Electronics and Communication Engineering (EC) 2007 GATE Paper without solution PDF Download

 Page 1


 EC? GATE Paper 2007  
Q.1 – Q.20 Carry One Mark Each 
1. If E denotes expectation, the variance of a random variable X is given by
(A) 
2 2
E X E X
? ?
- ? ?
? ?
? ?
 (B)
2 2
E X E X
? ?
+ ? ?
? ?
? ?
 (C)
2
E X
? ?
? ?
(D) 
2
E X ? ?
? ?
2. The following plot shows a function y which varies linearly with x . The value of
the integral 
2
1
 I y dx =
?
is
(A) 1.0 (B) 2.5 (C) 4.0 (D) 5.0 
3. For 1, x << ( ) coth x can be approximated as
(A) x (B) 
2
x (C) 
1
x
(D) 
2
1
x
4. 
( )
0
sin 2
lim
?
?
? ?
 is: 
(A) 0.5 (B) 1 (C) 2 (D) not defined 
5. Which one of the following functions is strictly bounded?
(A) 
2
1
x
(B) 
x
e (C) 
2
x (D) 
2
x
e
-
6. For the function ,
x
e
-
 the linear approximation around 2 x = is:
(A) ( )
2
3 x e
-
- (B) 1 x -
(C) 
( )
2
3 2 2 1 2 x e
-
? ?
+ - +
? ?
(D) 
2
e
-
7. An independent voltage source in series with an impedance
s s s
Z R jX = + delivers
a maximum average power to a load impedance
L
Z when 
(A)
L s s
Z R jX = + (B) 
L s
Z R =
(C)
L s
Z jX = (D) 
L s s
Z R jX = -
1 2 3 
x 
1 
2 
3 
y 
-1 
Page 2


 EC? GATE Paper 2007  
Q.1 – Q.20 Carry One Mark Each 
1. If E denotes expectation, the variance of a random variable X is given by
(A) 
2 2
E X E X
? ?
- ? ?
? ?
? ?
 (B)
2 2
E X E X
? ?
+ ? ?
? ?
? ?
 (C)
2
E X
? ?
? ?
(D) 
2
E X ? ?
? ?
2. The following plot shows a function y which varies linearly with x . The value of
the integral 
2
1
 I y dx =
?
is
(A) 1.0 (B) 2.5 (C) 4.0 (D) 5.0 
3. For 1, x << ( ) coth x can be approximated as
(A) x (B) 
2
x (C) 
1
x
(D) 
2
1
x
4. 
( )
0
sin 2
lim
?
?
? ?
 is: 
(A) 0.5 (B) 1 (C) 2 (D) not defined 
5. Which one of the following functions is strictly bounded?
(A) 
2
1
x
(B) 
x
e (C) 
2
x (D) 
2
x
e
-
6. For the function ,
x
e
-
 the linear approximation around 2 x = is:
(A) ( )
2
3 x e
-
- (B) 1 x -
(C) 
( )
2
3 2 2 1 2 x e
-
? ?
+ - +
? ?
(D) 
2
e
-
7. An independent voltage source in series with an impedance
s s s
Z R jX = + delivers
a maximum average power to a load impedance
L
Z when 
(A)
L s s
Z R jX = + (B) 
L s
Z R =
(C)
L s
Z jX = (D) 
L s s
Z R jX = -
1 2 3 
x 
1 
2 
3 
y 
-1 
 EC? GATE Paper 2007  
8. The RC circuit shown in the figure is
(A) a low-pass filter   (B) a high-pass filter  
(C) a band-pass filter (D) a band-reject filter 
9. The electron and hole concentrations in an intrinsic semiconductor are
i
n per 
3
cm at 300 K. Now, if acceptor impurities are introduced with a concentration of
A
N per 
3
cm ( ) where
A i
N n >> , the electron concentration per
3
cm at 300 K will 
be: 
(A) 
i
n (B) 
i A
n N + (C) 
A i
N n - (D) 
2
i
A
n
N
10. In a p n
+
junction diode under reverse bias, the magnitude of electric field is 
maximum at 
(A) the edge of the depletion region on the p-side 
(B) the edge of the depletion region on the n-side 
(C) the p n
+
 junction 
(D) the centre of the depletion region on the n-side 
11. The correct full wave rectifier circuit is:
(A) (B) 
R 
C 
R 
C 
i
V
o
V 
+ 
-
+ 
-
Input 
Output 
Input 
Output 
Page 3


 EC? GATE Paper 2007  
Q.1 – Q.20 Carry One Mark Each 
1. If E denotes expectation, the variance of a random variable X is given by
(A) 
2 2
E X E X
? ?
