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Let [{:(sin^4theta,,-1,-sin^2theta,),(1+cos^2theta,,,cos^4theta,):}]=alphaI+betaM^-1, where alpha=(theta) and beta=beta(theta) are real numbers, and I is then 2xx2 identify matrix. If alpha is the minimum of the set {alpha(theta):thetain[0,2pi)}and beta is the minimum of the set {beta(theta):thetain[0,2pi)}, then the value of alpha^**+beta^** is |
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Answer» `-(17)/(16)` `M=[{:(sintheta,,-1,-,sin^2theta),(1+cos^2theta,,cos^4theta,,):}]=alphaI+betaM^-1`, where `alpha=alpha(THETA) and beta =beta(theta)` are REAL NUMBER andI is the `2xx2` identify matrix. Now, det `(M)=|M|=sin^4theta cos^4theta +1sin^2theta +cos^2theta +sin^2theta cos^2theta =sin^4theta cos^4theta+sin^2theta+2` `and [{:(sintheta,,-1,-,sin^2theta),(1+cos^2theta,,cos^4theta,,):}]=[{:(alpha,0),(0,alpha):}]+(beta)/(|M|)(adj M)[because M^-1=(adjM)/(|M|)]` `rArr beta =-|M|` and `alpha =sin^4theta+cos^4theta` `rArr alpha=alpha (theta) =1-(1)/(2)sin^2(2theta)`, and `beta =beta(theta) =- {(sin^2theta cos ^2theta +(1)/(2))^2+(7)/(4)}=-{((sin^2(2theta))/(4)+(1)/(2))^2+(7)/(4)}` Now, `alpha^**=.^alphamin=(1)/(2)and beta^**=.^betamin=-(37)/(16)` `because alpha` is a minimum at `sin^2(2theta) =1and beta` is minimum at `sin^2(2theta)=1` So, `alpha^**=(1)/(2)-(37)/(16)=-(29)/(16)`. |
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