JEE Mains · Maths · STD 11 - 4.1 complex nubers
Let \(z\) be those complex numbers which satisfy \(|z+5| \leq 4\) and \(z(1+i)+\bar{z}(1-i) \geq-10, i=\sqrt{-1}\) If the maximum value of \(Iz +\left.1\right|^{2}\) is \(\alpha+\beta \sqrt{2}\), then the value of \((\alpha+\beta)\) is ...... .
- A \(56\)
- B \(48\)
- C \(24\)
- D \(36\)
Answer & Solution
Correct Answer
(B) \(48\)
Step-by-step Solution
Detailed explanation
\(|z+5| \leq 4\) \((x+5)^{2}+y^{2} \leq 16....(1)\) \(z(1+i)+\bar{z}(1-i) \geq-10\) \(( z +\overline{ z })+ i ( z -\overline{ z }) \geq-10\) \(x - y +5 \geq 0...(2)\) \(|z+1|^{2}=|z-(-1)|^{2}\) Let \(P (-1,0)\) \(| z +1|_{\text {Max. }}^{2}= PB ^{2} \quad\left(\right.\) where…
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
- If the solution of the differential equation \((2 x+3 y-2) d x+(4 x+6 y-7) d y=0, y(0)=3\), is \(\alpha x+\beta y+3 \log _e|2 x+3 y-\gamma|=6\), then \(\alpha+2 \beta+3 \gamma\) is equal toJEE Mains 2024 Hard
- Which of the following statements is incorrect for the function \(g(\alpha)\) for \(\alpha \in R\) such that \(g(\alpha)=\int_{\frac{\pi}{6}}^{\frac{\pi}{3}} \frac{\sin ^{\alpha} x}{\cos ^{\alpha} x+\sin ^{\alpha} x} d x\)JEE Mains 2021 Hard
- Let \(z_{1}, z_{2}\) be the roots of the equation \(z^{2}+a z+\) \(12=0\) and \(z _{1}, z _{2}\) form an equilateral triangle with origin. Then, the value of \(| a |\) isJEE Mains 2021 Hard
- For \(I(x)=\int \frac{\sec ^{2} x-2022}{\sin ^{2022} x} d x\), if \(I\left(\frac{\pi}{4}\right)=2^{1011}\), thenJEE Mains 2022 Hard
- Let \(g: R \rightarrow R\) be a non constant twice differentiable such that \(g^{\prime}\left(\frac{1}{2}\right)=g^{\prime}\left(\frac{3}{2}\right)\). If a real valued function \(f\) is defined as \(\mathrm{f}(\mathrm{x})=\frac{1}{2}[\mathrm{~g}(\mathrm{x})+\mathrm{g}(2-\mathrm{x})]\), thenJEE Mains 2024 Hard
- Let \((\alpha, \beta, \gamma)\) be the foot of perpendicular from the point \((1,2,3)\) on the line \(\frac{x+3}{5}=\frac{y-1}{2}=\frac{z+4}{3}\). then \(19(\alpha+\beta+\gamma)\) is equal to :JEE Mains 2024 Hard
More PYQs from JEE Mains
- If \(\alpha, \beta\) are natural numbers such that \(100^{\alpha}-199 \beta=(100)(100)+(99)(101)+(98)(102)\) \(+\ldots .+(1)(199),\) then the slope of the line passing through \((\alpha, \beta)\) and origin isJEE Mains 2021 Hard
- A rectangle is formed by the lines \( x=0, x=3, y=0 \) and \( y=4 \). Let the line L be perpendicular to \( 3x+y+6=0 \) and divide the area of the rectangle into two equal parts. Then the distance of the point \( (\frac{1}{2},-5) \) from the line L is equal to:JEE Mains 2026 Hard
- Let a line having direction ratios \(1,-4,2\) intersect the lines \(\frac{x-7}{3}=\frac{y-1}{-1}=\frac{z+2}{1}\) and \(\frac{x}{2}=\frac{y-7}{3}=\frac{z}{1}\) at the point \(A\) and \(B\). Then \(( AB )^{2}\) is equal toJEE Mains 2022 Hard
- For \(x\,\, \in \,R\,,x\, \ne \,0,\) let \({f_0}(x) = \frac{1}{{1 - x}}\) and \({f_{n + 1}}(x) = {f_0}({f_n}(x)),\) \(n\, = 0,1,2,....\) Then the value of \({f_{100}}(3) + {f_1}\left( {\frac{2}{3}} \right) + {f_2}\left( {\frac{3}{2}} \right)\) is equal toJEE Mains 2016 Hard
- Let \(\quad f(x)=x+\frac{a}{\pi^2-4} \sin x+\frac{b}{\pi^2-4} \cos x\) \(x \in R\) be a function which satisfies \(f(x)=x+\int \limits_0^{\pi / 2} \sin (x+y) f(y) d y\). Then \(( a + b )\) is equal to \(............\)JEE Mains 2023 Hard
- For \(k \in R\), let the solutions of the equation \(\cos \left(\sin ^{-1}\left(x \cot \left(\tan ^{-1}\left(\cos \left(\sin ^{-1} x\right)\right)\right)\right)\right)=k, 0\,<\,|x|<\,\frac{1}{\sqrt{2}}\) be \(\alpha\) and \(\beta\), where the inverse trigonometric functions take only principal values. If the solutions of the equation \(x ^{2}- bx -5=0\) are \(\frac{1}{\alpha^{2}}+\frac{1}{\beta^{2}}\) and \(\frac{\alpha}{\beta}\), then \(\frac{b}{k^{2}}\) is equal to\(......\)JEE Mains 2022 Hard