This question has Statement 1 and Statement 2. Out of the four choices given after the Statements, choose the one that best describes the two Statements.

Statement 1: When moment of inertia $$I$$ of a body rotating about an axis with angular speed $$\omega$$ increases, its angular momentum $$L$$ is unchanged but the kinetic energy $$K$$ increases if there is no torque applied on it. 

Statement 2: $$L = I\omega$$, kinetic energy of rotation $$= \dfrac{1}{2}I\omega^2$$

For a rigid body rotating about a fixed axis, the two standard relations are
angular momentum $$L = I\omega$$ and rotational kinetic energy $$K = \dfrac{1}{2}I\omega^{2}$$.
These are quoted in Statement 2.

Checking Statement 1
Suppose no external torque acts on the system. Then total angular momentum about that axis is conserved:

$$L = \text{constant}$$

If during the motion the moment of inertia changes from $$I_1$$ to $$I_2$$ (for example, by redistributing the mass), conservation gives

$$I_1\omega_1 = I_2\omega_2 \quad\Rightarrow\quad \omega_2 = \frac{I_1}{I_2}\,\omega_1$$

Rotational kinetic energy after the change will be

$$K_2 = \tfrac12 I_2\omega_2^{2} = \tfrac12 I_2\left(\frac{I_1}{I_2}\omega_1\right)^{2} = \tfrac12\frac{I_1^{2}}{I_2}\,\omega_1^{2}$$

Since $$I_2 \gt I_1$$ by assumption (moment of inertia increases), the factor $$\frac{I_1^{2}}{I_2}$$ is < $$I_1\,$$, so

$$K_2 \lt \tfrac12 I_1\omega_1^{2}=K_1$$

Thus the kinetic energy decreases when $$I$$ increases at constant external torque (zero torque).

Statement 1 claims that kinetic energy increases, which is opposite to what we have derived. Therefore Statement 1 is false.

Checking Statement 2
Statement 2 merely quotes the standard relations $$L = I\omega$$ and $$K = \tfrac12 I\omega^{2}$$. These formulas are correct, so Statement 2 is true.

We therefore have:
Statement 1 - false;  Statement 2 - true.

Option B which states “Statement 1 is false, Statement 2 is true” is the correct choice.