Title 17. Coordinate Geometry easy.pdf ✅

1. If the points (0, 1, 2), (2, –1, 3) and (1, –3, 1) are the vertices of a triangle, then the triangle is (a) Right angled (b) Isosceles right angled (c) Equilateral (d) None of these 2. If the points (–1, 3, 2), (–4, 2, –2) and (5, 5, ) are collinear, then  = (a) – 10 (b) 5 (c) – 5 (d) 10 3. The direction cosines of the normal to the plane 2x + 3y − 6z = 5 are (a) 2, 3, − 6 (b) 7 6 , 7 3 , 7 2 − (c) 5 6 , 5 3 , 5 2 − (d) None of these 4. The point dividing the line joining the points (1, 2, 3) and (3, − 5, 6 ) in the ratio 3 :− 5 is (a)       − 2 3 , 2 25 2, (b)       − − 2 3 , 2 25 2, (c)       2 3 , 2 25 2, (d) None of these 5. From which of the following the distance of the point (1, 2, 3) is 10 (a) Origin (b) x-axis (c) y-axis (d) z-axis 6. If , , be the angles which a line makes with the positive direction of co-ordinate axes, then  +  +  = 2 2 2 sin sin sin (a) 2 (b) 1 (c) 3 (d) 0 7. If ,, be the direction angles of a vector and 15 14 cos = , 3 1 cos  = then cos = (a) 15 2  (b) 5 1 (c) 15 1  (d) None of these 8. All the points on the x- axis have (a) x = 0 (b) y = 0 (c) x = 0, y = 0 (d) y = 0, z = 0 9. Distance between the points (1, 3, 2) and (2, 1, 3) is (a) 12 (b) 12 (c) 6 (d) 6 10. The direction cosines of the line x = y = z are (a) 3 1 , 3 1 , 3 1 (b) 3 1 , 3 1 , 3 1 (c) 1, 1, 1 (d) None of these 11. Distance of the point (1, 2, 3) from the co-ordinate axes are (a) 13, 10, 5 (b) 13 , 10 , 5 (c) 5, 13 , 10 (d) 5 1 , 10 1 , 13 1 12. If the centroid of triangle whose vertices are (a,1, 3), (– 2, b, –5) and (4, 7, c) be the origin, then the values of a, b, c are (a) – 2, –8, –2 (b) 2, 8, –2 (c) –2, –8, 2 (d) 7, –1, 0 13. Which of the following set of points are non- collinear (a) (1, –1, 1), (–1, 1, 1), (0, 0, 1) (b) (1, 2, 3), (3, 2, 1), (2, 2, 2) (c) (–2,4, –3), (4, –3, –2), (–3, –2, 4) (d) (2, 0, –1), (3, 2, –2), (5, 6, –4) 14. If a straight line in space is equally inclined to the co-ordinate axes, the cosine of its angle of inclination to any one of the axes is (a) 3 1 (b) 2 1 (c) 3 1 (d) 2 1 15. If a line makes angles of o 30 and o 45 with x-axis and y-axis, then the angle made by it with z − axis is (a) o 45 (b) o 60 (c) o 120 (d) None of these 16. Direction ratios of the normal to the plane passing through the points (0, 1, 1), (1, 1, 2)and (–1, 2, – 2) are (a) (1, 1, 1) (b) (2, 1, –1) (c) (1, 2, –1) (d) (1, – 2, – 1) 17. If the length of a vector be 21 and direction ratios be 2, – 3, 6 then its direction cosines are (a) 7 2 , 7 1 , 21 2 − (b) 7 6 , 7 3 , 7 2 − (c) 7 6 , 7 3 , 7 2 (d) None of these 18. If the co-ordinates of the points P, Q, R, S be (1, 2, 3), (4, 5, 7), (– 4, 3, – 6) and (2, 0, 2) respectively, then (a) PQ|| RS (b) PQ ⊥ RS (c) PQ = RS (d) None of these 19. If the co-ordinates of the points A, B, C, D be (2, 3, –1), (3, 5, – 3), (1, 2, 3) and (3, 5, 7) respectively, then the projection of AB on CD is (a) 0 (b) 1 (c) 2 (d) 3 20. If the co-ordinates of the points P and Q be (1, –2, 1) and (2, 3, 4) and O be the origin, then