- ? ?
? ?
? ?
 (B)
2 2
E X E X
? ?
+ ? ?
? ?
? ?
 (C)
2
E X
? ?
? ?
(D) 
2
E X ? ?
? ?
2. The following plot shows a function y which varies linearly with x . The value of
the integral 
2
1
 I y dx =
?
is
(A) 1.0 (B) 2.5 (C) 4.0 (D) 5.0 
3. For 1, x << ( ) coth x can be approximated as
(A) x (B) 
2
x (C) 
1
x
(D) 
2
1
x
4. 
( )
0
sin 2
lim
?
?
? ?
 is: 
(A) 0.5 (B) 1 (C) 2 (D) not defined 
5. Which one of the following functions is strictly bounded?
(A) 
2
1
x
(B) 
x
e (C) 
2
x (D) 
2
x
e
-
6. For the function ,
x
e
-
 the linear approximation around 2 x = is:
(A) ( )
2
3 x e
-
- (B) 1 x -
(C) 
( )
2
3 2 2 1 2 x e
-
? ?
+ - +
? ?
(D) 
2
e
-
7. An independent voltage source in series with an impedance
s s s
Z R jX = + delivers
a maximum average power to a load impedance
L
Z when 
(A)
L s s
Z R jX = + (B) 
L s
Z R =
(C)
L s
Z jX = (D) 
L s s
Z R jX = -
1 2 3 
x 
1 
2 
3 
y 
-1 
 EC? GATE Paper 2007  
8. The RC circuit shown in the figure is
(A) a low-pass filter   (B) a high-pass filter  
(C) a band-pass filter (D) a band-reject filter 
9. The electron and hole concentrations in an intrinsic semiconductor are
i
n per 
3
cm at 300 K. Now, if acceptor impurities are introduced with a concentration of
A
N per 
3
cm ( ) where
A i
N n >> , the electron concentration per
3
cm at 300 K will 
be: 
(A) 
i
n (B) 
i A
n N + (C) 
A i
N n - (D) 
2
i
A
n
N
10. In a p n
+
junction diode under reverse bias, the magnitude of electric field is 
maximum at 
(A) the edge of the depletion region on the p-side 
(B) the edge of the depletion region on the n-side 
(C) the p n
+
 junction 
(D) the centre of the depletion region on the n-side 
11. The correct full wave rectifier circuit is:
(A) (B) 
R 
C 
R 
C 
i
V
o
V 
+ 
-
+ 
-
Input 
Output 
Input 
Output 
 EC? GATE Paper 2007  
(C) (D) 
12. In a transconductance amplifier, it is desirable to have
(A) a large input resistance and a large output resistance
(B) a large input resistance and a small output resistance
(C) a small input resistance and a large output resistance
(D) a small input resistance and a small output resistance
13. X = 01110 and Y = 11001 are two 5-bit binary numbers represented in two’s
complement format. The sum of X and Y represented in two’s complement format
using 6 bits is:
(A) 100111 (B) 001000 (C) 000111 (D) 101001 
14. The Boolean function Y AB CD = + is to be realized using only 2-input NAND
gates. The minimum number of gates required is:
(A) 2 (B) 3 (C) 4 (D) 5
15. If the closed-loop transfer function of a control system is given as
( )
( )( )
5
,
2 3
s
T s
s s
-
=
+ +
 then it is 
(A) an unstable system  (B) an uncontrollable system  
(C) a minimum phase system (D) a non-minimum phase system 
16. If the Laplace transform of a signal ( ) y t is ( )
( )
1
,
1
Y s
s s
=
-
 then its final value is: 
(A) -1 (B) 0 (C) 1 (D) unbounded 
17. If ( ) R t is the autocorrelation function of a real, wide-sense stationary random
process, then which of the following is NOT true?
