List of top Engineering Mathematics Questions asked in GATE CH

A homogeneous azeotropic distillation process separates an azeotropic AB binary feed using a heavy entrainer, E, as shown in the figure. The loss of E in the two product streams is negligible so that E circulates around the process in a closed- circuit. For a distillation column with fully specified feed(s), given operating pressure, a single distillate stream and a single bottoms stream, the steady-state degrees of freedom equals 2. For the process in the figure with a fully specified AB feed stream and given column operating pressures, the steady-state degrees of freedom equals

Consider the function \( f(x,y,z) = x^4 + 2y^3 + z^2 \). The directional derivative of the function at the point \( P(-1,1,-1) \) along \( (\hat{i} + \hat{j}) \), where \( \hat{i} \) and \( \hat{j} \) are unit vectors in the \( x \) and \( y \) directions, respectively, rounded off to 2 decimal places, is ___________

Consider the line integral \( \int_C \mathbf{F}(\mathbf{r}) \cdot d\mathbf{r} \), where \( \mathbf{F}(\mathbf{r}) = x \hat{\mathbf{i}} + y \hat{\mathbf{j}} + z \hat{\mathbf{k}} \), with \( \hat{\mathbf{i}}, \hat{\mathbf{j}}, \) and \( \hat{\mathbf{k}} \) as unit vectors in the (x, y, z) Cartesian coordinate system. The path \( C \) is given by \( \mathbf{r}(t) = \cos(t) \hat{\mathbf{i}} + \sin(t) \hat{\mathbf{j}} + t \hat{\mathbf{k}} \), where \( 0 \leq t \leq \pi \). The value of the integral, rounded off to 2 decimal places, is _____

Let \( r \) and \( \theta \) be the polar coordinates defined by \( x = r \cos \theta \) and \( y = r \sin \theta \). The area of the cardioid \( r = a (1 - \cos \theta) \), \( 0 \leq \theta \leq 2\pi \), is:

Consider the ordinary differential equation \[x^2 \frac{d^2y}{dx^2} - x \frac{dy}{dx} - 3y = 0,\]
with the boundary conditions \( y(x = 1) = 2 \) and \( y(x = 2) = \frac{17}{2} \). The solution \( y(x) \) at \( x = \frac{3}{2} \), rounded off to 2 decimal places, is __________.

Consider a matrix \( A = \begin{pmatrix} -5 & a \\ -2 & -2 \end{pmatrix} \), where \( a \) is a constant. If the eigenvalues of \( A \) are \(-1\) and \(-6\), then the value of \( a \), rounded off to the nearest integer, is ___

Consider a linear homogeneous system of equations \( Ax = 0 \), where \( A \) is an \( n \times n \) matrix, \( x \) is an \( n \times 1 \) vector, and \( 0 \) is an \( n \times 1 \) null vector. Let \( r \) be the rank of \( A \). For a non-trivial solution to exist, which of the following conditions is/are satisfied?

If \( z_1 = -1 + i \) and \( z_2 = 2i \), where \( i = \sqrt{-1} \), then \(\text{Arg}\left(\frac{z_1}{z_2}\right)\) is:

Consider the normal probability distribution function \( f(x) = \frac{4}{\sqrt{2\pi}} e^{-8(x+3)^2} \). If \( \mu \) and \( \sigma \) are the mean and standard deviation of \( f(x) \) respectively, then the ordered pair \((\mu, \sigma)\) is:

The first non-zero term in the Taylor series expansion of \((1-x)-e^{-x}\) about \( x = 0 \) is

The first non-zero term in the taylor series expansion of (1 - x) - e^{- x}, about x = 0

Consider a Leniar Homogeneous system AX = 0, where A is  N × N matrix and X is N × 1 vector. When we solve this 0 is null matrix of N × 1. Let R be the rank matrix for a for a Non-trivial solution.

\(\frac{P}{Q}\) is not equal 1, \(\left(\frac{P}{Q}\right)^{\frac{P}{Q}} = P^{\frac{P}{Q} - 1}\)

F(x) = e^x - 5x and the value of X₀ = 1 then what will be the value of X₁?

What will the correct pair of \(μ\ \text{ and}\ σ\ ?\)
\(F(x) = \frac{4}{2\sqrt{\pi}} e^{-8(x + 3)^2}\)

What will be the argument of \(\frac{Z_1}{Z_2}\)=?
\(Z_1 = -1 + i\)
\(Z_2 = 2i\)