Find the transfer function of the state variable representation of the system given by the differential equation y” + 2y’ + 4y = 8u.

$\frac{2}{{\left( {{s^2} + 2s + 4} \right)}}$

$\frac{4}{{\left( {{s^2} + 2s + 4} \right)}}$

$\frac{8}{{\left( {{s^2} + 2s + 4} \right)}}$

$\frac{2}{{\left( {{s^2} + 2s + 4} \right)}}$

$\frac{6}{{\left( {{s^2} + 2s + 4} \right)}}$

If the signal ${\rm{x}}\left( {\rm{t}} \right) = \frac{{{\rm{sin}}\left( {\rm{t}} \right)}}{{{\rm{\pi t}}}}{\rm{\;*}}\frac{{{\rm{sin}}\left( {\rm{t}} \right)}}{{{\rm{\pi t}}}}$ with ∗ denoting the convolution operation, then x(t) is equal to

$\frac{{{\rm{sin}}\left( {2{\rm{t}}} \right)}}{{2{\rm{\pi t}}}}$

$\frac{{{\rm{sin}}\left( {\rm{t}} \right)}}{{{\rm{\pi t}}}}$

$\frac{{{\rm{sin}}\left( {2{\rm{t}}} \right)}}{{2{\rm{\pi t}}}}$

$\frac{{2{\rm{sin}}\left( {\rm{t}} \right)}}{{{\rm{\pi t}}}}$

${\left( {\frac{{{\rm{sin}}\left( {\rm{t}} \right)}}{{{\rm{\pi t}}}}} \right)^2}$

If the Fourier transform of x(t) is X(jω), obtain the Fourier transform of x(t – t0)

${e^{ - j\omega {t_0}}}X\left( {j{t_0}} \right)$

${e^{j\omega {t_0}}}X\left( {j\omega } \right)$

${e^{ - j\omega {t_0}}}X\left( {j\omega {t_0}} \right)$

${e^{ - j\omega {t_0}}}X\left( {j\omega } \right)$

Match the two lists and choose the correct answer from the code given below:List IList II(a) Sin 2t(i)(b) e-2t(ii) (c) e-2t sin 2t(iii) (d) 1 – e-2t(iv)

(a) – (iii), (b) – (iv), (c) – (i), (d) – (ii)

(a) – (ii), (b) – (iii), (c) – (iv), (d) – (i)

(a) – (iii), (b) – (iv), (c) – (i), (d) – (ii)

(a) – (iii), (b) – (iv), (c) – (ii), (d) – (i)

(a) – (ii), (b) – (i), (c) – (iv), (d) – (iii)

Consider a causal LTI system characterized by differential equation $\frac{{dy\left( t \right)}}{{dt}} + \frac{1}{6}y\left( t \right) = 3x\left( t \right)$. The response of the system to the input $x\left( t \right) = 3{e^{ - \frac{t}{3}u\left( t \right)}}$. Where u(t) denotes the unit step function is

$9{e^{ - \frac{t}{3}}}u\left( t \right)$

$9{e^{ - \frac{t}{6}}}u\left( t \right)$

$9{e^{ - \frac{t}{3}}}u\left( t \right) - 6{e^{ - \frac{t}{6}}}u\left( t \right)$

${54^{ - \frac{t}{6}}}u\left( t \right) - 54{e^{ - \frac{t}{3}}}u\left( t \right)$

Laplace transform of eθt sin (ωt) is:

$\frac{θ}{s^2-(\theta-ω)^2 }$

$\frac{\omega}{(s-\theta)^2+ω^2 }$

$\frac{θ}{s^2-(\theta-ω)^2 }$

$\frac{θ}{(s+\theta)^2-ω^2 }$

A function f (t) is shown in the figure.The Fourier transform F(ω) of f(t) is

imaginary and even function of ω

imaginary and odd function of ω

imaginary and even function of ω

An ideal square wave with period of 20 ms shown in the figure, is passed through an ideal low pass filter with cut-off frequency 120 Hz. Which of the following is an accurate description of the output?

Output is a square wave of fundamental frequency 50 Hz

Output consists of both 50 Hz and 100 Hz frequency components

Output is a pure sinusoid of frequency 50 Hz

Output is a square wave of fundamental frequency 50 Hz

In a series RLC circuit, the output is taken across the capacitor C, and the input is applied across the resistor R and ground Obtain the closed-loop transfer function.

$\frac{s}{{\left( {1 + sRC + {s^2}LC} \right)}}$

$\frac{{sC}}{{\left( {1 + sRC + {s^2}LC} \right)}}$

$\frac{1}{{\left( {1 + sRC + {s^2}LC} \right)}}$

A different non constant even function x(t) has a derivative y(t), and their respective Fourier Transforms are X(ω) and Y(ω). Which of the following statements is TRUE

X(ω) and Y(ω) are both imaginary

X(ω) is real and Y(ω) is imaginary

X(ω) and Y(ω) are both imaginary

X(ω) is imaginary and Y(ω) is real

Let u(t) denote the unit step function. The bilateral Laplace transform of the function f(t) = etu(−t) is ____.

$\frac{-1}{s-1}$ with real part of s > 1

$\frac{1}{s-1}$ with real part of s < 1

$\frac{1}{s-1}$ with real part of s > 1

$\frac{-1}{s-1}$with real part of s < 1

$\frac{-1}{s-1}$ with real part of s > 1

If f(x) represented by Fourier integral $f(x)=\int_0^{\infty} [A(ω) cos~ω x + B(ω) sin~ω x]dω $then A(ω) is defined as

$\frac{1}{\pi}\int_{-\infty}^{\infty} f(v) ~sin~\omega v ~dv$

$\frac{1}{\pi}\int_{-\infty}^{\infty} f(v) ~cos~\omega v ~dv$

$\frac{1}{\pi}\int_{-\infty}^{\infty} f(v) ~sin~\omega v ~dv$

$\int_{-\infty}^{\infty} f(\omega) ~cos~\omega v ~dv$

$\int_{-\infty}^{\infty} f(\omega) ~sin~\omega v ~dv$

Consider a linear time invariant system $\dot x = Ax$ with initial condition $x\left( 0 \right)$ at $t = 0$. Suppose $\alpha$ and $\beta$ are eigenvectors of $\left( {2 \times 2} \right)$ matrix A corresponding to distinct eigenvalues ${\lambda _1}\;and\;{\lambda _2}$ respectively. Then the response $x\left( t \right)$ of the system due to initial condition $x\left( 0 \right) = \alpha$ is

