$A=\left[\begin{array}{cc}\frac{1}{2}|[x]| & |\sin y| \ \cos z & 1\end{array} ight], B=\left[\begi

Question

$A=\left[\begin{array}{cc}\frac{1}{2}|[x]| & |\sin y| \ \cos z & 1\end{array}ight], B=\left[\begin{array}{cc}\{x\} & \{y\} \ \{z\} & 1\end{array}ight]$, if $x \in[-2,2], y, z \in(-\pi, \pi)$ ifnumber of triplets $(x, y, z)$ such that $A=B$ is $k$, then value of $k / 7$ is $\_\_\_\_$ , where [ ⋅ ] denote G.I.F. & $\{\cdot\}$ denotes fractional part function.

Professor PC Thomas Classes · Answer

$\frac{1}{2}|[x]|=\{x\} \quad x \in[-2,2] \Rightarrow x=0,-\frac{1}{2}, \frac{3}{2}$ using graph $|\sin y|=\{y\} \quad y \in(-\pi, \pi) \Rightarrow y$ can take 7 values, using graph $\cos z=\{z\} \quad z \in(-\pi, \pi) \Rightarrow z$ can take 2 values, using graph $\Rightarrow(x, y, z)$ can take $3 	imes 7 	imes 2$ i.e. 42 triplets