KVPY Sample Paper KVPY Stream-SX Model Paper-28

Let \[\omega \] be a complex cube root of unity with \[\omega \ne 1.\] A fair die is thrown three times. If \[{{r}_{1}},{{r}_{2}}\] and \[{{r}_{3}}\] are the numbers obtained on the die, then the Probability that \[{{\omega }^{{{r}_{1}}}}+{{\omega }^{{{r}_{2}}}}+{{\omega }^{{{r}_{3}}}}=0\]is

A) 1/18

B) 1/9

C) 2/9

D) 1/36

Correct Answer: C

Solution :

\[{{r}_{1}},{{r}_{2}},{{r}_{3}}\in \left\{ 1,2,3,4,5,6 \right\}\]

\[{{r}_{1}},{{r}_{2}},{{r}_{3}}\].are of the form \[3k,3k+1,3\text{ }k+2\]

Required probability

\[a=\frac{3!\times {{\,}^{2}}{{C}_{1}}\times {{\,}^{2}}{{C}_{1}}\times {{\,}^{2}}{{C}_{1}}}{6\times 6\times 6}=\frac{6\times 8}{216}=\frac{2}{9}.\]