${e^{{\lambda _1}t}}\alpha + {e^{{\lambda _2}t}}\beta$

Let the signal f(t) = 0 outside the interval [T1, T2], where T1 and T2 are finite. Furthermore, $\left| {{\rm{f}}\left( {\rm{t}} \right)} \right| < \infty$. The region of convergence (ROC) of the signal’s bilateral Laplace transform F(s) is

A half plane containing the jΩ axis

A parallel strip containing the jΩ axis

A parallel strip not containing the jΩ axis

A half plane containing the jΩ axis

A forcing function (t2 – 2t) u (t – 1) is applied to a linear system. The ${\cal L}$- transform of the forcing function is

$\frac{{2 - s}}{{{s^3}}}\epsilon{^{ - 2s}}$

$\left( {\frac{{1 - {s^2}}}{S}} \right)\epsilon{^{ - s}}$

$\frac{1}{s}{e^{ - s}} - \frac{1}{{{s^2}}}\epsilon{^{ - 2s}}$

$\left( {\frac{{2 - {s^2}}}{{{s^3}}}} \right)\epsilon{^{ - S}}$

Laplace transform of f(t) = t2 sin t is

$\frac{{\left( {3{s^2} + 1} \right)}}{{{{\left( {{s^2} + 1} \right)}^3}}}$

$\frac{{3{s^2} - 1}}{{{{\left( {{s^2} + 1} \right)}^3}}}$

$\frac{{2\left( {3{s^2} - 1} \right)}}{{{{\left( {{s^2} + 1} \right)}^3}}}$

$\frac{{\left( {3{s^2} + 1} \right)}}{{{{\left( {{s^2} + 1} \right)}^3}}}$

$\frac{{\left( {3{s^2} - 1} \right)}}{{{{\left( {{s^2} + 1} \right)}^3}}}$

A function, in the Laplace domain, is given by$F\left( s \right) = \frac{2}{s} - \frac{1}{{s + 3}}$Its value by final value theorem in 't' domain will be

$\mathop {\lim }\limits_{t \to \infty } f\left( t \right) = 1$

$\mathop {\lim }\limits_{t \to \infty } f\left( t \right) = 3$

$\mathop {\lim }\limits_{t \to \infty } f\left( t \right) = 2$

$\mathop {\lim }\limits_{t \to \infty } f\left( t \right) = 1$

$\mathop {\lim }\limits_{t \to \infty } f\left( t \right) = 4$

A signal $x(t) = \sin c(α t)$, where α is a real constant, is the input to a Linear Time-Invariant system whose impulse response is $h(t) = \sin c(β t)$, where β is a real constant. If min (α, β) denotes the minimum of α and β and similarly, max (α, β) denotes the maximum of α and β, and K is a constant, which one of the following statements is true about the (here, $\sin c\left( x \right) = \frac{{\sin \left( {\pi x} \right)}}{{\pi x}}$) output of the system?

It will be of the form $K\sin c\left( {γ t} \right)$ where γ = max (α, β)

It will be of the form $K\sin c\left( {γ t} \right)$ where γ = min (α, β)

It will be of the form $K\sin c\left( {γ t} \right)$ where γ = max (α, β)

It will be of the form $K\sin c\left( {\alpha t} \right)$

It cannot be a $\sin c$ type of signal

H_(Z)_IS_DISCRETE_RATIONAL_TRANSFER_FUNCTION._TO_ENSURE_THAT_BOTH_H(Z)_AND_ITS_INVERSE_ARE_STABLE:?$

Poles and zeroes must be outside the unit circle

Poles must be inside the unit circle and zeros must be outside the unit circle

Poles and zeroes must be inside the unit circle

Poles and zeroes must be outside the unit circle

Poles must be outside the unit circle and zeros must be inside the unit circle

What is the ROC of z-transform of finite duration anti-causal sequence?$

z=‚Äö√Ñ√∂‚àö‚Ä†‚àö¬™

Entire z-plane, except at z=‚Äö√Ñ√∂‚àö‚Ä†‚àö¬™

What is the ROC of the signal x(n)=‚âà√≠¬¨‚Ä¢(n-k),k>0?$

Entire z-plane, except at z=‚Äö√Ñ√∂‚àö‚Ä†‚àö¬™

z=‚Äö√Ñ√∂‚àö‚Ä†‚àö¬™

Entire z-plane, except at z=‚Äö√Ñ√∂‚àö‚Ä†‚àö¬™

150 + Mcqs in Z Transform in Control Systems Page 3 McqOptions

101.	Match List-I with List-II:List-IList-IIa)sin ωti) $\frac{\omega }{(s^2+ω^2 )}$b)cos ωtii) $\frac{s}{(s^2+ω^2 )}$c)sin h btiii) $\frac{s}{(s^2-b^2 )}$d)cos h btiv) $\frac{b}{(s^2-b^2 )}$Choose the correct option from those given below:
A.	a-i; b-ii; c-iii; d-iv
B.	a-ii; b-i; c-iv; d-iii
C.	a-i; b-ii; c-iv; d-iii
D.	a-ii; b-i; c-iii; d-iv
Answer» D. a-ii; b-i; c-iii; d-iv

Discussion

102.	Find the initial value of the signal x(t) whose unilateral Laplace transform is:$X\left( s \right) = \frac{{5s + 10}}{{s\left( {s + 4} \right)}}$
A.	5
B.	10
C.	14
D.	4
Answer» B. 10

Discussion

103.	If x(t) is of finite duration and is absolutely integrable, then the 'region of convergence' is:
A.	Entire s plane
B.	From σ = -1 to σ = + ∞
C.	From σ = +1 to σ = - ∞
D.	Entire right half plane
Answer» B. From σ = -1 to σ = + ∞

Discussion

104.	Decimation-in-Time is the class of
A.	Discrete Fourier Transform
B.	Z Transform
C.	Laplace Transform
D.	Fast Fourier Transform
Answer» E.

