GATE CS question library

Suppose that the eigenvalues of matrix $A$ are $1, 2, 4$. The determinant of $\left(A^{-1}\right)^{T}$ is _________.

Let $X$ be the number of distinct $16$-bit integers in $2's$ complement representation. Let $Y$ be the number of distinct $16$-bit integers in sign magnitude representation Then $X - Y$ is______.

A processor has $40$ distinct instruction and $24$ general purpose registers. A $32$-bit instruction word has an opcode, two registers operands and an immediate operand. The number of bits available for the immediate operand field is_______.

Consider the systems, each consisting of $m$ linear equations in $n$ variables. If $m < n$, then all such systems have a solution. If $m > n$, then none of these systems has a solution. If $m = n$, then there exists a system which has a solution. Which one of the following is CORRECT?

In an Ethernet local area network, which one of the following statements is TRUE?

Language $L_{1}$ is defined by the grammar: $S_{1} \rightarrow a S_{1} b \mid \varepsilon$ Language $L_{2}$ is defined by the grammar: $S_{2} \rightarrow a b S_{2} \mid \varepsilon$ Consider the following statements: P: $L_{1}$ is regular Q: $L_{2}$ is regular Which one of the following is TRUE?

Suppose that a shop has an equal number of LED bulbs of two different types. The probability of an LED bulb lasting more than $100$ hours given that it is of Type $1$ is $0.7$, and given that it is of Type $2$ is $0.4$. The probability that an LED bulb chosen uniformly at random lasts more than $100$ hours is _________.

Let, $x_{1} ⊕ x_{2} ⊕ x_{3} ⊕ x_{4}= 0$ where $x_{1}, x_{2}, x_{3}, x_{4}$ are Boolean variables, and $⊕$ is the XOR operator. Which one of the following must always be TRUE?

In a $2 \times 4$ rectangle grid shown below, each cell is rectangle. How many rectangles can be observed in the grid? $$\begin{array}{|c|c|c|c|c|}\hline{\;\;\;}&{\;\;\;}&{\;\;\;}&{\;\;\;}\\ \hline{}&{}&{}&\\ \hline\end{array}$$

Among $150$ faculty members in an institute, $55$ are connected with each other through Facebook and $85$ are connected through Whatsapp. $30$ faculty members do not have Facebook or Whatsapp accounts. The numbers of faculty members connected only through Facebook accounts is _______.

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All hill-stations have a lake. Ooty has two lakes. Which of the statement(s) below is/are logically valid and can be inferred from the above sentences? Ooty is not a hill-station. No hill-station can have more than one lake.

Computers were invented for performing only high-end useful computations. However, it is no understatement that they have taken over our world today. The internet, for example, is ubiquitous. Many believe that the internet itself is an unintended consequence of the original invention. With the advent of mobile computing on our phones, a whole new dimension …

In a quadratic function, the value of the product of the roots $(\alpha, \beta)$ is $4$. Find the value of $\dfrac{\alpha^{n}+\beta^{n}}{\alpha^{-n}+\beta^{-n}}$

Nobody knows how the Indian cricket team is going to cope with the difficult and seamer-friendly wickets in Australia. Choose the option which is closest in meaning to the underlined phrase in the above sentence.

Find the odd one in the following group of words. mock, deride, praise, jeer

The man who is now Municipal Commissioner worked as ________________.

Pick the odd one from the following options.