The number of distinct real roots of the equation, [| pmatrix x & x & x x & x & x x & x & …

Question

The number of distinct real roots of the equation,

$$\left| {\begin{pmatrix} {\cos x} & {\sin x} & {\sin x}  \
   {\sin x} & {\cos x} & {\sin x}  \
   {\sin x} & {\sin x} & {\cos x}  \ \end{pmatrix} } \right| = 0$$ in the interval $$\left[ { - {\pi  \over 4},{\pi  \over 4}} \right]$$ is :

Accepted Answer

Given,

$$\left| {\begin{pmatrix} {\cos x} & {\sin x} & {\sin x}  \
   {\sin x} & {\cos x} & {\sin x}  \
   {\sin x} & {\sin x} & {\cos x}  \

\end{pmatrix} } \right| = 0$$

R1  $$ \to $$  R1  -  R3

R1  $$ \to $$  R2  -  R3

$$\left| {\begin{pmatrix} {\cos x - \sin x} & 0 & {\sin x - \cos x}  \
   0 & {\cos x - \sin x} & {\sin x - \cos x}  \
   {\sin x} & {\sin x} & { \cos x}  \

\end{pmatrix} } \right| = 0$$

C3  $$ \to $$  C3  +  C2

$$\left| {\begin{pmatrix} {\cos x - \sin x} & 0 & {\sin x - \cos x}  \
   0 & {\cos x - \sin x} & 0  \
   {\sin x} & {\sin x} & {\sin x + \cos x}  \

\end{pmatrix} } \right| = 0$$

Expanding using  first column,

(cosx - sinx)(cos - sinx) (sinx + cos x)

+ sinx (cosx - sinx) (sinx - cosx)  =  0

$$ \Rightarrow $$   (cosx - sinx)2 (sinx + cosx)

$+$ sinx (cosx - sinx)2 = 0

$$ \Rightarrow $$    (cosx - sinx)2 (sinx + cosx $+$ sinx) =  0

$$ \Rightarrow $$    (2sinx + cosx )(cosx - sinx)2 = 0

$$ \therefore $$   cosx = -2sinx   or  cosx   =  sinx

$$ \Rightarrow $$ tanx = $$ - {1 \over 2}$$ or tanx = 1

$$ \therefore $$ x = $$ - {\tan ^{ - 1}}\left( {{1 \over 2}} \right)$$, $${\pi  \over 4}$$

$$ \therefore $$   Number of solutions   =  2

Solutions · 1