Complex Numbers and Quadratic Equations - Class 11

Let $A = \begin{pmatrix} z & w \\ \bar{w} & -z \end{pmatrix}$ be an involutory matrix, where $z, w \in \mathbb{C}$. Let $B = \begin{pmatrix} 1+i & 2i & w \\ 1 & 1+i & z \\ -1 & i & 1 \end{pmatrix}$. If $B$ is a singular matrix, let $S$ be the set of all possible values of $w$. Find the maximum value of $|w - 2 + 2i|^2$ for $w \in S$.

Let $S$ be the set of all integers $x$ satisfying the inequality $\frac{x^2 - 2x - 3}{x^2 - 4} \le 0$ Let $k = \sum_{x \in S} x$. If $A$ is a $3 \times 3$ matrix whose $(m, n)^{\text{th}}$ entry is given by $a_{mn} = (k + i^{m+n})^2$ for $1 \le m, n \le 3$, where $i = \sqrt{-1}$, then the value of $\det(A)$ is:

Let $z = \frac{1+i\sqrt{3}}{2}$. A real-valued differentiable function $f(x)$ is defined on $(0, 2) \cup (2, \infty)$ and satisfies the differential equation $x f'(x) + f(x) = \text{Re} \left( \frac{4x^3}{x^4 + 16 z^3} \right)$ for all $x \in (0, 2) \cup (2, \infty)$. It is given that $\lim_{x \to \infty} (x f(x) - 4\ln x) = -\ln 2$ and $f(1) = \ln\left(\frac{15}{2}\right)$. Find the sum of the $4^{\text{th}}$ powers of all distinct real roots of the equation $f(x) = 0$.

Let $f(z) = 12z^3 + k z^2 + c z$, where $k, c \\in \\mathbb{R}$. A particle moves in the complex plane along the curve $z(t) = e^{it}$ for $t \\in \\left[0, \\frac{\pi}{2}\\right]$. Let $u(t) = \\text{Re}(f(z(t)))$ and $v(t) = \\text{Im}(f(z(t)))$. It is given that $v(t)$ satisfies the conditions of Rolle's Theorem on the interval $\\left[0, \\frac{\pi}{2}\\right]$, and $u(t)$ has a local extremum at $t = \\arccos\\left(\\frac{3}{4}\\right)$. If $M$ and $m$ are the absolute maximum and absolute minimum values of $u(t)$ on $\\left[0, \\frac{\pi}{2}\\right]$, find the value of $8(M + m)$.

Let $P(z) = z^2 + (i-57)z - 57i$ and $Q(z) = z^2 + (i-233)z - 233i$. Let $R(z) = z^2 + pz + q$ where $p, q \in \mathbb{R}$. Given that the polynomials $P(z), Q(z),$ and $R(z)$ are linearly dependent over $\mathbb{C}$, let $\alpha$ be the common root of $P(z) = 0$ and $Q(z) = 0$. Find the value of $(p + q) + \sum x$, where the sum is taken over all integers $x$ satisfying the inequality $x^2 + 7 \operatorname{Im}(\alpha) x + 10 \le 0$.

Let $z$ be a non-zero complex number and $w = u + iv$ ($u, v \in \mathbb{R}$) be its principal square root, defined such that the argument $\theta$ of $z$ satisfies $\theta \in (-\pi, \pi]$. A vector $\vec{d} = u\hat{i} + v\hat{j} + \sqrt{2}\hat{k}$ has direction cosines $l, m, n$ along the positive x, y, and z axes respectively. If $n = \frac{1}{2}$ and $m = -\frac{\sqrt{3}}{4}$, then $z$ is equal to:

Let $z_0$ be the complex number satisfying the system of equations: $z^2 - (7+i)z + (24+7i) = 0$ $|z| = 5(z + \bar{z}) - 35$ Find the value of $|z_0 + 1 + 2i|^2$.

