Let integers a, b in [-3, 3]$a, b \in [-3, 3]$ be such that a + b ≠ 0$a + b \neq 0$. Then the number of all possible ordered pairs (a, b)$(a, b)$, for which | (z - a)/(z + b) | = 1$\left| \frac{z - a}{z + b} \right| = 1$ and | arraycccz + 1 & ω & ω² ω & z + ω² & 1 ω² & 1 & z + ω array | = 1, z in C$\left| \begin{array}{ccc}z + 1 & \omega & \omega^2\\ \omega & z + \omega^2 & 1\\ \omega^2 & 1 & z + \omega \end{array} \right| = 1, z \in \mathbb{C}$, where ω$\omega$ and ω²$\omega^2$ are the roots of x² + x + 1 = 0$x^{2} + x + 1 = 0$, is equal to

Related Formula

Core Logic

Represent z$z$ in polar form:

Calculate z²⁰¹$z^{201}$:

Step 1: Simplify Exponent

Note that (67π)/(2) = 33π + (π)/(2)$\frac{67\pi}{2} = 33\pi + \frac{\pi}{2}$.

Thus, z²⁰¹ = -i$z^{201} = -i$.

Step 2: Final Calculation

Substitute z²⁰¹$z^{201}$ into the expression:

Pattern Recognition

Convert standard coordinate complex numbers into Euler form immediately when large powers are present.

Chapter Mix

Class 11 Maths: Complex Numbers

Related Formula

Core Logic

First condition: |z - 6i| = |z - 2i|$|z - 6i| = |z - 2i|$. This means z$z$ lies on the perpendicular bisector of (0,6)$(0,6)$ and (0,2)$(0,2)$. Let z = x + iy$z = x + iy$. Thus, y = 4$y = 4$.

Step 1: Circle Equation

Second condition: 5|z - 8 + 2i| = 3|z + 2i|$5|z - 8 + 2i| = 3|z + 2i|$. Substitute y = 4$y = 4$ into z$z$: z = x + 4i$z = x + 4i$.

Squaring both sides:

Divide by 16:

Step 2: Roots and Modulus

Roots of x² - 25x + 136 = 0$x^2 - 25x + 136 = 0$:

Thus, the points in S$S$ are z₁ = 17 + 4i$z_1 = 17 + 4i$ and z₂ = 8 + 4i$z_2 = 8 + 4i$.

Pattern Recognition

Whenever an absolute value ratio equals 1, immediately map it to a line (perpendicular bisector) and substitute its constraint directly into the second curve equation to reduce dimensionality.

Chapter Mix

Class 11 Maths: Complex Numbers and Quadratic Equations

Related Formula

Core Logic

Given z = Πr=1ⁿ (1 + ri)$z = \prod_{r=1}^{n} (1 + ri)$. Take the modulus of both sides:

Square both sides:

Step 1: Factoring the Target Number

We are given |z|² = 44200$|z|^2 = 44200$. Factorizing 44200$44200$:

Step 2: Expanding the Product

Calculate the product series step-by-step for small values of n$n$: For r=1$r=1$: 1 + 1² = 2$1 + 1^2 = 2$ For r=2$r=2$: 1 + 2² = 5$1 + 2^2 = 5$ For r=3$r=3$: 1 + 3² = 10 = 2 × 5$1 + 3^2 = 10 = 2 \times 5$ For r=4$r=4$: 1 + 4² = 17$1 + 4^2 = 17$ For r=5$r=5$: 1 + 5² = 26 = 2 × 13$1 + 5^2 = 26 = 2 \times 13$

Now, multiply these first 5 terms together:

Step 3: Comparing and Concluding

The calculated product for n=5$n=5$ exactly matches the prime factorization of 44200$44200$. Therefore, n = 5$n = 5$.

Pattern Recognition

Modulus is multiplicative. In problems featuring chains of complex multiplications set equal to a huge real magnitude, instantly switch to magnitudes and map to integer factorization.

Chapter Mix

Class 11 Maths: Complex Numbers

Core Logic

Distance between the centers C₁$C_1$ and C₂$C_2$:

Notice that C₁C₂ = r₁ + r₂ = 5 + 5 = 10$C_1C_2 = r_1 + r_2 = 5 + 5 = 10$. This means the two circles touch each other externally at exactly one point.

Step 1: Finding z

Since the circles touch externally, the common point z$z$ is the midpoint of the line segment joining the centers C₁$C_1$ and C₂$C_2$.

z = 2 + 3i$z = 2 + 3i$

Step 2: Simplifying the Polynomial

We have z = 2 + 3i$z = 2 + 3i$. z - 2 = 3i$z - 2 = 3i$ Squaring both sides: (z - 2)² = -9$(z - 2)^2 = -9$

z² = 4z - 13$z^2 = 4z - 13$

Step 3: Evaluating the Expression

We need to evaluate z³ + 3z² - 15z + 141$z^3 + 3z^2 - 15z + 141$. First, find z³$z^3$:

Substitute z² = 4z - 13$z^2 = 4z - 13$ again:

Now substitute z³$z^3$ and z²$z^2$ into the target expression:

= 0 + 50 = 50$= 0 + 50 = 50$

Pattern Recognition

When given two complex distance modulus equations |z-z₁|=r₁$|z-z_1|=r_1$ and |z-z₂|=r₂$|z-z_2|=r_2$, always check the distance between centers |z₁ - z₂|$|z_1 - z_2|$. If it exactly equals r₁ + r₂$r_1 + r_2$, the single unique solution is the section formula midpoint.

Chapter Mix

Class 11 Mathematics: Complex Numbers and Quadratic Equations

Related Formula

For a quadratic equation ax² + bx + c = 0$ax^2 + bx + c = 0$: Sum of roots: α + β = -(b)/(a)$\alpha + \beta = -\frac{b}{a}$ Product of roots: αβ = (c)/(a)$\alpha\beta = \frac{c}{a}$

Core Logic

From the given equation λ x² - (λ + 3)x + 3 = 0$\lambda x^2 - (\lambda + 3)x + 3 = 0$:

We are given the condition:

Step 1: Square Identity

We know that (α + β)² - 4αβ = (β - α)²$(\alpha + \beta)^2 - 4\alpha\beta = (\beta - \alpha)^2$. Substitute the known values:

So, we substitute these into the identity:

Step 2: Solving for Lambda

Multiply the entire equation by λ²$\lambda^2$ (assuming λ ≠ 0$\lambda \neq 0$ because it's a quadratic leading coefficient):

Step 3: Finding Roots

So, λ = 2$\lambda = 2$ or λ = 4$\lambda = 4$. (The solution notes λ=0$\lambda=0$ derived from λ³ - 6λ² + 8λ = 0$\lambda^3 - 6\lambda^2 + 8\lambda = 0$ by cross multiplication, but λ=0$\lambda=0$ degrades the quadratic. Thus valid λ in 2, 4$\lambda \in \{2, 4\}$). Sum of possible values of λ = 2 + 4 = 6$\lambda = 2 + 4 = 6$.

Chapter Mix

Class 11 Mathematics: Complex Numbers and Quadratic Equations