1
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 $R=1$
2
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.
3
GATE ECE 2025
MCQ (Single Correct Answer)
+2
-0.67

Consider a system represented by the block diagram shown below. Which of the following signal flow graphs represent(s) this system? Choose the correct option(s).

GATE ECE 2025 Control Systems - Signal Flow Graph and Block Diagram Question 1 English
A
GATE ECE 2025 Control Systems - Signal Flow Graph and Block Diagram Question 1 English Option 1
B
GATE ECE 2025 Control Systems - Signal Flow Graph and Block Diagram Question 1 English Option 2
C
GATE ECE 2025 Control Systems - Signal Flow Graph and Block Diagram Question 1 English Option 3
D
GATE ECE 2025 Control Systems - Signal Flow Graph and Block Diagram Question 1 English Option 4
4
GATE ECE 2025
MCQ (Single Correct Answer)
+1
-0.33
A 3-input majority logic gate has inputs $X, Y$ and $Z$. The output $F$ of the gate is logic ' 1 ' if two or more of the inputs are logic ' 1 '. The output $F$ is logic ' 0 ' if two or more of the inputs are logic '0'. Which one of the following options is a Boolean expression of the output $F$ ?
A
$X Y+Y Z+Z X$
B
$X \oplus Y \oplus Z$
C
$X+Y+Z$
D
$X Y Z$
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