As per the Rankine’s earth pressure theory, which of the following statements is/are FALSE?

The correct option representing the FALSE statements is:
1. "For the active earth pressure, the inclination of failure plane is (45° - ϕ/2) with respect to the major principal plane."
2. "For the passive earth pressure, the inclination of failure plane is (45° - ϕ/2) with respect to the major principal plane."

Let's understand why these statements are false by deriving the inclination of the failure planes under Rankine's earth pressure theory.

1. Fundamental Concepts of Rankine's Earth Pressure Theory
In Rankine's theory, the soil is assumed to be cohesionless, homogenous, and in a state of plastic equilibrium. The relationship between the principal stresses at failure is given by:
$${\sigma }_1={\sigma }_3{\tan }^2\left(45°+\frac{\varphi }{2}\right)$$ where:
- ${\sigma }_1$ is the major principal stress.
- ${\sigma }_3$ is the minor principal stress.
- $\varphi$ is the angle of internal friction of the soil.

According to Mohr's stress circle, the failure plane is inclined at an angle $\alpha$ to the plane on which the major principal stress acts (the major principal plane), which is given by:
$$\alpha =45°+\frac{\varphi }{2}$$ This relation holds true universally with respect to the major principal plane in plastic equilibrium.

2. Active Earth Pressure State
In the active state, the retaining wall moves away from the backfill. As a result:
- The vertical stress remains the major principal stress (${\sigma }_1={\sigma }_v$). Therefore, the horizontal plane is the major principal plane.
- The horizontal active pressure is the minor principal stress (${\sigma }_3={\sigma }_a$).
Thus, the inclination of the failure plane with respect to the major principal plane (the horizontal plane) is:
$${\theta }_a=45°+\frac{\varphi }{2}$$ Therefore, the statement claiming that the inclination is $45°-\frac{\varphi }{2}$ with respect to the major principal plane is FALSE.

3. Passive Earth Pressure State
In the passive state, the retaining wall is pushed into the backfill. Consequently:
- The horizontal passive pressure becomes the major principal stress (${\sigma }_1={\sigma }_p$). Thus, the vertical plane is now the major principal plane.
- The vertical stress becomes the minor principal stress (${\sigma }_3={\sigma }_v$).
Since the failure plane is always inclined at $45°+\frac{\varphi }{2}$ to the major principal plane, the angle with respect to the horizontal (which is the minor principal plane in this case) becomes $45°-\frac{\varphi }{2}$.
However, with respect to the major principal plane itself, the inclination remains:
$${\theta }_p=45°+\frac{\varphi }{2}$$ Therefore, the statement claiming that the inclination is $45°-\frac{\varphi }{2}$ with respect to the major principal plane for passive pressure is also FALSE.