The time to cross the river depends on the component of velocity perpendicular to the river flow (\(v_{\text{normal}}\)). For simultaneous crossing, \(v_{\text{normal}}\) must be equal for both boats. If total speed relative to river is same, and \(v_{\text{normal}}\) is same, then the magnitude of the parallel component is also same. Different paths result from different directions of the parallel component. Thus, (A) is true and (R) is true and explains (A).

Assertion (A): Initial velocity \(\vec{u} = (10 cos 37^\circ)\hat{i} + (10 sin 37^\circ)\hat{j} approx 8\hat{i} + 6\hat{j}\). Velocity at time (t) is \(\vec{v}(t) = 8\hat{i} + (6 - gt)\hat{j}\). For perpendicularity, \(\vec{u} \cdot \vec{v} = 0 ⇒ 64 + 6(6 - gt) = 0 ⇒ 100 - 6gt = 0\). With \(g=10\text{ m/s}^2), (t = 100/60 = 5/3\text{ s}\). The assertion states \(4/3\text{ s})\, so (A) is False.
Reason (R): The dot product of two perpendicular vectors is indeed zero. So (R) is True.
Since (A) is false and (R) is true, none of the given options are strictly correct. However, if (A) is false, options (1), (2), (3) are ruled out, leaving (4) by elimination, despite (R) being true.

Assertion (A): If the separation \(\vec{r}_{rel}\) is constant, it implies \(\vec{r}_{rel} \cdot \vec{v}_{rel} = 0\), meaning                         \(\vec{v}_{rel}\) is perpendicular to \(\vec{r}_{rel}\), but not necessarily zero (e.g., two particles orbiting each other at constant distance). So (A) is False.

Reason (R): Relative velocity is defined as the time derivative of the relative position vector. So (R) is True.

Since (A) is false and (R) is true, none of the given options are strictly correct. However, if (A) is false, options (1), (2), (3) are ruled out, leaving (4) by elimination, despite (R) being true.

Assertion (A): The relative velocity is \(\vec{V}_{AB} = \vec{V}_A - \vec{V}_B\). If \(\vec{V}_B)\) is in the opposite direction to \(vec{V}_A\), then \(|vec{V}_{AB}| = |vec{V}_A| + |vec{V}_B|\), which is greater than (\|vec{V}_A|\). Thus, (A) is False.

Reason (R): The definition of relative velocity of A with respect to B is \(\vec{V}_{AB} = \vec{V}_A - \vec{V}_B\). So (R) is True.

Since (A) is false and (R) is true, none of the given options are strictly correct. However, if (A) is false, options (1), (2), (3) are ruled out, leaving (4) by elimination, despite (R) being true.

Assertion (A): The range formula is \(R = \frac{u^2 sin(2\theta)}{g}\). For angle \((90^\circ - \theta)\), \(R' = \frac{u^2 sin(2(90^\circ - \theta))}{g} = \frac{u^2 sin(180^\circ - 2\theta)}{g} = \frac{u^2 sin(2\theta)}{g} = R\). So (A) is True.
Reason (R): The horizontal range clearly depends on the angle of projection (theta) via (sin(2theta)). So (R) is False.
Since (A) is true and (R) is false, option (3) is correct.

Assertion (A): In ideal projectile motion, speed decreases on the way up and increases on the way down, reaching a minimum at the peak. It does not always decrease. So (A) is False.
Reason (R): In the presence of air drag, the drag force depends on velocity, making the net acceleration non-constant. Thus, it is not uniformly accelerated motion. So (R) is False.
Since both (A) and (R) are false, option (4) is correct.

Assertion (A): Time of flight \(T = \frac{2u sin\theta}{g}\). If (u) is replaced by (nu), (T' = nT). So (A) is True.

Reason (R): Horizontal range \(R = \frac{u^2 sin(2\theta)}{g}\). If (u) is replaced by (nu), (R' = n^2 R). So (R) is True.

Both statements are true. However, the scaling of range (R) does not explain the scaling of time of flight (A). They are independent consequences of initial speed scaling. So (R) is not the correct explanation for (A). Option (2) is correct.

Assertion (A): Maximum range occurs at (theta = 45^circ). The largest time of flight occurs at (theta = 90^circ). Since these angles are different, the assertion is false. So (A) is False.
Reason (R): Horizontal range is maximum at (theta = 45^circ) (when (sin(2theta)=1)). At (theta = 90^circ), the range is zero. So (R) is False.
Since both (A) and (R) are false, option (4) is correct.

Assertion (A) is false because positive acceleration doesn't always mean increasing speed; it depends on the direction of velocity. Speed increases only when \(vec{a}\) and \(vec{v}\) are in the same direction. Reason (R) is false because acceleration is the rate of change of velocity, not speed.

Assertion (A) is false; under constant acceleration, the trajectory can be a parabola (projectile motion) or a straight line depending on initial velocity. Reason (R) is false; the trajectory's shape depends on both initial velocity and acceleration.