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If \( \alpha \) satisfies the equation \( x^{2} + x + 1 = 0 \) and \( (1 + \alpha)^{7} = A + B\alpha + C\alpha^{2} \), \( A, B, C \ge 0 \), then \( 5(3A - 2B - C) \) is equal to ____________.

If \( z = \frac{1}{2} - 2i \) is such that \( |z + 1| = \alpha z + \beta (1 + i) \), \( i = \sqrt{-1} \) and \( \alpha, \beta \in \mathbb{R} \), then \( \alpha + \beta \) is equal to:

Let \( P \) be a point on the hyperbola \( H:\dfrac{x^{2}}{9}-\dfrac{y^{2}}{4}=1 \), in the first quadrant such that the area of triangle formed by \( P \) and the two foci of \( H \) is \( 2\sqrt{13} \). Then, the square of the distance of \( P \) from the origin is:

Let \( S=\{\, z\in \mathbb{C}:\ |z-1|=1 \ \text{and}\ (\sqrt{2}-1)(z+\bar{z})-i(z-\bar{z})=2\sqrt{2}\, \} \). Let \( z_{1},z_{2}\in S \) be such that \( |z_{1}|=\max\limits_{z\in S}|z| \) and \( |z_{2}|=\min\limits_{z\in S}|z| \). Then \( |\sqrt{2}\,z_{1}-z_{2}|^{2} \) equals:

If the set \( R = \{(a,b): a + 5b = 42,\, a,b \in \mathbb{N}\} \) has \( m \) elements and \( \displaystyle \sum_{n=1}^{m} (1 - i^{n!}) = x + i y \), where \( i = \sqrt{-1} \), then the value of \( m + x + y \) is:

Consider the following two statements:
Statement I : For any two non-zero complex numbers \( z_{1},z_{2} \) \( (|z_{1}|+|z_{2}|)\left|\frac{z_{1}}{|z_{1}|}+\frac{z_{2}}{|z_{2}|}\right|\le (|z_{1}|+|z_{2}|) \),
and
Statement II : If \( x,y,z \) are three distinct complex numbers and \( a,b,c \) are three positive real numbers such that \( \frac{a}{|y-z|}=\frac{b}{|z-x|}=\frac{c}{|x-y|} \), then \( \frac{a^{2}}{y-z}+\frac{b^{2}}{z-x}+\frac{c^{2}}{x-y}=1 \).
Between the above two statements:

The sum of all possible values of \( \theta \in [-\pi, 2\pi] \), for which \(\displaystyle \frac{1 + i\cos\theta}{1 - 2i\cos\theta} \) is purely imaginary, is equal to:

Let \( \alpha \) and \( \beta \) be the sum and the product of all the non-zero solutions of the equation \( (\bar{z})^{2}+|z|=0,\ z\in \mathbb{C} \). Then \( 4(\alpha^{2}+\beta^{2}) \) is equal to:

Let \( S_{1}=\{z\in \mathbb{C}:|z|\le 5\} \), \( S_{2}=\left\{z\in \mathbb{C}:\operatorname{Im}\!\left(\dfrac{z+1-\sqrt{3}\,i}{1-\sqrt{3}\,i}\right)\ge 0\right\} \) and \( S_{3}=\{z\in \mathbb{C}:\operatorname{Re}(z)\ge 0\} \). Then the area of the region \( S_{1}\cap S_{2}\cap S_{3} \) is:

If $\alpha$ denotes the number of solutions of $|1-i|^x = 2^x$ and $\beta = \left( \frac{|z|}{\arg(z)} \right)$, where $z = \frac{\pi}{4}(1+i)^4 \left[ \frac{1-\sqrt{\pi}i}{\sqrt{\pi}+i} + \frac{\sqrt{\pi}-i}{1+\sqrt{\pi}i} \right]$, $i = \sqrt{-1}$, then the distance of the point $(\alpha, \beta)$ from the line $4x - 3y = 7$ is _________

Let $z$ be a complex number such that $|z+2|=1$ and $\operatorname{Im}\left(\frac{z+1}{z+2}\right)=\frac{1}{5}$. Then the value of $\left|\operatorname{Re}(\overline{z+2})\right|$ is:

Let \( \alpha, \beta \) be the roots of the equation \( x^2 - x + 2 = 0 \) with \( \operatorname{Im}(\alpha) > \operatorname{Im}(\beta) \). Then \( \alpha^6 + \alpha^4 + \beta^4 - 5\alpha^2 \) is equal to __________.

