Question

If the max. and min. values of $△$= $\left|\begin{array}{ccc}1+si{n}^x& co{s}^{2}x& sin2x\\ si{n}^{2}x& 1+co{s}^{2}x& sin2x\\ sinx& co{s}^{2}x& 1+sinx\end{array}\right|$
are $\alpha$ and $\beta$, then

• A. $\alpha +{\beta }^{99}=4$
• B. ${\alpha }^3+{\beta }^{17}=26$
• C. ${\alpha }^{2n}-{\beta }^{2n}$ is always an even integer for n$\in$ N
• D. a triangle having as $\alpha ,\beta$, and $\alpha -\beta$.

Answer 1

Answer: (A), (B), (C)

The correct options are
A $\alpha +{\beta }^{99}=4$
B ${\alpha }^3+{\beta }^{17}=26$
C ${\alpha }^{2n}-{\beta }^{2n}$ is always an even integer for n$\in$ N

Apply $R_1-R_2$ and $R_2-R_3$ and expanding $△=2+sin2x$.
$\therefore$ Max. value = 2+1=3 =$\alpha$
and min. value = 2-1 =1=$\beta$
Hence (a), (b), (c) are correct.
(d) option is not possible as the sides will be 3, 1 and 2. Sum of two sides 2+1=3 = 3rd side whereas in a triangle sum of two sides is always greater than the third side.