1.3 Conjugate

Evaluate the following numbers.

(a) [math]\dfrac{1}{2+i}[/math]

(b) [math]\dfrac{5-2i}{4+3i}[/math]

(c) [math]\dfrac{6+12i}{5-5i}[/math]

Evaluate the following numbers.

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

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

(c) [math]\dfrac{(5i)^*}{(2-\sqrt{3}i)^*}[/math]

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

Let [math]z=a+bi[/math] be a complex number where [math]a<0[/math] and [math]b>0[/math] are constants. In which quadrant does the number [math]z^*[/math] lie?

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]x, y[/math] be complex numbers. Which statement(s) is/are true?

Let [math]z_1=3+5i[/math] and [math]z_2=-1+2i[/math], what is the value of [math]|z_1-z_2|[/math]?

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]?

If [math]x=a+bi[/math] is the solution to the equation [math](1+i)x-(4+i)=(5-2i)[/math], what is the value of [math]a[/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

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

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

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

(c) [math]x^2+(x^*)^2[/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]?

Let [math]z=a+bi[/math] be a complex number where [math]a[/math] and [math]b[/math] are real numbers. If [math]|z|=m[/math], express the following numbers in terms of [math]m[/math].

(a) [math]-a-bi[/math]

(b) [math]b-ai[/math]

(c) [math]2b+2ai[/math]

Let [math]z=5-9i[/math] be a complex number. Find the positive number [math]a[/math] such that [math]|z+a|=15[/math].

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

Evaluate the following numbers.

(a) [math]\dfrac{-2+3i}{-3-4i}[/math]

(b) [math](-3+2i)(i^{23})[/math]

Find the rationals [math]a[/math] and [math]b[/math] such that [math]\frac{2-3i}{2a+bi} =3+2i[/math].

If [math](-2-\sqrt{-32})-(4+\sqrt{-18})=a+b\sqrt{2}i[/math] where [math]a[/math] and [math]b[/math] are constants, what is the value of [math]a+b[/math]?

Evaluate [math]\displaystyle{\frac{10+2i}{-1-5i}}[/math].

If [math](6-\sqrt{-7})(-4+\sqrt{-7})=a+b\sqrt{c}[/math] where [math]a, b[/math], and [math]c[/math] are integers, what is the value of [math]a+b+c[/math]?

[math]\displaystyle{\frac{\sqrt{20}}{\sqrt{-4}}}=[/math]

Evaluate [math]\displaystyle{\frac{1-4i}{-2+6i}}[/math].

In the figure above, points [math]F[/math] and [math]G[/math] represent two complex numbers. Which point represents the sum of these two numbers?