To determine the correctness of the statements, we need to analyze the stability of the closed-loop system with different controllers. The system configuration is shown in the figure above.

Let's denote the transfer function of the system as \(G(s) = \frac{1}{10s^2}\). The open-loop transfer function with controller \(M(s)\) is given by:

\(L(s) = M(s)G(s)\)

For the system to be stable, the closed-loop characteristic equation \(1 + L(s) = 0\) must have poles with negative real parts.

Analysis of Statement (i):

Statement (i) asserts that there is no controller of the form \(M(s) = \frac{K_I}{s}\) (Integral controller) such that the closed-loop system is stable.

Substitute \(M(s) = \frac{K_I}{s}\) into the open-loop transfer function:

\(L(s) = \frac{K_I}{s} \cdot \frac{1}{10s^2} = \frac{K_I}{10s^3}\)

The characteristic equation becomes:

\(1 + \frac{K_I}{10s^3} = 0\)

This simplifies to:

\(10s^3 + K_I = 0\)

The roots of this characteristic equation are \(s = -\sqrt[3]{\frac{K_I}{10}}\). All roots are on the real axis, but one root will be positive, causing instability for \(K_I > 0\). Thus, statement (i) is TRUE.

Analysis of Statement (ii):

Statement (ii) claims there is at least one controller of the form \(M(s) = K_P + sK_D\) (Proportional-Derivative controller) which makes the system stable.

Substitute \(M(s) = K_P + sK_D\) into the open-loop transfer function:

\(L(s) = (K_P + sK_D) \cdot \frac{1}{10s^2} = \frac{K_P}{10s^2} + \frac{K_D}{10s}\)

The characteristic equation is:

\(10s^2 + K_Ds + K_P = 0\)

This is a standard second-order polynomial. For stability, the roots must have negative real parts. This occurs if the coefficients meet the Routh-Hurwitz criterion. Specifically, both \(K_D > 0\) and \(K_P > 0\) are necessary for stability.

By choosing appropriate positive values for \(K_P\) and \(K_D\), stability can be achieved, making statement (ii) TRUE.

Therefore, the correct option is: Both (i) and (ii) are TRUE.