$$IC$$ $$555$$ in the adjacent figure is configured as an Astable multivibrator. It is enabled to oscillator at $$t=0$$ by applying a high input to pin $$4.$$ The pin description is $$1$$ and $$8$$ $$-$$supply; $$2-$$trigger; $$4-$$reset; $$6$$-threshold; $$7$$-discharge. The wave form appearing across the capacitor starting from $$t=0$$ as observed on the storage $$CRO$$ is

The switch $$'S'$$ is the circuit of the figure is initially closed. It is opened at time $$t=0.$$ You may neglect the zener diode forward voltage drops. What is the behavior of $${V_{OUT}}$$ for $$t > 0$$

The input signal $${V_{in}}$$ shown in the figure is a $$1$$ $$kHz$$ square wave voltage that alternates between $$+7V$$ and $$-7V$$ with a $$50\% $$ duty cycle. Both transistors have the same current gain, which is large. The circuit delivers power to the load resistor $${R_L}.$$ What is the efficiency of this circuit for the given input? Choose the closest answer.

The circuit shown in figure is

The common emitter forward current gain of the transistor shown is $${\beta _p} = 100.$$ The transistor is operating in

The three terminal linear voltage regular is connected to a $$10\Omega $$ load resistor as shown in the figure. If $${V_{in}}$$ is $$10$$ $$V,$$ what is the power dissipated in the transistor

$$R-L-C$$ circuit shown in figure

If the above step response is to be observed on a non - storage $$CRO,$$ then it would be best have the $${e_i}$$ as a

The system $$900/s(s+1)(s+9)$$ is to be such that its gain crossover frequency becomes same as its uncompensated phase crossover frequency and provides at $${45^0}$$ phase margin . To achieve this, one may use

If $$X = {\mathop{\rm Re}\nolimits} G\left( {j\omega } \right),\,\,$$ and $$y = {\rm I}mG\left( {j\omega } \right)$$ then for $$\omega \to {0^ + },\,\,$$ the Nyquist plot for $$G\left( s \right) = 1/\left[ {s\left( {s + 1} \right)\left( {s + 2} \right)} \right]$$

The system shown in figure below
can be reduced to the form
With

The system shown in the figure is

If the loop gain $$K$$ of a negative feedback system having a loop transfer function $$K\left( {s + 3} \right)/{\left( {s + 8} \right)^2}$$ is to be adjusted to induce a sustained oscillation then

Consider the feedback system shown below which is subjected to a unit step input. The system is stable and has following parameters $${k_p} = 4,\,\,{k_i} = 10,\,\,\omega = 500\,\,$$ and $$\xi $$ $$=0.7.$$ The steady state value of $$z$$ is

$$R-L-C$$ circuit shown in figure

For a step input $${e_{i,}}$$ the overshoot in the output $${e_{0,}}$$ will be

In an $$8085$$ $$A$$ microprocessor based system, it is desired to increment the contents of memory location whose address is available in $$(D,E)$$ register pair and store the result in same location. The sequence of instructions is

The associated figure shows the two types of rotate right instruction $${R_1},\,{R_2}$$ available in a microprocessor where Register is a $$8$$-bit register and $$C$$ is the carry bit. The rotate left instructions $$L1$$ and $$L2$$ are similar except that $$C$$ now links the most significant bit of Register instead of the least significant one.

Suppose Register contains the $$2's$$ complement number $$11010110.$$ If this number is delivered by $$2$$ the answer should be

$$A,B,C$$ and $$D$$ are input bits, and $$Y$$ is the output bit in the $$XOR$$ gate circuit of the figure below. Which of the following statements about the sum $$S$$ of $$A,B,C,D$$ and $$Y$$ is correct?

Which one of the following statements regarding the $$INT$$ (interrupt) and the $$BRQ$$ (bus request) pins in a $$CPU$$ is true?

The associated figure shows the two types of rotate right instruction $${R_1},\,{R_2}$$ available in a microprocessor where Register is a $$8$$-bit register and $$C$$ is the carry bit. The rotate left instructions $$L1$$ and $$L2$$ are similar except that $$C$$ now links the most significant bit of Register instead of the least significant one.

Such a division can be correctly performed by the following set of operations

The $$R-L-C$$ series circuit shown is supplied from a variable frequency voltage source. The admittance $$-$$ locus of the $$R-L$$ $$-C$$ network at terminals $$AB$$ for increasing frequency $$\omega $$ is

In the figure given below, all phasors are with reference to the potential at point $$''O''.$$ The locus of voltage phasor $${V_{YX}}$$ as $$R$$ is varied from zero to infinity is shown by

In the figure, transformer $${T_1}$$ has two secondaries, all three windings having the same number of turns and with polarities having the same number of turns and with polarities as indicated. One secondary is shorted by a $$10\,\Omega $$ resistor $$R,$$ and the other by a $$15\mu F$$ capacitor. The switch $$SW$$ is opened $$(t=0)$$ when the capacitor is charged to $$5$$ $$V$$ with the left plate as positive. At $$t=0+$$ the voltage $${V_P}$$ and current $${I_R}$$ are

A three phase balanced star connected voltage source with frequency $$\omega \,\,rad/s$$ is connected to a star connected balanced load which is purely inductive. The instantaneous line currents and phase to neutral voltages are denoted by $$\left( {{i_a},{i_b},{i_c}} \right)$$ and $$\left( {{V_{an}},\,\,{V_{bn}},\,\,{V_{cn}}} \right)$$ respectively and their $$rms$$ values are denoted by $$V$$ and $$1.$$ If $$$R = \left[ {{V_{an}}\,\,{V_{bn}}\,\,{V_{cn}}} \right]\left[ {\matrix{ 0 & {{1 \over {\sqrt 3 }}} & { - {1 \over {\sqrt 3 }}} \cr { - {1 \over {\sqrt 3 }}} & 0 & {{1 \over {\sqrt 3 }}} \cr {{1 \over {\sqrt 3 }}} & { - {1 \over {\sqrt 3 }}} & 0 \cr } } \right]\left[ {\matrix{ {{i_a}} \cr {{i_b}} \cr {{i_c}} \cr } } \right],$$$
then the magnitude of $$R$$ is

In the circuit shown in Fig. switch $$S{w_1}$$ is initially CLOSED and $$S{w_2}$$ is OPEN. The inductor $$L$$ carries a current of $$10$$ $$A$$ and the capacitor is charged to $$10$$ $$V$$ with polarities as indicated. $$S{w_2}$$ in initially CLOSED at $$t = {0^ - }$$ and $$S{w_1}$$ is OPENED at $$t=0.$$ The current through $$C$$ and the voltage across $$L$$ at $$t = {0^ + }$$ is

A $$3$$ $$V$$ $$dc$$ supply with an internal resistance of $$2$$ $$\Omega $$ supplies a passive non-linear resistance characterized by the relation $${V_{NL}} = {{\rm I}^2}{}_{NL}$$. The power dissipated in the non-linear resistance is

The resonant frequency for the given circuit will be

The matrix $$A$$ given below is the node incidence matrix of a network. The columns correspond to braches of the network while the rows correspond to nodes. Let
$$V = {\left[ {{v_1}\,\,{v_2}....{v_6}} \right]^T}$$ denote the vector of branches voltages while
$${\rm I} = {\left[ {{i_1}\,{i_2}....{i_6}} \right]^T}$$ that of branch currents. The vector $$E = {\left[ {{e_1}\,{e_2}\,\,{e_3}\,{e_4}} \right]^T}$$ denotes the vector of node voltages relative to a common ground. $$$A = \left[ {\matrix{ 1 & 1 & 1 & 0 & 0 & 0 \cr 0 & { - 1} & 0 & { - 1} & 1 & 0 \cr { - 1} & 0 & 0 & 0 & { - 1} & { - 1} \cr 0 & 0 & { - 1} & 1 & 0 & 1 \cr } } \right]$$$

Which of the following statements is true?

The probes of a non $$-$$ isolated, two-channel oscilloscope are clipped to points $$A, B, $$ and $$C$$ in the circuit of the adjacent fig. $${V_{in}}$$ is a square wave of a suitable low frequency. The display on $$c{h_1}$$ and $$c{h_2}$$ are as shown on the right. Then the ''signal'' and ''Ground'' probes $${S_1},\,\,{G_1}$$ and $${S_2},\,\,{G_2}$$ of $$c{h_1}$$ and $$c{h_2}$$ respectively are connected to points.

