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: Dec 9, 2025
- an empty set
- $\{ 0 \}$
- $\left\{0 , \frac{1}{4} , - \frac{1}{4} \right\}$
- equal to R
Hide Solution
Verified By Collegedunia
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 $Re(z) \neq 1$
Was this answer helpful?
1
0
Top Questions on Sets
- Given the sets $ A = \{ x \mid |x - 2|<3 \} $ and $ B = \{ x \mid |x + 1| \leq 4 \} $, find $ A \cap B $.
- If \(A=\{1,2,3\}\), \(B=\{3,4\}\), then \(A\cup B\) is:
- If $A = \{x \in \mathbb{R} \mid \sin^{-1}(\sqrt{x^2+x+1}) \in [-\frac{\pi}{2}, \frac{\pi}{2}]\}$ and $B = \{y \in \mathbb{R} \mid y = \sin^{-1}(\sqrt{x^2+x+1}), x \in A\}$, then
- Let \( A = \{ n \in [100, 700] \cap \mathbb{N} : n \text{ is neither a multiple of 3 nor a multiple of 4} \} \).
Then the number of elements in \( A \) is: - Let $A = \{2, 3, 4, 5, \ldots, 16, 17, 18\}$. Let $R$ be the relation on the set $A$ of ordered pairs of positive integers defined by $(a, b) R (c, d)$ if and only if $ad = bc$ for all $(a, b), (c, d) \in A \times A$. Then the number of ordered pairs of the equivalence class of $(3, 2)$ is:
View More Questions
Questions Asked in JEE Main exam
In the given circuit diagram, $I_L = 5$ mA and the Zener voltage is $V_Z = 5$ V. If $i_z = 4 i_L$, find the value of $R_s$ (in $\Omega$).
- JEE Main - 2026
- Semiconductor electronics: materials, devices and simple circuits
- For the given statements below, mark the correct option. Statement-I: Work done by a conservative force \( f \), from \( r_1 \) to \( r_2 \) is given by: \[ W = \int_{r_1}^{r_2} f \, dr. \] Statement-II: Work done by conservative force is path dependent.
- JEE Main - 2026
- Semiconductor electronics: materials, devices and simple circuits
- A metallic conductor of length 2m and cross-sectional area \( 0.2 \, \text{mm}^2 \) carries a steady current of 1.2 A when a potential difference of 2 V is applied across it. (\( e = 1.6 \times 10^{-19} \), charge density = \( 7.5 \times 10^{28} \, \text{m}^{-3} \)) Then the mobility of the charge carrier is \( x \times 10^{-4} \) SI units. Find \( x \).
- JEE Main - 2026
- Semiconductor electronics: materials, devices and simple circuits
- For given logic gate circuit choose correct truth table.
- JEE Main - 2026
- Semiconductor electronics: materials, devices and simple circuits
Three silicon diodes connected parallel to each other as shown. Forward voltage of diode is 0.7 V. Find current through diode A :
- JEE Main - 2026
- Semiconductor electronics: materials, devices and simple circuits
View More Questions
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 "|".