Question

If $a$ is real and $\sqrt{2}ax+\sin By+\cos Bz=0,x+\cos By+\sin Bz=0,-x+\sin By-\cos Bz=0$ set of all values of a for which the system of linear equations has a non-trivial solution is

• A. $[1,2]$
• B. $[-1,1]$
• C. $[1,\infty ]$
• D. $[2^{-1/2},2^{1/2}]$

Answer 1

The correct option is B $[-1,1]$

For non trivial solution,

$\Delta =\left|\begin{array}{ccc}\sqrt{2}a& sinB& cosB\\ 1& cosB& sinB\\ -1& sinB& -cosB\end{array}\right|=0$

$⟹a\sqrt{2}[-co{s}^{2}B-si{n}^{2}B]-sinB[-cosB+sinB]+cosB[sinB+cosB]=0$

$⟹-a\sqrt{2}+sin2B+cos2B=0$

$⟹a\in [-1,1]$