Engineering Mathematics — GATE Previous Year Questions

What is the value of \(\displaystyle\lim_{n \to \infty}\left(1 - \frac{1}{n}\right)^{2n}\) ?

Each of the nine words in the sentence “The quick brown fox jumps over the lazy dog” is written on a separate piece of paper. These nine pieces of paper are kept in a box. One of the pieces is drawn at random from the box. The expected length of the word drawn is ______. (The answer should be rounded to one decimal place.)

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

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 ______.

The larger of the two eigenvalues of the matrix \(\begin{bmatrix} 4 & 5 \\ 2 & 1 \end{bmatrix}\) is ______.

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

Two eigenvalues of a \(3\times 3\) real matrix \(P\) are \((2+\sqrt{-1})\) and \(3\). The determinant of \(P\) is ______.

Consider the following experiment.

Step 1. Flip a fair coin twice.
Step 2. If the outcomes are (TAILS, HEADS) then output Y and stop.
Step 3. If the outcomes are either (HEADS, HEADS) or (HEADS, TAILS), then output N and stop.
Step 4. If the outcomes are (TAILS, TAILS), then go to Step 1.

The probability that the output of the experiment is Y is (up to two decimal places) ______.

If \(f(x) = R\sin\left(\dfrac{\pi x}{2}\right) + S\), \(f'\left(\dfrac{1}{2}\right) = \sqrt{2}\) and \(\displaystyle\int_{0}^{1} f(x)\,dx = \dfrac{2R}{\pi}\), then the constants \(R\) and \(S\) are respectively

If the characteristic polynomial of a \(3\times 3\) matrix \(M\) over \(\mathbb{R}\) (the set of real numbers) is \(\lambda^{3} - 4\lambda^{2} + a\lambda + 30\), \(a \in \mathbb{R}\), and one eigenvalue of \(M\) is \(2\), then the largest among the absolute values of the eigenvalues of \(M\) is ______.

Let \(X\) be a square matrix. Consider the following two statements on \(X\).

I. \(X\) is invertible.
II. Determinant of \(X\) is non-zero.

Which one of the following is TRUE?

Let \(A\) and \(B\) be two \(n \times n\) matrices over real numbers. Let \(\operatorname{rank}(M)\) and \(\det(M)\) denote the rank and determinant of a matrix \(M\), respectively. Consider the following statements.

I. \(\operatorname{rank}(AB) = \operatorname{rank}(A)\,\operatorname{rank}(B)\)
II. \(\det(AB) = \det(A)\,\det(B)\)
III. \(\operatorname{rank}(A+B) \le \operatorname{rank}(A) + \operatorname{rank}(B)\)
IV. \(\det(A+B) \le \det(A) + \det(B)\)

Which of the above statements are TRUE?