Here's the new source code with the new DI-injected instances (new/updated lines are highlighted):

using System;
using Microsoft.AspNetCore.Mvc;
using Newtonsoft.Json;
using System.Collections.Generic;
using System.Linq;
using TestMakerFreeWebApp.Data;
using Mapster;
using Microsoft.AspNetCore.Identity;
using Microsoft.Extensions.Configuration;

namespace TestMakerFreeWebApp.Controllers
{
    [Route("api/[controller]")]
    public class BaseApiController : Controller
    {
        #region Constructor
        public BaseApiController(
            ApplicationDbContext context,
            RoleManager<IdentityRole> roleManager,
            UserManager<ApplicationUser> userManager,
            IConfiguration configuration
            )
        {
            // Instantiate the required classes through DI
            DbContext = context;
            RoleManager = roleManager;
            UserManager = userManager;
            Configuration = configuration;

            // Instantiate a single JsonSerializerSettings object
            // that can be reused multiple times.
            JsonSettings = new JsonSerializerSettings()
            {
                Formatting = Formatting.Indented
            };

        }
        #endregion

        #region Shared Properties
        protected ApplicationDbContext DbContext { get; private set; }
        protected RoleManager<IdentityRole> RoleManager { get; private 
          set; }
        protected UserManager<ApplicationUser> UserManager { get; 
          private set; }
        protected IConfiguration Configuration { get; private set; }
        protected JsonSerializerSettings JsonSettings { get; private 
          set; }
        #endregion
    }
}

After all the DI we've been through, we can easily understand what we did here, so let's go ahead.