InterviewSolution
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InterviewSolution
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`M=[{:(sin^(4)theta,-1-sin^(2)theta),(1+cos^(2)theta,cos^(4)theta):}]=alphaI+betaM^(-1)` Where `alpha=alpha(theta)` and `beta=beta(theta)` ar real numbers and I is an identity matric of `2xx2` if `alpha^(**)=` min of set `{alpha(theta):thetain[0.2pi)}` and `beta^(**)=` min of set `{beta(theta):thetain[0.2pi)}` Then value of `alpha^(**)+beta^(**)` isA. `(-37)/(16)`B. `(-17)/(16)`C. `(-31)/(16)`D. `(-29)/(16)` |
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Answer» Correct Answer - D `msin^(4)theta.cos^(4)theta+(1+sin^(2)theta)(1+cos^(2)theta)` `2+sin^(4)cos^(4)theta+sin^(2)thetacos^(2)theta` `[{:(sin^(4)theta,-(1+sin^(2)theta)),(1+cos^(2)theta,cos^(4)theta):}]=[{:(alpha,0),(0,alpha):}]+beta=(1)/(|m|)[{:(cos^(4)theta,1+sin^(2)theta),(-1-cos^(2)theta,sin^(4)theta):}]` `sin^(4)theta=(alpha+beta)/(|m|)cos^(4)theta,-1-sin^(2)theta=(beta)/(|m|)(1+sin^(2)theta)` `beta=-|m|` `beta=-[sin^(4)thetacos^(4)theta+sin^(2)thetacos^(2)theta+2]=-[t^(2)+t+2]impliesbeta_(min)=-(37)/(16)` `alpha=sin^(2)theta+cos^(4)theta=1-2sin^(2)thetacos^(2)theta=1-(1)/(2)(sin^(2)2theta)impliesminalpha=(1)/(2)` `alpha+beta=-(37)/(16)+(1)/(2)=-(37)/(16)+(8)/(16)=-(29)/(16)` |
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