Question

If $x \in[0,2 \pi], a, b, c \in R(t>\,0)$ such that
$\sqrt{1+\sin 2 x}+\left|t+\frac{4}{t}-4\right|+(a+b+c)^{2} \leq 0$, then the value of $\begin{vmatrix}\sin x & -\cos x & t \\ t & 2 & \sin x \cos x \\ a^{3}+b^{3}+c^{3} & 3 a b c & a^{2}+b^{2}+c^{2}\end{vmatrix}$

• A. depend upon $a, b, c$
• B. depend upon $x$
• C. depend upon
• D. independent of $a, b, c, x$ and $t$

Answer 1

Answer: (D)

$1+\sin 2 x=0, t+\frac{4}{t}-4=0, a+b+c=0$
$\Rightarrow \sin x+\cos x=0, t=2$
use $C_{1} \rightarrow C_{1}-C_{2}$, we get
$\begin{vmatrix}{ccc}0 & -\cos x & t \\ 0 & 2 & \sin x \cos x \\ 0 & 3 a b c & a^{2}+b^{2}+c^{2}\end{vmatrix}=0$