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# Instrumentation (IN) 2013 GATE Paper with solution GATE Notes | EduRev

## GATE : Instrumentation (IN) 2013 GATE Paper with solution GATE Notes | EduRev

``` Page 1

|IN-GATE-2013 PAPER|
1
Q. No. 1 – 25 Carry One Mark Each
1. The dimension of the null space of the matrix
0 1 1
1 1 0
1 0 1
? ?
? ?
-
? ?
? ?
- -
? ?
is
(A) 0 (B) 1 (C) 2 (D) 3
Exp:
0 1 1
A 1 1 0 Rank A 2
1 0 1
? ?
? ?
= - ? =
? ?
? ?
- -
? ?
Dimension of the null space of A=3-2=1 ?
2. If the A- matrix of the state space model of a SISO linear time invariant system
is rank deficient, the transfer function of the system must have
(A) a pole with positive real part (B) a pole with negative real part
(C) a pole with positive imaginary part (D) a pole at the origin
3. Two systems with impulse responses
( )
1
h t and
( )
2
h t are connected in cascade.
Then the overall impulse response of the cascaded system is given by
(A) a product of
( )
1
h t and
( )
2
h t (B) sum of
( )
1
h t and
( )
2
h t
(C) convolution of
( )
1
h t and
( )
2
h t (D) subtraction of
( )
2
h t from
( )
1
h t
Exp:
4. The complex function tanh (s) is analytic over a region of the imaginary axis of
the complex s-plane if the following is TRUE everywhere in the region for all
integers n
(A)
( )
Re s 0 = (B)
( )
Im s n ? p
(C)
( )
n
Im s
3
p
? (D)
( )
( )
2n 1
Im s
2
+ p
?
Exp: Tanhs
s s
s s
e e
isanalytic
e e
-
-
-
=
+

s s
2s
m
if e e 0
(2n 1)
e 1 s i
2
(2n 1)
I (s)
2
-
+ ?
+
? ? - ? ? p
+
? ? p
x(t)
y(t)
1 2
h(t) *h(t)
x(t)
y(t) 1
h (t)
2
h (t)
Page 2

|IN-GATE-2013 PAPER|
1
Q. No. 1 – 25 Carry One Mark Each
1. The dimension of the null space of the matrix
0 1 1
1 1 0
1 0 1
? ?
? ?
-
? ?
? ?
- -
? ?
is
(A) 0 (B) 1 (C) 2 (D) 3
Exp:
0 1 1
A 1 1 0 Rank A 2
1 0 1
? ?
? ?
= - ? =
? ?
? ?
- -
? ?
Dimension of the null space of A=3-2=1 ?
2. If the A- matrix of the state space model of a SISO linear time invariant system
is rank deficient, the transfer function of the system must have
(A) a pole with positive real part (B) a pole with negative real part
(C) a pole with positive imaginary part (D) a pole at the origin
3. Two systems with impulse responses
( )
1
h t and
( )
2
h t are connected in cascade.
Then the overall impulse response of the cascaded system is given by
(A) a product of
( )
1
h t and
( )
2
h t (B) sum of
( )
1
h t and
( )
2
h t
(C) convolution of
( )
1
h t and
( )
2
h t (D) subtraction of
( )
2
h t from
( )
1
h t
Exp:
4. The complex function tanh (s) is analytic over a region of the imaginary axis of
the complex s-plane if the following is TRUE everywhere in the region for all
integers n
(A)
( )
Re s 0 = (B)
( )
Im s n ? p
(C)
( )
n
Im s
3
p
? (D)
( )
( )
2n 1
Im s
2
+ p
?
Exp: Tanhs
s s
s s
e e
isanalytic
e e
-
-
-
=
+

s s
2s
m
if e e 0
(2n 1)
e 1 s i
2
(2n 1)
I (s)
2
-
+ ?
+
? ? - ? ? p
+
? ? p
x(t)
y(t)
1 2
h(t) *h(t)
x(t)
y(t) 1
h (t)
2
h (t)
|IN-GATE-2013 PAPER|
2
5. For a vector E, which one of the following statements is NOT TRUE?
(A) If E=0, E is called solenoidal ?·
(B) If E=0, E is called conservative ?×
(C) If E=0, E is called irrotational ?×
(D) If E=0, E is called  irrotational ?·
Exp: .E 0, E ? = is called irrotational, which is not true
6. For a periodic signal
( )
v t 30sin100t 10cos300t 6sin 500t
4
p ? ?
