CBSE Board Important Questions Paper
Class – XII
Subject – Mathematics (Three Dimensional Geometry)
1. Find the direction cosines of X, Y and Z-axis.
2. Find the value of p so that the lines, (1 - x)/3 = (7y - 14)/2p = (z - 3)/2 and (7 – 7x)/3p = (y-5) = (6 - z)/5 all at right angles.
3. Find the equations of the line passing through (a, b, c) and perpendicular to lines x/l1 = y/m1 = z/n1 and x/l2 = y/m2 = z/n2.
4. The equation of a line is given by (4 – x)/2 = (y + 3)/3 = (z + 2)/6. Write the direction cosines of a line parallel to the above line. Ans: – 2/7, 3/7, 6/7
5. The equation of a line is (2x – 5)/4 = (y + 4)/3 = (6 – z)/6. Find the direction cosines of a line parallel to this line. Ans : 2/7, 3/7, – 6/7
6. Find the coordinates of the foot of the perpendicular drawn from the origin to the plane 2x – 3y + 4z -6 = 0.
7. Find the angle between the line (x + 1)/2 = y/3 = (z-3)/6 and the plane 10x + 2y – 11z = 3.
8. Find the image of the point (1, 6, 3) in the line x = (y - 1)/2 = (z - 2)/3.
9. Find the foot of the perpendicular drawn from the point P(1, 6, 3) on the line x/1 = (y – 1)/2 = (z – 2)/3. Also find the distance from P. Ans : √13 units.
10. Find the length and the foot of the perpendicular drawn from the point (2, – 1 , 5) to the line (x – 11)/10 = (y + 2)/– 4 = (z + 8)/ –11. Ans. = Point (1, 2, 3); distance = √14 units.
11. Prove that if a plane has the intercepts a, b, c and is at a distance of p units from the origin, then 1/a2 + 1/b2 + 1/c2 = 1/p2.
12. Show that the angles between the diagonals of a cube is cos-1(1/3).
13. Find the length of the perpendicular drawn from the point (2, 3, 7) to the plane 3x – y – z = 7 . Also find the coordinates of the foot of the perpendicular.
14. Find the point on the line : (x + 2)/3 = (y + 1)/2 = (z – 3)/2 at a distance 3√2 from the point (1, 2, 3).
15. Find the equation of the perpendicular drawn from the point (2, 4, – 1) to the line (x + 5)/1 = (y + 3)/4 = (z – 6)/–9 . Ans : (x – 2)/6 = (y – 4)/3 = (z + 1)/2
16. Find the equation of the line passing through the point (-1, 3, -2) and perpendicular to the lines x = y/2 = z/3 and (x + 2)/(-3) = (y - 1)/2 = (z + 1)/5.
17. Find the foot of the perpendicular drawn from the point (0, 2, 3) on the line (x + 3)/5 = (y - 1)/2 = (z + 4)/3 Also, find the length of the perpendicular.
18. Find the equation of the plane passing through the line intersection of the planes x – 2y + z = 1 and 2x + y + z = 8 and parallel to the line with direction ratios (1, 2, 1) Also, find the perpendicular distance of the point P(3, 1, 2) from this plane.
19. Find the shortest distance between the following lines :
(x – 3)/1 = (y – 5)/–2 = (z – 7)/1 and
(x + 1)/7 = (y + 1)/–6 = (z + 1)/1. Ans : 2√29
20. Find the equation of the plane passing through the points (1, 2, 3) and (0, – 1, 0) and parallel to the line (x – 1)/2 = (y + 2)/3 = z/–3. Ans : 6x – 3y + z = 3
21. Find the shortest distance between the following pairs of lines whose cartesian equations are : (x -1)/2 = (y + 1)/3 = z and (x + 1)/3 = (y - 2) , z = 2.
22. A plane meets the coordinate axis in A, B, C such that the centroid of triangle ABC is the point (p,q,r). Show that the equation of the plane is x/p = y/q = z/r = 3.
23. Find the distance of the point (1, -2, 3) from the plane x – y + z = 5 measured parallel to the line x/2 = y/3 = z/(-6)
24. Find the equation of the plane passing through the points (0, – 1, – 1), (4, 5, 1) and (3, 9, 4). Ans : 5x – 7y + 11z + 4 = 0
25. Find the equation of the plane passing through the point (–1, –1, 2) and perpendicular to each of the following planes :
2x + 3y – 3z = 2 and 5x – 4y + z = 6. Ans : 11x + 17y + 23z – 18 = 0
26. Find the equation of the plane passing through the point (1, 1, -1) and perpendicular to the planes x + 2y + 3z – 7=0 and 2x – 3y + 4z = 0.
27. Find the equation of the plane passing through the points (3, 4, 1) and (0, 1, 0) and parallel to the line (x + 3)/2 = (y – 3)/7 = (z – 2)/5. Ans : 8x – 13y + 15z + 13 = 0
28. Find the image of the point (1, 2, 3) in the plane x + 2y + 4z = 38. Ans : (3, 6, 11)
29. Find the distance of the point ( - 1, - 5, - 10 ) from the point of intersection of the line r = 2 i – j + 2 k + λ ( 3 i + 4 j + 2 k ) and the plane r . ( i – j + k ) = 5.
30. Find the co-ordinates of the image of the point (1, 3, 4) in the plane 2x – y + z + 3 = 0. Ans : (–3, 5, 2)
31. From the point P(1, 2, 4), a perpendicular is drawn on the plane 2x + y – 2z + 3 = 0. Find the equation, the length and the co-ordinates of the foot of the perpendicular.
32. Find the distance between the point P( 6,5,9) and the plane determined by the points, A ( 3,-1,2 ) , B ( 5,2,4 ) and C ( -1,-1,6 ).
33. Find the distance between the point P(6, 5, 9) and the plane determined by the points A(3, – 1 , 2), B(5, 2, 4) and C(– 1, – 1 , 6). Ans: 6/√(34).
34. Find the co-ordinates of the point where the line (x + 1)/2 = (y + 2)/3 = (z + 3)/4 meets the plane x + y + 4z = 6. Ans: P(1, 1, 1)
35. Find the distance of the point (– 2, 3 – 4) from the line
(x + 2)/3 = (2y + 3)/4 = (3z + 4)/5
measured parallel to the plane 4x + 12y – 3z + 1 = 0. Ans: 17/2