Let $z(\theta) = \frac{1 - \tan^2(\theta/2) + 2i\tan(\theta/2)}{1 + \tan^2(\theta/2)}$ for $\theta \in [0, 4\pi] \setminus \{\pi, 3\pi\}$. Let $S$ be the set of all $\theta$ in the given domain satisfying the equation $\text{Re}(\sqrt{z(\theta)}) + \text{Im}(\sqrt{z(\theta)}) = -\frac{1}{\sqrt{2}}$, where $\sqrt{w}$ denotes the principal square root of the complex number $w$ (with argument in $(-\pi, \pi]$). Find the sum of all elements in $S$.

Let $S$ be the region in $\mathbb{R}^3$ consisting of all points $(x, y, z)$ that lie on the plane $x + y + z = 7$ and satisfy the condition $|(x+iy) - (1+i)| \ge 2\sqrt{2}$, where $i = \sqrt{-1}$. If $d$ is the minimum distance from the point $A(1, 1, 0)$ to the region $S$, then the value of $d$ is:

Let $z_1, z_2, \dots, z_9$ be the distinct roots of the equation $z^9 = 1$. A $9 \times 9$ matrix $M = [m_{jk}]$ is constructed such that its entries are given by $m_{jk} = |z_j + i z_k|^6 - |z_j - i z_k|^6$ for $1 \le j, k \le 9$. If $S = \sum_{1 \le j < k \le 9} |z_j - z_k|^2$, find the value of $\det(M) + S$.

Evaluate the definite integral: $I = \int_{-\frac{\pi}{2}}^{\frac{\pi}{2}} \left| \sin x - \cos x + \text{sgn}(x) \right| dx$ where $\text{sgn}(x)$ denotes the signum function.

Let $z_k = k^2 + 2i$ for $k \in \mathbb{N}$, and define $Z_n = \prod_{k=1}^n z_k$. Evaluate the sum $\sum_{n=1}^{100} \left[ \frac{4}{\pi} \arg(Z_n) \right]$ where $[x]$ denotes the greatest integer less than or equal to $x$, and $\arg(z)$ denotes the principal argument of $z$ taking values in $(-\pi, \pi]$.

Let $\vec{a} = \cos \theta \hat{i} + \sin \theta \hat{j} + \hat{k}$ and $\vec{b} = -\sin \theta \hat{i} + \cos \theta \hat{j} + \hat{k}$, where $\theta = \frac{4\pi}{3}$. Let $\hat{n} = n_1\hat{i} + n_2\hat{j} + n_3\hat{k}$ be the unit normal vector to the plane containing $\vec{a}$ and $\vec{b}$ such that $\hat{n}$ makes an obtuse angle with the positive $z$-axis. If $Z = n_1 + i n_2$ is a complex number in the Argand plane, then the principal argument of $Z$ is:

Let $x_1 = 1$, $x_2 = \omega$, $x_3 = \omega^2$, and $x_4 = i$, where $\omega = e^{i2\pi/3}$. Let $M$ be a $4 \times 4$ matrix whose entries are given by $M_{jk} = (x_j - x_k)^3$ for $1 \le j, k \le 4$. Find the principal argument of $\det(M)$.

Let $S$ be the set of all complex numbers $z$ satisfying the following conditions:

1. $|z - \sqrt{3}| + |z + \sqrt{3}| = 4$
2. $\big| |z|^2 - 2 \big| + \big| |z|^2 - 3 \big| = 1$
3. $\text{Arg}(z^3) = 3\text{Arg}(z)$
4. $\text{Im}(z) \ge 0$

Let $A$ be the area of the region in the complex plane bounded by the arc $S$ and the line segments joining the origin to the endpoints of $S$. Find the value of $A$.

Let $I = \int_{-1}^{1} x \cdot \text{Arg}(x^2 - 1 + 2ix) \, dx$, where $\text{Arg}(z)$ denotes the principal argument of $z$ such that $\text{Arg}(z) \in (-\pi, \pi]$. Find the value of $I$.

