Column I Column II (a) If 2+√(3) i is a root of the equation x^4-4 x^2+8 x+ (p) 2+2 i 35=0 then

step 1 Let us solve this match the following problem. Let's see part A. In part A, 2 plus i root 3 is a

Column I Column II (a) If 2+√(3) i is a root of the equation...

Column I Column II (a) If $2+\sqrt{3} \mathrm{i}$ is a root of the equation $\mathrm{x}^{4}-4 \mathrm{x}^{2}+8 \mathrm{x}+$ (p) $2+2 \mathrm{i}$ $35=0$ then the other roots are (b) Conjugate of $(1+\mathrm{i})(2-3 \mathrm{i})\left(\frac{-11}{26}+\frac{3 \mathrm{i}}{26}\right)$ is equal to (q) $2-\sqrt{3}$ i (c) The complex number $\mathrm{z}$ satisfying the relation (r) $-2-i$ $|z-1|=|z-3|=|z-i|$ is (d) Modulus of the complex number $\frac{(1+i)(2-i)}{\frac{7}{5}-\frac{1}{5} i}$ is (s) $-2+\mathrm{i}$

Column I Column II
(a) If $2+\sqrt{3} \mathrm{i}$ is a root of the equation $\mathrm{x}^{4}-4 \mathrm{x}^{2}+8 \mathrm{x}+$
(p) $2+2 \mathrm{i}$
$35=0$ then the other roots are
(b) Conjugate of $(1+\mathrm{i})(2-3 \mathrm{i})\left(\frac{-11}{26}+\frac{3 \mathrm{i}}{26}\right)$ is equal to
(q) $2-\sqrt{3}$ i
(c) The complex number $\mathrm{z}$ satisfying the relation
(r) $-2-i$ $|z-1|=|z-3|=|z-i|$ is
(d) Modulus of the complex number $\frac{(1+i)(2-i)}{\frac{7}{5}-\frac{1}{5} i}$ is
(s) $-2+\mathrm{i}$

Key Concepts

Complex Number Arithmetic

This concept involves performing operations such as addition, subtraction, multiplication, and division on complex numbers, which are numbers of the form a + bi. Mastery of these operations is crucial for manipulating complex expressions, simplifying fractions, and reformatting terms, which form the basis for more advanced topics in complex analysis and algebra.

Complex Conjugates

The complex conjugate of a complex number a + bi is defined as a - bi. This concept is essential not only for simplifying expressions and finding moduli but also for solving equations, as non-real roots of polynomials with real coefficients appear in conjugate pairs. The properties of conjugates play a significant role in rationalizing denominators and in various proofs within complex analysis.

Complex Polynomial Equations

This key concept refers to equations where the unknown variable is a complex number and which have coefficients that are either real or complex. The fundamental theorem of algebra guarantees that every non-constant polynomial equation has at least one complex root, and when the coefficients are real, non-real roots occur in conjugate pairs. Understanding this concept is crucial for solving, factoring, and analyzing the roots of complex polynomials.

Geometry in the Complex Plane

This concept deals with interpreting complex numbers as points or vectors in a two-dimensional plane, where the horizontal axis represents the real part and the vertical axis represents the imaginary part. It enables one to solve problems involving distances, midpoints, and loci defined by specific conditions such as equidistance, thereby bridging algebraic properties of complex numbers with their geometric representations.

Modulus of a Complex Number

The modulus, or absolute value, of a complex number a + bi is the distance from the origin to the point (a, b) in the complex plane, computed as ?(a^2 + b^2). This measure is fundamental in understanding the magnitude of complex numbers, and it is widely used in problems involving distances, scaling, and polar form representations.

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