Discussion

105.	Find the transfer function of the state variable representation of the system given by the differential equation y” + 2y’ + 4y = 8u.
A.	$\frac{4}{{\left( {{s^2} + 2s + 4} \right)}}$
B.	$\frac{8}{{\left( {{s^2} + 2s + 4} \right)}}$
C.	$\frac{2}{{\left( {{s^2} + 2s + 4} \right)}}$
D.	$\frac{6}{{\left( {{s^2} + 2s + 4} \right)}}$
Answer» C. $\frac{2}{{\left( {{s^2} + 2s + 4} \right)}}$

Discussion

106.	L{sin2 at} equals
A.	2a2/s(s2+4a2)
B.	2a2/s(s2-4a2)
C.	2s/(s2+4a2)
D.	None of these
Answer» B. 2a2/s(s2-4a2)

Discussion

107.	Fourier transform of a real and odd function is
A.	Real and odd
B.	Real and even
C.	Imaginary and odd
D.	Imaginary and even
Answer» D. Imaginary and even

Discussion

108.	If the signal ${\rm{x}}\left( {\rm{t}} \right) = \frac{{{\rm{sin}}\left( {\rm{t}} \right)}}{{{\rm{\pi t}}}}{\rm{\;*}}\frac{{{\rm{sin}}\left( {\rm{t}} \right)}}{{{\rm{\pi t}}}}$ with ∗ denoting the convolution operation, then x(t) is equal to
A.	$\frac{{{\rm{sin}}\left( {\rm{t}} \right)}}{{{\rm{\pi t}}}}$
B.	$\frac{{{\rm{sin}}\left( {2{\rm{t}}} \right)}}{{2{\rm{\pi t}}}}$
C.	$\frac{{2{\rm{sin}}\left( {\rm{t}} \right)}}{{{\rm{\pi t}}}}$
D.	${\left( {\frac{{{\rm{sin}}\left( {\rm{t}} \right)}}{{{\rm{\pi t}}}}} \right)^2}$
Answer» B. $\frac{{{\rm{sin}}\left( {2{\rm{t}}} \right)}}{{2{\rm{\pi t}}}}$

Discussion

109.	In VHF spectrum Analyzer, two types of frequency instabilities which will cause difficulties when narrow frequency range are displayed are known as(a) Short-term instability(b) Phase-noise(c) Correlation-noise(d) Long-term instabilityChoose the correct option
A.	(a) and (c)
B.	(b) and (a)
C.	(c) and (d)
D.	(b) and (d)
Answer» E.

Discussion

110.	If the Fourier transform of x(t) is X(jω), obtain the Fourier transform of x(t – t0)
A.	${e^{ - j\omega {t_0}}}X\left( {j{t_0}} \right)$
B.	${e^{j\omega {t_0}}}X\left( {j\omega } \right)$
C.	${e^{ - j\omega {t_0}}}X\left( {j\omega {t_0}} \right)$
D.	${e^{ - j\omega {t_0}}}X\left( {j\omega } \right)$
Answer» E.

Discussion

111.	Evaluate $\mathop \smallint \limits_{ - 1}^1 \left( {3{t^2} + 1} \right)\delta \left( t \right)dt$
A.	4
B.	0
C.	1
D.	8
Answer» D. 8

Discussion

112.	Match the two lists and choose the correct answer from the code given below:List IList II(a) Sin 2t(i)(b) e-2t(ii) (c) e-2t sin 2t(iii) (d) 1 – e-2t(iv)
A.	(a) – (ii), (b) – (iii), (c) – (iv), (d) – (i)
B.	(a) – (iii), (b) – (iv), (c) – (i), (d) – (ii)
C.	(a) – (iii), (b) – (iv), (c) – (ii), (d) – (i)
D.	(a) – (ii), (b) – (i), (c) – (iv), (d) – (iii)
Answer» B. (a) – (iii), (b) – (iv), (c) – (i), (d) – (ii)

Discussion

113.	Consider Fourier representation of continuous and discrete-time systems. The complex exponentials (i.e., signals), which arise in such representation, have
A.	the same properties always
B.	different properties always
C.	non-specific properties
D.	mostly the same properties
Answer» C. non-specific properties

Discussion

114.	Consider a causal LTI system characterized by differential equation $\frac{{dy\left( t \right)}}{{dt}} + \frac{1}{6}y\left( t \right) = 3x\left( t \right)$. The response of the system to the input $x\left( t \right) = 3{e^{ - \frac{t}{3}u\left( t \right)}}$. Where u(t) denotes the unit step function is
A.	$9{e^{ - \frac{t}{3}}}u\left( t \right)$
B.	$9{e^{ - \frac{t}{6}}}u\left( t \right)$
C.	$9{e^{ - \frac{t}{3}}}u\left( t \right) - 6{e^{ - \frac{t}{6}}}u\left( t \right)$
D.	${54^{ - \frac{t}{6}}}u\left( t \right) - 54{e^{ - \frac{t}{3}}}u\left( t \right)$
Answer» E.

Discussion

115.	Laplace transform of eθt sin (ωt) is:
A.	$\frac{θ}{(s^2+ω^2 )}$
B.	$\frac{\omega}{(s-\theta)^2+ω^2 }$
C.	$\frac{θ}{s^2-(\theta-ω)^2 }$
D.	$\frac{θ}{(s+\theta)^2-ω^2 }$
Answer» C. $\frac{θ}{s^2-(\theta-ω)^2 }$

Discussion

116.	A function f (t) is shown in the figure.The Fourier transform F(ω) of f(t) is
A.	real and even function of ω
B.	real and odd function of ω
C.	imaginary and odd function of ω
D.	imaginary and even function of ω
Answer» D. imaginary and even function of ω

Discussion

117.	An ideal square wave with period of 20 ms shown in the figure, is passed through an ideal low pass filter with cut-off frequency 120 Hz. Which of the following is an accurate description of the output?
A.	Output is zero
B.	Output consists of both 50 Hz and 100 Hz frequency components
C.	Output is a pure sinusoid of frequency 50 Hz
D.	Output is a square wave of fundamental frequency 50 Hz
Answer» D. Output is a square wave of fundamental frequency 50 Hz

Discussion

118.	A system has impulse response h(t) = e-2tu(t). Find its system function if the input to the system is x(t) = e-tu(t).
A.	1/jω
B.	1/(jω + 2)
C.	1/(jω + 4)
D.	2/jω + 2)
Answer» C. 1/(jω + 4)

Discussion

119.	If the Laplace transform of function f(t) is given by $\frac{s+3}{(s+1)(s+2)}$, then f(0) is
A.	$\dfrac{3}{2}$
B.	$\dfrac{1}{2}$
C.	0
D.	1
Answer» E.