(A) ( ) ( ) R R t t = - (B) ( ) ( ) 0 R R t = (C) ( ) ( ) R R t t = - -
(D) The mean square value of the process is ( ) 0 R
Input 
Output 
Input 
Output 
Page 4


 EC? GATE Paper 2007  
Q.1 – Q.20 Carry One Mark Each 
1. If E denotes expectation, the variance of a random variable X is given by
(A) 
2 2
E X E X
? ?
- ? ?
? ?
? ?
 (B)
2 2
E X E X
? ?
+ ? ?
? ?
? ?
 (C)
2
E X
? ?
? ?
(D) 
2
E X ? ?
? ?
2. The following plot shows a function y which varies linearly with x . The value of
the integral 
2
1
 I y dx =
?
is
(A) 1.0 (B) 2.5 (C) 4.0 (D) 5.0 
3. For 1, x << ( ) coth x can be approximated as
(A) x (B) 
2
x (C) 
1
x
(D) 
2
1
x
4. 
( )
0
sin 2
lim
?
?
? ?
 is: 
(A) 0.5 (B) 1 (C) 2 (D) not defined 
5. Which one of the following functions is strictly bounded?
(A) 
2
1
x
(B) 
x
e (C) 
2
x (D) 
2
x
e
-
6. For the function ,
x
e
-
 the linear approximation around 2 x = is:
(A) ( )
2
3 x e
-
- (B) 1 x -
(C) 
( )
2
3 2 2 1 2 x e
-
? ?
+ - +
? ?
(D) 
2
e
-
7. An independent voltage source in series with an impedance
s s s
Z R jX = + delivers
a maximum average power to a load impedance
L
Z when 
(A)
L s s
Z R jX = + (B) 
L s
Z R =
(C)
L s
Z jX = (D) 
L s s
Z R jX = -
1 2 3 
x 
1 
2 
3 
y 
-1 
 EC? GATE Paper 2007  
8. The RC circuit shown in the figure is
(A) a low-pass filter   (B) a high-pass filter  
(C) a band-pass filter (D) a band-reject filter 
9. The electron and hole concentrations in an intrinsic semiconductor are
i
n per 
3
cm at 300 K. Now, if acceptor impurities are introduced with a concentration of
A
N per 
3
cm ( ) where
A i
N n >> , the electron concentration per
3
cm at 300 K will 
be: 
(A) 
i
n (B) 
i A
n N + (C) 
A i
N n - (D) 
2
i
A
n
N
10. In a p n
+
junction diode under reverse bias, the magnitude of electric field is 
maximum at 
(A) the edge of the depletion region on the p-side 
(B) the edge of the depletion region on the n-side 
(C) the p n
+
 junction 
(D) the centre of the depletion region on the n-side 
11. The correct full wave rectifier circuit is:
(A) (B) 
R 
C 
R 
C 
i
V
o
V 
+ 
-
+ 
-
Input 
Output 
Input 
Output 
 EC? GATE Paper 2007  
(C) (D) 
12. In a transconductance amplifier, it is desirable to have
(A) a large input resistance and a large output resistance
(B) a large input resistance and a small output resistance
(C) a small input resistance and a large output resistance
(D) a small input resistance and a small output resistance
13. X = 01110 and Y = 11001 are two 5-bit binary numbers represented in two’s
complement format. The sum of X and Y represented in two’s complement format
using 6 bits is:
(A) 100111 (B) 001000 (C) 000111 (D) 101001 
14. The Boolean function Y AB CD = + is to be realized using only 2-input NAND
gates. The minimum number of gates required is:
(A) 2 (B) 3 (C) 4 (D) 5
15. If the closed-loop transfer function of a control system is given as
( )
( )( )
5
,
2 3
s
T s
s s
-
=
+ +
 then it is 
(A) an unstable system  (B) an uncontrollable system  
(C) a minimum phase system (D) a non-minimum phase system 
16. If the Laplace transform of a signal ( ) y t is ( )
( )
1
,
1
Y s
s s
=
-
 then its final value is: 
(A) -1 (B) 0 (C) 1 (D) unbounded 
17. If ( ) R t is the autocorrelation function of a real, wide-sense stationary random
process, then which of the following is NOT true?
