JEE Mains · Maths · STD 11 - 4.1 complex nubers
Let \(z\) be a complex number such that \(|z|=1\). If \(\frac{2+\mathrm{k}^2 \mathrm{z}}{\mathrm{k}+\overline{\mathrm{z}}}=\mathrm{kz}, \mathrm{k} \in \mathbf{R}\), then the maximum distance of \(\mathrm{k}+\mathrm{ik}^2\) from the circle \(|\mathrm{z}-(1+2 \mathrm{i})|=1\) is:
- A \(\sqrt{5}+1\)
- B 2
- C 3
- D \(\sqrt{3}+1\)
Answer & Solution
Correct Answer
(A) \(\sqrt{5}+1\)
Step-by-step Solution
Detailed explanation
\(\begin{aligned} & \frac{2+\mathrm{k}^2 \mathrm{z}}{\mathrm{k}+\overline{\mathrm{z}}}=\mathrm{kz} \\ & |\mathrm{z}|^2 \mathrm{k}=2 \\ & \mathrm{k}=2\end{aligned}\) point \(\mathrm{p}(2,4)\); center \((1,2)\) distance from circle \((x-1)^2+(y-2)^2=1\) is max. if…
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
- The solution of the differential equation \(\frac{d y}{d x}-\frac{y+3 x}{\log _{e}(y+3 x)}+3=0\) is (where \(C\) is a constant of integration.)JEE Mains 2020 Medium
- If \(\mathrm{A}, \mathrm{B}\), and \(\left(\operatorname{adj}\left(\mathrm{A}^{-1}\right)+\operatorname{adj}\left(\mathrm{B}^{-1}\right)\right)\) are non-singular matrices of same order, then the inverse of \(\mathrm{A}\left(\operatorname{adj}\left(\mathrm{A}^{-1}\right)+\operatorname{adj}\left(\mathrm{B}^{-1}\right)\right)^{-1} \mathrm{~B}\), is equal toJEE Mains 2025 Medium
- If \(tan\, A\) and \(tan\, B\) are the roots of the quadratic equation, \(3x^2 - 10x - 25 = 0\) then the value of \(3\, sin^2\, (A +B)- 10\, sin\,(A +B). cos\,(A+ B)- 25\, cos^2\, (A+B)\) isJEE Mains 2018 Hard
- A group of \(40\) students appeared in an examination of \(3\) subjects - Mathematics, Physics Chemistry. It was found that all students passed in at least one of the subjects, \(20\) students passed in Mathematics, \(25\) students passed in Physics, \(16\) students passed in Chemistry, at most \(11\) students passed in both Mathematics and Physics, at most \(15\) students passed in both Physics and Chemistry, at most \(15\) students passed in both Mathematics and Chemistry. The maximum number of students passed in all the three subjects is ........... .JEE Mains 2024 Hard
- If the system of equations \(x-2 y+3 z=9\) \(2 x+y+z=b\) \(x-7 y+a z=24\) has infinitely many solutions, then \(a - b\) is equal toJEE Mains 2020 Medium
- For an initial screening of an admission test, a candidate is given fifty problems to solve. If the probability that the candidate can solve any problem is \(\frac{4}{5}\) , then the probability that he is unable to solve less than two problems isJEE Mains 2019 Hard
More PYQs from JEE Mains
- Let the image of the point \(P(1, 6, a)\) in the line \(L: \dfrac{x}{1} = \dfrac{y-1}{2} = \dfrac{z-a+1}{b}\), \(b > 0\), be \(\left(\dfrac{a}{3}, 0, a+c\right)\). If \(S(\alpha, \beta, \gamma)\), \(\alpha > 0\), is the point on \(L\) such that the distance of \(S\) from the foot of perpendicular from the point \(P\) on \(L\) is \(2\sqrt{14}\), then \(\alpha + \beta + \gamma\) is equal to:JEE Mains 2026 Hard
- If \(y=\tan ^{-1}\left(\sec x^{3}-\tan x^{3}\right) \cdot \frac{\pi}{2} < x^{3} < \frac{3 \pi}{2}\), thenJEE Mains 2022 Hard
- Let \(A (1,4)\) and \(B (1,-5)\) be two points. Let \(P\) be a point on the circle \((x-1)^{2}+(y-1)^{2}=1\) such that \(( PA )^{2}+( PB )^{2}\) have maximum value, then the points \(P , A\) and \(B\) lie on ...... .JEE Mains 2021 Hard
- Let \(a, b \in \mathbb{C}\). Let \(\alpha, \beta\) be the roots of the equation \(x^2 + ax + b = 0\). If \(\beta - \alpha = \sqrt{11}\) and \(\beta^2 - \alpha^2 = 3i\sqrt{11}\), then \((\beta^3 - \alpha^3)^2\) is equal to:JEE Mains 2026 Hard
- The region represented by \(\{z=x+i y \in C:|z|-\operatorname{Re}(z) \leq 1\}\) is also given by the inequalityJEE Mains 2020 Hard
- Let the domain of the function\(f(x)=\log _3 \log _5\left(7-\log _2\left(x^2-10 x+85\right)\right)+\sin ^{-1}\left(\left|\frac{3 x-7}{17-x}\right|\right)\)be \((\alpha, \beta]\). Then \(\alpha+\beta\) is equal to :JEE Mains 2026 Hard