= + + +
? ?
? ?
, the
(A) 100 (B) 300 (C) 500 (D) 1500
Exp:
o
100 rad / sec fundamental ? =
o
o
3 300 rad / sec third harmonic
5 500 rad / sec fifth harmonic
? =
? =
7. In the transistor circuit as shown below, the value of resistance
E
R in kO is
approximately,
(A) 1.0 (B) 1.5 (C) 2.0 (D) 2.5
Exp:
( )
B
6k
V 10 2.8V
6 15 k
= × =
+
E
E
E
E
V 2.8 0.7 2.1V
V 2.1
R 1k
I 2mA
? = - =
? = = = O
8. A source
( )
s
v t V cos100 t = p has an internal impedance of
( )
4 j3 + O . If a purely
resistive load connected to this source has to extract the maximum power out of
the source, its value in O should be
(A) 3 (B) 4 (C) 5 (D) 7
0.1 F µ
15kO
1.5kO
10V
c
I 2mA =
0.1 F µ
CE
V 5V =
6kO
E
R
out
v
Page 3

|IN-GATE-2013 PAPER|
1
Q. No. 1 – 25 Carry One Mark Each
1. The dimension of the null space of the matrix
0 1 1
1 1 0
1 0 1
? ?
? ?
-
? ?
? ?
- -
? ?
is
(A) 0 (B) 1 (C) 2 (D) 3
Exp:
0 1 1
A 1 1 0 Rank A 2
1 0 1
? ?
? ?
= - ? =
? ?
? ?
- -
? ?
Dimension of the null space of A=3-2=1 ?
2. If the A- matrix of the state space model of a SISO linear time invariant system
is rank deficient, the transfer function of the system must have
(A) a pole with positive real part (B) a pole with negative real part
(C) a pole with positive imaginary part (D) a pole at the origin
3. Two systems with impulse responses
( )
1
h t and
( )
2
h t are connected in cascade.
Then the overall impulse response of the cascaded system is given by
(A) a product of
( )
1
h t and
( )
2
h t (B) sum of
( )
1
h t and
( )
2
h t
(C) convolution of
( )
1
h t and
( )
2
h t (D) subtraction of
( )
2
h t from
( )
1
h t
Exp:
4. The complex function tanh (s) is analytic over a region of the imaginary axis of
the complex s-plane if the following is TRUE everywhere in the region for all
integers n
(A)
( )
Re s 0 = (B)
( )
Im s n ? p
(C)
( )
n
Im s
3
p
? (D)
( )
( )
2n 1
Im s
2
+ p
?
Exp: Tanhs
s s
s s
e e
isanalytic
e e
-
-
-
=
+

s s
2s
m
if e e 0
(2n 1)
e 1 s i
2
(2n 1)
I (s)
2
-
+ ?
+
? ? - ? ? p
+
? ? p
x(t)
y(t)
1 2
h(t) *h(t)
x(t)
y(t) 1
h (t)
2
h (t)
|IN-GATE-2013 PAPER|
2
5. For a vector E, which one of the following statements is NOT TRUE?
(A) If E=0, E is called solenoidal ?·
(B) If E=0, E is called conservative ?×
(C) If E=0, E is called irrotational ?×
(D) If E=0, E is called  irrotational ?·
Exp: .E 0, E ? = is called irrotational, which is not true
6. For a periodic signal
( )
v t 30sin100t 10cos300t 6sin 500t
4
p ? ?
= + + +
? ?
? ?