(A) ( ) ( ) R R t t = - (B) ( ) ( ) 0 R R t = (C) ( ) ( ) R R t t = - -
(D) The mean square value of the process is ( ) 0 R
Input 
Output 
Input 
Output 
 EC? GATE Paper 2007  
18. If ( ) S f is the power spectral density of a real, wide-sense stationary random
process, then which of the following is ALWAYS true? 
(A) ( ) ( ) 0 S S f = (B) ( ) 0 S f = (C) ( ) ( ) S f S f - = - (D) ( ) 0 S f df
8
-8
=
?
19. A plane wave of wavelength ? is traveling in a direction making an angle 30° with
positive x-axis and 90° with positive y-axis. The E
u r
 field of the plane wave can be
represented as ( ) is constant
o
E
(A) 
µ
3
0
j t x z
E yE e
p p
?
? ?
? ?
- - ? ?
? ?
? ?
=
u r
(B) 
µ
3
0
j t x z
E yE e
p p
?
? ?
? ?
- - ? ?
? ?
? ?
=
u r
(C) 
µ
3
0
j t x z
E yE e
p p
?
? ?
? ?
+ + ? ?
? ?
? ?
=
u r
(D) 
µ
3
0
j t x z
E yE e
p p
?
? ?
? ?
- + ? ?
? ?
? ?
=
u r
20. If C is a closed curve enclosing a surface S, then the magnetic field intensity , H
u r
the current density J
r
 and the electric flux density D
u r
are related by 
(A) . .
s c
D
H ds J dl
t
? ?
?
= + ? ?
? ?
?
? ?
?? ?
u r
u r r r r
Ñ
(B) . .
c s
D
H dl J d ds
t
? ?
?
= + ? ?
? ?
?
? ?
? ??
u r
r u r r r
Ò
(C) . .
s c
D
H ds J dl
t
? ?
?
= + ? ?
? ?
?
? ?
?? ?
u r
u r r r r
Ò
(D) . .
c s
D
H dl J ds
t
? ?
?
= + ? ?
? ?
?
? ?
? ??
u r
u r r r r
Ñ
Q.21 – Q.75 Carry Two Marks Each 
21. It is given that 
1 2
, ,....
M
X x X are M non-zero, orthogonal vectors. The dimension of 
the vector space spanned by the 2M vectors
1 2 1 2
, ,... , , ,...
M M
X X X X X X - - - is:
(A) 2M (B) M + 1 (C) M 
(D) dependent on the choice of
1 2
, ,...
M
X X X
22. Consider the function ( )
2
2. f x x x = - - The maximum value of ( ) f x in the closed
interval 4,4 - ? ?
? ?
 is:
(A) 18 (B) 10 
(C) -2.25 (D) indeterminate 
23. An examination consists of two papers, Paper 1 and Paper 2. The probability of
failing in Paper 1 is 0.3 and that in Paper 2 is 0.2. Given that a student has failed
in Paper 2, the probability of failing in Paper 1 is 0.6. The probability of a student
failing in both the papers is:
(A) 0.5 (B) 0.18 (C) 0.12 (D) 0.06 
Page 5


 EC? GATE Paper 2007  
Q.1 – Q.20 Carry One Mark Each 
1. If E denotes expectation, the variance of a random variable X is given by
(A) 
2 2
E X E X
? ?
- ? ?
? ?
? ?
 (B)
2 2
E X E X
? ?
+ ? ?
? ?
? ?
 (C)
2
E X
? ?
? ?
(D) 
2
E X ? ?
? ?
2. The following plot shows a function y which varies linearly with x . The value of
the integral 
2
1
 I y dx =
?
is
(A) 1.0 (B) 2.5 (C) 4.0 (D) 5.0 
3. For 1, x << ( ) coth x can be approximated as
(A) x (B) 
2
x (C) 
1
x
(D) 
2
1
x
4. 