Let \( \alpha , \beta \) be the roots of the equations \( x^{2} - \sqrt{6}\,x + 3 = 0 \) such that \( \operatorname{Im}(\alpha) > \operatorname{Im}(\beta) \). Let \( \alpha, \beta \) be integers not divisible by \( 3 \) and \( n \) be a natural number such that \( \dfrac{\alpha^{99}}{\beta} + \alpha^{98} = 3^{n}(a + ib),\; i = \sqrt{-1} \). Then \( n + a + b \) is equal to ______.

Let \( P=\{z\in\mathbb{C}:\;|z+2-3i|\leq 1\} \) and \( Q=\{z\in\mathbb{C}:\; z(1+i)+\bar{z}(1-i)\leq -8\}. \) Let in \( P\cap Q,\, |z-3+2i| \) be maximum and minimum at \( z_1 \) and \( z_2 \) respectively. If \( |z_1|^{2}+2| z_2|^{2}=\alpha+\beta\sqrt{2} \), where \( \alpha,\beta \) integers, then \( \alpha+\beta \) equals ______.

If \( z \) is a complex number such that \( |z| \ge 1 \), then the minimum value of \(\displaystyle \left| z + \frac{1}{2}(3 + 4i) \right| \) is:

Let \( z \) be a complex number such that the real part of \( \dfrac{z-2i}{z+2i} \) is zero. Then, the maximum value of \( \lvert z - (6 + 8i) \rvert \) is equal to

Let the complex numbers \( \alpha \) and \( \dfrac{1}{\alpha} \) lie on the circles \( |z-z_{0}|^{2}=4 \) and \( |z-z_{0}|^{2}=16 \) respectively, where \( z_{0}=1+i \). Then, the value of \( 100\,|\alpha|^{2} \) is __________ .

If \( z \) is a complex number, then the number of common roots of the equations \( z^{1985} + z^{100} + 1 = 0 \) and \( z^{3} + 2z^{2} + 2z + 1 = 0 \), is equal to:

If \( S = \{ z \in \mathbb{C} : |z-i| = |z+i| = |z-1| \} \), then, \( n(S) \) is:

Let \( r \) and \( \theta \) respectively be the modulus and amplitude of the complex number \( z = 2 - i\!\left( 2 \tan \dfrac{5\pi}{8} \right) \), then \( (r,\theta) \) is equal to:

If \( z_1 \) and \( z_2 \) are two distinct complex number such that \( \left|\dfrac{z_1-2z_2}{\dfrac{1}{2}-{z_1}\bar z_2}\right|=2 \), then

The sum of the square of the modulus of the elements in the set \( \left\{ z=a+ib:\; a,b\in\mathbb{Z},\; z\in \mathbb{C}\; ,\; |z-1|\le 1,\; |z-5|\le |z-5i| \right\} \) is _________.

The area (in sq. units) of the region \( S=\left\{ z\in\mathbb{C}:\ |z-1|\le 2;\ (z+\bar{z})+i(z-\bar{z})\le 2,\ \mathrm{Im}(z)\ge 0 \right\} \) is:

Let \( z_1 \) and \( z_2 \) be two complex numbers such that \( z_1+z_2=5 \) and \( z_1^3+z_2^3=20+15i \). Then, \( \left|z_1^4+z_2^4\right| \) equals:

If \( z = x + iy, \ xy \neq 0 \), satisfies the equation \( z^{2} + i\overline{z} = 0 \), then \( |z|^{2} \) is equal to:

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JEE Main 2024 8th April Morning Shift

If the set \( R = \{(a,b): a + 5b = 42,\, a,b \in \mathbb{N}\} \) has \( m \) elements and \( \displaystyle \sum_{n=1}^{m} (1 - i^{n!}) = x + i y \), where \( i = \sqrt{-1} \), then the value of \( m + x + y \) is: