Let -1- -2- -10 be positive valued angles in radian such that -1 - -2 - - -10 - 2 - Define the complex numbers z1 - ei-1- zk - zk-1ei-k for k - 2- 3-10- where i - -1- Consider the statements P and Q given below-P-z2-z1-z3-z2-z10-z9-z1-z10-2-Q-z22-z12-z32-z22-z102-z92-z12-z102-4-

|z_2 z_1|= nbsp;length of line AB≤ length of arc AB ⇒ |z_2 z_1|≤θ _2 Similarly, |z_3 z_2|= length of line BC ≤ length of arc BC ⇒ |z_3 z_2|≤θ _3 . . . ⇒ ...

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Byju's Answer

Standard XI

Mathematics

Properties of Argument

Let θ1, θ2,⋯,...

Question

Let $θ_{1}, θ_{2}, \dots, θ_{10}$ be positive valued angles (in radian) such that $θ_{1} + θ_{2} + \dots + θ_{10} = 2 π$ . Define the complex numbers $z_{1} = e^{i θ_{1}}, z_{k} = z_{k - 1} e^{i θ_{k}}$ for $k = 2, 3 \dots 10,$ where $i = \sqrt{- 1}$ . Consider the statements $P$ and $Q$ given below:
$P : | z_{2} - z_{1} | + | z_{3} - z_{2} | + \dots + | z_{10} - z_{9} | + | z_{1} - z_{10} | \leq 2 π$
$Q : | z_{2}^{2} - z_{1}^{2} | + | z_{3}^{2} - z_{2}^{2} | + \dots + | z_{10}^{2} - z_{9}^{2} | + | z_{1}^{2} - z_{10}^{2} | \leq 4 π$

P

is TRUE and

Q

is FALSE

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Q

is TRUE and

P

is FALSE

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Both

P

and

Q

are TRUE

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Both

P

and

Q

are FALSE

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Solution

The correct option is C Both $P$ and $Q$ are TRUE

$| z_{2} - z_{1} | =$ length of line $A B \leq$ length of arc $A B$
$\Rightarrow | z_{2} - z_{1} | \leq θ_{2}$
Similarly, $| z_{3} - z_{2} | =$ length of line $B C \leq$ length of arc $B C$
$\Rightarrow | z_{3} - z_{2} | \leq θ_{3}$
$.$
$.$
$.$
$\Rightarrow | z_{1} - z_{10} | \leq θ_{1}$
Now,

$\Rightarrow | z_{2} - z_{1} | + | z_{3} - z_{2} | + \dots + | z_{10} - z_{9} | + | z_{1} - z_{10} | \leq θ_{1} + θ_{2} + . . . . + θ_{10}$

$\Rightarrow | z_{2} - z_{1} | + | z_{3} - z_{2} | + \dots + | z_{10} - z_{9} | + | z_{1} - z_{10} | \leq 2 π$
$(∵ θ_{1} + θ_{2} + . . . . + θ_{10} = 2 π)$
And
$| z_{k}^{2} - z_{k - 1}^{2} | = | z_{k} - z_{k - 1} | | z_{k} + z_{k - 1} |$

As we know $| z_{k} + z_{k - 1} | \leq | z_{k} | + | z_{k - 1} | \leq 2$

$\Rightarrow | z_{2}^{2} - z_{1}^{2} | + | z_{3}^{2} - z_{2}^{2} | + \dots + | z_{10}^{2} - z_{9}^{2} | + | z_{1}^{2} - z_{10}^{2} | \leq 2 (| z_{2} - z_{1} | + | z_{3} - z_{2} | + \dots + | z_{10} - z_{9} | + | z_{1} - z_{10} |)$

$∴ | z_{2}^{2} - z_{1}^{2} | + | z_{3}^{2} - z_{2}^{2} | + \dots + | z_{10}^{2} - z_{9}^{2} | + | z_{1}^{2} - z_{10}^{2} | \leq 4 π$

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Similar questions

Q. Let

z_{1}, z_{2}, z_{3}

be complex numbers such that

z_{1}^{2} + z_{2}^{2} + z_{3}^{2} = z_{1} z_{2} + z_{2} z_{3} + z_{3} z_{1}

and

| z_{1} + z_{2} + z_{3} | = 21.

Given that

| z_{1} - z_{2} | = 2 \sqrt{3}, | z_{1} | = 3 \sqrt{3},

then the value of

| z_{2} |^{2} + | z_{3} |^{2}

Q. Let

z_{1}, z_{2}, z_{3} \in

C such that

| z_{1} | = | z_{2} | = | z_{3} | = 1

. If

z_{1} + z_{2} + z_{3} \neq 0

and

z_{1}^{2} + z_{2}^{2} + z_{3}^{2} = 0

, then

| z_{1} + z_{2} + z_{3} |

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z_{1}, z_{2}

and

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be three complex numbers such that

| z_{1} | = | z_{2} | = | z_{3} | = 1

and

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z_{1}, z_{2}, z_{3}

be three complex numbers such that

| z_{1} | = | z_{2} | = | z_{3} | = 1

and

\frac{z_{1}^{2}}{z_{2} z_{3}} + \frac{z_{2}^{2}}{z_{1} z_{3}} + \frac{z_{3}^{2}}{z_{1} z_{2}} = - 1.

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equals

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