$\begin{array}{1 1}(A)\;t \bigg(\frac{\sqrt 2 -1}{\sqrt 2}\bigg)\\(B)\;t^2 \bigg(\frac{\sqrt 2 -1}{\sqrt 2}\bigg) \\(C)\;t^3 \bigg(\frac{\sqrt 2 +1}{\sqrt 2}\bigg) \\(D)\;t \bigg(\frac{\sqrt 2 +1}{\sqrt 2}\bigg)\end{array} $

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Let the time taken to cover the first half portion of the journey be $t_1$ and for the second half portion be $ t_2$ . Then

$t=t_1+t_2$

For the first half-journey, putting

$S= L/2,u=0$ etc. in $S= ut+ \large\frac{1}{2} $$at^2$, we get $\large\frac{L}{2} $$=0+\frac{1}{2} $$at_1^2$

$\large\frac{L}{2}$$=0+ \large\frac{1}{2}$$at_1^2$

As the time to complete the whole journey is t, therefore ,

$L= 0+ \frac{1}{2}$$at^2$

Dividing both equations :

$L= 0+ \large\frac{1}{2}$$=> t_1 =\large\frac{t}{\sqrt{2}}$

$t_2= (t-t_1)=t- \frac{t}{\sqrt 2}=t \bigg[\large\frac{\sqrt 2-1}{\sqrt 2}\bigg]$

Hence A is the correct answer.

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