Discussion

120.	Laplace transform of 3 t4 is
A.	72 / s5
B.	24 / s4
C.	18 / s4
D.	12 / s5
Answer» B. 24 / s4

Discussion

121.	Find the Inverse Laplace transform of ${\rm{F}}\left( {\rm{s}} \right) = \frac{1}{{{\rm{s}} - 1}}$
A.	1
B.	eat
C.	cos at
D.	cosh at
Answer» C. cos at

Discussion

122.	In a series RLC circuit, the output is taken across the capacitor C, and the input is applied across the resistor R and ground Obtain the closed-loop transfer function.
A.	$\frac{s}{{\left( {1 + sRC + {s^2}LC} \right)}}$
B.	$\frac{{sC}}{{\left( {1 + sRC + {s^2}LC} \right)}}$
C.	1 + sRC + s2LC
D.	$\frac{1}{{\left( {1 + sRC + {s^2}LC} \right)}}$
Answer» E.

Discussion

123.	A different non constant even function x(t) has a derivative y(t), and their respective Fourier Transforms are X(ω) and Y(ω). Which of the following statements is TRUE
A.	X(ω) and Y(ω) are both real
B.	X(ω) is real and Y(ω) is imaginary
C.	X(ω) and Y(ω) are both imaginary
D.	X(ω) is imaginary and Y(ω) is real
Answer» C. X(ω) and Y(ω) are both imaginary

Discussion

124.	Let u(t) denote the unit step function. The bilateral Laplace transform of the function f(t) = etu(−t) is ____.
A.	$\frac{1}{s-1}$ with real part of s < 1
B.	$\frac{1}{s-1}$ with real part of s > 1
C.	$\frac{-1}{s-1}$with real part of s < 1
D.	$\frac{-1}{s-1}$ with real part of s > 1
Answer» D. $\frac{-1}{s-1}$ with real part of s > 1

Discussion

125.	Let $X(s) = \frac{{3{s^2} + 5s}}{{{s^2} + 10s + 21}}$ be the Laplace Transform of a signal x(t). Then X(0+) is
A.	0
B.	3
C.	5
D.	21
Answer» B. 3

Discussion

126.	If f(x) represented by Fourier integral $f(x)=\int_0^{\infty} [A(ω) cos~ω x + B(ω) sin~ω x]dω $then A(ω) is defined as
A.	$\frac{1}{\pi}\int_{-\infty}^{\infty} f(v) ~cos~\omega v ~dv$
B.	$\frac{1}{\pi}\int_{-\infty}^{\infty} f(v) ~sin~\omega v ~dv$
C.	$\int_{-\infty}^{\infty} f(\omega) ~cos~\omega v ~dv$
D.	$\int_{-\infty}^{\infty} f(\omega) ~sin~\omega v ~dv$
Answer» B. $\frac{1}{\pi}\int_{-\infty}^{\infty} f(v) ~sin~\omega v ~dv$

Discussion

127.	For distortionless transmission through LTI system phase of H(ω) is
A.	Constant
B.	One
C.	Zero
D.	Linearly dependent of ω
Answer» E.

Discussion

128.	Consider a linear time invariant system $\dot x = Ax$ with initial condition $x\left( 0 \right)$ at $t = 0$. Suppose $\alpha$ and $\beta$ are eigenvectors of $\left( {2 \times 2} \right)$ matrix A corresponding to distinct eigenvalues ${\lambda _1}\;and\;{\lambda _2}$ respectively. Then the response $x\left( t \right)$ of the system due to initial condition $x\left( 0 \right) = \alpha$ is
A.	${e^{{\lambda _1}t}}\alpha$
B.	${e^{{\lambda _2}t}}\beta$
C.	${e^{{\lambda _2}t}}\alpha$
D.	${e^{{\lambda _1}t}}\alpha + {e^{{\lambda _2}t}}\beta$
Answer» B. ${e^{{\lambda _2}t}}\beta$

Discussion

129.	Let the signal f(t) = 0 outside the interval [T1, T2], where T1 and T2 are finite. Furthermore, $\left\| {{\rm{f}}\left( {\rm{t}} \right)} \right\| < \infty$. The region of convergence (ROC) of the signal’s bilateral Laplace transform F(s) is
A.	A parallel strip containing the jΩ axis
B.	A parallel strip not containing the jΩ axis
C.	The entire s - plane
D.	A half plane containing the jΩ axis
Answer» D. A half plane containing the jΩ axis

Discussion

130.	A forcing function (t2 – 2t) u (t – 1) is applied to a linear system. The ${\cal L}$- transform of the forcing function is
A.	$\frac{{2 - s}}{{{s^3}}}\epsilon{^{ - 2s}}$
B.	$\left( {\frac{{1 - {s^2}}}{S}} \right)\epsilon{^{ - s}}$
C.	$\frac{1}{s}{e^{ - s}} - \frac{1}{{{s^2}}}\epsilon{^{ - 2s}}$
D.	$\left( {\frac{{2 - {s^2}}}{{{s^3}}}} \right)\epsilon{^{ - S}}$
Answer» E.

Discussion

131.	_______ is defined as the time delay that a signal component of frequency ω undergoes as it passes from the input to output of the system.
A.	Phase delay
B.	Group delay
C.	Frequency deviation
D.	Latency
Answer» B. Group delay

Discussion

132.	For a function g(t), it is given that $\mathop \smallint \limits_{ - \infty }^{ + \infty } g\left( t \right){e^{ - j\omega t}}dt = \omega {e^{ - 2{\omega ^2}}}$ for any real value $\omega $.If, $y\left( t \right) = \mathop \smallint \limits_{ - \infty }^t g\left( \tau \right)d\tau $, then $\mathop \smallint \limits_{ - \infty }^\infty y\left( t \right)dt$ is:
A.	0
B.	$-j$
C.	$-\frac{j}{2}$
D.	$\frac{j}{2}$
Answer» C. $-\frac{j}{2}$

Discussion

133.	Consider the function $F(s) = \frac{5}{{s({s^2} + 3s + 2)}}$, where F(s) is Laplace transform of function f(t). The initial value of f(t) is:
A.	5
B.	5/2
C.	5/3
D.	0
Answer» E.