A $$3$$ $$V$$ $$dc$$ supply with an internal resistance of $$2\,\,\Omega $$ supplies a passive non-linear resistance characterized by the relation $${V_{NL}} = {\rm I}_{NL}^2.$$ The power dissipated in the non-linear resistance is

A three phase squirrel cage induction motor has a starting current of seven times the full load current and full load slip of $$5\% $$

If a star-delta starter is used to start this induction motor, the per unit staring torque will be

A three phase squirrel cage induction motor has a starting current of seven times the full load current and full load slip of $$5\% $$

If an auto-transformer is used for reduced voltage starting to provide $$1.5$$ per unit starting torque, the auto-transformer ratio $$\left( \% \right)$$ should be

A $$3$$ phase, $$3$$ stack, variable reluctance stepper motor has $$20$$ poles on each rotor and stator stack. The step angle of this stepper motor is

A three phase squirrel cage induction motor has a starting current of seven times the full load current and full load slip of $$5\% $$

If a starting torque of $$0.5$$ per unit is required then the per unit starting current should be

A three phase synchronous motor connected to ac mains is running at full load and unity power factor. If its shaft load is reduced by half, with field current held constant, its new power factor will be

A three-phase squirrel cage induction motor has a starting torque of $$150\% $$ and a maximum torque of $$300\% $$ with respect to rated torque at rated voltage and rated frequency. Neglect the stator resistance and rotational losses. The value of slip for maximum torque is

A $$100$$ $$kVA$$, $$415V,$$ star connected synchronous machine generates rated open circuit voltage $$415V$$ at a field current of $$15A.$$ The short circuit armature current at a field current of $$10A$$ is equal to the rated armature current. The per unit saturated synchronous reactance is

A single-phase $$50$$ $$kVA,$$ $$250$$ $$V/500V$$ two winding transformer has an efficiency of $$95\% $$ at full load, unity power factor. If it is reconfigured as a $$500V/750V$$ auto-transformer, its efficiency at its new rated load at unity power factor will be

The dc motor, which can provide zero speed regulation at full load without any controller is

In a transformer, zero voltage regulation at full load is

A solid sphere made of insulating material has a radius $$R$$ and has a total charge $$Q$$ distributed uniformly in its volume. What is the magnitude of the electric field intensity, $$E,$$ at distribution $$r\left( {0 < r < R} \right)$$ inside the sphere?

An inductor designed with $$400$$ turns coil wound on an iron core of $$16\,\,c{m^2}$$ cross sectional area and with a cut of an air gap length of $$1$$ $$mm$$. The coil is connected to a $$230$$ $$V,$$ $$50$$ $$Hz$$ ac supply. Neglect coil resistance, core loss, iron reluctance and leakage inductance. $$\left( {{\mu _0} = 4\pi \times {{10}^{ - 7}}\,H/m} \right)$$

The average force on the core to reduce the air gap will be

An inductor designed with $$400$$ turns coil wound on an iron core of $$16\,\,c{m^2}$$ cross sectional area and with a cut of an air gap length of $$1$$ $$mm$$. The coil is connected to a $$230$$ $$V,$$ $$50$$ $$Hz$$ ac supply. Neglect coil resistance, core loss, iron reluctance and leakage inductance. $$\left( {{\mu _0} = 4\pi \times {{10}^{ - 7}}\,H/m} \right)$$

The current in the inductor is

Divergence of the vector field $$$\overrightarrow V\left(x,y,z\right)=-\left(x\cos xy\;+\;y\right)\;\widehat i\;+\;\left(y\cos xy\right)\;\widehat j\;+\;\left(\sin\;z^2\;+\;x^2\;+\;y^2\right)\widehat k$$$ is

Divergence of the vector field $$v\left( {x,y,z} \right) = - \left( {x\,\cos xy + y} \right)\widehat i + \left( {y\,\cos xy} \right)\widehat j + \left[ {\left( {\sin {z^2}} \right) + {x^2} + {y^2}} \right]\widehat k\,\,$$

Consider the function $$f\left( x \right) = {\left( {{x^2} - 4} \right)^2}$$ where $$x$$ is a real number. Then the function has

The integral $$\,\,{1 \over {2\pi }}\int\limits_0^{2\Pi } {Sin\left( {t - \tau } \right)\cos \tau \,d\tau \,\,\,} $$ equals

If $$A = \left[ {\matrix{ { - 3} & 2 \cr { - 1} & 0 \cr } } \right]\,$$ then $${A^9}$$ equals

Let $$x$$ and $$y$$ be two vectors in a $$3-$$ dimensional space and $$ < x,y > $$ denote their dot product. Then the determinant det $$\left[ {\matrix{ { < x,x > } & { < x,y > } \cr { < y,x > } & { < y,y > } \cr } } \right] = $$ ______.

