Column I (Equation)		Column II (number of Solutions)
A	$x^3+x^2+4x+2\sin x=0in0\le x\le 2\pi$	P	4
B	$\sin e^x\cos e^x=2^{x-2}+2^{-x-2}$	Q	1
C	$\sin 2x\text{}+\text{}\cos 4^x=2$	R	2
D	$30\left|\sin x\right|=xin0\le x\le 2\pi$	S	0

   

A) $x^3+{(x+2)}^2+2\sin x=4$  clearly x = 0 satisfy
 $\begin{array}{l}0<x\le \pi ,x^3+{\left(x+2\right)}^2+2\sin x>4\\ \pi <x\le 2\pi ,x^3+{(x+2)}^2+2\sin x>27-+25-2\end{array}$
x = 0 is only solution
B)  $\frac{1}{2}\sin 2e^x=\frac{2^x+2^{-x}}{4}\Rightarrow \sin \left(2e^x\right)\ge 1$$\sin \left(2e^x\right)=1\Rightarrow x=0,nottrue$
C) $\sin 2x+\cos 4x=2\Rightarrow \sin 2x=1,\cos 4x=1,nosolution$
D) $y=\left|\sin x\right|\Rightarrow y=\frac{x}{30}\Rightarrow 0\le x\le 2\pi$   4 points of intersection