΜ+ 3πΜ + 4πΜ) + 5 = 0 & point (1, 1, 1) .
7. Find the equation of plane passing through the points (3, 4, 1) & (0, 1, 0) & parallel to line
π₯+3
2
=
π¦β3
4
=
π§β2
5
.
8. Prove that the image of the point (3, -2, 1) in the plane 3x-y+4z = 2 lie on plane x+y+z+4 = 0.
9. Find the distance of points (-2, 3, -4) from the line
π₯+2
3
=
2π¦+3
4
=
3π§+4
5
measured parallel to plane
4x + 12y β 3z = 0.
10. Find the equation of plane through the line of intersection of planes x+y+z = 1 & 2x+3y+4z = 5. Which is
perpendicular to the plane x β y + z = 0.
11. Show that the lines
π₯+3
β3
=
π¦β1
1
=
π§β5
5
and
π₯+1
β1
=
π¦β2
2
=
π§β5
5
are coplanar. Also find the equation of plane
containing the lines.
3. 12. Find the equation of plane determined by the points A (3, -1, 2), B (5, 2, 4) & C (-1, -1, 6). Also find the distance
of point P(6,5, 9) from plane .[
6
β34
] .
13. Find the image of the point (1, 2, 3) in the plane x +2y + 4z = 38.
14. Show that the lines π = ( πΜ + πΜ β πΜ ) + πΎ(3πΜ β πΜ) and π = (4πΜ β π)Μ + π(2πΜ + 3πΜ) are coplanar. Also find the plane
containing these two lines.
15. Find the equation of plane which contains two parallel lines
π₯β3
3
=
π¦+4
2
=
π§β1
1
and
π₯+1
3
=
π¦β2
2
=
π§
1
.
16. Find the equation of plane through the points (1, 2, 3) & (0, -1, 0) and parallel to the line
π .(2 πΜ+ 3πΜ + 4πΜ) + 5 = 0 .
17. Find the Cartesian as well as vector equation of the planes , passing through the intersection of planes
π .(2 πΜ+ 6πΜ) + 12 = 0 &π .( 3πΜ β πΜ + 4πΜ) = 0 , which are at a unit distance from origin.
18. Find the distance of the point (2, 2, -1) from the plane x + 2y β z = 1 measured parallel to the line
π₯+1
1
=
π¦+1
2
=
π§
3
.
19. Show that the points with position vectors 6πΜ β 7πΜ , 16πΜ β 14πΜ - 4πΜ , 3πΜ -6πΜ and 2πΜ β 5πΜ + 10πΜ are
coplanar .
20. Find the co-ordinates of the foot of perpendicular and the length of the perpendicular drawn from the point
P (5, 4, 2) to the line π = - πΜ + 3πΜ + πΜ + π (2 πΜ + 3πΜ β πΜ) . Also find the image of P in this line.
Find the image of the point having position vector πΜ + 3πΜ + 4πΜ in the planer π (2 πΜ β πΜ + πΜ) +3 = 0 .
21. Show that the lines
π₯+3
β3
=
π¦β1
1
=
π§β5
5
,
π₯+1
β1
=
π¦β2
2
=
π§β5
5
are coplanar. Also find the equation of the plane
containing the lines.
22. Find the co-ordinates of the point where the line through (3, -4, -5) and (2, -3, 1) crosses the plane determined by
points A(1, 2, 3), B (2, 2, 1) and C (-1, 3, 6) .
-----------------------------------------------------------------------------------------------------------------------------------------
βAll you need is the plan, the road map, and the
courage to press on to your destination.β
βEarl Nightingale