(a) OP = OQ (b) OP ⊥ OQ (c) OP|| OQ (d) None of these 21. If the projections of a line on the co-ordinate axes be 2, –1, 2, then the length of the lines is (a) 3 (b) 4 (c) 2 (d) 2 1 22. xy-plane divides the line joining the points (2, 4, 5) and (– 4, 3, – 2) in the ratio (a) 3 : 5 (b) 5 : 2 (c) 1 : 3 (d) 3 : 4 23. If A(1, 2, −1) and B(−1, 0, 1) are given, then the co- ordinates of P which divides AB externally in the ratio 1 : 2 , are (a) (1, 4, 1) 3 1 − (b) (3, 4, –3) (c) (3, 4, 3) 3 1 − (d) None of these 24. The point of intersection of the line joining the points (3, 4, 1) and (5, 1, 6) and the xy-plane is (a) (13, 23, 0) (b)       ,0 5 23 , 5 13 (c) (–13, 23, 0) (d)       − ,0 5 23 , 5 13 25. If the co-ordinates of A and B be (1, 2, 3) and (7, 8, 7), then the projections of the line segment AB on the co-ordinate axes are (a) 6, 6, 4 (b) 4, 6, 4 (c) 3, 3, 2 (d) 2, 3, 2 26. The co-ordinates of the point P are (x, y, z) and the direction cosines of the line OP when O is the origin, are l,m, n . If OP = r , then (a) l = x,m = y,n = z (b) l = xr, m = yr, n = zr (c) x = lr, y = mr,z = nr (d) None of these 27. A line makes angles , ,  with the co-ordinate axes. If o  +  = 90 , then  = (a) 0 (b) 90 (c) 180  (d) None of these 28. Points (–2, 4, 7), (3, –6, –8) and (1, –2, –2) are (a) Collinear (b) Vertices of an equilateral triangle (c) Vertices of an isosceles triangle (d) None of these 29. If the points A(9, 8,−10), B(3, 2, − 4) and C(5, 4,− 6) be collinear, then the point C divides the line AB in the ratio (a) 2 : 1 (b) 3 : 1 (c) 1 : 2 (d) –1 : 2 30. The projections of a line on the co-ordinate axes are 4, 6, 12. The direction cosines of the line are (a) 7 6 , 7 3 , 7 2 (b) 2, 3, 6 (c) 11 6 , 11 3 , 11 2 (d) None of these 31. If the sum of the squares of the distance of a point from the three co- ordinate axes be 36,then its distance from the origin is (a) 6 (b) 3 2 (c) 2 3 (d) None of these 32. The line joining the points (−2, 1,− 8) and (a,b, c) is parallel to the line whose direction ratios are 6, 2, 3. The values of a, b, c are (a) 4, 3, –5 (b) 1, 2, –13/2 (c) 10, 5, –2 (d) None of these 33. The direction ratios of the line joining the points (4, 3, –5) and (–2, 1, –8) are (a) 7 3 , 7 2 , 7 6 (b) 6, 2, 3 (c) 2, 4, −13 (d) None of these 34. The co-ordinates of the point in which the line joining the points (3, 5, − 7) and (−2, 1, 8) is intersected by the plane yz are given by (a)       , 2 5 13 0, (b)       − , − 2 5 13 0, (c)       − 5 2 , 5 13 0, (d)       5 2 , 5 13 0, 35. The co-ordinates of a point which is equidistant from the points (0,0, 0),(a,0,0),(0, b, 0) and (0,0, c) are given by (a)       2 , 2 , 2 a b c (b)       − − 2 , 2 , 2 a b c (c)       − − 2 , 2 , 2 a b c (d)       − − 2 , 2 , 2 a b c 36. The projection of the line segment joining the points (–1, 0, 3) and (2, 5, 1) on the line whose direction ratios are 6, 2, 3 is (a) 7 10 (b) 7 22 (c) 7 18 (d) None of these 37. Points (1, 1, 1), (–2, 4, 1), (–1, 5, 5) and (2, 2, 5) are the vertices of a (a) Rectangle (b) Square (c) Parallelogram (d) Trapezium 38. If 1 1 1 l ,m ,n and 2 2 2 l ,m , n are the direction cosines of two perpendicular lines, then the direction cosine of the line which is perpendicular to both the lines, will be