, the
(A) 100 (B) 300 (C) 500 (D) 1500
Exp:
o
100 rad / sec fundamental ? =
o
o
3 300 rad / sec third harmonic
5 500 rad / sec fifth harmonic
? =
? =
7. In the transistor circuit as shown below, the value of resistance
E
R in kO is
approximately,
(A) 1.0 (B) 1.5 (C) 2.0 (D) 2.5
Exp:
( )
B
6k
V 10 2.8V
6 15 k
= × =
+
E
E
E
E
V 2.8 0.7 2.1V
V 2.1
R 1k
I 2mA
? = - =
? = = = O
8. A source
( )
s
v t V cos100 t = p has an internal impedance of
( )
4 j3 + O . If a purely
resistive load connected to this source has to extract the maximum power out of
the source, its value in O should be
(A) 3 (B) 4 (C) 5 (D) 7
0.1 F µ
15kO
1.5kO
10V
c
I 2mA =
0.1 F µ
CE
V 5V =
6kO
E
R
out
v
|IN-GATE-2013 PAPER|
3
Exp: For maximum power Transfer
L s
2 2
R Z
4 3
5
=
= +
= O
9. Which of the following statements is NOT TRUE for a continuous time causal and
stable LTI system?
(A) All the poles of the system must lie on the left side of the j?-axis
(B) Zeroes of the system can lie anywhere in the s-plane
(C) All the poles must lie within s 1 =
(D) All the roots of the characteristic equation must be located on the left side of
the j?-axis.
Exp: For an LTI system to be stable and causal all poles or roots of characteristic
equation must lie on LHS of s-plane i.e., left hand side of j axis ?-
[Refer Laplace transform].
10. The operational amplifier shown in the circuit below has a slew rate of 0.8V / s µ .
The input signal is 0.25sin t ? . The maximum frequency of input in kHz for which
there is no distortion in the output is
(A) 23.84 (B) 25.0 (C) 50.0 (D) 46.60
Exp:
Max 0 i 6
opk
SR 0.8 470
f ;where V V 5.34 sin t
2 V 22 2 5.34 10
23.84kHz
-
= = = = - ?
p p× ×
=
11. Assuming zero initial condition, the response
( )
y t of the system given below to a
unit step input
( )
u t is
(A)
( )
u t (B)
( )
t u t (C)
( )
2
t
u t
2
(D)
( )
t
e u t
-
( ) Y s
1
s
( ) U s
+
-
0.25sin t ?
22kO
470kO
o
V
Page 4

|IN-GATE-2013 PAPER|
1
Q. No. 1 – 25 Carry One Mark Each
1. The dimension of the null space of the matrix
0 1 1
1 1 0
1 0 1
? ?
? ?
-
? ?
? ?
- -
? ?
is
(A) 0 (B) 1 (C) 2 (D) 3
Exp:
0 1 1
A 1 1 0 Rank A 2
1 0 1
? ?
? ?
= - ? =
? ?
? ?
- -
? ?
Dimension of the null space of A=3-2=1 ?
2. If the A- matrix of the state space model of a SISO linear time invariant system
is rank deficient, the transfer function of the system must have
(A) a pole with positive real part (B) a pole with negative real part
(C) a pole with positive imaginary part (D) a pole at the origin
3. Two systems with impulse responses
( )
1
h t and
( )
2
h t are connected in cascade.
Then the overall impulse response of the cascaded system is given by
(A) a product of
( )
1
h t and
( )
2
h t (B) sum of
( )
1
h t and
( )
2
h t
(C) convolution of
( )
1
h t and
( )
2
h t (D) subtraction of
( )
2
h t from
( )
1
h t
Exp:
4. The complex function tanh (s) is analytic over a region of the imaginary axis of
the complex s-plane if the following is TRUE everywhere in the region for all
integers n
(A)
( )
Re s 0 = (B)
( )
Im s n ? p
(C)
( )
n
Im s
3
p
? (D)
( )
( )
2n 1
Im s
2
+ p
?
Exp: Tanhs
s s
s s
e e
isanalytic
e e
-
-
-
=
+

s s
2s
m
if e e 0
(2n 1)
e 1 s i
2
(2n 1)
I (s)
2
-
+ ?
+
? ? - ? ? p
+
? ? p
x(t)
y(t)
1 2
h(t) *h(t)
x(t)
y(t) 1
h (t)
2
h (t)
|IN-GATE-2013 PAPER|
2
5. For a vector E, which one of the following statements is NOT TRUE?
(A) If E=0, E is called solenoidal ?·
(B) If E=0, E is called conservative ?×
(C) If E=0, E is called irrotational ?×
(D) If E=0, E is called  irrotational ?·
Exp: .E 0, E ? = is called irrotational, which is not true
6. For a periodic signal
( )
v t 30sin100t 10cos300t 6sin 500t
4
p ? ?
