List of top Engineering Mathematics Questions asked in GATE ES

The value of $ \int_0^\infty \frac{\sin(4x)}{\pi x} \, dx $ is ________

The following is a system of linear equations
x - 2y + z = 34 (1)
2x + y + z = 102 (2)
x + y - 3z = 17 (3)
The value of $ x + y + z $ is ________. (rounded off to two decimal places)

A residential family is considering two cities for relocation. The data related to pollutant exposure and associated health cost per year are given in the following figure.

The pollutant exposure is characterized in high, mild and low exposure categories with respective probability values. The difference in expected value of health cost of City1 with respect to that of City 2 is ________ lakhs/year. (rounded off to two decimal places).

Evaluate the following limit:
\[ \lim_{x \to 0} \frac{\ln(1+x)}{2 \sin(x)} \quad \text{(rounded off to two decimal places).} \]

Consider the following two series:
\[ \textbf{P: } \sum_{n=1}^{\infty} \frac{1}{n} \qquad \textbf{Q: } \sum_{n=1}^{\infty} \frac{1}{n^2} \]
Choose the correct option from the following:

For the ordinary differential equation $ \frac{d^2y}{dx^2} + 4y = 0 $, the general solution is:

Assuming $ s>|a| $; the Laplace transform of $ f(x) = \cosh(ax) $ is:

A tank has inflow, outflow, and a stirring mechanism. Initially, the tank holds 500 L of a brine solution of concentration 200 g/L. At $ t = 0 $, an inflow of another brine solution of concentration 100 g/L starts entering the tank at the rate of 15 L/minute. At the same time, the outflow of thoroughly stirred mixture also takes place at the same rate so that the volume of brine in the tank remains constant. The brine concentration $ C $ (g/L) in the tank at any time $ t $ (minute) can be expressed by the following differential equation: \[ \frac{dC}{dt} + 0.03 C = 3 \] The brine concentration in the tank at $ t = 1.5 $ hour is ________ g/L. (rounded off to two decimal places)

Second order ordinary differential equation \[ \frac{d^2y}{dx^2} - \frac{dy}{dx} - 2y = 0 \] has values $y=2$ and $\frac{dy}{dx} = 1$ at $x=0$. The value of $y$ at $x=1$ is ............ (rounded off to three decimal places).

Consider two matrices \[ P = \begin{bmatrix} 2 & 3 \\ 1 & 4 \end{bmatrix}, Q = \begin{bmatrix} 5 & 4 \\ 0 & 2 \end{bmatrix}. \] If $R = (PQ)^T$, then $\det R$ is ............ (in integer).

For the function \[ f(x) = x \sqrt{4 - x^2}, \] the maximum value in the range $-2 \leq x \leq 2$ is ...................... (rounded off to two decimal places).

The value of \[ \frac{4}{\pi} \int_0^{\pi/2} \sin^2 x \, dx \] is ________________ (rounded off to two decimal places).

P and Q are square matrices. Consider the following:
X: $(P^{-1})^{-1} = P$
Y: Symmetric if $Q = -Q^T$
The correct choice is:

Given are two infinite series
P: $\displaystyle \sum \frac{n^2 + 1}{n^2}$
Q: $\displaystyle \sum \left(1 + \frac{1}{n}\right)^{-n}$
The correct choice is:

Given are two ordinary differential equations
P: $\dfrac{dy}{dx} + x = x \sin y$
Q: $\dfrac{dy}{dx} + x y = e^{x} y$
The correct choice is:

Evaluate the limit:

\[ \lim_{x \to 0} \frac{x^2}{\sin x} \]

= _______.

The 2×2 matrices P and Q satisfy the following relations:

The matrix Q is equal to _______.

The table below shows the carbon content of four samples of powdered coal. If these four samples are mixed completely, what would be the resultant carbon percentage of the mixture by weight?

Consider a function \[ y = f(x) \text{ which satisfies the following equation:} \] \[ \frac{d^2y}{dx^2} - \frac{dy}{dx} = 0. \] As $ x \to -\infty, y = 1 $, and at $ x = 0, y = 2 $. The value of \[ \frac{dy}{dx} \text{ at } x = 0 \text{ is } _______ \text{ (answer in integer).}

A biased die has six faces numbered as $ k = 1, 2, 3, 4, 5, 6 $. On rolling the die, the probability of the number $ k $ appearing is proportional to $ k^2 $. What is the probability that an even number will appear on rolling the die?

Consider the following equation: \[ x^3 - 10x^2 + 31x - 30 = 0 \] Which of the following is/are the root(s) of the above equation?

The ordinary differential equation \[ \frac{dy}{dx} = x^2 y \] has $ y $ as the dependent variable and $ x $ as the independent variable. Which of the following classification(s) is/are applicable to the equation?

A tangent is drawn on the curve of the function $ y = x^2 $ at the point $ (x, y) = (3, 9) $. The slope of the tangent is _______.