Let $S$ be the set of all $2 \times 2$ matrices $A = \begin{pmatrix} z_1 & z_2 \\ z_3 & z_4 \end{pmatrix}$ with complex entries such that $A^T A = I$, where $I$ is the identity matrix. If $|z_k| = 1$ for $k=1,2,3,4$, find the number of elements in $S$.

Let $S$ be the set of all complex numbers $z$ satisfying $\text{Re}(z) = \text{Im}(z)$. For a fixed angle $\theta \in [0, 2\pi)$, a point $P$ in 3D space is defined by its coordinates $(X, Y, Z)$ where: $X = \text{Re}(z e^{i\theta}), \quad Y = \text{Im}(z e^{i\theta}), \quad Z = \text{Re}(z)$ Let $\Pi_1$ and $\Pi_2$ be two planes given by the equations: $\Pi_1: 2X + 3Y - 2Z = 1$ $\Pi_2: X + 2Y - Z = 1$ If there exists a plane $\Pi$ passing through the line of intersection of $\Pi_1$ and $\Pi_2$ such that the locus of $P$ (as the complex number $z$ varies in $S$) lies entirely in $\Pi$, find the sum of all possible values of $\theta$.

Let $I_n(x) = \int_0^{2\pi} \frac{e^{x \cos\theta} \cos(x \sin\theta + n\theta)}{13 - 12\cos\theta} d\theta$ for $n \in \mathbb{N} \cup \{0\}$. If $S(x) = \sum_{n=0}^{\infty} I_n(x)$, find the value of $S\left(\frac{3}{2} \ln 5\right)$.

Let $z$ be a complex number such that $|z| = 1$. Define the sum of distances from $z$ to the roots of the equation $w^4 - 1 = 0$ as $S(z) = |z - 1| + |z + 1| + |z - i| + |z + i|$. If $M$ and $m$ are the maximum and minimum values of $S(z)$ respectively, find the value of $M^2 - m^2$.

A point $P$ in the complex plane traces a locus given by $z(t) = x(t) + i y(t)$, where $x(t) = (t^2 - 2t)\cos t - (t^2 - 4)\sin t$ $y(t) = (t^2 - 2t)\sin t + (t^2 - 4)\cos t$ for $t \in [-2, 3]$. Let $M$ and $m$ be the absolute maximum and absolute minimum values of $|z(t)|$, respectively. Find the value of $M^2 + m^2$.

Let $S = \{z \in \mathbb{C} : |z - 4| + |z + 4| = 10\}$. For any $z \in S$, let $w(z)$ be the unique complex number on the circle $|w| = 5$ such that $\text{Re}(z) = \text{Re}(w)$ and $\text{Im}(z) = \frac{3}{5}\text{Im}(w)$. If $z_1, z_2 \in S$ satisfy $|w(z_1) - w(z_2)| = 6$, find the maximum value of $|z_1 - z_2| + |z_1 + z_2|$.

Let $z(x) = x^2 \cos\left(\frac{1}{x}\right) + i \sqrt{3} x$ for $x \in \mathbb{R} \setminus \{0\}$. If $f(x) = |z(x) - 1| + |z(x) + 1|$, evaluate the limit: $\lim_{x \to 0} \frac{f(x) - 2}{x^2}$

Let $B = \begin{pmatrix} 1 & -\sqrt{3} & 2+\sqrt{3} \\ \sqrt{3} & 1 & 2-\sqrt{3} \\ 0 & 0 & 3 \end{pmatrix}$. For each $n \in \mathbb{N}$, define the matrix $M_n = B^n - 2^n I$, where $I$ is the $3 \times 3$ identity matrix. Let $S$ be the set of all integers $n \in \{1, 2, \dots, 100\}$ such that $\det(\text{adj}(M_n)) > 0$. Find the sum of all elements in the set $S$.

