We know that

(i) For a cylinder, B = 0 for 0 < r < R and $$B = {{{\mu _0}} \over {4\pi }} \times {{2I} \over r}$$ for r > R.

(ii) For a solenoid, B = $$\mu$$₀nI for 0 < r < R and B = 0 for r > R.

$$\bullet$$ Assuming that the magnetic induction to be B_C for cylinder and B_S for solenoid.

In the region R > r > 0, B_S = $$\mu$$₀nI while B_C = 0. Therefore, the total field is non-zero.

Hence, option (A) is correct.

$$\bullet$$ In the region 2R < r < R, the magnetic field is not along the axis of cylinder which is given as

$$B = \sqrt {B_S^2 + B_C^2} $$

Hence, option (B) is wrong.

$$\bullet$$ For checking option (C), in the region R < r < 2R, the magnetic field is tangential to the circle of radius r, centred on the axis. So this statement is wrong because magnetic field is not in the plane of circle.

Hence, option (C) is wrong.

$$\bullet$$ In the region r > 2, B_S = 0; while $${B_C} = {{{\mu _0}} \over {4\pi }} \times {{2I} \over r}$$. Therefore total field is non-zero.

Hence, option (D) is correct.