GATE 2022 EE Q.37: official answer

GATE 2022 · EEQ.37 · 2 marks · MCQ

An LTI system is shown in the figure where

G (s) = \frac{100}{s ^{2} + 0.1 s + 10} .

The steady state output of the system, to the input $r (t)$ , is given as $y (t) = a + b sin (10 t + θ)$ . The values of ' $a$ ' and ' $b$ ' will be

GATE 2022 EE question 37: A single block labelled G(s). — Figure from the GATE 2022 EE paper. It is part of the question.

(A)
$a = 1, b = 10$
(B)
$a = 10, b = 1$
(C)
$a = 1, b = 100$
(D)
$a = 100, b = 1$

Official answer

Official answer: ASource: Official answer key, GATE 2022 EE (IIT Kharagpur)

A NOTE ON THIS QUESTION

As printed, none of the four options is correct. The paper gives $G (s) = \frac{100}{s ^{2} + 0.1 s + 10}$ and $r (t) = 1 + 0.1 sin (10 t)$ . The DC term gives $a = G (0) = \frac{100}{10} = 10$ . At $ω = 10$ : $G (j 10) = \frac{100}{( 10 - 100 ) + j 1} = \frac{100}{- 90 + j 1}$ , so $∣ G (j 10) ∣ = \frac{100}{8101} \approx 1.111$ and $b = 0.1 \times 1.111 \approx 0.111$ . No option has $a = 10$ with $b \approx 0.111$ . The official key is (A), $a = 1$ and $b = 10$ , which is what the same working gives if the constant term of the denominator is $100$ instead of $10$ : then $G (0) = 1$ and $G (j 10) = \frac{100}{j 1}$ , so $∣ G (j 10) ∣ = 100$ and $b = 0.1 \times 100 = 10$ . The printed question appears to contain a misprint. We show the official answer, and we do not publish a worked solution that the printed numbers do not lead to.

From the GATE 2022 EE paper. Every question of this paper · Download the full paper · All 27 Bode & margins questions, with notes

GATE 2022 EE Q.37: official answer

Official answer

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