GATE CS Engineering Mathematics questions

A point is randomly selected with uniform probability in the $X-Y$ plane within the rectangle with corners at $(0,0), (1,0), (1,2)$ and $(0,2).$ If $p$ is the length of the position vector of the point, the expected value of $p^{2}$ is

How many graphs on $n$ labeled vertices exist which have at least $\frac{(n^2 - 3n)}{ 2}$ edges ?

Two $n$ bit binary strings, $S_1$ and $S_2$ are chosen randomly with uniform probability. The probability that the Hamming distance between these strings (the number of bit positions where the two strings differ) is equal to $d$ is

The minimum number of colours required to colour the following graph, such that no two adjacent vertices are assigned the same color, is

In an $M \times N$ matrix all non-zero entries are covered in $a$ rows and $b$ columns. Then the maximum number of non-zero entries, such that no two are on the same row or column, is

Mala has the colouring book in which each English letter is drawn two times. She wants to paint each of these $52$ prints with one of $k$ colours, such that the colour pairs used to colour any two letters are different. Both prints of a letter can also be coloured with the same colour. What is the minimum value of $k$ that satisfies this requirement?

An examination paper has $150$ multiple choice questions of one mark each, with each question having four choices. Each incorrect answer fetches $-0.25$ marks. Suppose $1000$ students choose all their answers randomly with uniform probability. The sum total of the expected marks obtained by all these students is

The inclusion of which of the following sets into $S = \left\{ \left\{1, 2\right\}, \left\{1, 2, 3\right\}, \left\{1, 3, 5\right\}, \left\{1, 2, 4\right\}, \left\{1, 2, 3, 4, 5\right\} \right\}$ is necessary and sufficient to make $S$ a complete lattice under the partial order defined by set containment?

The following is the incomplete operation table of a $4-$element group. …

How many solutions does the following system of linear equations have? $-x + 5y = -1$ $x - y = 2$ $x + 3y = 3$

The following propositional statement is $\left(P \implies \left(Q \vee R\right)\right) \implies \left(\left(P \wedge Q \right)\implies R\right)$

Let $A, B, C, D$ be $n \times n$ matrices, each with non-zero determinant. If $ABCD = I$, then $B^{-1}$ is

The number of different $n \times n$ symmetric matrices with each element being either 0 or 1 is: (Note: $\text{power} \left(2, X\right)$ is same as $2^X$)

If a fair coin is tossed four times. What is the probability that two heads and two tails will result?

Consider the binary relation: $S= \left\{\left(x, y\right) \mid y=x+1 \text{ and } x, y \in \left\{0, 1, 2\right\} \right\}$ The reflexive transitive closure is $S$ is

Identify the correct translation into logical notation of the following assertion. Some boys in the class are taller than all the girls Note: $\text{taller} (x, y)$ is true if $x$ is taller than $y$.

The following resolution rule is used in logic programming. Derive clause $(P \vee Q)$ from clauses $(P\vee R),(Q \vee ¬R)$ Which of the following statements related to this rule is FALSE?

A program consists of two modules executed sequentially. Let $f_1(t)$ and $f_2(t)$ respectively denote the probability density functions of time taken to execute the two modules. The probability density function of the overall time taken to execute the program is given by

Consider the following system of linear equations $$\left( \begin{array}{ccc} 2 & 1 & -4 \\ 4 & 3 & -12 \\ 1 & 2 & -8 \end{array} \right) \left( \begin{array}{ccc} x \\ y \\ z \end{array} \right) = \left( \begin{array}{ccc} \alpha \\ 5 \\ 7 \end{array} \right)$$ Notice that the second and the third columns of the coefficient matrix …

A graph $G=(V,E)$ satisfies $\mid E \mid \leq 3 \mid V \mid - 6$. The min-degree of $G$ is defined as $\min_{v\in V}\left\{ \text{degree }(v)\right \}$. Therefore, min-degree of $G$ cannot be