If $$A = \left[ {\matrix{ { - 3} & 2 \cr { - 1} & 0 \cr } } \right]$$ then $$A$$ satisfies the relation

$${q_1},\,{q_2},{q_3},.......{q_m}$$ are $$n$$-dimensional vectors with $$m < n.$$ This set of vectors is linearly dependent. $$Q$$ is the matrix with $${q_1},\,{q_2},{q_3},.......{q_m}$$ as the columns. The rank of $$Q$$ is

$$X = {\left[ {\matrix{ {{x_1}} & {{x_2}} & {.......\,{x_n}} \cr } } \right]^T}$$ is an $$n$$-tuple non-
zero vector. The $$n\,\, \times \,\,n$$ matrix $$V = X{X^T}$$

In the circuit of adjacent figure the diode connects the $$ac$$ source to pure inductance $$L.$$

The diode conducts for

A three - phase fully - controlled thyristor bridge converter is used as line commuted inverter to feed $$50kW$$ power $$420$$ $$V$$ $$dc$$ to a three phase, $$415$$ $$V$$ (line), $$50$$ $$Hz$$ $$ac$$ mains, Consider $$dc$$ link current to be constant. The $$rms$$ current of the thyristor is

A single - phase voltage source inverter is controlled in a single pulse - width modulated mode with a pulse width of $${150^0}$$ in each half cycle. Total harmonic distortion is defined as
$$THD = {{\sqrt {V_{rms}^2 - V_1^2} } \over {{V_1}}} \times 100,\,\,\,$$ Where $${{V_1}}$$ is the $$rms$$ value of the fundamental component of the voltage. The $$THD$$ of output $$ac$$ voltage waveform is

A single - phase inverter is operated in $$PWM$$ mode generating a single - pulse of width $$2d$$ in the center of each half cycle as shown in figure. It is found that the output voltage is free from $${5^{th}}$$ harmonic for pulse width $${144^0}.$$ What will be percentage of $${3^{rd}}$$ harmonic present in the output voltage $$\,\left( {{V_{03}}/{V_{01\,\,\,\max }}} \right)?$$

The circuit in the figure is current commutated $$dc$$ $$-$$ $$dc$$ chopper where, $$T{h_M}$$ is the main $$SCR$$ and $$T{h_AUX}$$ is the auxiliary $$SCR$$. The load current is constant at $$10$$ $$A.$$
$$T{h_M}$$ is $$ON$$
$$T{h_AUX}$$ is trigged at $$t=0.$$ $$T{h_M}$$ is turned $$OFF$$ between.

''Six $$MOSFETS$$ connected in a bridge configuration (having no other power device) MUST be operated as Voltage Source Inverter $$(VSI)''.$$ This statement is

A single phase full $$-$$ wave half $$-$$ controlled bridge converter feeds an inductive load. The two $$SCR's$$ in the converter are connected to a common $$DC$$ bus. The converter has to have a free wheeling diode

A single $$-$$ phase fully controlled thyristor bridge $$ac$$ $$-$$ $$dc$$ converter is operating at a firing angle of $${25^ \circ }$$ and an overlap angle of $${10^ \circ }$$ constant $$dc$$ output current of $$20$$ $$A$$. The fundamental power factor (displacement factor) at input $$ac$$ mains is

A $$1:1$$ pulse transformer $$(PT)$$ is used to trigger the $$SCR$$ in the adjacent figure. The $$SCR$$ is rated at $$1.5$$ $$kV$$, $$250$$ $$A$$ with $${{\rm I}_L} = 250\,mA,\,\,{{\rm I}_H} = 150\,mA,$$ and $${{\rm I}_{G\max }} = 150mA$$ with $${{\rm I}_L} = 250\,mA,$$ $${{\rm I}_{G\min }} = 100mA.$$ The $$SCR$$ is connected to an inductive load, where $$L$$ $$=150$$ $$mH$$ in series with a small resistance and the supply voltage is $$200$$ $$V$$ $$dc.$$ The forward drops of all transistors/diodes and gate $$-$$ cathode junction during $$ON$$ state are $$1.0V$$

