JEE Mains · Maths · STD 11 - 4.1 complex nubers
Let \(S=\{z=x+i y:|z-1+i| \geq|z|,|z|<2,|z+i|=\) \(|z-1|\}\). Then the set of all values of \(x\), for which \(w=2 x+i y \in S\) for some \(y \in R\), is.
- A \(\left(-\sqrt{2}, \frac{1}{2 \sqrt{2}}\right]\)
- B \(\left(-\frac{1}{\sqrt{2}}, \frac{1}{4}\right]\)
- C \(\left(-\sqrt{2}, \frac{1}{2}\right]\)
- D \(\left(-\frac{1}{\sqrt{2}}, \frac{1}{2 \sqrt{2}}\right]\)
Answer & Solution
Correct Answer
(B) \(\left(-\frac{1}{\sqrt{2}}, \frac{1}{4}\right]\)
Step-by-step Solution
Detailed explanation
\(|z-1+i| \geq|z| ;|z|<2 ;|z+i|=|z-1|\) Hence \(w=2 x+i y \in S\) \(2 x \leq \frac{1}{2} \therefore x \leq \frac{1}{4}\) Now \((2 x )^{2}+(2 x )^{2}<4\) \(x ^{2}<\frac{1}{2} \Rightarrow x \in\left(\frac{-1}{\sqrt{2}}, \frac{1}{\sqrt{2}}\right)\)…
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 the foot of perpendicular of the point \(P (3,-2,-9)\) on the plane passing through the points \((-1,-2,-3),(9,3,4),(9,-2,1)\) be \(Q(\alpha, \beta, \gamma)\). Then the distance of \(Q\) from the origin is:JEE Mains 2023 Hard
- Let \(A\) be a matrix of order \(3 \times 3\) and \(|A|=5\). If \(|2 \operatorname{adj}(3 \mathrm{~A} \operatorname{adj}(2 \mathrm{~A}))|=2^\alpha \cdot 3^\beta \cdot 5^\gamma \alpha, \beta, \gamma \in \mathrm{N}\) then \(\alpha+\beta+\gamma\) is equal toJEE Mains 2025 Medium
- Let \(f ( x )= x \cdot\left[\frac{ x }{2}\right],\) for \(-10< x <10,\) where \([ t ]\) denotes the greatest integer function. Then the number of points of discontinuity of \(f\) is equal toJEE Mains 2020 Hard
- Let the ellipse \(3 x^2+\mathrm{py}^2=4\) pass through the centre \(C\) of the circle \(x^2+y^2-2 x-4 y-11=0\) of radius \(r\). Let \(f_1, f_2\) be the focal distances of the point C on the ellipse. Then \(6 f_1 f_2-r\) is equal toJEE Mains 2025 Medium
- Let the values of p , for which the shortest distance between the lines \(\frac{x+1}{3}=\frac{y}{4}=\frac{z}{5}\) and \(\overrightarrow{\mathrm{r}}=(\mathrm{p} \hat{\mathrm{i}}+2 \hat{\mathrm{j}}+\hat{\mathrm{k}})+\lambda(2 \hat{\mathrm{i}}+3 \hat{\mathrm{j}}+4 \hat{\mathrm{k}})\) is \(\frac{1}{\sqrt{6}}\), be \(\mathrm{a}, \mathrm{b}\), \((a \lt b)\). Then the length of the latus rectum of the ellipse \(\frac{x^2}{a^2}+\frac{y^2}{b^2}=1\) is :-JEE Mains 2025 Medium
- The remainder when \((2021)^{2023}\) is divided by \(7\) isJEE Mains 2022 Hard
More PYQs from JEE Mains
- The mean and variance of \(7\) observations are \(8\) and \(16\) respectively. If one observation \(14\) is omitted a and \(b\) are respectively mean and variance of remaining \(6\) observation, then \(a+3 b-5\) is equal to \(..........\).JEE Mains 2023 Hard
- The value of \(\sum_{r=0}^{22}{ }^{22} C_r \cdot{ }^{23} C_r\) isJEE Mains 2023 Medium
- If \((x, y, z)\) be an arbitrary point lying on a plane \(P\) which passes through the point \((42,0,0) , (0,42,0)\) and \((0,0,42),\) then the value of expression \(3+\frac{x-11}{(y-19)^{2}(z-12)^{2}}+\frac{y-19}{(x-11)^{2}(z-12)^{2}}\)\( +\frac{z-12}{(x-11)^{2}(y-19)^{2}}-\frac{x+y+z}{14(x-11)(y-19)(z-12)} \)JEE Mains 2021 Hard
- If \(0<\theta, \phi<\frac{\pi}{2}, x =\sum_{ n =0}^{\infty} \cos ^{2 n } \theta, y =\sum_{ n =0}^{\infty} \sin ^{2 n } \phi\) and \(z =\sum_{ n =0}^{\infty} \cos ^{2 n } \theta \cdot \sin ^{2 n } \phi\) thenJEE Mains 2021 Hard
- The common difference of the \(A.P.:a_{1},a_{2},....,a_{m}\) is 13 more than the common difference of the \(A.P.: b_{1}, b_{2},...,b_{n}.\) If \(b_{31}=-277, b_{43}=-385\) and \(a_{78}=327,\) then \(a_{1}\) is equal toJEE Mains 2026 Hard
- If the sum of all the roots of the equation \(e^{2 x}-11 e^{x}-45 e^{-x}+\frac{81}{2}=0\) is \(\log _{ e } P\), then \(p\) is equal toJEE Mains 2022 Hard