Instantaneous Velocity in 1D

Q. A particle starts from origin $\text{O}$ from rest and moves with a uniform acceleration along the positive $x-$axis. Identify all figures that correctly represent the motion quantitatively.
($a=$ acceleration, $v=$ velocity, $x=$ displacement, $t=$ time)

1. $\text{(A), (B), (C)}$
2. $\text{(A)}$
3. $\text{(A), (B), (D)}$
4. $\text{(B), (C)}$

Q. A boy, standing inside a box sliding on a smooth inclined plane, throws a ball and catches it back at the same level. The unit vector along the velocity of the ball with respect to the box velocity will be

1. $\hat{j}$
2. $\frac{\sqrt{3}}{2}\hat{i}+\frac{\hat{j}}{2}$
3. $\frac{\hat{i}}{2}+\frac{\sqrt{3}}{2}\hat{j}$
4. $\frac{\hat{i}}{\sqrt{2}}+\frac{\hat{j}}{\sqrt{2}}$

Q. Two towns $\text{A}$ and $\text{B}$ are connected by a regular bus service with a bus leaving in either direction every $\text{T}\text{minutes}$. A man cycling with a speed $20\text{km/hr}$ in the direction $A$ to $B$ notices that a bus goes past him every $18\text{minutes}$ in the direction of his motion, and every $6\text{minutes}$ in the opposite direction. What is the period $T$ of the bus service and with what speed do the buses ply on the road. Assume the buses ply with constant speed on the road.

1. $T=6\text{min}$
2. $T=9\text{min}$
3. $V=40\text{km/hr}$
4. $V=60\text{km/hr}$

Q. The position of a particle as a function of time, $t$ is given by $x\left(t\right)=at+b{t}^2-c{t}^3$, where, $a,b$ and $c$ are constants. When the particle attains zero acceleration, then its velocity will be:

1. $a+\frac{b^2}{4c}$
2. $a+\frac{b^2}{c}$
3. $a+\frac{b^2}{2c}$
4. $a+\frac{b^2}{3c}$

Q. Two towns $\text{A}$ and $\text{B}$ are connected by a regular bus service with a bus leaving in either direction every $\text{T}\text{minutes}$. A man cycling with a speed $20\text{km/hr}$ in the direction $A$ to $B$ notices that a bus goes past him every $18\text{minutes}$ in the direction of his motion, and every $6\text{minutes}$ in the opposite direction. What is the period $T$ of the bus service and with what speed do the buses ply on the road. Assume the buses ply with constant speed on the road.

1. $T=6\text{min}$
2. $T=9\text{min}$
3. $V=40\text{km/hr}$
4. $V=60\text{km/hr}$