Exercise 13.2
QUESTION 1
Express the following complex numbers in the standard form a+ib :
(i)
Sol :
(ii)
Sol :
(iii)
Sol :
(iv)
Sol :
(v)
Sol :
(vi)
Sol :
(vii)
Sol :
(viii)
Sol :
(ix)
Sol :
(x)
Sol :
(xi)
Sol :
(xii)
Sool :
QUESTION 2
Find the real values of x and y, if
(i)
Sol :
Comparing both the sides :
Multiplying equation (1) by 3 and equation (2) by 2 :
Adding equations (3) and (4) :
Substituting the value of y in equation (1) :
(ii)
Sol :
Comparing both sides:
Multiplying equation (1) with 3 and equation (2) with 2 :
Adding equations (3) and (4) :
Substituting the value of y in equation (1):
(ii)
Sol :
Comparing both sides :
Multiplying equation (1) by 3 and equation (2) by 8 ,
Adding equations (3) and (4):
Substituting the value of x in equation (1) :
(iii)
Sol :
Comparing both the sides :
Multiplying equation (2) by 2 :
Subtracting equation (3) from 1 :
Substituting the value of y in equation (1) :
(iv)
Sol :
Comparing both the sides
Adding equations (1) and (2)
Substituting the value of x in equation (1)
QUESTION 3
Find the conjugates of the following complex numbers:
(i)
Sol :
(ii)
Sol :
(iii)
Sol :
(iv)
Sol :
(v)
Sol :
(vi)
Sol :
QUESTION 4
Find the multiplicative inverse of the following complex numbers:
(i)
Sol :
Let then,
(ii)
Sol :
(iii)
Sol :
(iv)
Sol :
QUESTION 5
If ,
, find
Sol :
Also ,
QUESTION 6
If ,
, find
(i)
(ii)
Sol :
(i)Given ,
(ii) Given ,
[\because~(i^2=-1)]
Since no term containing i is present
QUESTION 7
Find the modulus of
Sol :
QUESTION 8
If , prove that
Sol :
Taking mod on the both sides:
squaring both the sides
hence proved
QUESTION 9
Find the least positive integral value of n for which
Sol :
For to be real , the least positive value of n will be 2 .
As
QUESTION 10
Find the real values of for which the complex number
is purely real .
Sol :
For it to be purely real , the imaginary part must be zero .
This is true for odd multiples of
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QUESTION 11
Find the smallest positive integer value of m for which
Sol :
For this to be real , the smallest positive value of m will be 1
Thus, , which is real
QUESTION 12
If , find
Sol :
Also,
It is given that ,
[from (1) and (2)]
Thus ,
QUESTION 13
If find
Sol :
It is given that
[from (1)]
QUESTION 14
If find (a+b)
Sol :
It is given that
[from (i)]
Thus ,
QUESTION 15
If , find the value of
Sol :
on putting value of
QUESTION 16
Evaluate the following:
(i) when
Sol :
[squaring both the sides]
[putting value of
]
= 4
(ii) when
Sol :
putting the values of
= 5
(iii) when
Sol :
[putting the values of ]
[putting the value of
]
= 12
(iv) when
Sol :
[squaring both the sides]
[putting value of
]
[putting the value of
]
= 0
(v) when
Sol :
putting the value of
on putting the value of
= 6
QUESTION 17
For a positive integer n , find the value of
Sol :
Thus , the value of is