A particle starts from the origin at time \(t=0 \) with an initial velocity of \(5\hat{j}~\text{ms}^{-1}. \) It moves in the \(XY \text-\)plane under a constant acceleration of \(\left(10\hat{i}+4\hat{j}\right)~\text{ms}^{-2} .\) At some later time \(t,\) the coordinates of the particle are \((20~\text{m}, y_0~\text{m}). \) The values of \(t \) and \(y_0 \)​ are, respectively:
1. \(4~\text{s}\) and \(52~\text{m}\)
2. \(5~\text{s}\) and \(25~\text{m}\)
3. \(2~\text{s}\) and \(18~\text{m}\)
4. \(2~\text{s}\) and \(24~\text{m}\)

The position vector of a moving particle at time t is $\overrightarrow{\mathrm{r}}<mspace\; width="1em"></mspace>=<mspace\; width="1em"></mspace>\mathrm{a}\hat{\mathrm{i}}<mspace\; width="1em"></mspace>+<mspace\; width="1em"></mspace>4{\mathrm{t}}^2<mspace\; width="1em"></mspace>\widehat{\mathrm{j}<mspace\; width="1em"></mspace>}-<mspace\; width="1em"></mspace>{\mathrm{t}}^{3<mspace\; width="1em"></mspace>}\hat{\mathrm{k}}$. Its displacement during the time interval, t = 1 s to t = 3 s is :

1.  $\hat{\mathrm{i}}<mspace\; width="1em"></mspace>+<mspace\; width="1em"></mspace>\hat{\mathrm{j}}$

2.  $3\hat{\mathrm{i}}<mspace\; width="1em"></mspace>+<mspace\; width="1em"></mspace>4\hat{\mathrm{j}}-<mspace\; width="1em"></mspace>\hat{\mathrm{k}}$

3.  $9\hat{\mathrm{i}}<mspace\; width="1em"></mspace>+<mspace\; width="1em"></mspace>36\hat{\mathrm{j}}-<mspace\; width="1em"></mspace>27\hat{\mathrm{k}}$

4.  $32\hat{\mathrm{j}}-<mspace\; width="1em"></mspace>26\hat{\mathrm{k}}$

The position of a particle at time \(t\) is given by, \(x=3t^3\), \(y=2t^2+8t\), and \(z=6t-5\). The initial velocity of the particle is: