JEE Advance - Mathematics (2019 - Paper 1 Offline - No. 2)
Let $$M = \left[ {\matrix{
{{{\sin }^4}\theta } \cr
{1 + {{\cos }^2}\theta } \cr
} \matrix{
{ - 1 - {{\sin }^2}\theta } \cr
{{{\cos }^4}\theta } \cr
} } \right] = \alpha I + \beta {M^{ - 1}}$$,
where $$\alpha $$ = $$\alpha $$($$\theta $$) and $$\beta $$ = $$\beta $$($$\theta $$) are real numbers, and I is the 2 $$ \times $$ 2 identity matrix. If $$\alpha $$* is the minimum of the set {$$\alpha $$($$\theta $$) : $$\theta $$ $$ \in $$ [0, 2$$\pi $$)} and {$$\beta $$($$\theta $$) : $$\theta $$ $$ \in $$ [0, 2$$\pi $$)}, then the value of $$\alpha $$* + $$\beta $$* is
where $$\alpha $$ = $$\alpha $$($$\theta $$) and $$\beta $$ = $$\beta $$($$\theta $$) are real numbers, and I is the 2 $$ \times $$ 2 identity matrix. If $$\alpha $$* is the minimum of the set {$$\alpha $$($$\theta $$) : $$\theta $$ $$ \in $$ [0, 2$$\pi $$)} and {$$\beta $$($$\theta $$) : $$\theta $$ $$ \in $$ [0, 2$$\pi $$)}, then the value of $$\alpha $$* + $$\beta $$* is
$$ - {{17} \over {16}}$$
$$ - {{31} \over {16}}$$
$$ - {{37} \over {16}}$$
$$ - {{29} \over {16}}$$
Explanation
It is given that matrix
$$M = \left[ {\matrix{ {{{\sin }^4}\theta } \cr {1 + {{\cos }^2}\theta } \cr } \matrix{ { - 1 - {{\sin }^2}\theta } \cr {{{\cos }^4}\theta } \cr } } \right] = \alpha I + \beta {M^{ - 1}}$$, where
$$\alpha $$ = $$\alpha $$($$\theta $$) and $$\beta $$ = $$\beta $$($$\theta $$) are real numbers and I is the 2 $$ \times $$ 2 identity matrix.
Now,
$$(M) = \left| M \right| = {\sin ^4}\theta {\cos ^4}\theta + 1 + {\sin ^2}\theta + {\cos ^2}\theta + {\sin ^2}\theta {\cos ^2}\theta$$
= $${\sin ^4}\theta {\cos ^4}\theta + {\sin ^2}\theta {\cos ^2}\theta + 2$$
and$$\left[ {\matrix{ {{{\sin }^4}\theta } \cr {1 + {{\cos }^2}\theta } \cr } \matrix{ { - 1 - {{\sin }^2}\theta } \cr {{{\cos }^4}\theta } \cr } } \right] = \left[ {\matrix{ \alpha \cr 0 \cr } \,\matrix{ 0 \cr \alpha \cr } } \right]$$$$ + {\beta \over {\left| M \right|}}(adj\,M)$$
$$ \because $$ $$\left[ {{M^{ - 1}} = {{adj\,M} \over {\left| M \right|}}} \right]$$
$$ \Rightarrow $$ $$\left[ {\matrix{ {{{\sin }^4}\theta } \cr {1 + {{\cos }^2}\theta } \cr } \matrix{ { - 1 - {{\sin }^2}\theta } \cr {{{\cos }^4}\theta } \cr } } \right] = \left[ {\matrix{ \alpha \cr 0 \cr } \,\matrix{ 0 \cr \alpha \cr } } \right] + {\beta \over {\left| M \right|}}\left[ {\matrix{ {{{\cos }^4}\theta } \cr { - 1 - {{\cos }^2}\theta } \cr } \matrix{ {1 + {{\sin }^2}\theta } \cr {{{\sin }^4}\theta } \cr } } \right]$$
$$ \because $$ $$\left\{ {adj\left[ {\matrix{ a & b \cr c & d \cr } } \right] = \left[ {\matrix{ d & { - b} \cr { - c} & a \cr } } \right]} \right\}$$
$$ \Rightarrow \beta = - \left| M \right|$$ and $$\alpha = {\sin ^4}\theta + {\cos ^4}\theta $$
