Question 1: Express the following complex numbers in the form
i) ii)
iii)
iv)
v) vi)
vii)
viii)
ix) x)
xi) xii)
Answer:
i)
ii)
iii)
iv)
v)
vi)
vii)
viii)
ix)
x)
xi)
xii)
Question 2: Find the real values of latex y, if
i) ii)
iii)
iv)
Answer:
i)
Therefore comparing, we get
… … … … … i)
… … … … … ii)
Solving i) and ii), multiplying i) by and ii) by
and adding we get
Now substituting in ii) we get
Hence
and
ii)
Therefore comparing, we get
… … … … … i)
… … … … … ii)
Solving i) and ii), multiplying i) by and ii) by
and adding we get
Now substituting in i) we get
Hence
and
iii)
… … … … … i)
… … … … … ii)
Solving i) and ii), multiplying i) by and ii) by
and adding we get
Substituting in i) we get
Hence and
iv)
Comapring we get
… … … … … i)
… … … … … ii)
Adding i) and ii) we get
Substituting in i) we get
Hence
and
Question 3: Find the conjugates of the following complex numbers:
i) ii)
iii)
iv) v)
vi)
Answer:
Note: if , then conjugate of
is
i) If
ii) If
Therefore
iii) If
Therefore
iv) If
Therefore
v) If
Therefore
vi) If
Therefore
Question 4: Find the multiplicative inverse of the following complex numbers:
i) ii)
iii)
iv)
Answer:
If is the complex number, then multiplication inverse of
is
or
i)
ii)
iii)
iv)
Question 5: If , find
Answer:
If , then
Question 6: If , find i)
ii)
Answer:
i)
ii)
Question 7: Find the modulus of
Answer:
Question 8: If
, prove that
Answer:
Hence proved.
Therefore the general solution is given by
Question 9: Find the least positive integral value of for which
is real.
Answer:
For
which is not real.
For
which is real.
Therefore the least positive integral value of
Question 10: Find the real values of for which the complex number
is purely real.
Answer:
Let
For to be purely Real,
Question 11: Find the smallest positive integer value of for which
is a real number.
Answer:
For this to be real , the smallest positive value of will be
.
Thus , which is real.
Question 12: If
, find
Answer:
and
Question 13: If
, find
Answer:
Question 14: If
, find
Answer:
Question 15: If , find the value of
Answer:
Question 16:
i) , when
ii) , when
iii) , when
iv) , when
v) , when
Answer:
i)
Now
ii)
Now,
iii)
Now
iv)
Now,
v)
Now,
Question 17: For a positive integer, find the value of
Answer:
Question 18: If , then show that
Answer:
Hence proved.
Question 19: Solve the system of equations .
Answer:
Let
… … … … … i)
… … … … … ii)
Adding i) and ii) we get
Hence
For positive sign for
For negative sign for
Question 20: If is purely imaginary number
, find the value of
Answer:
Let
If it is purely imaginary,
Question 21: If is a complex number other than
such that
and
then show that the real parts of
is zero.
Answer:
Let and
… … … … … i)
Since there is no real part, is purely an imaginary numbers
Question 22: If , find
Answer:
Let
Comparing,
Question 23: Solve the equation
Answer:
Let
Comparing,
and
Question 24: What is the smallest positive integer for which
?
Answer:
is a multiple of
.
Hence the smallest positive number for which
is
Question 25: If , are complex numbers such that,
, then find the value of
.
Answer:
Question 26: Find the number of solutions of
Answer:
Let
Now
Therefore or
\Rightarrow z = 0 $
For
Thus there are infinite many solution of the form