Discussion

134.	Let f(t) be an even function i.e. f(-t) = f(t) for all t. Let the Fourier transform of f(t) be defined as $F(ω ) = \displaystyle\int_{-\infty}^\infty f(t) e^{-jω t}dt$. Suppose $\dfrac{dF(ω)}{dω} = -ω F(ω)$ for all ω, and F(0) = 1. Then
A.	f(0) < 1
B.	f(0) = 1
C.	f(0) = 0
D.	f(0) > 1
Answer» B. f(0) = 1

Discussion

135.	Laplace transform of f(t) = t2 sin t is
A.	$\frac{{3{s^2} - 1}}{{{{\left( {{s^2} + 1} \right)}^3}}}$
B.	$\frac{{2\left( {3{s^2} - 1} \right)}}{{{{\left( {{s^2} + 1} \right)}^3}}}$
C.	$\frac{{\left( {3{s^2} + 1} \right)}}{{{{\left( {{s^2} + 1} \right)}^3}}}$
D.	$\frac{{\left( {3{s^2} - 1} \right)}}{{{{\left( {{s^2} + 1} \right)}^3}}}$
Answer» C. $\frac{{\left( {3{s^2} + 1} \right)}}{{{{\left( {{s^2} + 1} \right)}^3}}}$

Discussion

136.	A function, in the Laplace domain, is given by$F\left( s \right) = \frac{2}{s} - \frac{1}{{s + 3}}$Its value by final value theorem in 't' domain will be
A.	$\mathop {\lim }\limits_{t \to \infty } f\left( t \right) = 3$
B.	$\mathop {\lim }\limits_{t \to \infty } f\left( t \right) = 2$
C.	$\mathop {\lim }\limits_{t \to \infty } f\left( t \right) = 1$
D.	$\mathop {\lim }\limits_{t \to \infty } f\left( t \right) = 4$
Answer» C. $\mathop {\lim }\limits_{t \to \infty } f\left( t \right) = 1$

Discussion

137.	A signal $x(t) = \sin c(α t)$, where α is a real constant, is the input to a Linear Time-Invariant system whose impulse response is $h(t) = \sin c(β t)$, where β is a real constant. If min (α, β) denotes the minimum of α and β and similarly, max (α, β) denotes the maximum of α and β, and K is a constant, which one of the following statements is true about the (here, $\sin c\left( x \right) = \frac{{\sin \left( {\pi x} \right)}}{{\pi x}}$) output of the system?
A.	It will be of the form $K\sin c\left( {γ t} \right)$ where γ = min (α, β)
B.	It will be of the form $K\sin c\left( {γ t} \right)$ where γ = max (α, β)
C.	It will be of the form $K\sin c\left( {\alpha t} \right)$
D.	It cannot be a $\sin c$ type of signal
Answer» B. It will be of the form $K\sin c\left( {γ t} \right)$ where γ = max (α, β)

Discussion

138.	In the Fourier transform, if the time domain signal x(t) is real and even, then the frequency domain signal X(jΩ) will be:
A.	imaginary and even
B.	imaginary and odd
C.	real and even
D.	real and odd
Answer» D. real and odd

Discussion

139.	Z_AND_LAPLACE_TRANSFORM_ARE_RELATED_BY:?$
A.	s = ln z
B.	s =ln z/T
C.	s =z
D.	s= T/ln z
Answer» C. s =z

Discussion

140.	H_(Z)_IS_DISCRETE_RATIONAL_TRANSFER_FUNCTION._TO_ENSURE_THAT_BOTH_H(Z)_AND_ITS_INVERSE_ARE_STABLE:?$
A.	Poles must be inside the unit circle and zeros must be outside the unit circle
B.	Poles and zeroes must be inside the unit circle
C.	Poles and zeroes must be outside the unit circle
D.	Poles must be outside the unit circle and zeros must be inside the unit circle
Answer» C. Poles and zeroes must be outside the unit circle

Discussion

141.	What is the ROC of z-transform of finite duration anti-causal sequence?$
A.	z=0
B.	z=‚Äö√Ñ√∂‚àö‚Ä†‚àö¬™
C.	Entire z-plane, except at z=0
D.	Entire z-plane, except at z=‚Äö√Ñ√∂‚àö‚Ä†‚àö¬™
Answer» C. Entire z-plane, except at z=0

Discussion

142.	A sequence x (n) with the z-transform X (z) = Z⁴ + Z² ‚Äö√Ñ√∂‚àö√ë‚àö¬® 2z + 2 ‚Äö√Ñ√∂‚àö√ë‚àö¬® 3Z^-4 is applied to an input to a linear time invariant system with the impulse response h (n) = 2‚âà√≠¬¨‚Ä¢ (n-3). The output at n = 4 will be?#
A.	-6
B.	Zero
C.	2
D.	-4
Answer» C. 2

Discussion

143.	If the region of convergence of x1[n]+x2[n] is 1/>\|z\|<2/3, the region of convergence of x1[n]-x2[n] includes:
A.	1/3>\|z\|<3
B.	2/3>\|z\|<3
C.	3/2>\|z\|<3
D.	1/3>\|z\|<2/3
Answer» E.

