Let P=zin C:|z+2-3i|≤1$P=\{z\in\mathbb{C}:|z+2-3i|\le1\}$ and Q=zin C:z(1+i)+ z(1-i)≤-8$Q=\{z\in\mathbb{C}:z(1+i)+\overline{z}(1-i)\le-8\}$. Let in P Q, |z-3+2i|$P \cap Q, |z-3+2i|$ be maximum and minimum at z₁$z_{1}$ and z₂$z_{2}$ respectively. If |z₁|²+2|z₂|²=α+β√(2)$|z_{1}|^{2}+2|z_{2}|^{2}=\alpha+\beta\sqrt{2}$, where α, β$\alpha, \beta$ are integers, then α+β$\alpha+\beta$ equals

Related Formula

A complex number w$w$ is purely real if w = w$w = \bar{w}$. A complex number w$w$ is purely imaginary if w + w = 0$w + \bar{w} = 0$.

Core Logic

Let's evaluate statement (S1):

If w$w$ is purely real, then w = w$w = \bar{w}$:

Since z + z = 2Re(z) = 0$z + \bar{z} = 2\text{Re}(z) = 0$, z$z$ must lie on the imaginary axis (y-axis). Given the condition |z| = 1$|z| = 1$, the only points are z = i$z = i$ and z = -i$z = -i$. However, the domain excludes z = -i$z = -i$. Let's test z = i$z = i$: For z = i$z = i$, (i - i)/(i + i) = 0$\frac{i - i}{i + i} = 0$, which is purely real. So it contains elements on the unit circle. But the condition z + z = 0$z + \bar{z} = 0$ alongside |z|=1$|z|=1$ explicitly limits it to z=i$z=i$ only, which is one element, not two. Thus, (S1) is incorrect.

Step 1: Evaluate Statement S2

Let's evaluate statement (S2):

If u$u$ is purely imaginary, then u + u = 0$u + \bar{u} = 0$:

This condition holds true for ALL points on the unit circle |z| = 1$|z| = 1$ except z = -1$z = -1$ (which makes the denominator zero). Because there are infinitely many points on the unit circle, the set contains infinitely many elements. Thus, (S2) is correct.

Pattern Recognition

Geometric shortcut: The transformation w = (z-1)/(z+1)$w = \frac{z-1}{z+1}$ maps the unit circle |z|=1$|z|=1$ directly onto the imaginary axis Re(w)=0$\text{Re}(w)=0$. Hence, any point on the unit circle (except the pole at z=-1$z=-1$) satisfies the condition naturally.

Chapter Mix

Class 11 Mathematics: Complex Numbers and Quadratic Equations

Related Formula

Core Logic

Isolate the real fractional component block by conjugating the complex quotient matrix expression, then resolve the structural wave equations across bounds boundaries.

Step 1: Expand Complex Real Operator

Step 2: Factor Trigonometric Expressions

Step 3: Collect Domain Solutions and Evaluate Squares

Since angular coordinate parameters scan [0, 2π]$[0, 2\pi]$, multi frequency vectors trace out:

Pattern Recognition

Transforming algebraic equations to clean forms like (2θ) = 0$\\cos(2\\theta) = 0$ guarantees evenly distributed coordinate solutions across standard periodicity ranges.

Chapter Mix

Class 11 Mathematics: Complex Numbers Class 11 Mathematics: Trigonometric Functions

Related Formula

Properties of cube roots of unity:

Core Logic

Simplify the determinant by performing row operation R₁ → R₁ + R₂ + R₃$R_1 \to R_1 + R_2 + R_3$:

Using 1 + ω + ω² = 0$1 + \omega + \omega^2 = 0$, the top row simplifies to vector [z, z, z]$[z, z, z]$. Factoring out z$z$:

Given modulus constraint |z³| = 1 |z| = 1$|z^3| = 1 \implies |z| = 1$. The root solutions are:

Step 1: Evaluate Geometric Magnitude Metric

The condition |(z - a)/(z + b)| = 1 |z - a| = |z + b|$\left|\frac{z - a}{z + b}\right| = 1 \implies |z - a| = |z + b|$. This equation represents the perpendicular bisector of the segment connecting real coordinate points a$a$ and -b$-b$ on the complex plane.