= + + +
? ?
? ?
, the
(A) 100 (B) 300 (C) 500 (D) 1500
Exp:
o
100 rad / sec fundamental ? =
o
o
3 300 rad / sec third harmonic
5 500 rad / sec fifth harmonic
? =
? =
7. In the transistor circuit as shown below, the value of resistance
E
R in kO is
approximately,
(A) 1.0 (B) 1.5 (C) 2.0 (D) 2.5
Exp:
( )
B
6k
V 10 2.8V
6 15 k
= × =
+
E
E
E
E
V 2.8 0.7 2.1V
V 2.1
R 1k
I 2mA
? = - =
? = = = O
8. A source
( )
s
v t V cos100 t = p has an internal impedance of
( )
4 j3 + O . If a purely
resistive load connected to this source has to extract the maximum power out of
the source, its value in O should be
(A) 3 (B) 4 (C) 5 (D) 7
0.1 F µ
15kO
1.5kO
10V
c
I 2mA =
0.1 F µ
CE
V 5V =
6kO
E
R
out
v
|IN-GATE-2013 PAPER|
3
Exp: For maximum power Transfer
L s
2 2
R Z
4 3
5
=
= +
= O
9. Which of the following statements is NOT TRUE for a continuous time causal and
stable LTI system?
(A) All the poles of the system must lie on the left side of the j?-axis
(B) Zeroes of the system can lie anywhere in the s-plane
(C) All the poles must lie within s 1 =
(D) All the roots of the characteristic equation must be located on the left side of
the j?-axis.
Exp: For an LTI system to be stable and causal all poles or roots of characteristic
equation must lie on LHS of s-plane i.e., left hand side of j axis ?-
[Refer Laplace transform].
10. The operational amplifier shown in the circuit below has a slew rate of 0.8V / s µ .
The input signal is 0.25sin t ? . The maximum frequency of input in kHz for which
there is no distortion in the output is
(A) 23.84 (B) 25.0 (C) 50.0 (D) 46.60
Exp:
Max 0 i 6
opk
SR 0.8 470
f ;where V V 5.34 sin t
2 V 22 2 5.34 10
23.84kHz
-
= = = = - ?
p p× ×
=
11. Assuming zero initial condition, the response
( )
y t of the system given below to a
unit step input
( )
u t is
(A)
( )
u t (B)
( )
t u t (C)
( )
2
t
u t
2
(D)
( )
t
e u t
-
( ) Y s
1
s
( ) U s
+
-
0.25sin t ?
22kO
470kO
o
V
|IN-GATE-2013 PAPER|
4
Exp: Integration of unit step function is ramp output
Writing in time domain
y(t) u(t)*u(t) tu(t) = =
12. The transfer function
( )
( )
2
1
V s
V s
of the circuit shown below is
(A)
0.5s 1
s 1
+
+
(B)
3s 6
s 2
+
+
(C)
s 2
s 1
+
+
(D)
s 1
s 2
+
+
Exp:
1 2 4 4
q
2
1 1 q
1 s 1
z (s) , z (s)
10 s 10 s
Z (s)
V (s) s 1
V (s) Z (s) Z (s) s 2
- -
+
= =
+
= =
+ +
13. The type of partial differential equation
2
2
f f
t x
? ?
=
? ?
is
(A) Parabolic (B) Elliptic (C) Hyperbolic (D) Nonlinear
Exp:
2
2
2
f f
HereB 4AC 0
t x
The equation is parabolic
? ?
= - =
? ?
?
14. The discrete-time transfer function
1
1
1 2z
1 0.5z
-
-
-
-
is
(A) Non-minimum phase and unstable (B) Minimum phase and unstable
(C) Minimum phase and stable (D) Non-minimum phase and stable
+
-
+
-
( )
2
V s ( )
1
V s
10kO
100 F µ
100 F µ
u(s) y(s)
1
s
u(t) y(t) u(t)
1
V (s)
+ +
- -
1
Z (s)
2
Z(s) 2
V (s)
Page 5

|IN-GATE-2013 PAPER|
1
Q. No. 1 – 25 Carry One Mark Each
1. The dimension of the null space of the matrix
0 1 1
1 1 0
1 0 1
? ?
