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KVPY Sample Paper KVPY Stream-SX Model Paper-23

  • question_answer
    Let w be a complex cube root of unity with \[w\ne 1.\] 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 \[{{w}^{{{r}_{1}}}}+{{w}^{{{r}_{2}}}}+{{w}^{{{r}_{3}}}}=0\]

    A) \[\frac{1}{18}\]

    B) \[\frac{1}{9}\]

    C) \[\frac{2}{9}\]

    D) \[\frac{1}{36}\]

    Correct Answer: C

    Solution :

    \[{{r}_{1}},{{r}_{2,}},{{r}_{3}}\in \{1,2,3,4,5,6\}\] \[{{r}_{1}},{{r}_{2,}},{{r}_{3}}\] are forms of \[\left( 3m,3m+1,3m+2 \right)\] \[\therefore \] Required probability\[=\frac{3!\times {}^{2}{{C}_{1}}\times {}^{2}{{C}_{1}}\times {}^{2}{{C}_{1}}}{{{6}^{3}}}=\frac{2}{9}\]


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