Since a$a$ and b$b$ are integers, the bisector is a vertical line: x = (a - b)/(2)$x = \frac{a - b}{2}$.

Step 2: Match Root Solutions and Count Pairs

For z=1$z=1$, it must lie on the line: (a-b)/(2) = 1 a - b = 2$\frac{a-b}{2} = 1 \implies a - b = 2$. For z = ω, ω²$z = \omega, \omega^2$, their real part is -(1)/(2)$-\frac{1}{2}$, so the line must be: (a-b)/(2) = -(1)/(2) a - b = -1$\frac{a-b}{2} = -\frac{1}{2} \implies a - b = -1$.

Counting integer pairs (a,b) in [-3, 3]²$(a,b) \in [-3, 3]^2$ with a+b ≠ 0$a+b \neq 0$: From a - b = 2$a - b = 2$: valid pairs are (3,1), (1,-1), (0,-2), (-1,-3)$(3,1), (1,-1), (0,-2), (-1,-3)$. Note: (2,0)$(2,0)$ is valid, but a+b=2 ≠ 0$a+b=2 \neq 0$. Total = 5 pairs. From a - b = -1$a - b = -1$: valid pairs match another 5 configurations.

Combining both groups gives a final count of 10 pairs.

Pattern Recognition

Using matrix summation properties (1+ω+ω²=0$1+\omega+\omega^2=0$) helps simplify large complex variable equations quickly.

Chapter Mix

Class 11 Mathematics: Complex Numbers Class 12 Mathematics: Matrices and Determinants

Related Formula

Complex rotation and scaling vector rule:

Core Logic

Given structural rotation conditions:

Evaluating the vectors yields coordinates showing |z₁ - z₂| = |z₂|$|z_1 - z_2| = |z_2|$.

Step 1: Analyzing Geometry Metrics

Since |z₁ - z₂| = |z₂|$|z_1 - z_2| = |z_2|$, Δ ABO$\Delta ABO$ forms an isosceles triangle with internal vertex angles evaluating explicitly to (π)/(6), (π)/(6)$\frac{\pi}{6}, \frac{\pi}{6}$, and (2π)/(3)$\frac{2\pi}{3}$.

Step 2: Conclusion

Since (2π)/(3) > (π)/(2)$\frac{2\pi}{3} > \frac{\pi}{2}$, the triangle is an obtuse-angled isosceles triangle.

Pattern Recognition

Complex argument shifts represent pure coordinate system rotations on the Argand plane diagram matrix.

Chapter Mix

Class 11 Maths: Complex Numbers

Related Formula

For a quadratic system equation az²+bz+c=0$az^2+bz+c=0$ with roots z₁, z₂$z_1, z_2$:

• z₁ + z₂ = -b/a$z_1 + z_2 = -b/a$
• z₁ z₂ = c/a$z_1 z_2 = c/a$
• z₁² + z₂² = (z₁+z₂)² - 2z₁z₂$z_1^2 + z_2^2 = (z_1+z_2)^2 - 2z_1z_2$

Core Logic

Cross-multiply the denominators to configure a linear equation layout:

Formulate the classic quadratic representation layout:

Step 1: Summing the squared roots

Identify coefficients from the structural template:

Evaluate sum of possible squared values (z₁² + z₂²$z_1^2 + z_2^2$):

= -19 - 2i$= -19 - 2i$

Pattern Recognition

The question asks for the sum of values of z²$z^2$, meaning z₁² + z₂²$z_1^2 + z_2^2$. Avoid using complex quadratic formulas to solve for z$z$ explicitly; structural expansions save massive computational effort.

Chapter Mix

Class 11 Mathematics: Complex Numbers