Discussion

144.	The region of convergence of the z-transform of a unit step function is:
A.	\|z\|>1
B.	\|z\|<1
C.	(Real part of z)>0
D.	(Real part of z)<0
Answer» B. \|z\|<1

Discussion

145.	Which one of the following is the correct statement? The region of convergence of z-transform of x[n] consists of the values of z for which x[n] is:
A.	Absolutely integrable
B.	Absolutely summable
C.	Unity
D.	<1
Answer» C. Unity

Discussion

146.	What is the ROC of the signal x(n)=‚âà√≠¬¨‚Ä¢(n-k),k>0?$
A.	z=0
B.	z=‚Äö√Ñ√∂‚àö‚Ä†‚àö¬™
C.	Entire z-plane, except at z=0
D.	Entire z-plane, except at z=‚Äö√Ñ√∂‚àö‚Ä†‚àö¬™
Answer» D. Entire z-plane, except at z=‚Äö√Ñ√∂‚àö‚Ä†‚àö¬™

Discussion

147.	Two sequences x1 (n) and x2 (n) are related by x2 (n) = x1 (- n). In the z- domain, their ROC‚Äö√Ñ√∂‚àö√ë‚àö¬•s are$
A.	The same
B.	Reciprocal of each other
C.	Negative of each other
D.	Complements of each other
Answer» C. Negative of each other

Discussion

148.	What is the set of all values of z for which X(z) attains a finite value?
A.	Radius of convergence
B.	Radius of divergence
C.	Feasible solution
D.	None of the mentioned
Answer» B. Radius of divergence

Discussion

149.	The frequency of a continuous time signal x (t) changes on transformation from x (t) to x (‚âà√≠¬¨¬± t), ‚âà√≠¬¨¬± > 0 by a factor$
A.	‚âà√≠¬¨¬±
B.	1/‚âà√≠¬¨¬±
C.	‚âà√≠¬¨¬±<sup>2</sup>
D.	‚âà√≠¬¨¬±
Answer» B. 1/‚âà√≠¬¨¬±

Discussion

150.	The discrete-time signal x (n) = (-1)ⁿ is periodic with fundamental period
A.	6
B.	4
C.	2
D.	0
Answer» D. 0

Discussion

108.	If the signal \({\rm{x}}\left( {\rm{t}} \right) = \frac{{{\rm{sin}}\left( {\rm{t}} \right)}}{{{\rm{\pi t}}}}{\rm{\;*}}\frac{{{\rm{sin}}\left( {\rm{t}} \right)}}{{{\rm{\pi t}}}}\) with ∗ denoting the convolution operation, then x(t) is equal to
A.	\(\frac{{{\rm{sin}}\left( {\rm{t}} \right)}}{{{\rm{\pi t}}}}\)
B.	\(\frac{{{\rm{sin}}\left( {2{\rm{t}}} \right)}}{{2{\rm{\pi t}}}}\)
C.	\(\frac{{2{\rm{sin}}\left( {\rm{t}} \right)}}{{{\rm{\pi t}}}}\)
D.	\({\left( {\frac{{{\rm{sin}}\left( {\rm{t}} \right)}}{{{\rm{\pi t}}}}} \right)^2}\)
Answer» B. \(\frac{{{\rm{sin}}\left( {2{\rm{t}}} \right)}}{{2{\rm{\pi t}}}}\)

126.	If f(x) represented by Fourier integral \(f(x)=\int_0^{\infty} [A(ω) cos~ω x + B(ω) sin~ω x]dω \)then A(ω) is defined as
A.	\(\frac{1}{\pi}\int_{-\infty}^{\infty} f(v) ~cos~\omega v ~dv\)
B.	\(\frac{1}{\pi}\int_{-\infty}^{\infty} f(v) ~sin~\omega v ~dv\)
C.	\(\int_{-\infty}^{\infty} f(\omega) ~cos~\omega v ~dv\)
D.	\(\int_{-\infty}^{\infty} f(\omega) ~sin~\omega v ~dv\)
Answer» B. \(\frac{1}{\pi}\int_{-\infty}^{\infty} f(v) ~sin~\omega v ~dv\)

128.	Consider a linear time invariant system \(\dot x = Ax\) with initial condition \(x\left( 0 \right)\) at \(t = 0\). Suppose \(\alpha\) and \(\beta\) are eigenvectors of \(\left( {2 \times 2} \right)\) matrix A corresponding to distinct eigenvalues \({\lambda _1}\;and\;{\lambda _2}\) respectively. Then the response \(x\left( t \right)\) of the system due to initial condition \(x\left( 0 \right) = \alpha\) is
A.	\({e^{{\lambda _1}t}}\alpha\)
B.	\({e^{{\lambda _2}t}}\beta\)
C.	\({e^{{\lambda _2}t}}\alpha\)
D.	\({e^{{\lambda _1}t}}\alpha + {e^{{\lambda _2}t}}\beta\)
Answer» B. \({e^{{\lambda _2}t}}\beta\)

136.	A function, in the Laplace domain, is given by\(F\left( s \right) = \frac{2}{s} - \frac{1}{{s + 3}}\)Its value by final value theorem in 't' domain will be
A.	\(\mathop {\lim }\limits_{t \to \infty } f\left( t \right) = 3\)
B.	\(\mathop {\lim }\limits_{t \to \infty } f\left( t \right) = 2\)
C.	\(\mathop {\lim }\limits_{t \to \infty } f\left( t \right) = 1\)
D.	\(\mathop {\lim }\limits_{t \to \infty } f\left( t \right) = 4\)
Answer» C. \(\mathop {\lim }\limits_{t \to \infty } f\left( t \right) = 1\)

137.	A signal \(x(t) = \sin c(α t)\), where α is a real constant, is the input to a Linear Time-Invariant system whose impulse response is \(h(t) = \sin c(β t)\), where β is a real constant. If min (α, β) denotes the minimum of α and β and similarly, max (α, β) denotes the maximum of α and β, and K is a constant, which one of the following statements is true about the (here, \(\sin c\left( x \right) = \frac{{\sin \left( {\pi x} \right)}}{{\pi x}}\)) output of the system?
A.	It will be of the form \(K\sin c\left( {γ t} \right)\) where γ = min (α, β)
B.	It will be of the form \(K\sin c\left( {γ t} \right)\) where γ = max (α, β)
C.	It will be of the form \(K\sin c\left( {\alpha t} \right)\)
D.	It cannot be a \(\sin c\) type of signal
Answer» B. It will be of the form \(K\sin c\left( {γ t} \right)\) where γ = max (α, β)

105.	Find the transfer function of the state variable representation of the system given by the differential equation y” + 2y’ + 4y = 8u.
A.	\(\frac{4}{{\left( {{s^2} + 2s + 4} \right)}}\)
B.	\(\frac{8}{{\left( {{s^2} + 2s + 4} \right)}}\)
C.	\(\frac{2}{{\left( {{s^2} + 2s + 4} \right)}}\)
D.	\(\frac{6}{{\left( {{s^2} + 2s + 4} \right)}}\)
Answer» C. \(\frac{2}{{\left( {{s^2} + 2s + 4} \right)}}\)

110.	If the Fourier transform of x(t) is X(jω), obtain the Fourier transform of x(t – t0)
A.	\({e^{ - j\omega {t_0}}}X\left( {j{t_0}} \right)\)
B.	\({e^{j\omega {t_0}}}X\left( {j\omega } \right)\)
C.	\({e^{ - j\omega {t_0}}}X\left( {j\omega {t_0}} \right)\)
D.	\({e^{ - j\omega {t_0}}}X\left( {j\omega } \right)\)
Answer» E.

