A) \[\frac{\Delta t}{t}+\frac{S(\Delta t)_{{}}^{2}}{(\Delta S)t_{{}}^{2}}=\pm 1\]
B) \[\frac{\Delta t}{t}+\frac{S\Delta t_{{}}^{{}}}{\Delta St_{{}}^{{}}}=\pm 1\]
C) \[\frac{\Delta t}{t}+\frac{(\Delta S)t_{{}}^{{}}}{S(\Delta t)_{{}}^{{}}}=\pm 1\]
D) \[\frac{\Delta t}{t}+\frac{S_{{}}^{2}\Delta t_{{}}^{{}}}{(\Delta S)_{{}}^{2}t_{{}}^{{}}}=\pm 1\]
Correct Answer: A
Solution :
[a] \[V=\frac{S}{t}\] \[\Delta V=\frac{1}{t}.\frac{\partial s}{\partial s}.\Delta S+\frac{S}{{{t}^{2}}}\Delta t=\left( \frac{\Delta S}{t}+\frac{S\Delta t}{{{t}^{2}}} \right)\] So \[\frac{\Delta S}{t}+\frac{S\Delta t}{{{t}^{2}}}=\frac{\Delta S}{\Delta t}.\frac{\Delta t}{t}+\frac{(\Delta S).S(\Delta {{t}^{2}})}{(\Delta S).{{t}^{2}}(\Delta t)}\] Or \[\frac{\Delta S}{\Delta t}\left[ \frac{\Delta t}{t}+\frac{S{{(\Delta t)}^{2}}}{(\Delta S){{t}^{2}}} \right]=\pm \frac{\Delta S}{\Delta t}\](given) So, \[\frac{\Delta t}{t}+\frac{S{{(\Delta t)}^{2}}}{(\Delta S){{t}^{2}}}=\pm 1\]
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