? ?
-
? ?
? ?
- -
? ?
is
(A) 0 (B) 1 (C) 2 (D) 3
Exp:
0 1 1
A 1 1 0 Rank A 2
1 0 1
? ?
? ?
= - ? =
? ?
? ?
- -
? ?
Dimension of the null space of A=3-2=1 ?
2. If the A- matrix of the state space model of a SISO linear time invariant system
is rank deficient, the transfer function of the system must have
(A) a pole with positive real part (B) a pole with negative real part
(C) a pole with positive imaginary part (D) a pole at the origin
3. Two systems with impulse responses
( )
1
h t and
( )
2
h t are connected in cascade.
Then the overall impulse response of the cascaded system is given by
(A) a product of
( )
1
h t and
( )
2
h t (B) sum of
( )
1
h t and
( )
2
h t
(C) convolution of
( )
1
h t and
( )
2
h t (D) subtraction of
( )
2
h t from
( )
1
h t
Exp:
4. The complex function tanh (s) is analytic over a region of the imaginary axis of
the complex s-plane if the following is TRUE everywhere in the region for all
integers n
(A)
( )
Re s 0 = (B)
( )
Im s n ? p
(C)
( )
n
Im s
3
p
? (D)
( )
( )
2n 1
Im s
2
+ p
?
Exp: Tanhs
s s
s s
e e
isanalytic
e e
-
-
-
=
+

s s
2s
m
if e e 0
(2n 1)
e 1 s i
2
(2n 1)
I (s)
2
-
+ ?
+
? ? - ? ? p
+
? ? p
x(t)
y(t)
1 2
h(t) *h(t)
x(t)
y(t) 1
h (t)
2
h (t)
|IN-GATE-2013 PAPER|
2
5. For a vector E, which one of the following statements is NOT TRUE?
(A) If E=0, E is called solenoidal ?·
(B) If E=0, E is called conservative ?×
(C) If E=0, E is called irrotational ?×
(D) If E=0, E is called  irrotational ?·
Exp: .E 0, E ? = is called irrotational, which is not true
6. For a periodic signal
( )
v t 30sin100t 10cos300t 6sin 500t
4
p ? ?
= + + +
? ?
? ?
, the
(A) 100 (B) 300 (C) 500 (D) 1500
Exp:
o
100 rad / sec fundamental ? =
o
o
3 300 rad / sec third harmonic
5 500 rad / sec fifth harmonic
? =
? =
7. In the transistor circuit as shown below, the value of resistance
E
R in kO is
approximately,
(A) 1.0 (B) 1.5 (C) 2.0 (D) 2.5
Exp:
( )
B
6k
V 10 2.8V
6 15 k
= × =
+
E
E
E
E
V 2.8 0.7 2.1V
V 2.1
R 1k
I 2mA
? = - =
? = = = O
8. A source
( )
s
v t V cos100 t = p has an internal impedance of
( )
4 j3 + O . If a purely
resistive load connected to this source has to extract the maximum power out of
the source, its value in O should be
(A) 3 (B) 4 (C) 5 (D) 7
0.1 F µ
15kO
1.5kO
10V
c
I 2mA =
0.1 F µ
CE
V 5V =
6kO
E
R
out
v
|IN-GATE-2013 PAPER|
3
Exp: For maximum power Transfer
L s
2 2
R Z
4 3
5
=
= +
= O
9. Which of the following statements is NOT TRUE for a continuous time causal and
stable LTI system?
(A) All the poles of the system must lie on the left side of the j?-axis
(B) Zeroes of the system can lie anywhere in the s-plane
(C) All the poles must lie within s 1 =
(D) All the roots of the characteristic equation must be located on the left side of
the j?-axis.
Exp: For an LTI system to be stable and causal all poles or roots of characteristic
equation must lie on LHS of s-plane i.e., left hand side of j axis ?-
[Refer Laplace transform].
