Page 156 - Engineering Mathematics Workbook_Final

Complex Variables

140. Consider the circle z − − 2 (a) 1 (b) e x − y 2
2
5 5i = in
the complex number plane (x, y) with (c) e (d) e − y
y
z = x iy . The minimum distance
+
from the origin to the circle is [GATE-2009 (IN)]
3
j
145. One of the roots of equation x = ,
2
(a) 5 2 − (b) 54
where j is the positive square roots of
(c) 34 (d) 5 2 -1 is
[GATE-2005 (IN)] 3 j

3
141. Let z = , where z is a complex (a) j (b) 2 + 2
z
number not equal to zero. Then z is a 3 j 3 j
solution of (c) − (d) − −
2 2 2 2
2
3
(a) z = 1 (b) z = 1 [GATE-2009 (IN)]
4
9
(c) z = 1 (d) z = 1
146. If x = − 1 , then the value of x is
x
[GATE-2005 (IN)]
(a) e −  /2 (b) e  /2
3 1
142. If a complex number z = + i (c) x (d) 1
2 2
4
then z is [GATE-2012-EC, EE, IN]
147. Square roots of -i, where i = − 1,
1 3
(a) 2 2 + 2i (b) + i are
2 2
3 1 3 1 (a) i, -1
(c) i − (d) i −      


2 2 8 8 (b) cos −  i + sin −  ,
 
 
 
 
[GATE-2007 (PI)]  4   4 
 3   3 
143. The equation sin (z) = 10 has cos   i + sin  
 
 
 
 
 4   4 
(a) no real (or) complex solution     3  
(c) cos   i + sin   ,
 
 
 
 
(b) exactly two distinct complex  4   4 
solution  3    
cos   i + sin  
 
 
 
 
(c) a unique solution  4   4 
(d) an infinite number of complex  3   3 




solutions (d) cos   4   i + sin −   4   ,




[GATE-2008 (ME)]  3    3  
cos −   i + sin  
 
 
 
 
144. If Z = x + jy where x, y are real  4   4 
when the value of e jz is [GATE-2013-EE]

154