Question

The values of $\theta$ lying between $\theta =0$ and $\theta =\frac{\pi }{2}$ and satisying the equation
$\left|\begin{array}{ccc}1+si{n}^2\theta & co{s}^2\theta & 4sin4\theta \\ si{n}^2\theta & 1+co{s}^2\theta & 4sin4\theta \\ si{n}^2\theta & co{s}^2\theta & 1+4sin4\theta \end{array}\right|=0$ are

• A. $\frac{7\pi }{24}$
  
• B. $\frac{5\pi }{24}$
  
• C. $\frac{11\pi }{24}$
  
• D. $\frac{\pi }{24}$

Answer 1

Answer: (A), (C)

The correct options are
A

$\frac{7\pi }{24}$

C

$\frac{11\pi }{24}$

We have
$\left|\begin{array}{ccc}1+si{n}^2\theta & co{s}^2\theta & 4sin4\theta \\ si{n}^2\theta & 1+co{s}^2\theta & 4sin\theta \\ si{n}^2\theta & co{s}^2\theta & 1+4sin4\theta \end{array}\right|=0$
Operating $c_1\to C_1+C_2$, we get
$\left|\begin{array}{ccc}2& co{s}^2\theta & 4sin4\theta \\ 2& 1+co{s}^2\theta & 4sin4\theta \\ 1& co{s}^2\theta & 1+4sin4\theta \end{array}\right|=0$
Operating $R_1\to R_1-R_2\Rightarrow R_2-R_3$, we get.
$\left|\begin{array}{ccc}0& -1& 0\\ 1& 1& -1\\ 1& co{s}^2\theta & 1+4sin4\theta \end{array}\right|=0$
Expanding along $R_1,weget1+4sin4\theta +1=0$
or $2(1+2sin4\theta )=0$
or $sin4\theta =-\frac{1}{2}\Rightarrow 4\theta =\pi +\frac{\pi }{6}$
or $2\pi -\frac{\pi }{6}$
$\Rightarrow 4\theta =7\frac{\pi }{6}or11\frac{\pi }{6}\Rightarrow \theta =7\frac{\pi }{24}or11\frac{\pi }{24}$