Let S <math><mrow><mo>=</mo><mrow><mo>{</mo><mrow><mi>z</mi><mo>∈</mo><mi>C</mi><mo>:</mo><msup><mrow><mi>z</mi></mrow><mrow><mn>2</mn></mrow></msup><mo>+</mo><mover accent="true"><mi>Z</mi><mo>¯</mo></mover><mo>=</mo><mn>0</mn></mrow><mo>}</mo></mrow><mo>.</mo></mrow></math> Then <math><mrow><mstyle displaystyle="true"><munderover><mo>∑</mo><mrow><mi>z</mi><mo>∈</mo><mi>R</mi></mrow><mrow></mrow></munderover><mrow><mrow><mo>(</mo><mrow><mi>R</mi><mi>e</mi><mrow><mo>(</mo><mrow><mi>z</mi></mrow><mo>)</mo></mrow><mo>+</mo><mi>I</mi><mi>m</mi><mrow><mo>(</mo><mrow><mi>z</mi></mrow><mo>)</mo></mrow></mrow><mo>)</mo></mrow></mrow></mstyle></mrow></math> is equal to

take z = x + iyz2+z¯=0⇒x2−y2+x+i 2xy−yi=0⇒x2−y2+x=0 and y (2x−1)=0if y = 0 x = 0, 1 i f x = 1 2 ⇒ y = ± 3 2 Σ ( R e ( z ) + l m ( z ) ) = ( 0 − 1 + 1 2 + 1 2 ) + ( 0 + 0 + 3 2 − 3 2 ) = 0

Ask & Answer Question

Ncert Solutions Maths class 12th Maths Class 12th

Let S $= {z \in C : z^{2} + \bar{Z} = 0} .$ Then $\sum_{z \in R}^{} (R e (z) + I m (z))$ is equal to

3 Views | Posted 2 months ago

Asked by Shiksha User

1 Answer

Answered by

Payal Gupta | Contributor-Level 10

2 months ago

take z = x + iy

$z^{2} + \bar{z} = 0$

$\Rightarrow x^{2} - y^{2} + x + i 2 x y - y i = 0$

$\Rightarrow x^{2} - y^{2} + x = 0 a n d y (2 x - 1) = 0$

if y = 0 x = 0, 1

$i f x = \frac{1}{2} \Rightarrow y = \pm \frac{\sqrt[]{3}}{2}$

$Σ (R e (z) + l m (z)) = (0 - 1 + \frac{1}{2} + \frac{1}{2}) + (0 + 0 + \frac{\sqrt[]{3}}{2} - \frac{\sqrt[]{3}}{2}) = 0$

take z = x + iy<math><mrow><msup><mrow><mi>z</mi></mrow><mrow><mn>2</mn></mrow></msup><mo>+</mo><mover accent="true"><mi>z</mi><mo>¯</mo></mover><mo>=</mo><mn>0</mn></mrow></math><math><mrow><mo>⇒</mo><msup><mrow><mi>x</mi></mrow><mrow><mn>2</mn></mrow></msup><mo>−</mo><msup><mrow><mi>y</mi></mrow><mrow><mn>2</mn></mrow></msup><mo>+</mo><mi>x</mi><mo>+</mo><mi>i</mi><mtext> </mtext><mtext> </mtext><mn>2</mn><mi>x</mi><mi>y</mi><mo>−</mo><mi>y</mi><mi>i</mi><mo>=</mo><mn>0</mn></mrow></math><math><mrow><mo>⇒</mo><msup><mrow><mi>x</mi></mrow><mrow><mn>2</mn></mrow></msup><mo>−</mo><msup><mrow><mi>y</mi></mrow><mrow><mn>2</mn></mrow></msup><mo>+</mo><mi>x</mi><mo>=</mo><mn>0</mn><mtext> </mtext><mtext> </mtext><mi>a</mi><mi>n</mi><mi>d</mi><mtext> </mtext><mtext> </mtext><mi>y</mi><mrow><mo> (</mo><mrow><mn>2</mn><mi>x</mi><mo>−</mo><mn>1</mn></mrow><mo>)</mo></mrow><mo>=</mo><mn>0</mn></mrow></math>if y = 0 x = 0, 1 <math> <mrow> <mi>i</mi> <mi>f</mi> <mtext> </mtext> <mtext> </mtext> <mi>x</mi> <mo>=</mo> <mfrac> <mrow> <mn>1</mn> </mrow> <mrow> <mn>2</mn> </mrow> </mfrac> <mo>⇒</mo> <mi>y</mi> <mo>=</mo> <mo>±</mo> <mfrac> <mrow> <mroot> <mrow> <mn>3</mn> </mrow> <mrow></mrow> </mroot> </mrow> <mrow> <mn>2</mn> </mrow> </mfrac> </mrow> </math> <math> <mrow> <mi>Σ</mi> <mrow> <mo> (</mo> <mrow> <mi>R</mi> <mi>e</mi> <mrow> <mo> (</mo> <mrow> <mi>z</mi> </mrow> <mo>)</mo> </mrow> <mo>+</mo> <mi>l</mi> <mi>m</mi> <mrow> <mo> (</mo> <mrow> <mi>z</mi> </mrow> <mo>)</mo> </mrow> </mrow> <mo>)</mo> </mrow> <mo>=</mo> <mrow> <mo> (</mo> <mrow> <mn>0</mn> <mo>−</mo> <mn>1</mn> <mo>+</mo> <mfrac> <mrow> <mn>1</mn> </mrow> <mrow> <mn>2</mn> </mrow> </mfrac> <mo>+</mo> <mfrac> <mrow> <mn>1</mn> </mrow> <mrow> <mn>2</mn> </mrow> </mfrac> </mrow> <mo>)</mo> </mrow> <mo>+</mo> <mrow> <mo> (</mo> <mrow> <mn>0</mn> <mo>+</mo> <mn>0</mn> <mo>+</mo> <mfrac> <mrow> <mroot> <mrow> <mn>3</mn> </mrow> <mrow></mrow> </mroot> </mrow> <mrow> <mn>2</mn> </mrow> </mfrac> <mo>−</mo> <mfrac> <mrow> <mroot> <mrow> <mn>3</mn> </mrow> <mrow></mrow> </mroot> </mrow> <mrow> <mn>2</mn> </mrow> </mfrac> </mrow> <mo>)</mo> </mrow> <mo>=</mo> <mn>0</mn> </mrow> </math>

0 Upvotes 0 Downvotes

Get authentic answers from experts, students and alumni that you won't find anywhere else

On Shiksha, get access to

65k Colleges
1.2k Exams
687k Reviews
1800k Answers

Community GuidelinesUser Points System

QuestionsDiscussionsTags

Let S $= {z \in C : z^{2} + \bar{Z} = 0} .$ Then $\sum_{z \in R}^{} (R e (z) + I m (z))$ is equal to

1 Answer

Learn more about...

Most viewed information

Share Your College Life Experience

Didn't find the answer you were looking for?

Need guidance on career and education? Ask our experts

Your Question

Let S ={z∈C:z2+Z¯=0}. Then ∑z∈R(Re(z)+Im(z)) is equal to

1 Answer

Learn more about...

Most viewed information

Share Your College Life Experience

Didn't find the answer you were looking for?

Need guidance on career and education? Ask our experts

Your Question

Let S $= {z \in C : z^{2} + \bar{Z} = 0} .$ Then $\sum_{z \in R}^{} (R e (z) + I m (z))$ is equal to