Question:
The set of all $\alpha \epsilon R$, for which $w = \frac{1 + (1 - 8 \alpha)z}{1 - z}$ is a purely imaginary number, for all $z \neq 1$, is :
The set of all $\alpha \epsilon R$, for which $w = \frac{1 + (1 - 8 \alpha)z}{1 - z}$ is a purely imaginary number, for all $z \neq 1$, is :
Updated On: Feb 14, 2025
- an empty set
- $\{ 0 \}$
- $\left\{0 , \frac{1}{4} , - \frac{1}{4} \right\}$
- equal to R
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The Correct Option is B
Solution and Explanation
As $ \omega$ is purely imaginary
$\omega+\bar{\omega}=0$
$\frac{1+(1-8 \alpha) z}{1-z}+\frac{1+(1-8 a) \bar{z}}{1-z}=0$
$\frac{1-\bar{z}+(1-8 a)(z-1)+a-z+(1-8 \alpha)(\bar{z}-1)}{(1-z)(1-\bar{z})}=0$
$1-\bar{z}+z-1-8 a z+8 a+1-z+\bar{z}-1-8 \bar{z}-1-8 \bar{z} \alpha+8 \alpha=0$
$-8 a(z+\bar{z})+16 \alpha=0$
$8 a[2-(2+2)]=0$
if $Re(z) \neq 1$
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Concepts Used:
Sets
Set is the collection of well defined objects. Sets are represented by capital letters, eg. A={}. Sets are composed of elements which could be numbers, letters, shapes, etc.
Example of set: Set of vowels A={a,e,i,o,u}
Representation of Sets
There are three basic notation or representation of sets are as follows:
Statement Form: The statement representation describes a statement to show what are the elements of a set.
- For example, Set A is the list of the first five odd numbers.
Roster Form: The form in which elements are listed in set. Elements in the set is seperatrd by comma and enclosed within the curly braces.
- For example represent the set of vowels in roster form.
A={a,e,i,o,u}
Set Builder Form:
- The set builder representation has a certain rule or a statement that specifically describes the common feature of all the elements of a set.
- The set builder form uses a vertical bar in its representation, with a text describing the character of the elements of the set.
- For example, A = { k | k is an even number, k ≤ 20}. The statement says, all the elements of set A are even numbers that are less than or equal to 20.
- Sometimes a ":" is used in the place of the "|".