GATE CS Engineering Mathematics questions

Consider an undirected graph $G$ where self-loops are not allowed. The vertex set of $G$ is $\{(i,j) \mid1 \leq i \leq 12, 1 \leq j \leq 12\}$. There is an edge between $(a,b)$ and $(c,d)$ if $|a-c| \leq 1$ and $|b-d| \leq 1$. The number of edges in this graph is______.

Let ܵ$S$ denote the set of all functions $f:\{0,1\}^4 \to \{0,1\}$. Denote by $N$ the number of functions from S to the set $\{0,1\}$. The value of $\log_2 \log_2N$ is _______.

A pennant is a sequence of numbers, each number being $1$ or $2$. An $n-$pennant is a sequence of numbers with sum equal to $n$. For example, $(1,1,2)$ is a $4-$pennant. The set of all possible $1-$pennants is ${(1)}$, the set of all possible $2-$pennants is ${(2), (1,1)}$ and the set of all $3-$pennants is ${(2,1), (1,1,1), (1,2)}$. …

Four fair six-sided dice are rolled. The probability that the sum of the results being $22$ is $\dfrac{X}{1296}$. The value of $X$ is _______

A function $f(x)$ is continuous in the interval $[0,2]$. It is known that $f(0) = f(2) = -1$ and $f(1) = 1$. Which one of the following statements must be true?

The function $f(x) =x \sin x$ satisfies the following equation: $$f''(x) + f(x) +t \cos x = 0$$The value of $t$ is______.

Let the function $$f(\theta) = \begin{vmatrix} \sin\theta & \cos\theta & \tan\theta \\ \sin(\frac{\pi}{6}) & \cos(\frac{\pi}{6}) & \tan(\frac{\pi}{6}) & \\ \sin(\frac{\pi}{3}) & \cos(\frac{\pi}{3}) & \tan(\frac{\pi}{3}) \end{vmatrix}$$ where $\theta \in \left[ \frac{\pi}{6},\frac{\pi}{3} \right]$ and $f'(\theta)$ denote the …

The value of the dot product of the eigenvectors corresponding to any pair of different eigenvalues of a $4-by-4$ symmetric positive definite matrix is ___________

Consider the following system of equations: $3x + 2y = 1$ $4x + 7z = 1$ $x + y + z = 3$ $x - 2y + 7z = 0$ The number of solutions for this system is ______________

Suppose you break a stick of unit length at a point chosen uniformly at random. Then the expected length of the shorter stick is ________ .

Consider the statement "Not all that glitters is gold” Predicate glitters$(x)$ is true if $x$ glitters and predicate gold$(x)$ is true if $x$ is gold. Which one of the following logical formulae represents the above statement?

Which one of the following Boolean expressions is NOT a tautology?

A cycle on $n$ vertices is isomorphic to its complement. The value of $n$ is _____.

Consider the following relation on subsets of the set $S$ of integers between $1$ and $2014$. For two distinct subsets $U$ and $V$ of $S$ we say $U\:<\:V$ if the minimum element in the symmetric difference of the two sets is in $U$. Consider the following two statements: $S1$: There is a subset of $S$ that is larger than every other …

The number of distinct positive integral factors of $2014$ is _____________

The probability that a given positive integer lying between $1$ and $100$ (both inclusive) is NOT divisible by $2$, $3$ or $5$ is ______ .

The product of the non-zero eigenvalues of the matrix is ____ $\begin{pmatrix} 1 & 0 & 0 & 0 & 1 \\ 0 & 1 & 1 & 1 & 0 \\ 0 & 1 & 1 & 1 & 0 \\ 0 & 1 & 1 & 1 & 0 \\ 1 & 0 & 0 & 0 & 1 \end{pmatrix}$

A non-zero polynomial $f(x)$ of degree 3 has roots at $x=1$, $x=2$ and $x=3$. Which one of the following must be TRUE?

If the matrix $A$ is such that $$A= \begin{bmatrix} 2\\ −4\\7\end{bmatrix}\begin{bmatrix}1& 9& 5\end{bmatrix}$$ then the determinant of $A$ is equal to ______.

The maximum number of edges in a bipartite graph on $12$ vertices is____