The wall of a house is made of two different materials of same thickness. The temperature of the outer wall is  $$T_2$$

and that of inner wall is $$T_{1\ }<\ T_2$$ . the temperature variation inside the wall is as shown in the figure in steady state, then

The heat flow through both materials is the same in steady state.

For a material of thickness $$L$$, the temperature gradient is related to thermal conductivity by  $$\frac{dT}{dx}=\frac{q}{k}$$

Since the thicknesses of the two materials are equal, the material having the larger temperature gradient has the smaller thermal conductivity.

Let the thermal conductivities of the inner and outer walls be $$k_i$$ and $$k_o$$ respectively.

From the graph, the temperature gradient is greater in the inner wall than in the outer wall.

Hence,  $$k_i<k_o$$

Hence, the correct option is B.