Position of a point

Q. Instantaneous velocity of an object varies with time as $v=\alpha -\beta t^2$. Find its position, $x$ as a function of time, $t$. Also find the object`s maximum positive displacement, $x_{max}$ from the origin.

1. $x=\alpha t-\frac{\beta t^3}{3}$ and $x_{max}=\frac{2}{3}\frac{{\alpha }^{3/2}}{{\beta }^{3/2}}$
2. $x=\alpha t-\frac{\beta t^3}{3}$ and $x_{max}=\frac{2\alpha }{{\beta }^{1/2}}$
3. $x=2\alpha t-\beta$ and $x_{max}=2\alpha \beta$
4. $x=2{\alpha }^{2}t-\beta t^3$ and $x_{max}=\frac{2{\alpha }^{1/2}}{{\beta }^{3/2}}$

Q. The position of point $B$ with respect to point $A$ is

1. $-13$
2. $13$
3. $6$
4. $-7$

Q. The figure shows the graph of the $x-$coordinate of a particle moving along the $x-$axis as a function of time. Average velocity during $t=0$ to $t=6\text{s}$ is

1. $10\text{m/s}$
2. $60\text{m/s}$
3. $5\text{m/s}$
4. $0$

Q. A particle is moving along positive $x-$ axis with velocity $v=A-Bt$, where $A$ and $B$ are positive constants and '$t$' is in seconds. If the particle is at the origin initially, then find the coordinates of the position of the particle at $t=\frac{3A}{B}$.

1. $(0,\frac{3A^2}{2B})$
2. $(\frac{3A^2}{2B},0)$
3. $(0,\frac{-3A^2}{2B})$
4. $(\frac{-3A^2}{2B},0)$

Q. A body of mass $2\text{kg}$ moves under a force of $(2\hat{i}+3\hat{j}+5\hat{k})\text{N}$. It starts from rest and was at the origin initially. After $4\text{s}$, its new coordinates are $(8,b,20)$. The value of $b$ is .
(Round off to the Nearest Integer)

Q. Initially the origin was at $X=0$ and position of a particle at point $P$ was given as $X=-3\text{m}$, if origin is shifted by $1$ unit to the left, then position of point $P$ with respect to new origin will be:

1. $4\text{m}$
2. $-4\text{m}$
3. $-2\text{m}$
4. $-3\text{m}$

Q. The position of point $B$ with respect to point $A$ is

1. $-13$
2. $13$
3. $6$
4. $-7$

Q. At $t=0$, a particle starts from $(4,0)$ and moves towards positive $x-$axis with speed of $v=8t+3\text{m/s}$. The final position of the particle as a function of time is

1. $4t^2-3t-4\text{m}$
2. $4t^2-3t+4\text{m}$
3. $4t^2+3t+4\text{m}$
4. $4t^2-3t\text{m}$

Q. In the given figure, what is the position of point A from origin ?