The total input power is \(P_{\text{in}} = \frac{dm}{dt} gh = 15 \times 10 \times 60 = 9000\text{ W} = 9\text{ kW}\). Given that \(10%\) of energy is lost to friction, the efficiency is \(90%\), so the power generated is \(P_{\text{out}} = 0.9 \times 9\text{ kW} = 8.1\text{ kW}\).

According to the work-energy theorem, \(W = \Delta K = \frac{1}{2}m(v_f^2 - v_i^2)\). Substituting \(v_i = 10\text{ m/s}\) and \(v_f = 30\sqrt{3}\text{ m/s}\), we obtain \(W = \frac{1}{2} \times 1 \times (2700 - 100) = 1300\text{ J}\).

Since velocity is constant, the external force must balance gravity along the incline: \(F = mg sin theta\). The work done is \(W = F \cdot d = 10 \times 10 \times sin(30^\circ) \times 10 = 500\text{ J}\).

At the equilibrium position of a simple harmonic oscillator, the net restoring force acting on the particle is zero. Setting \( F = 15 - 3x = 0 \) yields \( 3x = 15 ⇒ x = 5 \text{ m} \).

Since linear momentum \(p = \sqrt{2mK}\) and kinetic energy \(K\) is constant, \(p \propto \sqrt{m}\). Therefore, the ratio of momenta is \(\frac{p_1}{p_2} = \sqrt{\frac{1}{9}} = 1:3\).

The energy stored in a spring is \(U = \frac{1}{2}kx^2\). If stretched by 1 cm, \(U_1 = 50\text{ J}\). If further stretched by 1 cm, total stretch becomes 2 cm, so the stored energy is \(U_2 = \frac{1}{2}k(2x)^2 = 4U_1 = 200\text{ J}\).

Work done by conservative forces in a closed loop is zero. For all forces, it depends on energy change and non-conservative forces. Force is required to move a body, especially to overcome inertia or friction. Therefore, both Assertion (A) and Reason (R) are false.

Assertion is true by definition of a conservative force, which does zero work over any closed path.

Reason is also true as displacement is zero for a closed path, i.e., \(\Delta \vec{r} = 0\). However, the reason is not the correct explanation for why work by a conservative force is zero; this is due to path independence and \(\Delta U = 0\) for a closed path.

Thus, both are true, but R is not the correct explanation of A.

Assertion is true: Non-conservative forces like friction or air resistance dissipate mechanical energy into other forms (e.g., heat).

Reason is false: The work done by a non-conservative force is not always negative. For example, an applied external force can be non-conservative and do positive work on a system.

Thus, A is true but R is false.

Assertion is false: Mechanical energy \(E = K + U\) is conserved when only conservative forces do work. External conservative forces (e.g., gravity) can act and do work, yet mechanical energy can be conserved.

Reason is false: If no non-conservative forces act, then \(\Delta K + \Delta U = 0\), meaning mechanical energy is conserved. In this scenario, the work done by external forces \(W_{ext}\) is not necessarily zero. For instance, gravity does work when an object falls, but \(E\) is conserved.

Therefore, both Assertion and Reason are false.