The resistance $$R$$ should be

A single $$-$$ phase, $$230$$ $$V$$, $$50$$ $$Hz$$ $$ac$$ mains fed step down transformer $$(4:1)$$ is supplying power to a half $$-$$ wave uncontrolled $$ac$$ $$-$$ $$dc$$ converter used for charging a battery ($$12$$ $$V$$ $$dc$$) with the series current limiting resistor being $$19.04\,\Omega $$. The charging current is

A $$1:1$$ pulse transformer $$(PT)$$ is used to trigger the $$SCR$$ in the adjacent figure. The $$SCR$$ is rated at $$1.5$$ $$kV$$, $$250$$ $$A$$ with $${{\rm I}_L} = 250\,mA,\,\,{{\rm I}_H} = 150\,mA,$$ and $${{\rm I}_{G\max }} = 150mA$$ with $${{\rm I}_L} = 250\,mA,$$ $${{\rm I}_{G\min }} = 100mA.$$ The $$SCR$$ is connected to an inductive load, where $$L$$ $$=150$$ $$mH$$ in series with a small resistance and the supply voltage is $$200$$ $$V$$ $$dc.$$ The forward drops of all transistors/diodes and gate $$-$$ cathode junction during $$ON$$ state are $$1.0V$$

The minimum approximate volt $$-$$second rating of the pulse transformer suitable for triggering the $$SCR$$ should be: (Volt - second rating is the maximum of product of the voltage and the width of the pulse that may be applied)

Consider a synchronous generator connected to an infinite bus by two identical parallel transmission lines. The transient reactance $$x'$$ of the generator $$0.1$$ pu. Due to some previous disturbance, the rotor angle $$(d)$$ is undergoing an undamped oscillation, with the maximum value of $$\delta \left( t \right)$$ equal to $$\,{130^ \circ }\,.$$ One of the parallel lines trip due to relay mal-operation at an instant $$\,\,\,\,\,$$ when $$\,\delta \left( t \right)\,\, = {130^ \circ }\,\,$$ as shown in the figure. The maximum value of the per unit line reactance $$x,$$ such that the system does not lose synchronism subsequent to this tripping is

Two regional system, each having several synchronous generators and loads are interconnected by an ac line and a HVDC link as shown in the figure. Which of the following statements is true in the steady state:

Consider the protection system shown in the figure below. The circuit breakers numbered from $$1$$ to $$7$$ are of identical type. A single line to ground fault with zero fault impedance occurs at the midpoint of the line (at point F), but circuit breaker $$4$$ fails to operate (''Stuck breaker''). If the relays are coordinated correctly, a valid sequence of circuit breaker operation is b

Suppose we define a sequence transformation between ''a-b-c'' and ''p-n-0''' variables as follows:
$$\left[ {\matrix{ {{f_a}} \cr {{f_b}} \cr {{f_c}} \cr } } \right] = k\left[ {\matrix{ 1 & 1 & 1 \cr {{\alpha ^2}} & \alpha & 1 \cr \alpha & {{\alpha ^2}} & 1 \cr } } \right]\left[ {\matrix{ {{f_p}} \cr {{f_n}} \cr {{f_o}} \cr } } \right]$$ where $$\,\alpha = {e^{j{{2\pi } \over 3}}}\,\,$$ and $$k$$ is a constant
Now, if it is given that:
$$\left[ {\matrix{ {{V_p}} \cr {{V_n}} \cr {{V_o}} \cr } } \right] = k\left[ {\matrix{ {0.5} & 0 & 0 \cr 0 & {0.5} & 0 \cr 0 & 0 & {2.0} \cr } } \right]\left[ {\matrix{ {{i_p}} \cr {{I_n}} \cr {{i_o}} \cr } } \right]\,\,$$ and $$\left[ {\matrix{ {{V_a}} \cr {{V_b}} \cr {{V_c}} \cr } } \right] = z\left[ {\matrix{ {{i_a}} \cr {{I_b}} \cr {{i_c}} \cr } } \right]\,\,$$ then,

Consider the transformer connections in a part of a power system shown in the figure. The nature of transformer connections and phase shifts are indicated for all but one transformer. Which of the following connections, and the corresponding phase shift $$\theta ,$$ should be used for the transformer between $$A$$ and $$B$$?

$$230$$ $$V$$ (phase) $$50$$ Hz, three-phase, $$4$$-wire, system has a sequence $$ABC$$. A unity power-factor load of $$4$$ kW is connected between phase A and neutral $$N$$. It is desired to achieve zero neutral current through the use of a pure inductor and pure capacitor in the other two phases. The Value of inductor and capacitor is......

