JEE Mains · Maths · STD 11 - 4.1 complex nubers
Let the curve \(z(1+i)+\bar{z}(1-i)=4, z \in \mathrm{C}\), divide the region \(|z-3| \leq 1\) into two parts of areas \(\alpha\) and \(\beta\). Then \(|\alpha-\beta|\) equals :
- A \(1+\frac{\pi}{2}\)
- B \(1+\frac{\pi}{3}\)
- C \(1+\frac{\pi}{6}\)
- D \(1+\frac{\pi}{4}\)
Answer & Solution
Correct Answer
(A) \(1+\frac{\pi}{2}\)
Step-by-step Solution
Detailed explanation
\(\begin{aligned} & \text { Let } z=x+i y \\ & (x+i y)(1+i)+(x-i y)(1-i)=4 \\ & x+i x+i y-y+x-i x-i y-y=4 \\ & 2 x-2 y=4 \\ & x-y=2 \\ & |z-3| \leq 1 \\ & (x-3)^2+y^2 \leq 1 \end{aligned}\) Area of shaded region…
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 \(\vec a = 2\hat i + \hat j - 2\hat k\) and \(\vec b = \hat i + \hat j\) . Let \(\vec c\) be vector such that \(\left| {\vec c - \vec a} \right| = 3,\;\left| {\left( {\vec a \times \vec b} \right) \times \vec c} \right| = 3\) and the angle between \(\vec c\) and \(\vec a \times \vec b\) be \(30^\circ \) . Then \(\vec a \cdot \vec c\) is equal to :JEE Mains 2017 Hard
- Let \(M\) and \(m\) be respectively the local maximum and the local minimum values of the function, \(f(x) = \,2{x^3} - 9{x^2} + 12x + 5\) in the interval \([0, 3].\) Then \(M-m\) is equal toJEE Mains 2018 Hard
- If \(x=f(y)\) is the solution of the differential equation
\(\left(1+y^2\right)+\left(x-2 \mathrm{e}^{\tan ^{-1} y}\right) \frac{\mathrm{d} y}{\mathrm{~d} x}=0, y \in\left(-\frac{\pi}{2}, \frac{\pi}{2}\right)\)
with \(f(0)=1\), then \(f\left(\frac{1}{\sqrt{3}}\right)\) is equal to :JEE Mains 2025 Medium - If \(\lim _{x \rightarrow 1^{+}} \frac{(x-1)(6+\lambda \cos (x-1))+\mu \sin (1-x)}{(x-1)^3}=-1\), where \(\lambda, \mu \in \mathbb{R}\), then \(\lambda+\mu\) is equal toJEE Mains 2025 Medium
- \(\smallint \frac{{dx}}{{{x^2}{{\left( {{x^4} + 1} \right)}^{\frac{3}{4}}}}} = \)JEE Mains 2015 Hard
- \(\lim \limits_{x \rightarrow \frac{\pi}{2}}(\tan ^{2} x((2 \sin ^{2} x+3 \sin x+4)^{\frac{1}{2}}\) \(-(\sin ^{2} x+6 \sin x+2)^{\frac{1}{2}}))\) is equal toJEE Mains 2022 Hard
More PYQs from JEE Mains
- Let \(y=y(x)\) be the solution of the differential equation \(\operatorname{cosec}^{2} x d y+2 d x=(1+y \cos 2 x) \operatorname{cosec}^{2} x d x\), with \(y\left(\frac{\pi}{4}\right)=0\). Then, the value of \((y(0)+1)^{2}\) is equal to :JEE Mains 2021 Hard
- Let \(a_1, a_2, \ldots . a_{10}\) be \(10\) observations such that \(\sum_{\mathrm{k}=1}^{10} \mathrm{a}_{\mathrm{k}}=50\) and \(\sum_{\forall \mathrm{k}<\mathrm{j}} \mathrm{a}_{\mathrm{k}} \cdot \mathrm{a}_{\mathrm{j}}=1100\). Then the standard deviation of \(a_1, a_2, \ldots, a_{10}\) is equal to :JEE Mains 2024 Hard
- Let \(m, n \in N\) and \(\operatorname{gcd}(2, n)=1\). If \(30\left(\begin{array}{l}30 \\ 0\end{array}\right)+29\left(\begin{array}{l}30 \\ 1\end{array}\right)+\ldots+2\left(\begin{array}{l}30 \\ 28\end{array}\right)+1\left(\begin{array}{l}30 \\ 29\end{array}\right)= n .2^{ m }\) then \(n + m\) is equal to (Here \(\left.\left(\begin{array}{l} n \\ k \end{array}\right)={ }^{ n } C _{ k }\right)\)JEE Mains 2021 Hard
- Given below are two statements:
Statement I: The function \(f:R\rightarrow R\) defined by \(f(x)=\frac{x}{1+|x|}\) is one-one.
Statement II: The function \(f:R\rightarrow R\) defined by \(f(x)=\frac{x^{2}+4x-30}{x^{2}-8x+18}\) is many-one.
In the light of the above statements, choose the correct answer from the options given below :JEE Mains 2026 Easy - Let the equation of the circle, which touches \(x\)-axis at the point \((a, 0), a\gt0\) and cuts off an intercept of length \(b\) on \(y\)-axis be \(x^2+y^2-\alpha x+\beta y+\gamma=0\). If the circle lies below \(x\)-axis, then the ordered pair \(\left(2 a, b^2\right)\) is equal toJEE Mains 2025 Medium
- Let \(y=y(x)\) be the solution of the differential equation \(\frac{d y}{d x}+\frac{\sqrt{2} y}{2 \cos ^{4} x-\cos 2 x}= Xe ^{\tan ^{-1}(\sqrt{2} \cot 2 x )}, 0 < x < \pi / 2\) with \(y\left(\frac{\pi}{4}\right)=\frac{\pi^{2}}{32}\). If \(y\left(\frac{\pi}{3}\right)=\frac{\pi^{2}}{18} e^{-\tan ^{-1}(\alpha)}\), then the value of \(3 \alpha^{2}\) is equal toJEE Mains 2022 Hard