A) \[\frac{-\,29}{16}\]
B) \[-\,\frac{37}{16}\]
C) \[\,-\,\frac{17}{16}\]
D) \[-\,\frac{31}{16}\]
Correct Answer: A
Solution :
| \[\text{M}\,=\,\left[ \begin{matrix} {{\sin }^{4}}\theta & -\,1\,-\,{{\sin }^{2}}\theta \\ 1+{{\cos }^{2}}\theta & {{\cos }^{4}}\theta \\ \end{matrix} \right]\,=\,\alpha \,I\,+\,\beta {{\text{M}}^{\,-\,1}}\] |
| \[\text{M}\,=\alpha \text{I}\,+\,\beta \,{{\text{M}}^{\,-\,1}}\] |
| \[{{\text{M}}^{2}}\,=\alpha \text{M}\,+\beta I\] |
| \[\left[ \begin{matrix} \text{si}{{\text{n}}^{4}} & -\,1\,-\,\text{si}{{\text{n}}^{2}}\theta \\ 1+{{\cos }^{2}} & {{\cos }^{4}}\theta \\ \end{matrix} \right]\,\left[ \begin{matrix} \text{si}{{\text{n}}^{4}}\theta & -\,1\,-\,\text{si}{{\text{n}}^{2}}\theta \\ 1+\text{co}{{\text{s}}^{2}}\theta & 1+\text{co}{{\text{s}}^{4}}\theta \\ \end{matrix} \right]\] |
| \[=\,\beta \,\left[ \begin{matrix} 1 & 0 \\ 0 & 1 \\ \end{matrix} \right]+a\left[ \begin{matrix} \text{si}{{\text{n}}^{4}}\theta & -\,1\,-\,\text{si}{{\text{n}}^{2}}\theta \\ 1+\text{co}{{\text{s}}^{2}}\theta & \text{co}{{\text{s}}^{4}}\theta \\ \end{matrix} \right]\] |
| \[\text{si}{{\text{n}}^{8}}\theta \,-\,1\,-\,\text{si}{{\text{n}}^{2}}\theta \,-\,\text{co}{{\text{s}}^{2}}\theta \,\,-\,\text{co}{{\text{s}}^{2}}\theta \text{si}{{\text{n}}^{2}}\,\theta \,=\,\beta +\alpha \text{si}{{\text{n}}^{4}}\theta \]\[\text{si}{{\text{n}}^{8}}\theta \,-\,2\,-\text{co}{{\text{s}}^{2}}\theta \,\,\text{si}{{\text{n}}^{2}}\theta \,=\,\beta \,+\alpha \,\text{si}{{\text{n}}^{4}}\theta \] ? (i) |
| \[\text{si}{{\text{n}}^{2}}\theta \,+\text{co}{{\text{s}}^{2}}\theta \,\text{si}{{\text{n}}^{4}}\theta +\text{co}{{\text{s}}^{4}}+\text{co}{{\text{s}}^{6}}\theta \,=\,\alpha (1+\text{co}{{\text{s}}^{2}}\theta )\] |
| \[\alpha \,=\,\frac{\text{si}{{\text{n}}^{4}}\theta \,(1+\text{co}{{\text{s}}^{2}}\theta )+\text{co}{{\text{s}}^{4}}\theta \,(1+\text{co}{{\text{s}}^{2}}\theta )}{(1+\text{co}{{\text{s}}^{2}}\theta )}\] |
| \[\alpha \,=\,\text{si}{{\text{n}}^{4}}\theta +\text{co}{{\text{s}}^{4}}\theta \,=\,1\,-\,\frac{1}{2}\text{si}{{\text{n}}^{2}}2\theta \] |
| \[{{\alpha }_{\min \,\,}}=\,1\,-\,\frac{1}{2}\,=\,-\,\frac{1}{2}\,\] |
| for equation, (i), \[\text{si}{{\text{n}}^{8}}\theta \,-\,2\,-\,\text{co}{{\text{s}}^{2}}\theta \,\text{si}{{\text{n}}^{2}}\theta \,-\,\alpha \,\,\text{si}{{\text{n}}^{4}}\theta \,=\,\beta \] |
| \[\beta \,=\,\text{si}{{\text{n}}^{2}}\theta \,-\,2\,-\,\text{si}{{\text{n}}^{2}}\theta \,\text{co}{{\text{s}}^{2}}\theta -\,\text{si}{{\text{n}}^{4}}\theta \,(\text{si}{{\text{n}}^{4}}+\text{co}{{\text{s}}^{4}}\theta )\] |
| \[\beta \,=\,-\,2\,-\,\text{si}{{\text{n}}^{2}}\theta \,\text{co}{{\text{s}}^{2}}\theta \,-\,\text{si}{{\text{n}}^{4}}\theta \,\text{co}{{\text{s}}^{4}}\theta \] |
| \[\beta \,=\,-\,2\,-\,\frac{1}{4}\text{si}{{\text{n}}^{2}}2\theta \,-\,\frac{1}{16}\,{{(\text{sin}2\theta )}^{4}}\] |
| \[\beta \,=\,-\,2\,-\,\frac{1}{16}\left\{ {{(\text{sin}2\theta )}^{4}}+4\,(\text{si}{{\text{n}}^{2}}2\theta )+4 \right\}+\frac{1}{4}\] |
| \[\beta \,=\,-\,\frac{7}{4}\,-\,\frac{1}{16}\,{{\left\{ \text{sin}\,2\theta +2 \right\}}^{2}}\] |
| \[\beta \,=\,-\,\frac{7}{4}\,-\,\frac{1}{6}.9\,=\,\frac{-7}{4}\,-\frac{9}{16}\,=\,\frac{-28-9}{16}\,=\,-\,\frac{37}{16}\] |
| \[a_{\min }^{*}\,+\beta _{\min }^{*}\,=\,\frac{-\,37+8}{16}\,=\,\frac{-29}{16}\] |
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