JEE Mains · Maths · STD 11 - 4.1 complex nubers
If \(z\) is a complex number such that \(\left| z \right| \ge 2\) , then the minimum value of \(\left| {z + \frac{1}{2}} \right|\):
- A is strictly greater than \(\frac{5}{2}\)
- B is strictly greater than \(\;\frac{3}{2}\) but less than \(\frac{5}{2}\)
- C is equal to \(\frac{5}{2}\)
- D lie in the interval \((1,2)\)
Answer & Solution
Correct Answer
(D) lie in the interval \((1,2)\)
Step-by-step Solution
Detailed explanation
\(|z| \geq 2\) is the region on or outside circle whose centre is \((0,0)\) and the radius is \(2\) . Minimum \(\left|z+\frac{1}{2}\right|\) is distance of \(z\), which lies on circle \(|z|=2\) from \(\left(-\frac{1}{2}, 0\right)\) therefore, minimum…
See the Complete Solution
Get step-by-step explanations for this and 2.5 Lakh+ more JEE, NEET & CET questions.
- Unlock all solutions
- Practice the full chapter
- Track accuracy across PYQs
4.8 rated on Google Play · 14,000+ reviews
More questions from Maths
- Let \(S=\left\{p_1, p_2 \ldots ., p_{10}\right\}\) be the set of first ten prime numbers. Let \(A=S \cup P\), where \(P\) is the set of all possible products of distinct elements of \(S\). Then the number of all ordered pairs ( \(x, y\) ), \(x \in S\), \(y \in A\), such that \(x\) divides \(y\), is ______.JEE Mains 2025 Medium
- Two marbles are drawn in succession from a box containing \(10\) red, \(30\) white, \(20\) blue and \(15\) orange marbles, with replacement being made after each drawing. Then the probability, that first drawn marble is red and second drawn marble is white, isJEE Mains 2024 Medium
- If \(\int \frac{\cos x d x}{\sin ^{3} x\left(1+\sin ^{6} x\right)^{2 / 3}}=f(x)\left(1+\sin ^{6} x\right)^{1 / \lambda}+c\) where \(c\) is a constant of integration, then \(\lambda f\left(\frac{\pi}{3}\right)\) is equal toJEE Mains 2020 Hard
- Let a circle \(C_1\) be obtained on rolling the circle \(x^2+y^2-4 x-6 y+11=0\) upwards 4 units on the tangent \(T\) to it at the point \((3,2)\). Let \(C _2\) be the image of \(C _1\) in T. Let A and B be the centers of circles \(C _1\) and \(C _2\) respectively, and \(M\) and \(N\) be respectively the feet of perpendiculars drawn from \(A\) and \(B\) on the \(x\)-axis. Then the area of the trapezium \(AMNB\) is :JEE Mains 2023 Hard
- Consider the function \(f: \mathbb{R} \rightarrow \mathbb{R}\) defined by \(f(x)=\frac{2 x}{\sqrt{1+9 x^2}}\). If the composition of \(f, \underbrace{(f \circ f \circ f \circ \ldots \circ f)}_{10 \text { times }}(x)=\frac{2^{10} x}{\sqrt{1+9 \alpha x^2}}\), then the value of \(\sqrt{3 \alpha+1}\) is equal to ...........JEE Mains 2024 Hard
- Let \(A\) be a \(3 \times 3\) real matrix such that \(A^2(A-2 I)-\) \(4(\mathrm{~A}-\mathrm{I})=\mathrm{O}\), where I and O are the identity and null matrices, respectively. If \(A^5=\alpha A^2+\beta A+\gamma I\), where \(\alpha, \beta\) and \(\gamma\) are real constants, then \(\alpha+\beta+\gamma\) is equal to:JEE Mains 2025 Medium
More PYQs from JEE Mains
- Let \(a_{n}\) be the \(n^{\text {th }}\) term of a G.P. of positive terms. If \(\sum\limits_{n=1}^{100} a_{2 n+1}=200\) and \(\sum\limits_{n=1}^{100} a_{2 n}=100,\) then \(\sum\limits_{n=1}^{200} a_{n}\) is equal toJEE Mains 2020 Hard
- The number of points, where the curve \(f(x)=e^{8 x}-e^{6 x}-3 e^{4 x}-e^{2 x}+1, x \in R\) cuts \(x\)-axis, is equal toJEE Mains 2023 Hard
- Let \(\overrightarrow{ a }=2 \hat{ i }+\hat{ j }+\hat{ k }\), and \(\overrightarrow{ b }\) and \(\overrightarrow{ c }\) be two nonzero vectors such that \(|\vec{a}+\vec{b}+\vec{c}|=|\vec{a}+\vec{b}-\vec{c}| \quad\) and \(\vec{b} \cdot \vec{c}=0\). Consider the following two statement: \((A)\) \(|\overrightarrow{ a }+\lambda \overrightarrow{ c }| \geq|\overrightarrow{ a }|\) for all \(\lambda \in R\). \((B)\) \(\overrightarrow{ a }\) and \(\overrightarrow{ c }\) are always parallelJEE Mains 2023 Hard
- Let the system of linear equations \(x +2 y + z =2\), \(\alpha x +3 y - z =\alpha,-\alpha x + y +2 z =-\alpha\) be inconsistent. Then \(\alpha\) is equal toJEE Mains 2022 Medium
- For \(x \in R -\{0,1\},\) \({f_1}\left( x \right) = \frac{1}{x},{f_2}\left( x \right) = 1 - x\) and \(f_{3}(x)=\frac{1}{1-x}\) be three given functions. If a function, \(J ( x )\) satisfies \(\left( {{f_2}oJo{f_1}} \right)\left( x \right)= f _{3}( x )\) then \(J ( x )\) is equal toJEE Mains 2019 Hard
- The mean of a set of \(30\) observations is \(75\). If each other observation is multiplied by a nonzero number \(\lambda \) and then each of them is decreased by \(25\), their mean remains the same. The \(\lambda \) is equal toJEE Mains 2018 Hard