InterviewSolution
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InterviewSolution
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Consider the system of equations `x cos^(3) y+3x cos y sin^(2) y=14` `x sin^(3) y+3x cos^(2) y sin y=13` The value of `sin^(2) y+2 cos^(2) y` isA. `4//5`B. `9//5`C. 2D. none of these |
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Answer» Correct Answer - B The given equations are `x cos^(3)y+3x cos y sin^(2) y=14` ...(i) and `x sin^(2) y+3x cos^(2) y sin y=13` ...(ii) Adding Eqs. (i) and (ii), we have `x(cos^(3) y+3 cos y sin^(2) y+3 cos^(2) y sin y+ sin^(3) y)=27` or `x(cos y+ sin y)^(3)=27` or `x^(1//3) (cos y + sin y) =3` ...(iii) Subtracting Eq. (ii) from Eq. (i), we have `x(cos^(3)y+3 cos y sin^(2) y-3 cos^(2) y sin y- sin^(3) y)=1` or `x(cos y- sin y)^(3)=1` or `x^(1//3) (cos y- sin y)=1` ...(iv) Dividing Eq. (iii) by (iv), we get `cos y+sin y=3 cos y-3 sin y` or `tan y=1//2` Case I : `sin y=1//sqrt(5) and cos y =2//sqrt(5)` `y=2n pi +alpha`, where `0 lt alpha lt pi//2` and `sin alpha =1//sqrt(5)` i.e., y lies in the first quadrant From Eqs. (iii) `x^(1//3) (3//sqrt(5))=3 or x=5 sqrt(5)` Case II : `sin y=-1//sqrt(5) and cos y=-2//sqrt(5)` `y=2npi+(pi+alpha)`, where `0 lt alpha lt pi//2` and `sin alpha = -1 //sqrt(5)` i.e., y lies in the third quadrant. Therefore, from Eq. (iii), `x^(1//3) (-3//sqrt(5))=3 or x=-5sqrt(5)`. Thus, `sin^(2) y+2 cos^(@) y=1//5+8//5=9//5`. Also there are exactly six values of `y in [0, 6pi]`, there in 1st quadrant and three in 3rd quadrant. |
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