( )
0
sin 2
lim
?
?
? ?
 is: 
(A) 0.5 (B) 1 (C) 2 (D) not defined 
5. Which one of the following functions is strictly bounded?
(A) 
2
1
x
(B) 
x
e (C) 
2
x (D) 
2
x
e
-
6. For the function ,
x
e
-
 the linear approximation around 2 x = is:
(A) ( )
2
3 x e
-
- (B) 1 x -
(C) 
( )
2
3 2 2 1 2 x e
-
? ?
+ - +
? ?
(D) 
2
e
-
7. An independent voltage source in series with an impedance
s s s
Z R jX = + delivers
a maximum average power to a load impedance
L
Z when 
(A)
L s s
Z R jX = + (B) 
L s
Z R =
(C)
L s
Z jX = (D) 
L s s
Z R jX = -
1 2 3 
x 
1 
2 
3 
y 
-1 
 EC? GATE Paper 2007  
8. The RC circuit shown in the figure is
(A) a low-pass filter   (B) a high-pass filter  
(C) a band-pass filter (D) a band-reject filter 
9. The electron and hole concentrations in an intrinsic semiconductor are
i
n per 
3
cm at 300 K. Now, if acceptor impurities are introduced with a concentration of
A
N per 
3
cm ( ) where
A i
N n >> , the electron concentration per
3
cm at 300 K will 
be: 
(A) 
i
n (B) 
i A
n N + (C) 
A i
N n - (D) 
2
i
A
n
N
10. In a p n
+
junction diode under reverse bias, the magnitude of electric field is 
maximum at 
(A) the edge of the depletion region on the p-side 
(B) the edge of the depletion region on the n-side 
(C) the p n
+
 junction 
(D) the centre of the depletion region on the n-side 
11. The correct full wave rectifier circuit is:
(A) (B) 
R 
C 
R 
C 
i
V
o
V 
+ 
-
+ 
-
Input 
Output 
Input 
Output 
 EC? GATE Paper 2007  
(C) (D) 
12. In a transconductance amplifier, it is desirable to have
(A) a large input resistance and a large output resistance
(B) a large input resistance and a small output resistance
(C) a small input resistance and a large output resistance
(D) a small input resistance and a small output resistance
13. X = 01110 and Y = 11001 are two 5-bit binary numbers represented in two’s
complement format. The sum of X and Y represented in two’s complement format
using 6 bits is:
(A) 100111 (B) 001000 (C) 000111 (D) 101001 
14. The Boolean function Y AB CD = + is to be realized using only 2-input NAND
gates. The minimum number of gates required is:
(A) 2 (B) 3 (C) 4 (D) 5
15. If the closed-loop transfer function of a control system is given as
( )
( )( )
5
,
2 3
s
T s
s s
-
=
+ +
 then it is 
(A) an unstable system  (B) an uncontrollable system  
(C) a minimum phase system (D) a non-minimum phase system 
16. If the Laplace transform of a signal ( ) y t is ( )
( )
1
,
1
Y s
s s
=
-
 then its final value is: 
(A) -1 (B) 0 (C) 1 (D) unbounded 
17. If ( ) R t is the autocorrelation function of a real, wide-sense stationary random
process, then which of the following is NOT true?
(A) ( ) ( ) R R t t = - (B) ( ) ( ) 0 R R t = (C) ( ) ( ) R R t t = - -
(D) The mean square value of the process is ( ) 0 R
Input 
Output 
Input 
Output 
 EC? GATE Paper 2007  
18. If ( ) S f is the power spectral density of a real, wide-sense stationary random
process, then which of the following is ALWAYS true? 
(A) ( ) ( ) 0 S S f = (B) ( ) 0 S f = (C) ( ) ( ) S f S f - = - (D) ( ) 0 S f df
8
-8
=
?
19. A plane wave of wavelength ? is traveling in a direction making an angle 30° with
positive x-axis and 90° with positive y-axis. The E
u r
 field of the plane wave can be
represented as ( ) is constant
o
E
(A) 
µ
3
0
j t x z
E yE e
p p
?
