Download pathania institue of mathematics

PATHANIA INSTITUE OF MATHEMATICS
PATHANIA INSTITUE OF MATHEMATICS
S.C.F 13 PHASE: 2 MOHALI, PH- 98145- 06093
S.C.F 13 PHASE: 2 MOHALI, PH- 98145- 06093
COMPLEX NUMBER
COMPLEX NUMBER
1.
Find the square root of i.
(a  i)
(a  1)
 p  iq, show that p 2  q 2 
.
2a  i
4a 2  1
2
2
1.
Find the square root of i.
2.
If
3.
If z = x + iy and  
2
(a  i) 2
(a 2  1) 2
 p  iq, show that p 2  q 2 
.
2a  i
4a 2  1
2.
If
3.
If z = x + iy and  
4.
1 i 
Find the least value of positive integer n if 
  1.
1 i 
4.
Find the least value of positive integer n if 
  1.
1 i 
5.
If z = x + iy, x , y real, prove that | x |  | y |  2 | z | .
5.
If z = x + iy, x , y real, prove that | x |  | y |  2 | z | .
6.
If (x + iy)3 = u + iv, show that
6.
If (x + iy)3 = u + iv, show that
7.
25

1 
Evaluate: i18     .
 i  

7.
 18  1  25 
Evaluate: i     .
 i  

|1  iz |
, show that |  | 1  z is purely real.
|z i|
|1  iz |
, show that |  | 1  z is purely real.
|z i|
n
u v
  4(x 2  y2 ).
x y
3
1 i 
n
u v
  4(x 2  y2 ).
x y
3
8.
Find the real numbers x and y if (x – iy) (3 + 5i) is conjugate of
-6 – 24i.
8.
Find the real numbers x and y if (x – iy) (3 + 5i) is conjugate of
-6 – 24i.
9.
If (a + ib) (c + id) (e + if) (g + ih) = A + iB, then show that
9.
If (a + ib) (c + id) (e + if) (g + ih) = A + iB, then show that
(a 2  b2 )(c2  d2 )(e2  f 2 )(g2  h 2 )  A2  B2 .
10.
Reduce the following to the standard form (a + ib):
(3  i 5)(3  i 5)
( 3  2i)  ( 3  i 2)
.
(a 2  b2 )(c2  d2 )(e2  f 2 )(g2  h 2 )  A2  B2 .
10.
Reduce the following to the standard form (a + ib):
(3  i 5)(3  i 5)
( 3  2i)  ( 3  i 2)
.
Solution
1.
4.
5.
2.
6.
3.
7.
8.
9.
10.
(a + ib) (c + id) (e + if) (g + ih) = A + iB