Step 1: Think of the Mohr circle as a wheel of radius \(\dfrac{\sigma_1-\sigma_3}{2}\) sitting on the normal stress axis. The failure envelope is a straight ramp leaning upward from the origin at angle \(\phi\).
Step 2: Failure is the first moment this wheel bumps into the ramp as it grows, a tangency condition, and geometry shows the tangent point sits at an angle \(2\theta = 90^\circ + \phi\) measured from the major principal stress direction on the circle, giving a failure plane inclined at \(\theta = 45^\circ + \phi/2\) to the major principal plane in real space.
Step 3: If \(\phi\) were zero, the ramp becomes flat, the tangent point sits exactly at the top of the wheel, and both the top-of-the-wheel idea and the 45 degree plane idea would be true together. But the question is posed as a general statement about the Mohr failure hypothesis, not restricted to frictionless soils.
Step 4: For any soil that actually has friction, the ramp is tilted, so the tangent point slides away from the top of the wheel. The shear stress there is less than the wheel's maximum radius value, so Statement (I) breaks down.
Step 5: The failure plane angle also shifts away from \(45^\circ\) by half the friction angle, so Statement (II) breaks down too.
Step 6: Since the general hypothesis, with friction present, does not support either claim, both must be marked false.
\[\boxed{\text{Both Statement (I) and Statement (II) are false}}\]