If $\left| {{a_k}} \right| < 1,\;{\lambda _k} \ge 0\;\,{\rm{for}}\,\;k = 1,\;2, \ldots ,n\;\,{\rm{and}}\,\;{\lambda _1} + {\lambda _2} + ....{\lambda _n} = 1,$ $|{\lambda _1}{a_1} + {\lambda _2}{a_2} + ... + {\lambda _n}{a_n}|$ is

JEE MATHEMATICS MCQ: COMPLEX NUMBERS

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Application +4 | -1

\left| {{a_k}} \right| < 1,\;{\lambda _k} \ge 0\;\,\mathrm{for}\,\;k = 1,\;2, \ldots ,n\;\,\mathrm{and}\,\;{\lambda _1} + {\lambda _2} + ....{\lambda _n} = 1,

|{\lambda _1}{a_1} + {\lambda _2}{a_2} + ... + {\lambda _n}{a_n}|

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Correct option: D

Less than one

Option D (Less than one) is correct because the absolute value of the sum is less than the sum of the absolute values, and since each |a_k| < 1 and the sum of λ_k is 1, the result is less than 1.

JEE MATHEMATICS MCQ: COMPLEX NUMBERS

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