1
GATE CSE 2025 Set 2
MCQ (Single Correct Answer)
+1
-0.33

Let $L, M$, and $N$ be non-singular matrices of order 3 satisfying the equations $L^2=L^{-1}, M=L^8$ and $N=L^2$. Which ONE of the following is the value of the determinant of $(M-N)$ ?

A
0
B
1
C
2
D
3
2
GATE CSE 2025 Set 2
MCQ (Single Correct Answer)
+1
-0.33

Let $P(x)$ be an arbitrary predicate over the domain of natural numbers. Which ONE of the following statements is TRUE?

A
$(P(0) \wedge(\forall x[P(x+1)])) \Rightarrow(\forall x P(x))$
B
$(P(0) \wedge(\forall x[P(x) \Rightarrow P(x-1)])) \Rightarrow(\forall x P(x))$
C
$(P(1000) \wedge(\forall x[P(x) \Rightarrow P(x-1)])) \Rightarrow(\forall x P(x))$
D
$(P(1000) \wedge(\forall x[P(x) \Rightarrow P(x+1)])) \Rightarrow(\forall x P(x))$
3
GATE CSE 2025 Set 2
MCQ (More than One Correct Answer)
+2
-0

Let $F$ be the set of all functions from $\{1, \ldots, n\}$ to $\{0,1\}$. Define the binary relation $\preccurlyeq$ on $F$ as follows:

$\forall f . g \in F, f \preccurlyeq g$ if and only if $\forall x \in\{1, \ldots, n\}, f(x) \leq g(x)$, where $0=1$.

Which of the following statement(s) is/are TRUE? re TRUE?

A
$\preccurlyeq$ is a symmetric relation
B
$(F, \preccurlyeq)$ is a partial order
C
$(F, \preccurlyeq)$ is a lattice
D
$\preccurlyeq$ is an equivalence relation
4
GATE CSE 2025 Set 2
MCQ (More than One Correct Answer)
+2
-0

Consider a system of linear equations $P X=Q$ where $P \in \mathbb{R}^{3 \times 3}$ and $Q \in \mathbb{R}^{3 \times 3}$. Suppose $P$ has an $L U$ decomposition, $P=L U$, where

$$L=\left[\begin{array}{ccc} 1 & 0 & 0 \\ l_{21} & 1 & 0 \\ l_{31} & l_{32} & 1 \end{array}\right] \text { and } u=\left[\begin{array}{ccc} u_{11} & u_{12} & u_{13} \\ 0 & u_{22} & u_{23} \\ 0 & 0 & u_{33} \end{array}\right]$$

Which of the following statement(s) is/are TRUE?

A
The system $P X=Q$ can be solved by first solving $L Y=Q$ and then $U X=Y$.
B
If $P$ is invertible, then both $L$ and $U$ are invertible.
C
If $P$ is singular, then at least one of the diagonal elements of $U$ is zero.
D
If $P$ is symmetric, then both $L$ and $U$ are symmetric.
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