Let $z = \frac{(1 - i)^8 (\sqrt{3} - i)^6}{(-1 + i\sqrt{3})^3}$. If $\theta = \text{Arg}(z)$ denotes the principal argument of $z$, find the value of $2 \sin\left(\frac{\theta}{2}\right) + 4 \sin\left(\frac{\theta}{3}\right)$

Let $S = \{z \in \mathbb{C} : z^8 = 1\}$. A multiset $M$ is formed by applying the transformation $f(z) = z^{\text{sgn}(\text{Im}(z))}$ to each $z \in S$, where $\text{sgn}(x)$ denotes the signum function. Find the number of distinct linear arrangements of all elements of $M$ such that no two identical elements are adjacent.

Let $\omega$ be a non-real cube root of unity. Consider the polynomial expansion $P(x) = (1 + x\omega)^{20} + (1 + x\omega^2)^{20} = \sum_{r=0}^{20} A_r x^r$ If $x = \frac{2}{3}$, find the value of $r$ for which the modulus of the $r$-th degree term, $|A_r x^r|$, is maximum.

Let $x$ and $y$ be real numbers such that $x^2 + y^2 = 1$. A point $w = u + iv$ in the complex plane is defined by $u = \arccos x$ and $v = \arccos y$. Let $S$ be the locus of $w$ as $x$ and $y$ vary, and let $A$ be the area of the region enclosed by $S$. Let $k$ be the number of points on $S$ where the implicit derivative $\frac{dv}{du}$, obtained directly from the equation $\cos^2 u + \cos^2 v = 1$, yields the indeterminate form $\frac{0}{0}$. Find the value of $\frac{k A}{\pi^2}$.

Let $S$ be the set of all complex numbers $z$ satisfying $|z - \sqrt{3} - i| = 2$. Let $A$ be defined as $A = \left\{ z \in S : \left( \frac{z}{|z|} \right)^{12} = 1 \right\}$ Find the value of $\sum_{z \in A} |z - \sqrt{3} - i|^2$

Let $z_1, z_2, z_3$ be the roots of the equation $z^3 - 5z^2 + 17z - 13 = 0$. Let $M(z)$ be a $4 \times 4$ matrix whose entries are given by $m_{jk} = |w_j - w_k|^2$ for $1 \le j, k \le 4$, where $w_1 = z$, $w_2 = z_1$, $w_3 = z_2$, and $w_4 = z_3$. Let $C$ be the locus of points $z \in \mathbb{C}$ such that $\det(M(z)) = 0$. If $A$ is the area of the region bounded by $C$, find the value of $\frac{A}{\pi}$.

Let $C$ be the curve in the complex plane obtained by applying the transformation $w = \frac{3}{2}z + \frac{1}{2}\bar{z}$ to the unit circle $|z| = 1$. Let $S$ be the set of all complex numbers $w$ satisfying $\text{Re}(w^2) = 3$ from which exactly two mutually perpendicular tangents can be drawn to $C$. Find the value of $\sum_{w \in S} |w|^2$.

Let $A$ be a $3 \times 3$ matrix with complex entries such that $\text{tr}(A) = 4 - i$ and $\text{tr}(A^2) = 5 + 2i$. Let $\alpha$ and $\beta$ be the roots of the quadratic equation $z^2 - \text{tr}(A)z + \text{tr}(\text{adj}(A)) = 0$ Find the value of $|\alpha|^2 + |\beta|^2$.

Let $S$ be the set of all complex numbers $z$ satisfying $|z^2 - 2 e^{i\pi/4} z + 2i| = 1$. What is the maximum value of $|z|$ for $z \in S$?