111.	Evaluate \(\mathop \smallint \limits_{ - 1}^1 \left( {3{t^2} + 1} \right)\delta \left( t \right)dt\)
A.	4
B.	0
C.	1
D.	8
Answer» D. 8

114.	Consider a causal LTI system characterized by differential equation \(\frac{{dy\left( t \right)}}{{dt}} + \frac{1}{6}y\left( t \right) = 3x\left( t \right)\). The response of the system to the input \(x\left( t \right) = 3{e^{ - \frac{t}{3}u\left( t \right)}}\). Where u(t) denotes the unit step function is
A.	\(9{e^{ - \frac{t}{3}}}u\left( t \right)\)
B.	\(9{e^{ - \frac{t}{6}}}u\left( t \right)\)
C.	\(9{e^{ - \frac{t}{3}}}u\left( t \right) - 6{e^{ - \frac{t}{6}}}u\left( t \right)\)
D.	\({54^{ - \frac{t}{6}}}u\left( t \right) - 54{e^{ - \frac{t}{3}}}u\left( t \right)\)
Answer» E.

115.	Laplace transform of eθt sin (ωt) is:
A.	\(\frac{θ}{(s^2+ω^2 )}\)
B.	\(\frac{\omega}{(s-\theta)^2+ω^2 }\)
C.	\(\frac{θ}{s^2-(\theta-ω)^2 }\)
D.	\(\frac{θ}{(s+\theta)^2-ω^2 }\)
Answer» C. \(\frac{θ}{s^2-(\theta-ω)^2 }\)

119.	If the Laplace transform of function f(t) is given by \(\frac{s+3}{(s+1)(s+2)}\), then f(0) is
A.	\(\dfrac{3}{2}\)
B.	\(\dfrac{1}{2}\)
C.	0
D.	1
Answer» E.

121.	Find the Inverse Laplace transform of \({\rm{F}}\left( {\rm{s}} \right) = \frac{1}{{{\rm{s}} - 1}}\)
A.	1
B.	eat
C.	cos at
D.	cosh at
Answer» C. cos at

122.	In a series RLC circuit, the output is taken across the capacitor C, and the input is applied across the resistor R and ground Obtain the closed-loop transfer function.
A.	\(\frac{s}{{\left( {1 + sRC + {s^2}LC} \right)}}\)
B.	\(\frac{{sC}}{{\left( {1 + sRC + {s^2}LC} \right)}}\)
C.	1 + sRC + s2LC
D.	\(\frac{1}{{\left( {1 + sRC + {s^2}LC} \right)}}\)
Answer» E.

124.	Let u(t) denote the unit step function. The bilateral Laplace transform of the function f(t) = etu(−t) is ____.
A.	\(\frac{1}{s-1}\) with real part of s < 1
B.	\(\frac{1}{s-1}\) with real part of s > 1
C.	\(\frac{-1}{s-1}\)with real part of s < 1
D.	\(\frac{-1}{s-1}\) with real part of s > 1
Answer» D. \(\frac{-1}{s-1}\) with real part of s > 1

125.	Let \(X(s) = \frac{{3{s^2} + 5s}}{{{s^2} + 10s + 21}}\) be the Laplace Transform of a signal x(t). Then X(0+) is
A.	0
B.	3
C.	5
D.	21
Answer» B. 3

130.	A forcing function (t2 – 2t) u (t – 1) is applied to a linear system. The \({\cal L}\)- transform of the forcing function is
A.	\(\frac{{2 - s}}{{{s^3}}}\epsilon{^{ - 2s}}\)
B.	\(\left( {\frac{{1 - {s^2}}}{S}} \right)\epsilon{^{ - s}}\)
C.	\(\frac{1}{s}{e^{ - s}} - \frac{1}{{{s^2}}}\epsilon{^{ - 2s}}\)
D.	\(\left( {\frac{{2 - {s^2}}}{{{s^3}}}} \right)\epsilon{^{ - S}}\)
Answer» E.

132.	For a function g(t), it is given that \(\mathop \smallint \limits_{ - \infty }^{ + \infty } g\left( t \right){e^{ - j\omega t}}dt = \omega {e^{ - 2{\omega ^2}}}\) for any real value \(\omega \).If, \(y\left( t \right) = \mathop \smallint \limits_{ - \infty }^t g\left( \tau \right)d\tau \), then \(\mathop \smallint \limits_{ - \infty }^\infty y\left( t \right)dt\) is:
A.	0
B.	\(-j\)
C.	\(-\frac{j}{2}\)
D.	\(\frac{j}{2}\)
Answer» C. \(-\frac{j}{2}\)

133.	Consider the function \(F(s) = \frac{5}{{s({s^2} + 3s + 2)}}\), where F(s) is Laplace transform of function f(t). The initial value of f(t) is:
A.	5
B.	5/2
C.	5/3
D.	0
Answer» E.

135.	Laplace transform of f(t) = t2 sin t is
A.	\(\frac{{3{s^2} - 1}}{{{{\left( {{s^2} + 1} \right)}^3}}}\)
B.	\(\frac{{2\left( {3{s^2} - 1} \right)}}{{{{\left( {{s^2} + 1} \right)}^3}}}\)
C.	\(\frac{{\left( {3{s^2} + 1} \right)}}{{{{\left( {{s^2} + 1} \right)}^3}}}\)
D.	\(\frac{{\left( {3{s^2} - 1} \right)}}{{{{\left( {{s^2} + 1} \right)}^3}}}\)
Answer» C. \(\frac{{\left( {3{s^2} + 1} \right)}}{{{{\left( {{s^2} + 1} \right)}^3}}}\)

Explore topic-wise MCQs in Control Systems.

