Yes. On the D5200, exposures longer than 30 seconds are done in Manual mode using Bulb. Turn the mode dial to M, then adjust shutter speed until Bulb appears. In Bulb, the shutter stays open while the shutter button is held down.

You do not strictly need a remote, but using the camera’s shutter button can introduce vibration, especially for long exposures. A remote release or, better yet, an intervalometer is strongly recommended because it lets you trigger and hold the exposure without touching the camera.

Also keep in mind that very long digital exposures can build up sensor noise. Depending on what you’re photographing, another option is to shoot multiple shorter exposures and combine them later, which can produce a similar result with less noise.