The incremental cost curves in Rs/MWhr for two generators supplying a common load of $$700$$ MW are shown in the figures. The maximum and minimum generation limits are also indicated. The optimum generation schedule is:

Consider a bundled conductor of an overhead line consisting of three identical sub-conductors placed at the corners of an equilateral triangle as shown in figure. If we neglect the charges on the other phase conductors and ground, and assume that spacing between sub-conductors is much larger than their radius, the maximum electric field intensity is experienced at

Single line diagram of a $$4$$-bus single source distribution system is shown below. Branches $${e_1},\,\,$$ $${e_2},\,\,$$ $${e_3}\,\,$$ and $${e_4}\,\,$$ have equal impedences. The load current values indicated in the figure are in per unit.

Distribution Company's policy requires radial system operation with minimum loss. This can be achieved by opening of the branch

The total reactance and total susceptance of a lossless overhead $$EHV$$ line, operating at $$50$$ $$Hz,$$ are given by $$0.045$$ pu and $$1.2$$ pu respectively. If the velocity of wave propagation is $$3\,\, \times \,\,{10^5}$$ km/s, then the approximate length of line is

The frequency spectrum of a signal is shown in the figure. If this is ideally sampled at intervals of $$1$$ $$ms,$$ then the frequency spectrum of the sampled signal will be

A signal is processed by a causal filter with transfer function $$G(s).$$ For a distortion free output signal waveform, $$G(s)$$ must

A signal is processed by a causal filter with transfer function $$G(s).$$ For a distortion free output signal waveform, $$G(s)$$ must.

$$G\left( z \right) = a{z^{ - 1}} + \beta \,\,{z^{ - 3}}$$ is a low-pass digital filter with a phase characteristic same as that of the above question if

$$X\left( z \right) = 1 - 3\,\,{z^{ - 1}},\,\,Y\left( z \right) = 1 + 2\,\,{z^{ - 2}}$$ are $$Z$$-transforms of two signals $$x\left[ n \right],\,\,y\left[ n \right]$$ respectively. A linear time invariant system has the impulse response $$h\left[ n \right]$$ defined by these two signals as $$h\left[ n \right] = x\left[ {n - 1} \right] * y\left[ n \right]$$ where $$ * $$ denotes discrete time convolution. Then the output of the system for the input $$\delta \left[ {n - 1} \right]$$

A signal $$x(t)$$ is given by
$$x\left( t \right) = \left\{ {\matrix{ {1, - {\raise0.5ex\hbox{$\scriptstyle T$} \kern-0.1em/\kern-0.15em \lower0.25ex\hbox{$\scriptstyle 4$}} < t \le {\raise0.5ex\hbox{$\scriptstyle {3T}$} \kern-0.1em/\kern-0.15em \lower0.25ex\hbox{$\scriptstyle 4$}}} \cr { - 1,{\raise0.5ex\hbox{$\scriptstyle {3T}$} \kern-0.1em/\kern-0.15em \lower0.25ex\hbox{$\scriptstyle 4$}} < t \le {\raise0.5ex\hbox{$\scriptstyle {7T}$} \kern-0.1em/\kern-0.15em \lower0.25ex\hbox{$\scriptstyle 4$}},\,\,\,} \cr { - x\left( {t + T} \right)} \cr } } \right.$$ Which among the following gives the fundamental Fourier term of $$x(t)$$?

Consider the discrete-time system shown in the figure where the impulse response of $$G\left( z \right)$$ is
$$g\left( 0 \right) = 0,\,\,g\left( 1 \right) = g\left( 2 \right) = 1,\,g\left( 3 \right) = g\left( 4 \right) = .... = 0$$

This system is stable for range of values of $$K$$

Let a signal $${a_1}\,\sin \left( {{\omega _1}t + {\phi _1}} \right)$$ be applied to a stable linear time-invariant system. Let the corresponding steady state output be represented as $${a_2}F\left( {{\omega _2}t + {\phi _2}} \right).$$ Then which of the following statements is true?

If u(t), r(t) denote the unit step and unit ramp functions respectively and u(t)*r(t) their convolution, then the function u(t+1)*r(t-2) is given by