1.

For all real values of `theta`, choose the correct options.

Answer» Correct Answer - 4
a. `A=sin^2theta+cos^4theta`
`=1-cos^2theta+cos^4theta`
`=(cos^2theta-1//2)^2+3//4`
`0le cos^2thetale1`
`rArr-1//2lecos^2theta-1//2le1//2`
`rArr 0le (cos^2theta-1//2)^2le1//4`
`rArr 0le (cos^2theta-1//2)^2le1//4`
`rArr 3//4 le (cos^2theta-1//2)^2+3//4le1`
b. `3cos^2theta+sin^4theta=3-3sin^2theta+sin^4theta`
`=(sin^2theta-3//2)^2+3//4`
`0le sin^2thetale1`
`rArr -3//2 lesin^2theta-3//2le-1//2`
`rArr 1//4 le(sin^2theta-3//2)^2+3//4le3`
`rArr 1le(sin^2theta-3//2)^2+3//4le3`
c. `A=sin^2theta-cos^4theta`
`=1-cos^2theta-cos^4theta`
`=5//4-(cos^2theta+1//2)^2`
`0lecos^2thetale1`
`rArr 1//2lecos^2theta+1//2le3//2`
`rArr 1//4le(cos^2+1//2)^2le9//4`
`rArr -9//4le-(cos^2theta+1//2)^2le-1//4`
`rArr-1le5//4-(cos^2theta+1//2)^2le1`
d. `A=tan^2theta+2cot^2theta`
`=(tantheta-sqrt2cottheta)^2+2sqrt2ge2sqrt2`


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