Question

The number of distinct real roots of $\left|\begin{array}{ccc}sinx& cosx& cosx\\ cosx& sinx& cosx\\ cosx& cosx& sinx\end{array}\right|=0$
in the interval - $\frac{\pi }{4}\le x\le \frac{\pi }{4}$is

• A. 0.0
• B. 2
• C. 1
• D. 3

Answer 1

The correct option is C 1

To simplify the det. Let $sinx=a;cosx=b$ the equation becoms
$\left|\begin{array}{ccc}a& b& b\\ b& a& b\\ b& b& a\end{array}\right|=0Operating{C}_2\to C_2-C_1;C_3\to C_3-C_{2}weget\left|\begin{array}{ccc}a& b-a& 0\\ b& a-b& b-a\\ b& 0& a-b\end{array}\right|=0\Rightarrow a(a-b{)}^2-(b-a\left)\right[b(a-b)-b(b-a)]=0\Rightarrow a(a-b{)}^2-2b(b-a)(a-b)=0\Rightarrow (a-b{)}^2(a-2b)=0\Rightarrow (a=b)ora=2b\Rightarrow \frac{a}{b}=1or\frac{a}{b}=2\Rightarrow tanx=1ortanx=2Butwehave-\pi /4\le x\le \pi /4\Rightarrow tan(-\pi /4)\le tanx\le tan(\pi /4)\Rightarrow -1\le tanx\le 1\therefore tanx=1\Rightarrow x=\pi /4$
$\therefore$ Only one real root is there.