A) \[\text{p}{{\text{T}}^{\text{2}}}=\text{constant}\]
B) \[\left[ \frac{nRT}{V} \right]{{T}^{2}}=\text{constant}\]
C) \[{{T}^{3}}{{V}^{-1}}=\text{constant}\]
D) \[\frac{3{{T}^{2}}}{V}dT-\frac{{{T}^{3}}}{{{V}^{2}}}dV=0\]
Correct Answer: B
Solution :
Let \[l\], m, n be the direction cosines. Then,\[l=\cos \theta ,m=\cos \beta \,\,and\,\,n=\cos \theta \] \[\because \] \[{{l}^{2}}+{{m}^{2}}+{{n}^{2}}=1\] \[\therefore \] \[{{\cos }^{2}}\theta +{{\cos }^{2}}\beta +{{\cos }^{2}}\theta =1\] \[\Rightarrow \] \[2{{\cos }^{2}}\theta +1-{{\sin }^{2}}\beta =1\] \[\Rightarrow \] \[2{{\cos }^{2}}\theta -{{\sin }^{2}}\beta =0\] \[\Rightarrow \] \[2{{\cos }^{2}}\theta -3{{\sin }^{2}}\theta =0\] \[[\because {{\sin }^{2}}\beta =3{{\sin }^{2}}\theta ,given]\] \[\Rightarrow \] \[{{\tan }^{2}}\theta =\frac{2}{3}\] \[\therefore \] \[{{\cos }^{2}}\theta =\frac{1}{1+{{\tan }^{2}}\theta }=\frac{1}{1+\frac{2}{3}}=\frac{3}{5}\]
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