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Location of Roots

Let r⃗⃗1⃗, ...

Question

Let $_{1},_{2},_{3}, . . . . . . ._{n}$ be the position vectors of points $P_{1}, P_{2}, P_{3}, . . . . . ., P_{n}$ relative to the origin $O$ . If the vector equation $a_{1}_{1} + a_{2}_{2} + . . . . . . + a_{n}_{n} = 0$ holds, then a similar equation will also hold w.r.t. to any other origin provided

A

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

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B

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

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C

a_{1} + a_{2} + . . . . + a_{n} = 0

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D

a_{1} = a_{2} = a_{3} = . . . . = a_{n} = 0

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Solution

The correct option is C $a_{1} + a_{2} + . . . . + a_{n} = 0$
Given $a_{1}_{1} + a_{2}_{2} + . . . . . + a_{n}_{n} = 0$
Now $\to a + \to r_{1}^{'} =_{1}$ and so on
Hence, $a_{1} (\to a + \to r_{1}^{'}) + a_{2} (\to a + \to r_{2}^{'}) + . . . . + a_{n} (\to a + \to r_{n}^{'}) = 0$
$a_{1} \to r_{1}^{'} + a_{2} \to r_{2}^{'} + . . . . + a_{n} \to r_{n}^{'} + \to a (a_{1} + a_{2} + . . . . + a_{n}) = 0$
Hence, $a_{1} \to r_{1}^{'} + a_{2} \to r_{2}^{'} + . . . . + a_{n} \to r_{n}^{'} = 0$
if $a_{1} + a_{2} + . . . . + a_{n} = 0.$

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Similar questions

_{1},_{2},_{3} . . . . . . . . . ._{n},

be the position vectors of points

P_{1^{'}} P_{2^{'}} P_{3^{'}} . . . . . . . . P_{n}

relative to the origin O. If the vector equation

a_{1}_{1} + a_{2}_{2} + . . . . . . . . . + a_{n}_{n} = 0

holds, then a similar equation will also hold w.r.t. to any other origin provided:

a_{1} > a_{2} > a_{3} > . . . . > a_{n} > 1.

p_{1} > p_{2} > p_{3} > . . . . > p_{n} > 0

p_{1} + p_{2} + p_{3} + . . . + p_{n} = 1.

F (x) = (p_{1} a_{1}^{x} + p_{2} a_{2}^{x} + . . . + p_{n} a_{n}^{x})^{1 / x}

{lim}_{x \to 0^{+}} F (x)

a_{1} > a_{2} > a_{3} > . . . . > a_{n} > 1.

p_{1} > p_{2} > p_{3} > . . . . > p_{n} > 0

p_{1} + p_{2} + p_{3} + . . . + p_{n} = 1.

F (x) = (p_{1} a_{1}^{x} + p_{2} a_{2}^{x} + . . . + p_{n} a_{n}^{x})^{1 / x}

lim x \to - \infty F (x)

a_{1} > a_{2} > a_{3} > . . . . > a_{n} > 1.

p_{1} > p_{2} > p_{3} > . . . . > p_{n} > 0

p_{1} + p_{2} + p_{3} + . . . + p_{n} = 1.

F (x) = (p_{1} a_{1}^{x} + p_{2} a_{2}^{x} + . . . + p_{n} a_{n}^{x})^{1 / x}

{lim}_{x \to \infty} F (x)

If $a_{1}, a_{2}, a_{3} . . . . . a_{n}$ are in A.P. Where $a_{i} > 0$ for all i, then the value of
$\frac{1}{\sqrt{a_{1}} + \sqrt{a_{2}}} + \frac{1}{\sqrt{a_{2}} + \sqrt{a_{3}}} + . . . . . . . . . + \frac{1}{\sqrt{a_{n - 1}} + \sqrt{a_{n}}} =$

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