Two infinitely long straight wires lie in the xy -plane along the line  $x=\pm R$. The wire located at $x=+R$  carries a constant current I1 and the wire located at x = -R carries a constant current I₂. A circular loop of radius R suspended with its center at  $(0,0,\sqrt{3}R)$ and in a plane parallel to the xy – plane. This loop carries a constant current I in the clockwise direction as seen from above the loop. The current in the wire is taken to be positive if is in the  $+\widehat{j}$ direction. Which of the following statements regarding the magnetic field  $\overrightarrow{B}$ is (are) true?

Fig below shows the situation described in the question. If $I_1=I_2$, then the magnetic fields due to $I_1$ and $I_2$ at origin O will cancel out each other. But the magnetic field at O due to the circular loop will be present hence option(A) is correct.
If $I_1>0$ and $I_2<0$, then the magnetic field due to both current will be in +z direction and will be added – up. The magnetic field due to current I will be in – Z direction and if its magnitude is equal to the combined magnitudes of $I_1$ and $I_2$, then magnetic field can be zero the origin. Hence option (B) is correct.

   
If $I_1<0and{I}_2>0$ then their resulting magnetic field at origin will be in -Z direction and the magnetic field due to I at origin will also be in -Z direction. Thus, magnetic field at origin cannot be zero. Hence option (C) is NOT correct.
If $I_1=I_2$ then the resultant of the magnetic field at the center of the circular loop at point P is along +X direction as shown in fig below. Thus, the magnetic field at P is only due to the current I which is in -Z direction and is given as  $\overrightarrow{B}=\frac{{\mu }_{0}I}{2R}(-\widehat{k})$

 Hence option (D) is correct.