If a line segment OP makes angles of π4 and π3&nb

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71.

If a line segment OP makes angles of π4 and π3 with X-axis and Y-axis, respectively. Then, the direction cosines are

  • 12, 32, 12

  • 12, 12, 12

  • 1, 3, 1

  • 1, 13, 1


B.

12, 12, 12

Let α, β and γ be the angles made by the line segment OP with X-axis, Y-axis and Z-axis, respectively

Given, α = π4 and β = π3 c      os2α + cos2β + cos2γ = 1 cos2π4 + cos2π3 + cos2γ = 1          122 + 122 + cos2γ = 1 12 + 14 + cos2γ = 1          34 + cos2γ = 1                   cos2γ = 14                     cosγ = 12                            γ = π4

Hence, direction cosines are cosα, cosβ, cosγ i.e., 12, 12, 12


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72.

If a plane passing through the point (2, 2, 1) and is perpendicular to the planes 3x + 2y + 4z + 1 = 0 and 2x + y + 3z + 2 = 0. Then, the equation of the plane is

  • 2x - y - z - 1 = 0

  • 2x + 3y + z - 1 = 0

  • 2x + y + z + 3 = 0

  • x - y + z - 1 = 0


73.

If the points (1, 2, 3) and (2, - 1, 0) lie on the opposite sides of the plane 2x + 3y - 2z = k, then

  • k < 1

  • k > 2

  • k < 1 or k > 2

  • 1 < k < 2


74.

The triangle formed by the tangent to the curve f (x) = x2 + bx - b at the point (1, 1) and the coordinate axes lies in the first quadrant. If its area is 2, then the value of b is

  • - 1

  • 3

  • - 3

  • 1


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75.

If a plane meets the coordinate axes at A, B and C such that the centroid of the triangle is (1, 2, 4), then the equation of the plane is

  • x + 2y + 4z = 12

  • 4x + 2y + z = 12

  • x + 2y + 4z = 3

  • 4x + 2y + z = 3


76.

The volume of the tetrahedron included between the plane 3x + 4y - 5z - 60 = 0 and the coordinate planes is

  • 60

  • 600

  • 720

  • 400


77.

The length of longer diagonal of the parallelogram constructed on 5a + 2b and a - 3b, if it is given that a = 22, b = 3and the angle between a and b is π4, is

  • 15

  • 113

  • 593

  • 369


78.

If the gradient of the tangent at any point (x, y) of acurve which passes through the point 1, π4 is yx - sin2yx, then the equation of the curve is

  • y = cot-1logex

  • y = cot-1logexe

  • y = xcot-1logeex

  • y = cot-1logeex


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79.

If a plane meets the coordinate axes at A, B and C in such a way that the centroid of ABC is at the point (1, 2, 3), then equation of the plane is

  • x1 + y2 + z3 = 1

  • x3 + y6 + z9 = 1

  • x1 + y2 + z3 = 13

  • None of these


80.

If 3p and 4p are resultant of force 5p,then angle between 3p and 5p is

  • sin-135

  • sin-145

  • 90°

  • None of these


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