Match List-I with List-II:List-IList-IIa)sin ωti) \(\frac{\omega }{(s^2+ω^2 )}\)b)cos ωtii) \(\frac{s}{(s^2+ω^2 )}\)c)sin h btiii) \(\frac{s}{(s^2-b^2 )}\)d)cos h btiv) \(\frac{b}{(s^2-b^2 )}\)Choose the correct option from those given below:

Find the initial value of the signal x(t) whose unilateral Laplace transform is:\(X\left( s \right) = \frac{{5s + 10}}{{s\left( {s + 4} \right)}}\)

If x(t) is of finite duration and is absolutely integrable, then the 'region of convergence' is:

Decimation-in-Time is the class of

Find the transfer function of the state variable representation of the system given by the differential equation y” + 2y’ + 4y = 8u.

L{sin2 at} equals

Fourier transform of a real and odd function is

If the signal \({\rm{x}}\left( {\rm{t}} \right) = \frac{{{\rm{sin}}\left( {\rm{t}} \right)}}{{{\rm{\pi t}}}}{\rm{\;*}}\frac{{{\rm{sin}}\left( {\rm{t}} \right)}}{{{\rm{\pi t}}}}\) with ∗ denoting the convolution operation, then x(t) is equal to

In VHF spectrum Analyzer, two types of frequency instabilities which will cause difficulties when narrow frequency range are displayed are known as(a) Short-term instability(b) Phase-noise(c) Correlation-noise(d) Long-term instabilityChoose the correct option

If the Fourier transform of x(t) is X(jω), obtain the Fourier transform of x(t – t0)

Evaluate \(\mathop \smallint \limits_{ - 1}^1 \left( {3{t^2} + 1} \right)\delta \left( t \right)dt\)

Match the two lists and choose the correct answer from the code given below:List IList II(a) Sin 2t(i)(b) e-2t(ii) (c) e-2t sin 2t(iii) (d) 1 – e-2t(iv)

Consider Fourier representation of continuous and discrete-time systems. The complex exponentials (i.e., signals), which arise in such representation, have

Laplace transform of eθt sin (ωt) is:

A function f (t) is shown in the figure.The Fourier transform F(ω) of f(t) is

An ideal square wave with period of 20 ms shown in the figure, is passed through an ideal low pass filter with cut-off frequency 120 Hz. Which of the following is an accurate description of the output?

A system has impulse response h(t) = e-2tu(t). Find its system function if the input to the system is x(t) = e-tu(t).

If the Laplace transform of function f(t) is given by \(\frac{s+3}{(s+1)(s+2)}\), then f(0) is

Laplace transform of 3 t4 is

Find the Inverse Laplace transform of \({\rm{F}}\left( {\rm{s}} \right) = \frac{1}{{{\rm{s}} - 1}}\)​

In a series RLC circuit, the output is taken across the capacitor C, and the input is applied across the resistor R and ground Obtain the closed-loop transfer function.

A different non constant even function x(t) has a derivative y(t), and their respective Fourier Transforms are X(ω) and Y(ω). Which of the following statements is TRUE

Let u(t) denote the unit step function. The bilateral Laplace transform of the function f(t) = etu(−t) is ____.

Let \(X(s) = \frac{{3{s^2} + 5s}}{{{s^2} + 10s + 21}}\) be the Laplace Transform of a signal x(t). Then X(0+) is

If f(x) represented by Fourier integral \(f(x)=\int_0^{\infty} [A(ω) cos~ω x + B(ω) sin~ω x]dω \)then A(ω) is defined as

For distortionless transmission through LTI system phase of H(ω) is

Let the signal f(t) = 0 outside the interval [T1, T2], where T1 and T2 are finite. Furthermore, \(\left| {{\rm{f}}\left( {\rm{t}} \right)} \right| < \infty\). The region of convergence (ROC) of the signal’s bilateral Laplace transform F(s) is

A forcing function (t2 – 2t) u (t – 1) is applied to a linear system. The \({\cal L}\)- transform of the forcing function is

_______ is defined as the time delay that a signal component of frequency ω undergoes as it passes from the input to output of the system.

Consider the function \(F(s) = \frac{5}{{s({s^2} + 3s + 2)}}\), where F(s) is Laplace transform of function f(t). The initial value of f(t) is:

Let f(t) be an even function i.e. f(-t) = f(t) for all t. Let the Fourier transform of f(t) be defined as \(F(ω ) = \displaystyle\int_{-\infty}^\infty f(t) e^{-jω t}dt\). Suppose \(\dfrac{dF(ω)}{dω} = -ω F(ω)\) for all ω, and F(0) = 1. Then

Laplace transform of f(t) = t2 sin t is

A function, in the Laplace domain, is given by\(F\left( s \right) = \frac{2}{s} - \frac{1}{{s + 3}}\)Its value by final value theorem in 't' domain will be

In the Fourier transform, if the time domain signal x(t) is real and even, then the frequency domain signal X(jΩ) will be:

Z_AND_LAPLACE_TRANSFORM_ARE_RELATED_BY:?$

H_(Z)_IS_DISCRETE_RATIONAL_TRANSFER_FUNCTION._TO_ENSURE_THAT_BOTH_H(Z)_AND_ITS_INVERSE_ARE_STABLE:?$

What is the ROC of z-transform of finite duration anti-causal sequence?$

If the region of convergence of x1[n]+x2[n] is 1/>|z|<2/3, the region of convergence of x1[n]-x2[n] includes:

The region of convergence of the z-transform of a unit step function is:

Which one of the following is the correct statement? The region of convergence of z-transform of x[n] consists of the values of z for which x[n] is:

What is the ROC of the signal x(n)=‚âà√≠¬¨‚Ä¢(n-k),k>0?$

Two sequences x1 (n) and x2 (n) are related by x2 (n) = x1 (- n). In the z- domain, their ROC‚Äö√Ñ√∂‚àö√ë‚àö¬•s are$

What is the set of all values of z for which X(z) attains a finite value?

The frequency of a continuous time signal x (t) changes on transformation from x (t) to x (‚âà√≠¬¨¬± t), ‚âà√≠¬¨¬± > 0 by a factor$

The discrete-time signal x (n) = (-1)n is periodic with fundamental period

Find the Inverse Laplace transform of \({\rm{F}}\left( {\rm{s}} \right) = \frac{1}{{{\rm{s}} - 1}}\)

The discrete-time signal x (n) = (-1)ⁿ is periodic with fundamental period