How to know if a photo is B & W

the key point is: Grayscale mode does Not accept colors

while RGB or CMYK do accept colors and black and white at the same time,

I would think no.

A grayscale image is B&W has a single color channel.  A  RGB image can have many color but the colors in a particular RGB image may be all shade of Gray.  A B&W adjustment layer can also be added to a colorful image to convert the image's composite to B&W.   I would think any Image format supports B&W and  CMYK printing should be able to print all shades of gray I would think.  To know if an image other than a grayscal image Is B&W my require you to examine every pixels in the image.