List of top Engineering Mathematics Questions asked in GATE Environmental Science and Engineering

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)