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Complex Plane

21. Let M be ...

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21. Let M be the maximum value of (6x−3y−8z), subject to 2x²+3y²+4z²= 1.Find [M].

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Q. Find the value of

{(\frac{1}{2} x - \frac{1}{3} y)}^{3} + {(\frac{1}{3} y - \frac{1}{4} z)}^{3} + {(\frac{1}{4} z - \frac{1}{2} x)}^{3} = m (\frac{1}{2} x - \frac{1}{3} y) (\frac{1}{3} y - \frac{1}{4} z) (\frac{1}{4} z - \frac{1}{2} x)

m

Q. Find the value of the addition of

(x - 3 y + 4 z), (y - 2 x - 8 z), (5 x - 2 y - 3 z)

p_{1}, p_{2}, p_{3}

denote the distance of the plane

2 x - 3 y + 4 z + 2 = 0

from the planes

2 x - 3 y + 4 z + 6 = 0, 4 x - 6 y + 8 z + 3 = 0

2 x - 3 y + 4 z - 6 = 0

respectively, then

p_{1}, p_{2}, p_{3}

denote the distances of the plane 2x - 3y + 4z + 2 = 0 from the planes 2x - 3y + 4z + 6 = 0, 4x - 6y + 8z + 3 = 0 and 2x - 3y + 4z - 6 = 0 respectively, then

P_{1}, P_{2}, P_{3}

denotes the perpendicular distances of the plane

2 x - 3 y + 4 z + 2 = 0

from the parallel planes

2 x - 3 y + 4 z + 6 = 0, 4 x - 6 y + 8 z + 3 = 0

2 x - 3 y + 4 z - 6 = 0

respectively, then

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