Let -1-2-n-1 where - i -0 for i -1-2- - n - If - a i -1 and - is a complex cube root of unity- then -1 a 1-2 a 2-2-n a n-n- cannot exceedA- 0B- 1C- 2D- - a 1- a 2- a n -

The correct options are B 1 C 2 D |a_1|+|a_2|+...+|a_n||λ_1a_1ω+λ_2a_2ω^2+...+λ_na_nω^n| ≤ |λ_1a_1ω|+|λ_2a_2ω^2|+...+|λ_na_nω^n| =|λ_1||a_1||ω|+|λ_2||a_2||ω^2|+ ...

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Byju's Answer

Standard XII

Chemistry

First Order Reaction

Let λ1+λ2+…+λ...

Question

Let $λ_{1} + λ_{2} + . . . + λ_{n} = 1$ where $λ_{i} > 0$ for $i = 1, 2, . . ., n$ . If $| a_{i} | < 1$ and $ω$ is a complex cube root of unity, then $| λ_{1} a_{1} ω + λ_{2} a_{2} ω^{2} + . . . + λ_{n} a_{n} ω^{n} |$ cannot exceed

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| a_{1} | + | a_{2} | + . . . + | a_{n} |

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Solution

The correct options are
B $1$
C $2$
D $| a_{1} | + | a_{2} | + . . . + | a_{n} |$
$| λ_{1} a_{1} ω + λ_{2} a_{2} ω^{2} + . . . + λ_{n} a_{n} ω^{n} |$
$\leq | λ_{1} a_{1} ω | + | λ_{2} a_{2} ω^{2} | + . . . + | λ_{n} a_{n} ω^{n} |$
$= | λ_{1} | | a_{1} | | ω | + | λ_{2} | | a_{2} | | ω^{2} | + . . . + | λ^{n} | | a_{n} | | ω^{n} |$
As $| ω | = 1$ and $λ_{i} \geq 0$ , we get
$= λ_{1} | a_{1} | + λ_{2} | a_{2} | + . . . + λ_{n} | a_{n} | . . . (1)$
$< λ_{1} + λ_{2} + . . . + λ_{n}$
$= 1$

Also, as $λ_{1} + λ_{2} + . . . + λ_{n} = 1$ and $λ_{i} \geq 0$ , we can infer that none of the $λ_{i}$ can exceed $1$ , thus $0 \leq λ_{i} \leq 1$ .
$∴$ from $(1)$
$| λ_{1} a_{1} ω + λ_{2} a_{2} ω^{2} + . . . + λ_{n} a_{n} ω^{n} | < | a_{1} | + | a_{2} | + . . . | a_{n} |$

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Let λ1+λ2+...+λn=1 where λi>0 for i=1,2,...,n. If |ai|<1 and ω is a complex cube root of unity, then |λ1a1ω+λ2a2ω2+...+λnanωn| cannot exceed

Let $λ_{1} + λ_{2} + . . . + λ_{n} = 1$ where $λ_{i} > 0$ for $i = 1, 2, . . ., n$ . If $| a_{i} | < 1$ and $ω$ is a complex cube root of unity, then $| λ_{1} a_{1} ω + λ_{2} a_{2} ω^{2} + . . . + λ_{n} a_{n} ω^{n} |$ cannot exceed