Let $\alpha = \frac{{ - 1 + i\sqrt 3 }}{2}.$ If $a\, = (1 + \alpha )\sum\limits_{k = 0}^{100} {{\alpha ^{2k}}\,\,{\rm{and}}\,\,b\, = \,} \sum\limits_{k = 0}^{100} {{\alpha ^{3k}}\,\,} ,$ then $a$ and $b$ the roots of the quadratic equation.

JEE MATHEMATICS MCQ: QUADRATIC EQUATIONS

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Analyzing +4 | -1

Let

\alpha = \frac{{ - 1 + i\sqrt 3 }}{2}.

a\, = (1 + \alpha )\sum\limits_{k = 0}^{100} {{\alpha ^{2k}}\,\,\mathrm{and}\,\,b\, = \,} \sum\limits_{k = 0}^{100} {{\alpha ^{3k}}\,\,} ,

then

a

and

b

the roots of the quadratic equation.

View answer and explanation

Correct option: C

{x^2} - 102x + 101 = 0

Option C (

{x^2} - 102x + 101 = 0

) is correct because it is the quadratic equation formed with roots a=1 and b=101, derived from the properties of the complex cube root of unity.

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