1.3 Conjugate (in class)

Evaluate the following numbers.

(a) [math]\left(\dfrac{-3+3i}{3+i}\right)^*[/math]

(b) [math]\dfrac{(2i)^*}{1-2i}[/math]

(c) [math]((4+7i)^{3})^*-((4+7i)^*)^3[/math]

Let [math]z[/math] be a complex number and [math]z\neq0[/math]. If [math]z^*=-z[/math], which of the following statements about [math]z[/math] is true?

Let [math]a, b[/math] be real numbers. If [math]\dfrac{6-7i}{a+bi}=2+i[/math], what is the value of [math]a+b[/math]?

Let [math]x[/math] be a complex number.

(a) [math]i(x+x^*)[/math] is a
(A) real number
(B) pure imaginary number
(C) neither

(b) [math](x-x^*)^2[/math] is a
(A) real number
(B) pure imaginary number
(C) neither

(c) [math]\dfrac{x}{x^*}[/math] is a
(A) real number
(B) pure imaginary number
(C) neither

Consider three points [math]O, A[/math], and [math]B[/math] on the complex plane. [math]O[/math] is at the origin. If [math]A=-13i[/math] and [math]B=8+7i[/math], what is the area of triangle [math]OAB[/math]?

Let [math]a[/math] be a positive real number that satisfying [math]|2+ai|\leq3[/math]. If the maximum possible of [math]a[/math] is [math]\sqrt{b}[/math] where [math]b[/math] is an integer, what is the value of [math]b[/math]?

[math]|3-4i|=[/math]