1
GATE ECE 2025
MCQ (Single Correct Answer)
+1
-0.33

Consider the unity-negative-feedback system shown in Figure (i) below, where gain $K \geq 0$. The root locus of this system is shown in Figure (ii) below. For what value(s) of $K$ will the system in Figure (i) have a pole at $-1+j 1$ ?

GATE ECE 2025 Control Systems - Root Locus Diagram Question 1 English 1 GATE ECE 2025 Control Systems - Root Locus Diagram Question 1 English 2
A
$K=5$
B
$K=\frac{1}{5}$
C
For no positive value of $K$
D
For all positive values of $K$
2
GATE ECE 2025
MCQ (Single Correct Answer)
+2
-0.67

Let $G(s)=\frac{1}{10 s^2}$ be the transfer function of a second-order system. A controller $M(s)$ is connected to the system $G(s)$ in the configuration shown below. Consider the following statements.

(i) There exists no controller of the form $M(s)=\frac{K_I}{s}$, where $K_I$ is a positive real number, such that the closed loop system is stable.

(ii) There exists at least one controller of the form $M(s)=K_P+s K_D$, where $K_P$ and $K_D$ are positive real numbers, such that the closed loop system is stable.

Which one of the following options is correct?

GATE ECE 2025 Control Systems - Compensators Question 1 English
A
(i) is TRUE and (ii) is FALSE
B
(i) is FALSE and (ii) is TRUE
C
Both (i) and (ii) are FALSE
D
Both (i) and (ii) are TRUE
3
GATE ECE 2025
MCQ (Single Correct Answer)
+2
-0.67
Consider the polynomial $p(s)=s^5+7 s^4+3 s^3-33 s^2+2 s-40$. Let $(L, I, R)$ be defined as follows. $L$ is the number of roots of $p(s)$ with negative real parts. $I$ is the number of roots of $p(s)$ that are purely imaginary. $R$ is the number of roots of $p(s)$ with positive real parts. Which one of the following options is correct?
A
$L=2, I=2$ and $R=1$
B
$L=3, I=2$ and $R=0$
C
$L=1, I=2$ and $R=2$
D
$L=0, I=4$ and $R=1$
4
GATE ECE 2025
MCQ (Single Correct Answer)
+2
-0.67

Consider a system where $x_1(t), x_2(t)$, and $x_3(t)$ are three internal state signals and $u(t)$ is the input signal. The differential equations governing the system are given by

$$ \frac{d}{d t}\left[\begin{array}{l} x_1(t) \\ x_2(t) \\ x_3(t) \end{array}\right]=\left[\begin{array}{ccc} 2 & 0 & 0 \\ 0 & -2 & 0 \\ 0 & 0 & 0 \end{array}\right]\left[\begin{array}{l} x_1(t) \\ x_2(t) \\ x_3(t) \end{array}\right]+\left[\begin{array}{l} 1 \\ 1 \\ 1 \end{array}\right] u(t) $$

Which of the following statements is/are TRUE?

A
The signals $x_1(t), x_2(t)$, and $x_3(t)$ are bounded for all bounded inputs.
B
There exists a bounded input such that at least one of the signals $x_1(t), x_2(t)$, and $x_3(t)$ is unbounded.
C
There exists a bounded input such that the signals $x_1(t), x_2(t)$ and $x_3(t)$ are unbounded.
D
The signals $x_1(t), x_2(t)$ and $x_3(t)$ are unbounded for all bounded inputs.
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