SATHEE: Complex Numbers 2 Question 30

Complex Numbers 2 Question 30

31.

Match the statements of Column I with those of Column II.

Here, $z$ takes values in the complex plane and $Im (z)$ and $Re (z)$ denote respectively, the imaginary part and

	Column I	Column II
A.	The set of points $z$ satisfying $\| z - i \| z \| \| = \| z + i \| z \| \|$ is contained in or equal to	p. an ellipse with eccentricity 4/5
B.	The set of points $z$ satisfying $\| z + 4 \| + \| z - 4 \| = 0$ is contained in or equal to	q. the set of points $z$ satisfying $Im (z) = 0$
C.	If $\| w \| = 2$ , then the set of points $z = w - \frac{1}{w}$ is contained in or equal to	r. the set of points $z$ satisfying $\| lm (z) \| \leq 1$
D.	If $\| w \| = 1$ , then the set of points $z = w + \frac{1}{w}$ is contained in or equal to	s. the set of points t. satisfying $\| Re (z) \| \leq 2$ the set of points $z$ satisfying $\| z \| \leq 3$

(2010)

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Answer:

Correct Answer: 31. $A \to q, r; B \to p; C \to p, s, t; D \to q, r, s, t$

Solution:

A. Let $z = x + i y$

$\Rightarrow$ we get $y \sqrt{x^{2} + y^{2}} = 0$

$\Rightarrow$ $y = 0$

$\Rightarrow$ $I_{m} (z) = 0$

B. We have

$\Rightarrow 2 a e = 8, 2 a = 10$

$\Rightarrow 10 e = 8$

$\Rightarrow e = \frac{4}{5}$

$\Rightarrow b^{2} = 25 (1 - \frac{16}{25}) = 9$

$∴ \frac{x^{2}}{25} + \frac{y^{2}}{9} = 1$

C. Let $w = 2 (\cos θ + i \sin θ)$

$∴ z = 2 (\cos θ + i \sin θ) - \frac{1}{2 (\cos θ + i \sin θ)}$

$\begin{aligned} = 2 (\cos θ + i \sin θ) - \frac{1}{2} (\cos θ - i \sin θ) \\ = \frac{3}{2} \cos θ + \frac{5}{2} i \sin θ \end{aligned}$

Let $z = x + i y$

$\Rightarrow x = \frac{3}{2} \cos θ$ and $y = \frac{5}{2} \sin θ$

$\Rightarrow \frac{2 x^{2}}{3} + {\frac{2 y}{5}}^{2} = 1$

$\Rightarrow \frac{x^{2}}{9 / 4} + \frac{y^{2}}{25 / 4} = 1$

$∴ e = \sqrt{1 - \frac{9 / 4}{25 / 4}} = \frac{4}{5}$

D. Let $w = \cos θ + i \sin θ$

Then, $z = x + i y = \cos θ + i \sin θ + \frac{1}{\cos θ + i \sin θ}$

$= 2 \cos θ$

$\Rightarrow x = 2 \cos θ, y = 0$

Complex Numbers 2 Question 30

31.

Answer:

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Complex Numbers 2 Question 30

31.

Answer:

Engineering Preparation

Medical Preparation

NCERT Books & Solutions

Important Resources

Other Resources

Syllabus

Test Platform

Sample Mock Test

Mentorship

NCERT Exercise Video Solution

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