List of top Engineering Mathematics Questions asked in GATE AE

The system of equations \[ \begin{cases} x-2y+\alpha z=0,
2x+y-4z=0,
x-y+z=0 \end{cases} \] has a non-trivial solution for \(\alpha=\;\underline{\hspace{1cm}}\). \;(Answer in integer)

The direction in which a scalar field \(\phi(x,y,z)\) has the largest rate of change is along:

Consider the one–dimensional wave (advection) equation \[ \frac{\partial u}{\partial t}+\frac{\partial u}{\partial x}=0, -\infty<x<\infty,\ t\ge 0. \] For the initial condition \(u(x,0)=e^{-x^{2}}\), the solution at \(t=1\) is:

If a monotonic and continuous function \(y=f(x)\) has exactly one root in the interval \(x_1<x<x_2\), then:

Let $x$ be a real number and $i=\sqrt{-1}$. Then the real part of $\cos(ix)$ is

The vector field \( \vec{v} = x^3 \hat{i} + y^3 \hat{j} + z^3 \hat{k} \) is a vector field where \( \hat{i}, \hat{j}, \hat{k} \) are the base vectors of a Cartesian coordinate system.
Using the Gauss divergence theorem, the value of the outward flux of the vector field over the surface of a sphere of unit radius centered at the origin is _________ (rounded off to one decimal place).

The largest eigenvalue of the given matrix is: \[ \begin{bmatrix} 0 & 1 & 1 \\ 1 & 0 & 1 \\ 1 & 1 & 0 \\ \end{bmatrix} \]

The real function \( y = \sin^2(|x|) \) is

Consider the differential equation \(\frac{d^2y}{dx^2} - 2 \frac{dy}{dx} + y = 0\). The boundary conditions are \(y = 0\) and \(\frac{dy}{dx} = 1\) at \(x = 0\). Then the value of \(y\) at \(x = \frac{1}{2}\) is

Consider the partial differential equation \(\frac{\partial^2 f}{\partial x^2} + \frac{\partial^2 f}{\partial y^2} = 0\) where \(x, y\) are real. If \(f(x, y) = a(x)b(y)\), where \(a(x)\) and \(b(y)\) are real functions, which one of the following statements can be true?

The height of a right circular cone of maximum volume that can be enclosed within a hollow sphere of radius \(R\) is

The arc length of the parametric curve: \( x=\cos\theta,\; y=\sin\theta,\; z=\theta \) from \( \theta=0 \) to \( \theta=2\pi \) is equal to _________ (round off to one decimal place).

If $\hat{a}, \hat{b}, \hat{c}$ are three mutually perpendicular unit vectors, then $\hat{a} \cdot (\hat{b} \times \hat{c})$ can take the value(s):

Let \(\hat{i}, \hat{j}, \hat{k}\) be the unit vectors in the x, y, and z directions, respectively. If the vector \(\hat{i} + \hat{j}\) is rotated about positive \(\hat{k}\) by \(135^\circ\), one gets

The equation of the straight line representing the tangent to the curve \(y = x^2\) at the point (1, 1) is