(a) ( ), ( ),( ) 1 2 2 1 1 2 2 1 1 2 2m1 m n −m n n l −n l l m −l (b) ( ),( ),( ) 1 2 1 2 1 2 1 2 1 2 1 2 l l −m m m m −n n n n −l l (c) 3 1 , 1 , 1 2 2 2 2 2 2 2 1 2 1 2 l1 + m + n l + m + n (d) 3 1 , 3 1 , 3 1 39. If a line makes the angle , ,  with three dimensional co- ordinate axes respectively, then cos 2 + cos 2 + cos 2 = (a) – 2 (b) – 1 (c) 1 (d) 2 40. Perpendicular distance of the point (3, 4, 5) from the y-axis, is (a) 34 (b) 41 (c) 4 (d) 5 41. The direction cosines of a line equally inclined to three mutually perpendicular lines having direction cosines as 1 1 1 2 2 2 l ,m , n ;l ,m , n and 3 3 3 l ,m , n are (a) 1 2 3 1 2 3 1 2 3 l + l + l ,m + m + m , n + n + n (b) 3 , 3 , 3 1 2 3 1 2 3 1 2 3 l + l + l m + m + m n + n + n (c) 3 , 3 , 3 1 2 3 m1 m2 m3 n1 n2 n3 l + l + l + + + + (d) None of these 42. A point (x, y, z) moves parallel to x-axis. Which of the three variable x, y,z remain fixed (a) x (b) y and z (c) x and y (d) z and x 43. If the direction cosines of a line are       c c c 1 , 1 , 1 , then (a) c  0 (b) c =  3 (c) 0  c 1 (d) c  2 44. The plane XOZ divides the join of (1,−1, 5) and (2, 3, 4) in the ratio  : 1 , then  is (a) – 3 (b) 3 (c) 3 1 − (d) 3 1 45. The co-ordinates of a point P are (3, 12, 4) with respect to origin O, then the direction cosines of OP are (a) 3, 12, 4 (b) 2 1 , 3 1 , 4 1 (c) 13 2 , 13 1 , 13 3 (d) 13 4 , 13 12 , 13 3 46. The locus of a first degree equation in x, y,z is a (a) Straight line (b) Sphere (c) Plane (d) None of these 47. The direction cosines of the normal to the plane x + 2y − 3z + 4 = 0 are (a) 14 3 , 14 2 , 14 1 − − (b) 14 3 , 14 2 , 14 1 (c) 14 3 , 14 2 , 14 1 − (d) 14 3 , 14 2 , 14 1 − 48. The direction cosines of the line 6 1 3 2 3 3 1 − = + = − x + y z are (a)       , 0 3 2 , 3 1 (b)       − ,1 3 2 1, (c)       − − 2 1 , 1, 2 1 (d)         − − 6 1 , 6 2 , 6 1 49. The co-ordinates of the point which divides the join of the points (2, –1, 3) and (4, 3, 1) in the ratio 3 : 4 internally are given by (a) 7 10 , 7 20 , 7 2 (b) 7 3 , 7 20 , 7 15 (c) 7 2 , 7 15 , 7 10 (d) 7 15 , 7 5 , 7 20 50. If the direction ratios of a line are 1,−3, 2 , then the direction cosines of the line are (a) 14 2 , 14 3 , 14 1 − (b) 14 3 , 14 2 , 14 1 (c) 14 2 , 14 3 , 14 −1 − (d) 14 3 , 14 2 , 14 −1 − − 51. A line makes angles of 45 and 60 with the positive axes of X and Y respectively. The angle made by the same line with the positive axis of Z, is (a) 30 or 60 (b) 60 or 90 (c) 90 or 120  (d) 60 or 120  52. The direction cosines of the normal to the plane 3x + 4y + 12 z = 52 will be (a) 3, 4, 12 (b) – 3, – 4, – 12 (c) 13 12 , 13 4 , 13 3 (d) 13 12 , 13 4 , 13 3 53. The shortest distance of the point (a, b, c) from the x-axis is (a) ( ) 2 2 a + b (b) ( ) 2 2 b + c (c) ( ) 2 2 c + a (d) ( ) 2 2 2 a + b + c 54. The direction ratios of the line x − y + z − 5 = 0 = x − 3y − 6 are (a) 3, 1, – 2 (b) 2, – 4, 1 (c) 14 2 , 14 1 , 14 3 − (d) 41 1 , 41 4 , 41 2 − 55. If O is the origin and OP = 3 with direction ratios −1, 2,−2 , then co- ordinates of P are