$$ \Rightarrow \alpha = \alpha (\theta ) = 1 - {1 \over 2}{\sin ^2}(2\theta )$$, and $$\beta = \beta (\theta ) = - \left\{ {{{\left( {{{\sin }^2}\theta {{\cos }^2}\theta + {1 \over 2}} \right)}^2} + {7 \over 4}} \right\} = - \left\{ {{{\left( {{{{{\sin }^2}(2\theta )} \over 4} + {1 \over 2}} \right)}^2} + {7 \over 4}} \right\}$$
Now, $${\alpha ^*} = {\alpha _{\min }} = {1 \over 2}$$
and $${\beta ^*} = {\beta _{\min }} = - {{37} \over {16}}$$
$$ \because $$ $$\alpha $$ is minimum at sin2(2$$\theta $$) = 1 and $$\beta $$ is minimum at sin2(2$$\theta $$) = 1
So, $${\alpha ^*} + {\beta ^*} = {1 \over 2} - {{37} \over {16}} = - {{29} \over {16}}$$
$$M = \left[ {\matrix{ {{{\sin }^4}\theta } \cr {1 + {{\cos }^2}\theta } \cr } \matrix{ { - 1 - {{\sin }^2}\theta } \cr {{{\cos }^4}\theta } \cr } } \right] = \alpha I + \beta {M^{ - 1}}$$, where
$$\alpha $$ = $$\alpha $$($$\theta $$) and $$\beta $$ = $$\beta $$($$\theta $$) are real numbers and I is the 2 $$ \times $$ 2 identity matrix.
Now,
$$(M) = \left| M \right| = {\sin ^4}\theta {\cos ^4}\theta + 1 + {\sin ^2}\theta + {\cos ^2}\theta + {\sin ^2}\theta {\cos ^2}\theta$$
= $${\sin ^4}\theta {\cos ^4}\theta + {\sin ^2}\theta {\cos ^2}\theta + 2$$
and$$\left[ {\matrix{ {{{\sin }^4}\theta } \cr {1 + {{\cos }^2}\theta } \cr } \matrix{ { - 1 - {{\sin }^2}\theta } \cr {{{\cos }^4}\theta } \cr } } \right] = \left[ {\matrix{ \alpha \cr 0 \cr } \,\matrix{ 0 \cr \alpha \cr } } \right]$$$$ + {\beta \over {\left| M \right|}}(adj\,M)$$
$$ \because $$ $$\left[ {{M^{ - 1}} = {{adj\,M} \over {\left| M \right|}}} \right]$$
$$ \Rightarrow $$ $$\left[ {\matrix{ {{{\sin }^4}\theta } \cr {1 + {{\cos }^2}\theta } \cr } \matrix{ { - 1 - {{\sin }^2}\theta } \cr {{{\cos }^4}\theta } \cr } } \right] = \left[ {\matrix{ \alpha \cr 0 \cr } \,\matrix{ 0 \cr \alpha \cr } } \right] + {\beta \over {\left| M \right|}}\left[ {\matrix{ {{{\cos }^4}\theta } \cr { - 1 - {{\cos }^2}\theta } \cr } \matrix{ {1 + {{\sin }^2}\theta } \cr {{{\sin }^4}\theta } \cr } } \right]$$
$$ \because $$ $$\left\{ {adj\left[ {\matrix{ a & b \cr c & d \cr } } \right] = \left[ {\matrix{ d & { - b} \cr { - c} & a \cr } } \right]} \right\}$$
$$ \Rightarrow \beta = - \left| M \right|$$ and $$\alpha = {\sin ^4}\theta + {\cos ^4}\theta $$
$$ \Rightarrow \alpha = \alpha (\theta ) = 1 - {1 \over 2}{\sin ^2}(2\theta )$$, and $$\beta = \beta (\theta ) = - \left\{ {{{\left( {{{\sin }^2}\theta {{\cos }^2}\theta + {1 \over 2}} \right)}^2} + {7 \over 4}} \right\} = - \left\{ {{{\left( {{{{{\sin }^2}(2\theta )} \over 4} + {1 \over 2}} \right)}^2} + {7 \over 4}} \right\}$$
Now, $${\alpha ^*} = {\alpha _{\min }} = {1 \over 2}$$
and $${\beta ^*} = {\beta _{\min }} = - {{37} \over {16}}$$
$$ \because $$ $$\alpha $$ is minimum at sin2(2$$\theta $$) = 1 and $$\beta $$ is minimum at sin2(2$$\theta $$) = 1
So, $${\alpha ^*} + {\beta ^*} = {1 \over 2} - {{37} \over {16}} = - {{29} \over {16}}$$
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