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A line makes angles α, β, γ with the co-ordinate axis. If α + β = 90°, then γ =

1) 0 2) 90° 3) 180° 4) None of these Solution: (2) 90°... View Article

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

1) l = x, m = y, n = z 2) l = xr, m = yr, n = zr 3) x = lr, y = mr, z = nr 4) None of these Solution: (3) x = lr, y = mr, z = nr l = x... View Article

If the coordinates of A and B be (1, 2, 3) and (7, 8, 7), then the projections of the line segment AB on the co-ordinate axis are

1) 6, 6, 4 2) 4, 6, 4 3) 3, 3, 2 4) 2, 3, 2 Solution: (1) 6, 6, 4... View Article

The point of intersection of the line joining the points (3, 4, 1) and (5, 1, 6) and the xy-plane is

1) (13, 23, 0) 2) (13 / 5, 23 / 5, 0) 3) (- 13, 23, 0) 4) (- 13 / 5, 23 / 5, 0) Solution: (2) (13 / 5, 23 / 5, 0)... View Article

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

1) (1 / 3) (1, 4, - 1) 2) (3, 4, - 3) 3) (1 / 3) (3, 4, - 3) 4) None of these Solution: (2) (3, 4, - 3)... View Article

If θ is the angle between the lines AB and CD, then the projection of line segment AB on line CD, is

1) AB sin θ 2) AB cos θ 3) AB tan θ 4) CD cos θ Solution: (2) AB cos θ It is a fundamental concept.... View Article

The number of straight lines that are equally inclined to the three-dimensional co-ordinate axis, is

1) 2 2) 4 3) 6 4) 8 Solution: (2) 4... View Article

If the coordinates of the points P (2, 3, 4) and Q be (1, – 2, 1) and O be the origin, then

1) OP = OQ 2) OP ⟂ OQ 3) OP ∥ OQ 4) None of these Solution: (2) OP ⟂ OQ O (0, 0, 0), P (2, 3, 4) and Q (1, - 2, 1) a1a2 + b1b2 +... View Article

The shortest distances of the point (1, 2, 3) from the axis x, y and z respectively are

1) 1, 2, 3 2) √5, √13, √10 3) √10, √13, √5 4) √13, √10, √5 Solution: (4) √13, √10, √5... View Article

If the length of a vector by 21 and direction ratios be 2, – 3, 6 then its direction cosines are

1) 2 / 21, - 1 / 7, 2 / 7 2) 2 / 7, - 3 / 7, 6 / 7 3) 2 / 7, 3 / 7, 6 / 7 4) None of these Solution: (2) 2 / 7, - 3 / 7, 6 / 7... View Article

Direction ratios of the normal to the plane passing through the points (0, 1, 1), (1, 1, 2) and (-1, 2, – 2) are

1) (1, 1, 1) 2) (2, 1,-1) 3) (1, 2, -1) 4) (1,-2,-1) Solution: (4) (1,-2,-1)... View Article

If a line makes angles of 30° and 45° with x-axis and y-axis, then the angle made by it with z-axis is

1) 45° 2) 60° 3) 120° 4) None of these Solution: (4) None of these cos = √1 - (3 / 4) - (1 / 2) = √- 1 / 4, which is not possible.... View Article

Let = { ∈ : 1 ≤ ≤ 50}. If A = { ∈ : is a multiple of 2} and B = { ∈ : is a multiple of 7}, then the number of elements in the smallest subset of containing both A and B is

A = {x: x is multiple of 2} = {2,4,6,8, … } B = {x: x is multiple of 7} = {7,14,21, … } X = {x ∶ 1 ≤ x ≤ 50, x ∈N} Smallest subset of X which... View Article

If the mean and variance of eight numbers 3, 7, 9, 12, 13, 20, x and y be 10 and 25 respectively then xy is equal to

Mean = 10 (64 + x + y)/8 = 10 x + y = 80 - 64 = 16 Variance = 25 = ((32 + 72 + 92 + 122 + 132 + 202 + x2 + y2 )/8 )- 100 1000 = 852 + x2 + y2... View Article

If the function f defined on (-1/3, 1/3) by

If the function f defined on (-1/3, 1/3) by is continuous, then k is equal to Solution: As f(x) is continous 3+2 = k k = 5 Answer: (5)... View Article

If the foot of perpendicular drawn from the point (1,0,3) on a line passing through (α, 7,1) is (5/3, 7/3, 17/3) then α is equal to:

Given points (1, 0, 3) and Q (5/3, 7/3, 17/3) Direction ratios of line L : (α-5/3, 7-7/3, 1-17/3) = ((3α-5)/3, 14/3, -14/3) Direction ratios of... View Article

If system of linear equations: x + y + z = 6,x + 2y + 3z = 10 and 3x + 2y + z = has more than two solutions, then μ – λ² is equal to

The system of equations has more than 2 solutions ∴ D = D3 = 0 2 λ− 6 −λ + 9 + 2 − 6 = 0 λ= 1 ⇒ μ = 14 So, μ − λ2 = 13 Answer: (13)... View Article

Let the tangents drawn from the origin to the circle, x² + y² − 8x − 4y + 16 = 0 touch it at the points A and B. The (AB)² is equal to

a. 32/5 b. 64/5 c. 52/5 d. 56/5 Solution: x2 + y2 − 8x − 4y + 16 = 0 (x − 4)2 + (y − 2)2 = 4 ⇒ Centre (4, 2) , radius = 2 OA = 4 = OB In... View Article

The area (in sq. units) of the region {(x, y)ϵR|4x² ≤ y ≤ 8x + 12} is

a. 125/3 b. 128/3 c. 124/3 d. 127/3 Solution: For point of intersection, 4x2 = 8x + 12 ⇒ x2 − 2x − 3 = 0 ⇒ x = 3, −1 Area bounded is... View Article

Let α and β are the roots of the equation x² − x − 1 = 0. If p_k = (α)^k + (β)^k, k ≥ 1 then which one of the following statements is not true?

a. (p1 + p2 + p3 + p4 + p5) = 26 b. p5 = 11 c. p5 = p2 ⋅ p3 d. p3 = p5 − p4 Solution: Given α, β are the roots of x2 − x − 1 = 0 ⇒ α + β = 1... View Article