(a) (1, 2, 2) (b) (−1, 2, − 2) (c) (–3, 6, –9) (d) (−1 / 3, 2 / 3, − 2 / 3) 56. If x co-ordinates of a point P of line joining the points Q(2, 2,1) and R(5, 2,−2) is 4, then the z-coordinates of P is (a) – 2 (b) –1 (c) 1 (d) 2 57. The points A(5, − 1, 1) ; B(7,−4,7); C(1, − 6,10) and D(−1,−3, 4) are vertices of a (a) Square (b) Rhombus (c) Rectangle (d) None of these 58. The direction cosines of the line joining the points (4, 3, – 5) and (– 2, 1, – 8) are (a)       7 3 , 7 2 , 7 6 (b)       7 6 , 7 3 , 7 2 (c)       7 2 , 7 3 , 7 6 (d) None of these 59. If a line lies in the octant OXYZ and it makes equal angles with the axes, then (a) 3 1 l = m = n = (b) 3 1 l = m = n =  (c) 3 1 l = m = n = − (d) 2 1 l = m = n =  60. The triangle formed by the points (0, 7, 10), (–1, 6, 6), (– 4, 9, 6) is (a) Equilateral (b) Isosceles (c) Right angled (d) Right angled Isosceles 61. If A(1, 2, 3), B(−1,−1,−1) be the points, then the distance AB is (a) 5 (b) 21 (c) 29 (d) None of these 62. If , ,  be the angles which a line makes with the co- ordinate axes, then (a) sin cos sin 1 2 2 2  +  +  = (b) cos cos cos 1 2 2 2  +  +  = (c) sin sin sin 1 2 2 2  +  +  = (d) cos cos sin 1 2 2 2  +  +  = 63. If P(3, 4, 5), Q(4, 6, 3), R(−1, 2, 4), S(1, 0, 5) then the projection of RS on PQ is (a) – 2/3 (b) – 4/3 (c) 1/2 (d) 2 64. If a line makes , ,  with the positive direction of x, y and z-axis respectively. Then,    2 2 2 cos + cos + cos is (a) 1/2 (b) –1/2 (c) –1 (d) 1 65. The projection of a line on a co-ordinate axes are 2, 3, 6. Then the length of the line is (a) 7 (b) 5 (c) 1 (d) 11 66. A line which makes angle o 60 with y-axis and z-axis, then the angle which it makes with x-axis is (a) 45 (b) 60 (c) 75 (d) 30 67. The points (5, – 4, 2), (4, –3, 1), (7, – 6, 4) and (8, –7, 5) are the vertices of (a) A rectangle (b) A square (c) A parallelogram (d) None of these 68. If       , n 3 1 , 2 1 are the direction cosines of a line, then the value of n is (a) 6 23 (b) 6 23 (c) 3 2 (d) 2 3 69. The distance of the point (4, 3, 5) from the y-axis is (a) 34 (b) 5 (c) 41 (d) 15 70. If projection of any line on co-ordinate axis 3, 4, and 5, then its length is (a) 12 (b) 50 (c) 5 2 (d) 3 2 71. If a line makes angles , ,  ,  with four diagonals of a cube, then the value of  +  + 2 2 sin sin   2 2 sin + sin is (a) 3 4 (b) 1 (c) 3 8 (d) 3 7 72. If  is the angle between the lines AB and CD , then projection of line segment AB on line CD , is (a) AB sin (b) AB cos (c) AB tan  (d) CD cos 73. The co-ordinates of points A, B,C, D are (a, 2, 1), (1, –1, 1), (2, – 3, 4) and (a+1, a+2, a+3) respectively. If AB = 5 and CD = 6 , then a = (a) 2 (b) 3 (c) – 2 (d) – 3 74. If the co-ordinates of the points A, B,C be (−1, 3, 2), (2, 3, 5) and (3, 5,–2) respectively, then A = (a) 0 (b) 45 (c) 60 (d) 90