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 lowpass filter (B) a highpass filter
(C) a bandpass filter (D) a bandreject 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 pside
(B) the edge of the depletion region on the nside
(C) the p n
+
junction
(D) the centre of the depletion region on the nside
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 lowpass filter (B) a highpass filter
(C) a bandpass filter (D) a bandreject 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 pside
(B) the edge of the depletion region on the nside
(C) the p n
+
junction
(D) the centre of the depletion region on the nside
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 5bit 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 2input NAND
gates. The minimum number of gates required is:
(A) 2 (B) 3 (C) 4 (D) 5
15. If the closedloop 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 nonminimum 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, widesense 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 lowpass filter (B) a highpass filter
(C) a bandpass filter (D) a bandreject 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 pside
(B) the edge of the depletion region on the nside
(C) the p n
+
junction
(D) the centre of the depletion region on the nside
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 5bit 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 2input NAND
gates. The minimum number of gates required is:
(A) 2 (B) 3 (C) 4 (D) 5
15. If the closedloop 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 nonminimum 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, widesense 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, widesense 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 xaxis and 90° with positive yaxis. 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 nonzero, 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 lowpass filter (B) a highpass filter
(C) a bandpass filter (D) a bandreject 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 pside
(B) the edge of the depletion region on the nside
(C) the p n
+
junction
(D) the centre of the depletion region on the nside
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 5bit 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 2input NAND
gates. The minimum number of gates required is:
(A) 2 (B) 3 (C) 4 (D) 5
15. If the closedloop 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 nonminimum 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, widesense 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, widesense 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 xaxis and 90° with positive yaxis. 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 nonzero, 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 NewtonRaphson
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 semicircular 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
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