Xavier Aptitude Test 2016 Paper

(c)${\mathit{a}}^3+\mathrm{}{\mathit{b}}^3+\mathrm{}{\mathit{c}}^3-3\mathrm{}\mathit{a}\mathit{b}\mathit{c}=\left(\mathrm{}\mathit{a}+\mathrm{}\mathit{b}+\mathrm{}\mathit{c}\right)$
$\mathrm{}\frac{1}{2}\mathrm{}\left\{{\left(\mathit{a}-\mathrm{}\mathit{b}\right)}^2+{\left(\mathrm{}\mathit{b}-\mathrm{}\mathit{c}\right)}^2+{\left(\mathrm{}\mathit{c}-\mathrm{}\mathit{a}\right)}^2\mathrm{}\right\}......$ (1)
For three confuse integers, (n – 1), n and n + 1
(a – b)${}^2$+ (b – c)${}^2$+ (c – a)${}^2$= 6 .........(2)
a + b + c = 3x ........(3)
abc = x${}^3$– x .........(4)
Now, $\mathrm{}\frac{{\mathit{a}}^3+\mathrm{}{\mathit{b}}^3+{\mathit{c}}^3+3\mathrm{}\mathit{a}\mathit{b}\mathit{c}}{{\left(\mathit{a}+\mathrm{}\mathit{b}+\mathrm{}\mathit{c}\right)}^2}$
$=\mathrm{}\frac{\mathrm{}{\mathit{a}}^3+\mathrm{}{\mathit{b}}^3+\mathrm{}{\mathit{c}}^3-3\mathit{a}\mathit{b}\mathit{c}+6\mathit{a}\mathit{b}\mathit{c}}{{\left(\mathrm{}\mathit{a}+\mathrm{}\mathit{b}+\mathrm{}\mathit{c}\right)}^2}$
$=\mathrm{}\frac{3\mathit{x}\times \frac{1}{2}\times 6+6\mathrm{}\left({\mathit{x}}^3-\mathit{x}\right)}{{\left(3\mathit{x}\right)}^2}$
$=\frac{9\mathit{x}+6{\mathit{x}}^3-6\mathit{x}}{9{\mathit{x}}^2}=\mathrm{}\frac{1}{3\mathit{x}}+\mathrm{}\frac{2\mathit{x}}{3}$
For the expression, integer value is possible only ifx = 1 and x = – 1.
∴ Possible values of a, b, c are (0, 1, 2) and (–2, –1, 0)