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. 2
• B. 1
• C. 3

Answer 1

The correct option is B 1

To simplify the determinant, let sinx =a; cos x =b. Then the equation becomes
$\left|\begin{array}{ccc}a& b& b\\ b& a& b\\ b& b& a\end{array}\right|=0$
Operating $C_2\to C_1;C_3\to C_3-C_2$, we get
$\left|\begin{array}{ccc}a& b-a& 0\\ b& a-b& b-a\\ b& 0& a-b\end{array}\right|=0$
or $a(a-b{)}^2-(b-a\left)\right[b(a-b)-b(b-a)]=0$
or $a(a-b{)}^2-2b(b-a\left)\right(a-b)=0$
or $(a-b{)}^2(a-2b)=0$
or a=b or a =2b
or $\frac{a}{b}=1or\frac{a}{b}=2$
$\Rightarrow tan=1ortanx=2$
But we have $-\frac{\pi }{4}\le tanx\le \frac{\pi }{4}$
$\Rightarrow tan\left(\frac{\pi }{4}\right)\le tanx\le tan\left(\frac{\pi }{4}\right)\Rightarrow -1\le tanx\le 1\therefore tanx=1\Rightarrow x=\frac{\pi }{4}$
Therefore, there is only one real root.