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Cosider the cubic equation : x^3-(1+cost...

Cosider the cubic equation : `x^3-(1+costheta+sintheta)x^2+(costhetasintheta+costheta+sintheta)x-sinthetacostheta=0` whose roots are `x_1,x_2,x_3`. The value of `(x_1)^2+(x_2)^2+(x_3)^2` equals

A

1

B

2

C

`2 cos theta`

D

`sin theta (sin theta+ cos theta)`

Text Solution

Verified by Experts

The correct Answer is:
B
`x^(3)-(1+cos theta + sin theta) x^(2) +(cos theta sin theta + cos theta + sin theta)x-sin theta cos theta=0`
Given cubic function is
`f(x)=(x-1)(x-cos theta) (x- sin theta)`
Therefore, roots are `1, sin theta`, and `cos theta`.
Hence, `x_(1)^(2)+x_(2)^(2)+x_(3)^(2)=1+sin^(2) theta+cos^(2) theta=2`
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cosider the cubic equation : x^(3)-(1+cos theta+sin theta)x^(2)+(cos theta sin theta+cos theta+sin theta)x-sin theta cos theta=0 whose roots are x_(1),x_(2),x_(3) .The value of (x_(1))^(2)+(x_(2))^(2)+(x_(3))^(2) equals

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Knowledge Check

  • If sintheta+costheta=x then sintheta-costheta=?

    A
    `pmsqrt(2-x^(2))`
    B
    `pmsqrt(2+x^(2))`
    C
    `pmsqrt(4-x^(2))`
    D
    `pmsqrt(4+x^(2))`

  • If (sintheta)/(x)=(costheta)/(y) then sintheta-costheta is equal to

    A
    x-y
    B
    x+y
    C
    `(x-y)/(sqrt(x^(2)+y^(2)))`
    D
    `(y-x)/(sqrt(x^(2)+y^(2)))`

  • Consider the cubic equation in x , x ^(3) - x^(2) + (x- x ^(2)) sin theta + (x -x ^(2)) cos theta + (x-1) sin theta cos theta =0 whose roots are alpha, beta , gamma. The value of ((alpha)/(2))^(2) + ((beta)/(2 )) ^(2) + ((gamma)/(2 ))^(2) =

    A
    1
    B
    `1/2`
    C
    `2 cos theta`
    D
    `1/2(sin theta + cos theta + sin theta cos theta)`

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