If z ≠ 0 be a complex number such that\(|z-\frac{1}{z}|=2\), then the maximum value of |z| is
- \(\sqrt2\)
- 1
- \(\sqrt2-1\)
- \(\sqrt2+1\)
The Correct Option is D
Solution and Explanation
\(|z-\frac{1}{z}|\)≥||z\(|\frac{-1}{z}|\)
⇒ \(||z|-\frac{1}{|z|}|\)≤2
Let |z| = r
\(|r-\frac{1}{r}|\)≤2
−2≤\(r-\frac{1}{r}\)≤2
\(r-\frac{1}{r}\)≥−2 and \(r-\frac{1}{r}\)≤2
\(r^2\)+2r–1≥0 and \(r^2\)–2r–1≤0
r∈[−∞,−1–\(\sqrt2\)]∪[−1+\(\sqrt2\),∞] and r∈[1−\(\sqrt2\), 1+\(\sqrt2\)]
Taking intersection r∈[\(\sqrt2-1,\sqrt2+1\)]
So, the correct option is (D): \(\sqrt2+1\).
Top Questions on Algebra of Complex Numbers
Solve for \( x \):
\( \log_{10}(x^2) = 2 \).- BITSAT - 2025
- Mathematics
- Algebra of Complex Numbers
- Find the roots of the quadratic equation \( 2x^2 - 4x - 6 = 0 \).
- MHT CET - 2025
- Mathematics
- Algebra of Complex Numbers
- If the roots of the quadratic equation \( x^2 - 5x + 6 = 0 \) are \( p \) and \( q \), then what is the value of \( p + q \)?
- MHT CET - 2025
- Mathematics
- Algebra of Complex Numbers
Let \( K \) be an algebraically closed field containing a finite field \( F \). Let \( L \) be the subfield of \( K \) consisting of elements of \( K \) that are algebraic over \( F \).
Consider the following statements:
S1: \( L \) is algebraically closed.
S2: \( L \) is infinite.
Then, which one of the following is correct?- GATE MA - 2025
- Mathematics
- Algebra of Complex Numbers
- If the expression \( 7 + 6x - 3x^2 \) attains its extreme value \( \beta \) at \( x = \alpha \), then the sum of the squares of the roots of the equation \( x^2 + ax - \beta = 0 \) is:
- TS EAMCET - 2024
- Mathematics
- Algebra of Complex Numbers
Questions Asked in JEE Main exam
- Suppose that the number of terms in an A.P. is \( 2k, k \in \mathbb{N} \). If the sum of all odd terms of the A.P. is 40, the sum of all even terms is 55, and the last term of the A.P. exceeds the first term by 27, then \( k \) is equal to:
- JEE Main - 2025
- Arithmetic Progression
- If \( \alpha + i\beta \) and \( \gamma + i\delta \) are the roots of the equation \( x^2 - (3-2i)x - (2i-2) = 0 \), \( i = \sqrt{-1} \), then \( \alpha\gamma + \beta\delta \) is equal to:
- JEE Main - 2025
- Complex numbers
- Let \( [x] \) denote the greatest integer function, and let \( m \) and \( n \) respectively be the numbers of the points, where the function \( f(x) = [x] + |x - 2| \), \( -2<x<3 \), is not continuous and not differentiable. Then \( m + n \) is equal to:
- JEE Main - 2025
- Differential Equations
- Let \( f(x) = \frac{x - 5}{x^2 - 3x + 2} \). If the range of \( f(x) \) is \( (-\infty, \alpha) \cup (\beta, \infty) \), then \( \alpha^2 + \beta^2 \) equals to.
- JEE Main - 2025
- Functions
- Let \( P_n = \alpha^n + \beta^n \), \( P_{10} = 123 \), \( P_9 = 76 \), \( P_8 = 47 \), and \( P_1 = 1 \). The quadratic equation whose roots are \( \frac{1}{\alpha} \) and \( \frac{1}{\beta} \) is:
- JEE Main - 2025
- Sequences and Series
Concepts Used:
Complex Number
A Complex Number is written in the form
a + ib
where,
- “a” is a real number
- “b” is an imaginary number
The Complex Number consists of a symbol “i” which satisfies the condition i^2 = −1. Complex Numbers are mentioned as the extension of one-dimensional number lines. In a complex plane, a Complex Number indicated as a + bi is usually represented in the form of the point (a, b). We have to pay attention that a Complex Number with absolutely no real part, such as – i, -5i, etc, is called purely imaginary. Also, a Complex Number with perfectly no imaginary part is known as a real number.