Recall the moments of inertia of a uniform circular disc about axes through its center:

• About any diameter in its plane: $I_x = I_y = \frac{1}{4}MR^2$
• About the axis perpendicular to the plane through the center: $I_z = \frac{1}{2}MR^2$

Use the definition of radius of gyration:

If moment of inertia about an axis is $I$, then radius of gyration $k$ is defined by $I = Mk^2 \implies k = \sqrt{\frac{I}{M}}$

Find radii of gyration about the three axes:

• About $x$-axis: $k_x = \sqrt{\frac{I_x}{M}} = \sqrt{\frac{1}{4}R^2} = \frac{R}{2}$

• About $y$-axis: $k_y = \sqrt{\frac{I_y}{M}} = \sqrt{\frac{1}{4}R^2} = \frac{R}{2}$

• About $z$-axis: $k_z = \sqrt{\frac{I_z}{M}} = \sqrt{\frac{1}{2}R^2} = \frac{R}{\sqrt{2}}$

Compare them:

$k_x = k_y = \frac{R}{2}, \qquad k_z = \frac{R}{\sqrt{2}}$

Clearly, $\frac{R}{2} \ne \frac{R}{\sqrt{2}}$

So the radii of gyration about the three axes are not the same.

Hence, Assertion A is false.

Check Reason R:

Reason says: A rigid body making rotational motion has fixed mass and shape.

This is true: by definition, a rigid body has fixed shape and size, and its mass is fixed.

Hence, Reason R is true.

Choose the correct option:

• Assertion A: False
• Reason R: True

Therefore, the correct option is: $\boxed{\text{B}}$