If a and b are the roots of \[6{{x}^{2}}-6x+1=0,\] then the value of \[\frac{1}{2}\left[ \,a+b\alpha +c{{\alpha }^{2}}+d{{\alpha }^{3}}\, \right]\] \[\frac{1}{2}\left[ \,a+b\alpha +c{{\alpha }^{2}}+d{{\alpha }^{3}}\, \right]+\frac{1}{2}\left[ \,a+b\beta +c{{\beta }^{2}}+d{{\beta }^{3}}\, \right]\] is [RPET 2000]

• A. \[\frac{1}{4}(a+b+c+d)\]
• B. \[\frac{a}{1}+\frac{b}{2}+\frac{c}{3}+\frac{d}{4}\]
• C. \[\frac{a}{2}-\frac{b}{2}+\frac{c}{3}-\frac{d}{4}\]
• D. None of these