InterviewSolution
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InterviewSolution
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`" if " |{:(sin x,,sin y,,sin z),(cos x,,cos y,,cos z),(cos^(3) x,,cos^(3)y,,cos^(3)z):}|=0` then which of the following is`//`are possible ?A. `x=y`B. `y=z`C. `x=z`D. `x+y+z=pi//2` |
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Answer» Correct Answer - A::B::C::D `|{:(sin x ,,sin y,,sin z),(cos x,,cos y,,cos z),(cos^(3)x,,cos^(3)y,,cos^(3)z):}|` Taking cos x cos y and z common from `C_(1) ,C_(2) " and " C_(3)` respectively. `Delta = " cos x cos y cos z " |{:(tan x ,,tan y,,tan z),(1,,1,,1),(cos^(2),,cos^(2) y,,cos^(2)z):}|` Applyin `C_(1) to C_(1)-C_(2),C_(2) to C_(2) -C_(3)` we get `Delta = " cos x cos y cos z " |{:(tan x - tan y,,tan y - tan x,,tan z),(0,,0,,1),(cos^(2)x-cos^(2)y,,cos^(2)y-cos^(2)z,,cos^(2)z):}|` `=-" cos x cos y cos z "|{:(tan x-tan y,,tan y- tan z),(cos^(2) x-cos^(2) y,,cos^(2) y -cos^(2) z):}|` `=- cos x cos y cos z` `|{:((sin(x-y))/(cos x cos y),,(sin(y-z))/(cos ycos x)),(-sin (x-y) sin (x+y),,-sin (y-z) sin (y+z)):}|` `=sin (x-y) sin (y-z) |{:(cos z,,cos x ),(sin (x+y),,sin (y+z)):}|` `=sin (x-y) sin(y-z) [cos z sin (y+z)- cos x sin (x+y)]` `=sin (x-y) sin (y-z) [sin y cos^(2) z+sin z cos y cos z- sin x cos y cos x -sin y cos^(2) x]` `=sin (x-y) sin (y-z) [sin y(cos^(2) z-cos^(2)x) +cos y(sin zcos z-sin x cos x)]` `=sin (x-y) sin (y-z)[sin y sin (x-z) sin (x-z) +cos y (sin 2z-sin 2x)//2]` `=sin (x-y) sin (y-z)[sin y sin (x+z) sin (x-z) + cos y sin (z-x) cos (z+x)]` `=sin (x-y) sin (y-z) sin (z-x) -sin y sin (x+z) + cos y cos (z+x)]` `=sin (x-y) sin (y-z) sin (z-x) cos (x+y+z)` For `Delta =0, x=y " or " y=z " or " x+y+z=pi//2` |
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