The velocity-time graph of a particle in one dimensional motion is shown in the figure. Which of the following formulae is correct for describing the motion of the particle over the time interval t₁ to t₂? 

From graph it is clear that the slope of the Velocity-time ( acceleration) is changing. So the motion is a non-uniformly accelerated motion.

$\mathrm{x}\left({\mathrm{t}}_2\right)=\mathrm{x}\left({\mathrm{t}}_1\right)+\mathrm{v}\left({\mathrm{t}}_1\right)\left({\mathrm{t}}_2-{\mathrm{t}}_1\right)+\left(\frac{1}{2}\right)\mathrm{a}{\left({\mathrm{t}}_2-{\mathrm{t}}_1\right)}^2$,

$\mathrm{v}\left({\mathrm{t}}_2\right)=\mathrm{v}\left({\mathrm{t}}_1\right)+\mathrm{a}\left({\mathrm{t}}_2-{\mathrm{t}}_1\right)$,

${\mathrm{v}}_{\text{average }}=\frac{\left(\mathrm{x}\left({\mathrm{t}}_2\right)+\mathrm{x}\left({\mathrm{t}}_1\right)\right)}{\left({\mathrm{t}}_2-{\mathrm{t}}_1\right)}$

 are valid only for motion under constant acceleration. 

Only ${\mathrm{a}}_{\text{average }}=\frac{\left(\mathrm{v}\left({\mathrm{t}}_2\right)-\mathrm{v}\left({\mathrm{t}}_1\right)\right)}{\left({\mathrm{t}}_2-{\mathrm{t}}_1\right)}$ will be valid for non-uniform acceleration.