? ?
? ?
- - ? ?
? ?
? ?
=
u r
(B) 
µ
3
0
j t x z
E yE e
p p
?
? ?
? ?
- - ? ?
? ?
? ?
=
u r
(C) 
µ
3
0
j t x z
E yE e
p p
?
? ?
? ?
+ + ? ?
? ?
? ?
=
u r
(D) 
µ
3
0
j t x z
E yE e
p p
?
? ?
? ?
- + ? ?
? ?
? ?
=
u r
20. If C is a closed curve enclosing a surface S, then the magnetic field intensity , H
u r
the current density J
r
 and the electric flux density D
u r
are related by 
(A) . .
s c
D
H ds J dl
t
? ?
?
= + ? ?
? ?
?
? ?
?? ?
u r
u r r r r
Ñ
(B) . .
c s
D
H dl J d ds
t
? ?
?
= + ? ?
? ?
?
? ?
? ??
u r
r u r r r
Ò
(C) . .
s c
D
H ds J dl
t
? ?
?
= + ? ?
? ?
?
? ?
?? ?
u r
u r r r r
Ò
(D) . .
c s
D
H dl J ds
t
? ?
?
= + ? ?
? ?
?
? ?
? ??
u r
u r r r r
Ñ
Q.21 – Q.75 Carry Two Marks Each 
21. It is given that 
1 2
, ,....
M
X x X are M non-zero, orthogonal vectors. The dimension of 
the vector space spanned by the 2M vectors
1 2 1 2
, ,... , , ,...
M M
X X X X X X - - - is:
(A) 2M (B) M + 1 (C) M 
(D) dependent on the choice of
1 2
, ,...
M
X X X
22. Consider the function ( )
2
2. f x x x = - - The maximum value of ( ) f x in the closed
interval 4,4 - ? ?
? ?
 is:
(A) 18 (B) 10 
(C) -2.25 (D) indeterminate 
23. An examination consists of two papers, Paper 1 and Paper 2. The probability of
failing in Paper 1 is 0.3 and that in Paper 2 is 0.2. Given that a student has failed
in Paper 2, the probability of failing in Paper 1 is 0.6. The probability of a student
failing in both the papers is:
(A) 0.5 (B) 0.18 (C) 0.12 (D) 0.06 
 EC? GATE Paper 2007  
24. The solution of the differential equation
2
2
2
2
d y
k y y
dx
= - under the boundary
conditions (i) 
1
y y = at 0 x = and (ii)
2
y y = at x = 8 , where , k
1
y and 
2
y are 
constants, is 
(A) ( )
( )
2
1 2 2
exp y y y x k y = - - + (B) ( ) ( )
2 1 1
exp y y y x k y = - - +
(C) ( ) ( )
1 2 1
sinh y y y x k y = - + (D) ( ) ( )
1 2 2
exp y y y x k y = - - +
25. The equation
3 2
4 4 0 x x x - + - = is to be solved using the Newton-Raphson
method. If 2 x = is taken as the initial approximation of the solution, then the
next approximation using this method will be: 
(A) 
2
3
(B) 
4
3
(C) 1 (D) 
3
2
26. Three functions ( ) ( ) ( )
1 2 3
, and , f t f t f t
which are zero outside the interval [0,T],
are shown in the figure. Which of the
following statements is correct?
(A) ( ) ( )
1 2
 and  f t f t are orthogonal 
(B) ( ) ( )
1 3
 and  f t f t are orthogonal 
(C) ( ) ( )
2 3
 and  f t f t are orthogonal 
(D) ( ) ( )
1 2
 and  f t f t are orthonormal 
27. If the semi-circular contour D of radius 2 is as shown in the figure, then the value
of the integral
( )
2
1
1
D
ds
s -
? Ñ
is: 
(A) jp (B) -jp (C) -p (D) p
t 
t 
t 
2 
-2 
1 
2 
-1 
1 
-3 
( )
1
f t
( )
2
f t
( )
3
f t
0 
3
T
2
3
T
T 
j?
j2 
D 
2 0 
0 
s 
-j2