If a line in the space makes angles α, β and γ with the coordinates axes, the cos 2α + cos 2β + cos 2γ + sin2 α + sin2 β + sin2 γ equals 1) -1 2) 0 3) 1 4) 2 Solution: (3) 1... View Article
The perimeter of the triangle with vertices at (1, 0, 0), (0, 1, 0) and (0, 0, 1) is 1) 3 2) 2 3) 2√2 4) 3√2 Solution: (4) 3√2... View Article
The xy-plane divides the line joining the points (-1, 3, 4) and (2, -5, 6) 1) internally in the ratio 2 : 3 2) externally in the ratio 2 : 3 3) internally in the ratio 3 : 2 4) externally in the ratio 3 : 2 Solution:... View Article
The distance cosines of a line are (1 / c, 1 / c, 1 / c), then 1) 0 < c < 1 2) c > 2 3) c = ± √2 4) c = ± √3 5) c = ± 3 Solution: (4) c = ± √3... View Article
Let A (1, -1, 2) and B (2, 3, -1) be two points. If a point P divides AB internally in the ratio 2 : 3, then the position vector of P is 1) (1 / √5) (i + j + k) 2) (1 / √3) (i + 6j + k) 3) (1 / √3) (i + j + k) 4) (1 / 5) (7i + 3j + 4k) 5) None of these Solution: (4) (1 / 5)... View Article
The vectors of magnitude a, 2a, 3a meet at a point and their directions are along the diagonals of three adjacent faces of a cube. Then, the magnitude of their resultant is 1) 5a 2) 6a 3) 10a 4) 9a Solution: (1) 5a... View Article
If (1, -2, -2) and (0, 2, 1) are direction ratios of two lines, then the direction cosines of a perpendicular to both the lines are 1) (1 / 3), (- 1 / 3), (2 / 3) 2) (2 / 3), (- 1 / 3), (2 / 3) 3) (- 2 / 3), (- 1 / 3), (2 / 3) 4) (2 / √14), (- 1 / √14), (3 / √14) Solution:... View Article
If a line makes angles α, β and γ with the coordinates axes, then cos 2α + cos 2β + cos 2γ is 1) -1 2) -2 3) 2 4) -3 Solution: (1) -1... View Article
A line makes the same angle θ with each of the X and Z – axes. If it makes the angle β with y-axis such that sin2 β = 3 sin2θ, then cos2 θ equals 1) 3/5 2) 1/5 3) 2/5 4) 2/3 Solution: (1) â…—... View Article
Find the length of the diagonal of the parallelepiped formed by planes drawn through the points (2, 3, 5) and (5, 9, 7) parallel to the coordinate planes. 1) √38 units 2) 7 units 3) √155 units 4) 13 units Solution: (2) 7 units... View Article
The direction cosines of the line joining the points (4, 3, -5) and (-2, 1, -8) are 1) (6 / 7), (2 / 7), (3 / 7) 2) (2 / 7), (3 / 7), (- 6 / 7) 3) (6 / 7), (3 / 7), (2 / 7) 4) None of these Solution: (1) (6 / 7), (2 / 7), (3... View Article
The projection of any line on coordinate to the respectively 3, 4 and 5, then its length is 1) 50 2) 12 3) 5√2 4) None of these Solution: (3) 5√2 Let d be the length of line, then projection on x-axis = dl = 3, projection... View Article
The angle between the lines, whose direction ratios are (1, 1, 2), (√3, -1, -√3 – 1, 4), is 1) cos-1 (1/65) 2) Ï€/6 3) Ï€/3 4) Ï€/2 Solution: (3) Ï€/3... View Article
If A (3, 4, 5), B(4, 6, 3), C (-1, 2, 4) and D(1, 0, 5) are such that the angle between the lines DC and AB is θ, then cos θ is equal to 1) 7/9 2) 2/9 3) 4/9 4) 5/9 Solution: (3) 4/9... View Article
If the direction cosines of two lines are given by l + m + n = 0 and l2 – 5m2 + n2 = 0, then the angle between them is 1) Ï€/2 2) Ï€/6 3) Ï€/4 4) Ï€/3 Solution: (4) Ï€/3... View Article
If the line joining A(1, 3, 4) and B is divided by the point (-2, 3, 5) in the ration 1 : 3, then B is equal to 1) (-11, 3, 8) 2) (-11, 3, -8) 3) (-8, 12, 20) 4) (13, 6, -13) Solution: (1) (-11, 3, 8)... View Article
The distance between the x-axis and the point (3, 12, 5) is 1) 3 2) 13 3) 14 4) 12 5) 5 Solution: (2) 13 (x, y, z) = (3, 12, 5) x = 0 d = √x2 + y2 + z2 = √0 + 122 + 52 = √144 + 25... View Article
The angle between a normal to the plane 2x – y + 2z – 1 = 0 and the z-axis is 1) cos-1 (1/3) 2) sin-1 (2/3) 3) cos-1 (2/3) 4) sin-1 (1/3) 5) sin-1 (3/5) Solution: (3) cos-1 (2/3) The direction ratios of... View Article
If the direction cosines of a vector of magnitude 3 are (2 / 3), (- a / 3), (2 / 3) and a > 0, then the vector is 1) 2i + j + 2k 2) 2i - j + 2k 3) i - 2j + 2k 4) i + 2j + 2k Solution: (2) 2i - j + 2k (4 / 9) + (a2 / 9) + (4 / 9) = 1 (a2 / 9) =... View Article
The angle between two diagonals of a cube is 1) cos-1 (1 / 3) 2) 30o 3) cos-1 (1 / √3) 4) 45o Solution: (1) cos-1 (1 / 3)... View Article