10. The operational amplifier shown in the circuit below has a slew rate of 0.8V / s µ .
The input signal is 0.25sin t ? . The maximum frequency of input in kHz for which
there is no distortion in the output is
(A) 23.84 (B) 25.0 (C) 50.0 (D) 46.60
Exp:
Max 0 i 6
opk
SR 0.8 470
f ;where V V 5.34 sin t
2 V 22 2 5.34 10
23.84kHz
-
= = = = - ?
p p× ×
=
11. Assuming zero initial condition, the response
( )
y t of the system given below to a
unit step input
( )
u t is
(A)
( )
u t (B)
( )
t u t (C)
( )
2
t
u t
2
(D)
( )
t
e u t
-
( ) Y s
1
s
( ) U s
+
-
0.25sin t ?
22kO
470kO
o
V
|IN-GATE-2013 PAPER|
4
Exp: Integration of unit step function is ramp output
Writing in time domain
y(t) u(t)*u(t) tu(t) = =
12. The transfer function
( )
( )
2
1
V s
V s
of the circuit shown below is
(A)
0.5s 1
s 1
+
+
(B)
3s 6
s 2
+
+
(C)
s 2
s 1
+
+
(D)
s 1
s 2
+
+
Exp:
1 2 4 4
q
2
1 1 q
1 s 1
z (s) , z (s)
10 s 10 s
Z (s)
V (s) s 1
V (s) Z (s) Z (s) s 2
- -
+
= =
+
= =
+ +
13. The type of partial differential equation
2
2
f f
t x
? ?
=
? ?
is
(A) Parabolic (B) Elliptic (C) Hyperbolic (D) Nonlinear
Exp:
2
2
2
f f
HereB 4AC 0
t x
The equation is parabolic
? ?
= - =
? ?
?
14. The discrete-time transfer function
1
1
1 2z
1 0.5z
-
-
-
-
is
(A) Non-minimum phase and unstable (B) Minimum phase and unstable
(C) Minimum phase and stable (D) Non-minimum phase and stable
+
-
+
-
( )
2
V s ( )
1
V s
10kO
100 F µ
100 F µ
u(s) y(s)
1
s
u(t) y(t) u(t)
1
V (s)
+ +
- -
1
Z (s)
2
Z(s) 2
V (s)
|IN-GATE-2013 PAPER|
5
Exp:
1
1
1 2z
H(z)
1 0.5z
-
-
-
=
-
For minimum phase system, all poles and zeros must lie inside the unit circle.
For stable system, all poles must be inside the unit circle
For the given system,  Zero is at 2,  pole is at 0.5
This system is stable but non-minimum phase
15. Match the following biomedical instrumentation techniques with their application.
P. Otoscopy U. Respiratory volume measurement
Q. Ultrasound Technique V. Ear diagnostics
R. Spirometry W. Echo-cardiography
S. Thermodilution Technique X. Heart-volume measurement
(A) P-U;Q-V;R-X;S-W (B) P-V;Q-U;R-X;S-W
(C) P-V;Q-W;R-U;S-X (D) P-V;Q-W;R-X;S-U
16. A continuous random variable X has a probability density function
( )
x
f x e , 0<x<
-
= 8 , then
( )
P X 1 > is
(A) 0.368 (B) 0.5 (C) 0.632 (D) 1.0
Exp:
x x 1
1 1
p(x 1) e dx e e 0.368
8 8
- - -
? ? > = = - = =
? ? ?

17. A band limited signal with a maximum frequency of 5 kHz is to be sampled.
According to the sampling theorem, the sampling frequency in kHz which is not
valid is
(A) 5 (B) 12 (C) 15 (D) 20
Exp: Given:
m
f 5kHz =
According to sampling frequency
s m s
f 2f ; f 10 kHz = =
So, only in option (a) it is less than 10KHz ie., (5KHz)
18. A differential pressure transmitter of a flow meter using venture tube reads
5
2.5 10 Pa × for a flow rate of
3
0.5 m / s . The approximate flow rate in
3
m / s for
a differential pressure of
5
0.9 10 Pa × is
(A) 0.30 (B) 0.18 (C) 0.83 (D) 0.60
Exp:
2 2
1 1
2
Q p
Q p
Q p
Q 0.30
a ? =
=
```
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## GATE Past Year Papers for Practice (All Branches)

380 docs|127 tests

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