GATE CS Engineering Mathematics questions

Which of the following propositions is a tautology?

Let $\left(Z, *\right)$ be an algebraic structure where $Z$ is the set of integers and the operation $*$ is defined by $n*m = \max(n,m)$. Which of the following statements is true for $\left(Z, *\right)$?

The determinant of the matrix $\begin{bmatrix} 6 & -8 & 1 & 1 \\ 0 & 2 & 4 & 6 \\ 0 & 0 & 4 & 8 \\ 0 & 0 & 0 & -1 \end{bmatrix}$

The probability that it will rain today is $0.5$. The probability that it will rain tomorrow is $0.6$. The probability that it will rain either today or tomorrow is $0.7$. What is the probability that it will rain today and tomorrow?

The probability that top and bottom cards of a randomly shuffled deck are both aces is

The matrices $\begin{bmatrix} \cos\theta && -\sin\theta \\ \sin\theta && \cos\theta \end{bmatrix}$ and $\begin{bmatrix} a && 0\\ 0&& b \end{bmatrix}$ commute under multiplication

Which one of the following is false?

Which of the following is NOT True? (Read $\wedge$ as AND, $\vee$ as OR, $\neg$ as NOT, $\rightarrow$ as one way implication and $\leftrightarrow$ as two way implication)

Let $R$ be a non-empty relation on a collection of sets defined by $_{A}R_ B$ if and only if $A \cap B = \phi$. Then, (pick the true statement)

Let $R$ denote the set of real numbers. Let $f:R\times R \rightarrow R \times R$ be a bijective function defined by $f(x,y) = (x+y, x-y)$. The inverse function of $f$ is given by

Let $Ax = b$ be a system of linear equations where $A$ is an $m \times n$ matrix and $b$ is a $m \times 1$ column vector and $X$ is an $n \times1$ column vector of unknowns. Which of the following is false?

The formula used to compute an approximation for the second derivative of a function $f$ at a point $x_0$ is

Two dice are thrown simultaneously. The probability that at least one of them will have $6$ facing up is

Which of the following statements is FALSE?

Suppose $X$ and $Y$ are sets and $|X| \text{ and } |Y|$ are their respective cardinality. It is given that there are exactly $97$ functions from $X$ to $Y$. From this one can conclude that

Let $X = \{2, 3, 6, 12, 24\}$, Let $\leq$ be the partial order defined by $X \leq Y$ if $x$ divides $y$. Number of edges in the Hasse diagram of $(X, \leq)$ is

Let $A$ and $B$ be sets and let $A^c$ and $B^c$ denote the complements of the sets $A$ and $B$. The set $(A-B) \cup (B-A) \cup (A \cap B)$ is equal to

Compute without using power series expansion $\displaystyle \lim_{x \to 0} \frac{\sin x}{x}.$

Determine the number of positive integers $(\leq 720)$ which are not divisible by any of $2,3$ or $5.$

Find the minimum value of $3-4x+2x^2$.