A first-order system has a time constant of 2 seconds and is subjected to a unit step input. The system gain is unity, then1.The output reaches approximately 63 percent of its final value at 2 seconds.2.The steady-state value of the output is equal to1.3. The response is exponential in nature and does not exhibit oscillations.4.The output reaches 95 percent of final value in about 2 seconds.Which of the above statements is/are correct?

Correct Answer: B - 1, 2 and 3 only

Quick Concept Explanation

A standard first-order system has one real pole and an exponential step response controlled by a time constant τ. For τ=2 s, y(t)=1−e^(−t/2). At t=2 s, y≈0.632. Key relation: G(s)=1/(τs+1); y(t)=1−e^(−t/τ).

Exam focus: 63% at τ, ~95% at 3τ, ~99% at 5τ. Common trap: Do not confuse one time constant with 95% settling.

Statement-wise Verification

• Statement 1 - Correct.
  The output reaches approximately 63 percent of its final value at 2 seconds.
  Starting from zero with unit DC gain, the output reaches 63.2% at one time constant, about 95% near 3τ and over 99% near 5τ.
• Statement 2 - Correct.
  The steady-state value of the output is equal to
  Starting from zero with unit DC gain, the output reaches 63.2% at one time constant, about 95% near 3τ and over 99% near 5τ.
• Statement 1 - Correct.
  3. The response is exponential in nature and does not exhibit oscillations.
  The output reaches approximately 63 percent of its final value at 2 seconds.
• Statement 4 - Incorrect.
  The output reaches 95 percent of final value in about 2 seconds.
  A first-order response reaches about 95% at roughly 3τ; for τ=2 s this is about 6 s

Core Concept

A standard first-order system has one real pole and an exponential step response controlled by a time constant τ. Starting from zero with unit DC gain, the output reaches 63.2% at one time constant, about 95% near 3τ and over 99% near 5τ.

Formula / Key Relationship

G(s)=1/(τs+1); y(t)=1−e^(−t/τ).

Step-by-Step Check

For τ=2 s, y(t)=1−e^(−t/2). At t=2 s, y≈0.632. A first-order response reaches about 95% at ≈3τ≈6 s, not 2 s.

Why the Other Options Are Wrong

Option A is incorrect because it omits correct statement(s) 2. Option C is incorrect because it omits correct statement(s) 1. Option D is incorrect because it includes incorrect statement(s) 4.

Exam Shortcut / Approach

63% at τ, ~95% at 3τ, ~99% at 5τ.

Common Mistake

Do not confuse one time constant with 95% settling.

Quick Revision

Time Constant and Step Response is linked with First-Order System, Time Response and Control Systems. 63% at τ, ~95% at 3τ, ~99% at 5τ.