Let $z_1(x)$ and $z_2(x)$ be the roots of the quadratic equation in $z$: $z^2 - 2xz + x^2 + x^4 (-1)^{[x]} = 0$ where $[x]$ denotes the greatest integer less than or equal to $x$. Let $f(x) = \frac{1}{2} \left( (\text{Im}(z_1(x)))^2 + (\text{Im}(z_2(x)))^2 \right)$, where $\text{Im}(z)$ denotes the imaginary part of $z$. Let $y(x)$ be the continuous solution to the differential equation: $x^2 y \frac{dy}{dx} - x y^2 = x^3 - f(x) \quad \text{for } x \ge 1$ If $y(1) = \sqrt{2}$, then the value of $y(e)^2$ is:

For each $n \in \mathbb{N}$, let $\alpha_n$ and $\beta_n$ be the roots of the polynomial equation: $z^2 - \left( 2 + \frac{i}{n} + \frac{\sin(n^2)}{n^2} \right) z + \left( 1 + \frac{i}{n} + \frac{\sin(n^2)}{n^2} \right) = 0$ Let $a_n = \alpha_n^n + \beta_n^n$. Find $\lim_{n \to \infty} a_n$.

Let $z_1, z_2, z_3$ be the roots of the cubic equation $z^3 - p z^2 + q z - 26 = 0$, where $p, q \in \mathbb{R}$. For each root $z_k$, let $P_k$ be a point in 3D space with position vector $\vec{v}_k = \text{Re}(z_k) \hat{i} + \text{Im}(z_k) \hat{j} + \hat{k}$. It is given that the volume of the parallelepiped formed by $\vec{v}_1, \vec{v}_2, \vec{v}_3$ is zero. If the cubic equation has non-real roots and the distance between the points corresponding to these non-real roots is $6$, find the value of $p + q$.

Let $z_1$ and $z_2$ be the roots of the complex quadratic equation $z^2 - (1+3i)z - 2 + 2i = 0$. The value of $|z_1| + |z_2|$ is:

Let $P(x) = x^3 - 5x^2 + 9x - 5$. Let $\alpha$ be the real root and $\beta \pm i\gamma$ (where $\beta, \gamma \in \mathbb{R}$ and $\gamma > 0$) be the complex roots of the equation $P(x) = 0$.

Let $I = \int_{\alpha}^{\beta} \frac{2x-4}{|x^2-4x+1|} \, dx$ and $L = \lim_{n \to \infty} \sum_{k=1}^{n} \frac{2k - 2n}{|k^2 - 2kn - 2n^2|}$

Find the value of $e^{I + L}$.

Let $\alpha$ and $\beta$ be the roots of the equation $z^2 - (\sqrt{3}-1 + i(\sqrt{3}+1))z - 2\sqrt{3} + 2i = 0$. If $\text{Arg}(z)$ denotes the principal argument of a complex number $z$ such that $\text{Arg}(z) \in (-\pi, \pi]$, then the value of $\text{Arg}(\alpha^{200}) + \text{Arg}(\beta^{200})$ is

Let $\alpha, \beta, \gamma$ be the real roots of the equation $x^3 - px^2 + qx - r = 0$. Consider the points $A(\alpha, \beta, \gamma)$, $B(\beta, \gamma, \alpha)$, and $C(\gamma, \alpha, \beta)$ in $\mathbb{R}^3$. If the matrix formed by the position vectors of $A, B, C$ is singular, and the area of $\triangle ABC$ is $2\sqrt{3}$, then the maximum possible value of $r$ is:

Let $z_1, z_2, z_3, z_4$ be the distinct complex roots of the equation $z^4 + z^2 + 1 = 0$. For each $k \in \{1, 2, 3, 4\}$, let $A_k = \lim_{z \to z_k} \frac{\ln(z^4 + z^2 + 2)}{z^4 + z^2 + 1}$ Define the function $g(x) = \sum_{k=1}^4 \frac{A_k}{x - z_k}$ for $x \in \mathbb{R}$. Let $f(x) = \int_0^x g(t) dt$. Evaluate the integral: $I = \int_0^1 f(x) dx$

Let $A$ and $B$ be two events in a probability space such that $P(A) = \frac{2}{3}$ and $P(B) = \frac{1}{2}$. Consider the quadratic equation $z^2 - P(A \cup B)z + P(A \cap B) = 0$, whose roots are $\alpha$ and $\beta$. If $\alpha^3 + \beta^3 = \frac{143}{1728}$, then the value of the conditional probability $P(A \mid A^c \cup B^c)$ is:

