Suppose z is any root of 11 z 8-20 iz 7-10 iz -22-0- where i -1- Then S - z -2- z -1 satisfiesA- S - 3B- 3- S -7C- S - 13D- 7 - S -13

Let z_1,z_2,z_3,⋯ ,z_8 be the roots of the equation. Then ∏_i=1^8 z_i= 2 ⇒∏_i=1^8 |z_i|=2 ⋯ (1) nbsp; Let a=min|z_i| and b=max|z_i| Then |a| lt;2 a ...

You visited us 1 times! Enjoying our articles? Unlock Full Access!

Byju's Answer

Standard XII

Mathematics

Properties of nth Root of a Complex Number

Suppose z is ...

Question

Suppose $z$ is any root of $11 z^{8} + 20 i z^{7} + 10 i z - 22 = 0$ , where $i = \sqrt{- 1}$ . Then $S = | z |^{2} + | z | + 1$ satisfies

S \leq 3

No worries! We‘ve got your back. Try BYJU‘S free classes today!

3 < S < 7

Right on! Give the BNAT exam to get a 100% scholarship for BYJUS courses

7 \leq S < 13

No worries! We‘ve got your back. Try BYJU‘S free classes today!

S \geq 13

No worries! We‘ve got your back. Try BYJU‘S free classes today!

Open in App

Solution

The correct option is B $3 < S < 7$
Let $z_{1}, z_{2}, z_{3}, \dots, z_{8}$ be the roots of the equation.
Then $8 \prod i = 1 z_{i} = - 2$
$\Rightarrow 8 \prod i = 1 | z_{i} | = 2 \dots (1)$
Let $a = min | z_{i} | and b = max | z_{i} |$
Then $| a | < 2 and | b | > 1 [From (1)]$
From $| a | < 2,$ we get
$| a |^{2} + | a | + 1 < 2^{2} + 2 + 1 = 7$
From $| b | > 1,$ we get
$| b |^{2} + | b | + 1 > 1^{2} + 1 + 1 = 3$
$∴ 3 < S < 7$

Suggest Corrections

Similar questions

Q. Let

a, b \in R

and

a^{2} + b^{2} \neq 0

. Suppose

S = {z \in C : z = \frac{1}{a + i b t}, t \in R, t \neq 0}

, where

i = \sqrt{- 1}

. If

z = x + i y

and

z \in S

, then

(x, y)

lies on

Q. Let

a, b ϵ R

and

a^{2} + b^{2} \neq 0

. Suppose

S = {z ϵ C : z = \frac{1}{a + i b t}, t ϵ R, t \neq 0}

, where

i = \sqrt{- 1}

. If

z = x + i y

and

z ϵ S

, then

(x, y)

lies on

Q. Let

α, β \in R

and

(1 - cos α)^{2} + {sin}^{2} β \neq 0

. Suppose

S = {z \in C : z = \frac{1}{(1 - cos α) + i k sin β}, k \in R - {0}},

where

i = \sqrt{- 1} .

z = x + i y, z \in S

and

(x, y)

lies on a circle, then

Q. Let a complex number

α (α \neq 1)

be a root of the equation

z^{12} - z^{7} - z^{5} + 1 = 0

and

S_{n} = n - 1 \sum k = 0 α^{k} .

Then which of the following is FALSE?

Q. Let

z = \frac{- 1 + \sqrt{3} i}{2},

where

i = \sqrt{- 1}

and

r, s \in {1, 2, 3} .

Let

P = [\begin{matrix} (- z)^{r} & z^{2 s} z^{2 s} & z^{r} \end{matrix}]

and

I

be the identity matrix of order

2

. Then the total number of ordered pairs

(r, s)

for which

P^{2} = - I

Join BYJU'S Learning Program

Suppose z is any root of 11z8+20iz7+10iz–22=0, where i=√−1 . Then S=|z|2+|z|+1 satisfies

The correct option is B 3<S<7Let z1,z2,z3,⋯,z8 be the roots of the equation. Then 8∏i=1 zi=−2 ⇒8∏i=1 |zi|=2 ⋯(1) Let a=min|zi| and b=max|zi| Then |a|<2 and |b|>1 [From (1)] From |a|<2, we get |a|2+|a|+1<22+2+1=7 From |b|>1, we get |b|2+|b|+1>12+1+1=3 ∴3<S<7

Suppose $z$ is any root of $11 z^{8} + 20 i z^{7} + 10 i z - 22 = 0$ , where $i = \sqrt{- 1}$ . Then $S = | z |^{2} + | z | + 1$ satisfies