Consider a continuous-time system with input x(t) and output y(t) given by

y(t) = x(t)cos(t)                                      

This system is

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  1. linear and time-invariant
  2. non-linear and time-invariant
  3. linear and time-varying
  4. non – linear and time-varying

Answer (Detailed Solution Below)

Option 3 : linear and time-varying
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Detailed Solution

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Concept:

Linear system

The system is said to be linear if it follows Homogenous and Superposition property.

1) The system is homogenous if 

y(t) = α1x(t) 

y(t) = α2x(t)

2) The system follows Superposition if

y(t) = α1x(t) + α2x(t)

Time invariant system

if y(t - t0) = x(t - t0)

and y(t') = x(t - t0)  

Analysis:

For the given signal check linearity property

y(t) = x(t)cos(t)

1) check homogeneity

y(t) = α1x(t)cos(t)

y(t) = α2x(t)cos(t)

2) check superposition 

y(t) = α1x(t)cos(t) + α2x(t)cos(t)

The system follows both homogeneity and superposition thus the system is Linear.

check the time invariance

y(t - t0) = x(t - t0)cos(t - t0)

and y(t') = x(t - t0)cos(t)

since y(t - t0) ≠ y(t') 

System is not time invariant.

Hence option (3) is correct

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