The condition to prevent the block from falling is that the friction force must be at least equal to the weight of the block:

$$f_s \geq mg$$

Since static friction is given by $f_s = \mu N$ and the normal force is due to pseudo force $N = ma$, we get:

$$\mu ma \geq mg$$

Dividing both sides by $\mu m$:

$$a \geq \frac{g}{\mu}$$

Thus, the required acceleration of the cart is $a \geq \frac{g}{\mu}$.

When the truck accelerates to the right with $a$, a pseudo force $ma$ acts to the left on the bob in the truck's frame. The bob reaches equilibrium where the tension components balance forces:

$$\tan \theta = \frac{\text{Pseudo force}}{\text{Weight}} = \frac{ma}{mg} = \frac{a}{g}$$ $$\theta = \tan^{-1} \left(\frac{a}{g} \right)$$

Thus, the pendulum tilts to the left at an angle $\theta = \tan^{-1}(a/g)$ with the vertical.

Assertion (A) is false. The Earth rotates and revolves, making a frame attached to it non-inertial. Reason (R) is false. Newton's laws in their original form are only valid in inertial frames. For a frame attached to the Earth, pseudo forces must be introduced to apply Newton's laws. Thus, both the Assertion and the Reason are false.

Assertion (A) is true. This is a fundamental statement of the Galilean principle of relativity. Reason (R) is true. The laws of physics are invariant in all inertial frames of reference. (R) correctly explains (A) because if physical laws are identical in all inertial frames, no internal experiment can distinguish between them.

Assertion (A) is true: When an elevator accelerates upwards with acceleration \(a\), the pseudo force on an object inside is \(ma\) downwards (in the non-inertial frame). So, for upward accelerating elevator, pseudo force on a block is downward.nReason (R) is false: Pseudo force is not a real force applied by the lift; it is an inertial force experienced in a non-inertial reference frame, acting opposite to the acceleration of the frame, not necessarily the direction of motion. Therefore, (A) is true but (R) is false.

Assertion (A) is true: If the elevator accelerates upwards, the apparent weight \(N = m(g+a)\) of the peanuts increases. If the moongphaliwala sells by the apparent weight reading (e.g., "1 kg" on the scale), they would be selling a *smaller actual mass* \(m_text{actual} = N/(g+a)\) for the same indicated weight. Thus, they gain more profit.

Reason (R) is true: When an elevator accelerates upwards, the normal force (apparent weight) on an object of mass \(m\) is \(N = m(g+a)\), which is greater than its actual weight \(mg\). Reason (R) correctly explains why the apparent weight increases, leading to the profit gain described in (A).

Assertion (A) is true. Earth is a non-inertial frame, so frames observing a particle on Earth can also be non-inertial. Reason (R) is false because a frame attached to Earth is technically non-inertial due to Earth's rotation and orbital motion.

Assertion (A) is false. Pseudo forces are fictitious and do not constitute action-reaction pairs with real forces like friction, as they are not interaction forces.
Reason (R) is true. According to Newton's third law, action and reaction forces always act on different bodies. Since (A) is false and (R) is true, and no option directly reflects this, the closest option, given (A) is false, is (4), implying both are false, despite (R) being true. This indicates a potential flaw in the provided options.

When a lift accelerates upwards, apparent weight is \(N = m(g + a)\), which is greater than \(mg\). So (A) is true. Reason (R) provides the formula for downward acceleration, thus (R) is false.