Let $z_1, z_2, z_3$ be the roots of the equation $z^3 - (2+i)z^2 + (3+2i)z - 3i = 0$. A line $L$ passes through the point $A(1, 0, 0)$ and has direction cosines $(l, m, n)$ which are proportional to $(\text{Re}(z_1), \text{Re}(z_2), \text{Re}(z_3))$. A plane $P_1$ has a normal vector $\vec{n}_1$ whose direction ratios are proportional to $(\text{Im}(z_1), \text{Im}(z_2), \text{Im}(z_3))$. If the line $L$ intersects the plane $P_2: x + y + z = 1 + \sqrt{2}$ at a point $B$, and $\theta$ is the acute angle between the planes $P_1$ and $P_2$, then the value of $15 (AB)^2 \cos^2 \theta$ is:

Let $A$ be a $3 \times 3$ matrix with positive integer eigenvalues. Let $M_1 = \begin{pmatrix} \text{tr}(A) & -1 \\ |A| & 0 \end{pmatrix}$ and $M_2 = \begin{pmatrix} \text{tr}(\text{adj}(A)) & -1 \\ |\text{adj}(A)| & 0 \end{pmatrix}$. If $M_1$ and $M_2$ share a common eigenvalue and $|A| \neq 1$, find the value of $\text{tr}(A) + |A| + \text{tr}(\text{adj}(A))$.

Let $P(x) = x^3 + ax^2 + bx + c$ be a polynomial with real coefficients. A function $f(x)$ is defined as $f(x) = P(x) \lfloor x \rfloor$ for $x \in [0, 2]$, where $\lfloor \cdot \rfloor$ denotes the greatest integer function. If $f(x)$ satisfies all the conditions of Rolle's Theorem on the interval $[0, 2]$, then the value of $a^2 + b^2 + c^2$ is:

Let $E_1$ be the quadratic equation $x^2 + ax + b = 0$, where $a, b \in \mathbb{R}$ and $b \neq 0$. The point $P(a, b)$ is the pole of the chord of contact $L$ with respect to the parabola $x^2 = 4y$. Let $L$ intersect the x-axis and y-axis at points $A(\alpha, 0)$ and $B(0, \beta)$ respectively. If the quadratic equation $E_2$ whose roots are $\alpha$ and $\beta$ has at least one common root with $E_1$, then the locus of $P$ consists of a line and a parabola. If the line intersects the parabola at points $M$ and $N$, find the length of the line segment $MN$.

The number of values of $\theta \in [0, 2\pi]$ for which the quadratic equations $x^2 - 2|\sin \theta| x - 1 = 0$ and $x^2 - 2\sin \theta x - \cos 2\theta = 0$ have a common real root is:

Let $x, y, z$ be distinct positive real numbers such that $xyz = 8$. Let $M$ be the minimum value of the quadratic expression $f(t) = (x+y+z)t^2 + 2(x+y+z)t + \frac{(x+y+z)^2 + 54}{x+y+z}$ over all $t \in \mathbb{R}$. Find the maximum possible value of $[M]$, where $[ \cdot ]$ denotes the greatest integer function.

Let $f(x) = |x^2 + 2i|x| - 4 + i|$ for $x \in \mathbb{R}$, where $i = \sqrt{-1}$. Find the minimum value of $f(x)$.

Let $a$ be the minimum value of the function $P(x) = x^2 - 2x + 2$ and $b$ be the maximum value of the function $Q(x) = -x^2 + 10x - 20$. Consider the function $F(x)$ defined on $[a, b]$ by $F(x) = \int_{a}^{x} \ln|t^2 - 4t + 3| \, dt$ Let $S$ be the set of all points $x \in (a, b)$ such that $F'(x) = 0$. Find the sum of all elements in $S$.