List of top Engineering Mathematics Questions asked in GATE Mechanical Engineering

The directional derivative of the function \( f \) given below at the point \( (1, 0) \) in the direction of \( \frac{1}{2} (\hat{i} + \sqrt{3} \hat{j}) \) is (rounded off to 1 decimal place). \[ f(x, y) = x^2 + xy^2 \]

If \( C \) is the unit circle in the complex plane with its center at the origin, then the value of \( n \) in the equation given below is (rounded off to 1 decimal place). \[ \int_C \frac{z^3}{(z^2 + 4)(z^2 - 4)} \, dz = 2 \pi i n \]

In the closed interval [0, 3], the minimum value of the function \( f \) given below is: \[ f(x) = 2x^3 - 9x^2 + 12x \]

The values of a function \( f \) obtained for different values of \( x \) are shown in the table below.

Using Simpson’s one-third rule, approximate the integral \[ \int_0^1 f(x) \, dx \quad \text{(rounded off to 2 decimal places)}. \]

Let \(A\) and \(B\) be real symmetric matrices of the same size. Which one of the following options is correct?

Let \( y \) be the solution of the differential equation with the initial conditions given below. If \( y(x = 2) = A \ln 2 \), then the value of \( A \) is ___________ (rounded off to 2 decimal places). \[ x^2 \frac{d^2y}{dx^2} + 3x \frac{dy}{dx} + y = 0 \] \[ y(x = 1) = 0, \quad \frac{dy}{dx}(x = 1) = 1 \]

The divergence of the curl of a vector field is:

For the differential equation given below, which one of the following options is correct? \[ \frac{\partial^2 u}{\partial x^2} + \frac{\partial^2 u}{\partial y^2} = 0, \quad 0 \leq x \leq 1, 0 \leq y \leq 1 \]

If two unbiased coins are tossed, then what is the probability of having at least one head?