A meter bridge set up as shown to determine end correction at $A$ and $B$ . When a resistance of $15\,\Omega $ is used in left gap and of $20\,\Omega $ in right gap, then null point comes at a distance $42\ cm$ from $A$ . When these resistances are interchanged null point comes at a distance $57\ cm$ from $A$ . Values of end corrections are
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Let and corrections at $\mathrm{A}$ and $\mathrm{B}\, \alpha$ and $\beta$
$\frac{15}{42+\alpha}=\frac{20}{58+\beta} \Rightarrow \frac{58+\beta}{42+\alpha}=\frac{4}{3}$
$\Rightarrow 3 \beta+174=168+4 \alpha$
$4 \alpha-3 \beta=6$ ............$(i)$
$\frac{20}{57+\alpha}=\frac{15}{43+\beta} \Rightarrow \frac{57+\alpha}{43+\beta}$
$=\frac{4}{3} \Rightarrow 171+3 \alpha=172+4 \beta$
$3 \alpha-4 \beta=1$ ............$(ii)$
from $(i)$ and $(ii)$
$\alpha=3 \mathrm{\,cm}, \beta=2 \mathrm{\,cm}$
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