The angle between the planes 3x + 4y + 5z = 3 and 4x - 3y + 5z =

Subject

Mathematics

Class

JEE Class 12

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41.

The angle between the planes 3x + 4y + 5z = 3 and 4x - 3y + 5z = 9 is equal to

  • π2

  • π4

  • π6

  • π3


D.

π3

Given planes are 3x + 4y + 5z = 3 and 4x - 3y + 5z = 9

Since, DR's of normal to the planes are

       a1, b1, c1 = 3, 4, 5and a2, b2, c2 = 4, - 3, 5

  Angle between the planes = Angle between their normals

  cosθ = a1a2 + b1b2 + c1c2a12 + b12 + c12 a22 + b22 + c22                = 3 × 4 + 4 × - 3 + 5 × 532 + 42 + 52 42 + - 32 + 52                = 12 - 12 + 259 + 16 + 2516 + 9 + 25 = 2550 cosθ = 12  θ = π3


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42.

The vector equation of the plane through the point (2, 1, - 1) and parallel to the plane r - (i + 3j - k) = 0 is

  • r . (i + 9j + 11k) = 6

  • r . (i - 9j + 11k) = 4

  • r . (i + 3j - k) = 6

  • r . (i + 3j - k) = 4


43.

If the foot of the perpendicular drawn from the point (5, 1, - 3) to a plane is (1, - 1, 3), then the equation of the plane is

  • 2x + y - 3z + 8 = 0

  • 2x + y + 3z + 8 = 0

  • 2x - y - 3z + 8 = 0

  • 2x - y + 3z + 8 = 0


44.

The equation of the plane through the line of intersection of the planes x - y + z + 3 = 0 and x + y + 22 + 1 = 0 and parallel to x-axis is

  • 2y - z = 2

  • 2y + z = 2

  • 4y + z = 4

  • y - 2z = 3


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45.

5x dx1 - x3 is equal to

  • 52x - 12 - 5x - 1 + C

  • 52x - 12 + 5x - 1 + C

  • 53x - 12 + 52x - 1 + C

  • 53x - 12 - 52x - 1 + C


46.

dxx - x is equal to

  • 2logx - 1 + C

  • 2logx + 1 + C

  • logx - 1 + C

  • 12logx + 1 + C


47.

dx4sin2x + 3cos2x

  • 34tan-12tanx3 + C

  • 123tan-1tanx3 + C

  • 23tan-12tanx3 + C

  • 123tan-12tanx3 + C


48.

secxdxcos2x is equal to

  • 2sin-1tanx

  • tan-1tanx2 + C

  • sin-1tanx

  • 32tan-1tanx3 + C


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49.

exxxlogx + 1dx is equal to

  • exx + C

  • xexlogx + C

  • exlogx + C

  • x(exlogx) + C


50.

1 + logx1 + x logx2dx is equal to

  • 11 + xlogx + C

  • 11 + logx + C

  • - 11 + xlogx + C

  • log11 + logx + C


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