Complex Numbers Problems with Solutions - Grade 12
Explore a variety of complex number problems with step-by-step solutions. Learn how to solve complex numbers, including operations, polar form, and applications. Complex numbers play a crucial role in applied mathematics, physics, electrical engineering, and other technical fields.
In what follows, is the imaginary unit.
Question 1
Evaluate the following expressions:
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Solution:
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Question 2
If , where and are real, what is the value of ?
Solution:
Question 3
Determine all complex numbers that satisfy the equation:
where denotes the complex conjugate of .
Solution:
Let , and its conjugate ; and real numbers
Substituting and in the given equation obtain
Simplify by grouping terms on theleft side:
Two complex numbers are equal if both their real and imaginary parts are equal.. Hence
Question 4
Find all complex numbers of the form , where and are real numbers, such that:
where, represents the complex conjugate of .
Solution:
Let
Hence its conjugate
therefore
Substitute above in the equation
Solve the above quadratic function for and use to find .
or
The complex numbers and satisfy .
Check that:
and have the property
Question 5
The complex number is one of the roots of the quadratic equation
where and are real numbers.
a) Find and
b) Write down the second root and check it.
Solution:
a) Substitute the root in the equation:
Expand terms in equation and rewrite as:
Real part and imaginary part are both equal to zero.
Solve for :
Substitute and solve for :
b) Since the given equation has real numbers, the second root is the complex conjugate of the given root:
is the second solution.
Check:
Expand:
Question 6
Find all complex numbers such that:
Solution:
Let
Substitute into given equation:
Expand:
Real part and imaginary parts must be equal.
Equation gives:
Substitute:
Solve above equation and select only real roots: and
Substitute to find and write the two complex numbers that satisfy the given equation.
Question 7
Find all complex numbers such that
where is the complex conjugate of .
Solution:
Let where and are real numbers. The complex conjugate is written in terms of and as follows: .
Substitute and in the given equation
Expand and separate real and imaginary parts.
Two complex numbers are equal if their real parts and imaginary parts are equal. Group like terms.
Solve the system of the unknown and to find:
Question 8
Given that the complex number is a root to the equation:
find the real root to the equation.
Solution:
Since is a root of the equation and all the coefficients in the terms of the equation are real numbers, then , the complex conjugate of , is also a solution. Hence we may factor the left side as follows:
is a factor of and therefore is the real root of the given equation.
Question 9
a) Show that the complex number is a root of the equation
b) Find all the roots of this equation.
Solution:
a) Substitute by in the left side of the expression:
which shows that is a root of the given equation.
b) Since is a root and all coefficients are real, is also a root (complex conjugate). Hence we may factor the left side of the given equation as follows:
The other two roots of the equation are the roots of and are given by: .
Question 10
is a polynomial where , , , and are real numbers.
Find , , , and if two zeros of polynomial are the following complex numbers: and .
Solution:
Since all coefficients of polynomial P are real, the complex conjugate
to the given zeros are also zeros of the polynomial . Hence in factored form: