List of top Engineering Mathematics Questions asked in GATE Agricultural Engineering

If \( u = x^4 + y^4 + 3x^2 y^2 \), the value of \( \left( x \frac{\partial u}{\partial x} + y \frac{\partial u}{\partial y} \right)^{-1} \) is:
The value of \( \lim_{x \to 0} \frac{x \cos x - \sin x}{x^2 \sin x} \) is:
The median of the following set of numbers: 154, 130, 144, 137, 156, 146, 138, 149, 160, 138 is:

The equation \[ y'' + p(x)y' + q(x)y = r(x) \] is a _________ ordinary differential equation.

If the matrix \[ \begin{pmatrix} x & -3 \\ 2 & x - 5 \end{pmatrix} \] is singular, sum of the values of \( x \) is:
The value of \( y \) at \( x = 0.3 \) and \( y(x = 0) = 0 \) for the differential equation \[ \frac{dy}{dx} = 3y + 2e^x \] is _________. (Rounded off to 3 decimal places)
The series \[ \sum_{n=0}^{r} q^n = 1 + q + q^2 + \cdots + q^r \] has the sum:
The Laplace transform of \( \frac{s}{s^2 + a^2} \) is:
In a locality 'A', the probability of a convective storm event is 0.7 with a density function \[ f_{X_1}(x_1) = e^{-x_1}, \; x_1 > 0 \] The probability of a tropical cyclone-induced storm in the same location is given by the density function \[ f_{X_2}(x_2) = 2 e^{-2x_2}, \; x_2 > 0 \] The probability of occurring more than 1 unit of storm event is \underline{\hspace{3cm}} (rounded off to 2 decimal places).
Given that \[ \frac{dy}{dx} = 2x + y, y(0) = 1 \] Using Runge-Kutta fourth order method, the value of \(y\) at \(x = 0.2\) is \underline{\hspace{2cm}} (rounded off to 3 decimal places).
A vector \(\vec{F} = 5\hat{i} - 10\hat{j} + 8\hat{k}\) is passing through the origin of a 3-D frame. Considering the tendency of rotation in the counter clockwise direction as positive, the moment about a point \(A: (3, 4, 8)\) is:
The probability that a storm event with a return period of 20 years will occur once in a 5-year period is \underline{\hspace{6cm}} (rounded off to 2 decimal places).
If \(A = \begin{bmatrix} 1 & -1 \\ 2 & -1 \end{bmatrix}, \; B = \begin{bmatrix} a & 1 \\ b & -1 \end{bmatrix}\) and \((A + B)^2 = A^2 + B^2\), then the values of \(a\) and \(b\) are:
The value of \[ I=\int_0^{\pi/2}\frac{(\sin x+\cos x)^2}{\sqrt{1+\sin 2x}}\,dx \] is:
If \(A\) and \(B\) are square matrices of order \(3\) such that \(|A|=-1\) and \(|B|=3\), then \(|3AB|\) equals:
\(y=ae^{mx}+be^{-mx}\) is the solution of the differential equation:
The probability of having 53 Sundays in a randomly selected leap year is __________.

The root of the equation \( \sin x - 4x + 1 = 0 \) after its first iteration, using the Newton-Raphson method with an initial guess of \( x_0 = 0.2 \), is _____.\( \textit{[Round off to three decimal places.]}\)

The slope of the function \( f(x) = 2x^4 - 3x^2 + 5x \) at \( x = 2 \) is _____. \(\textit{[Answer in integer.]}\)

The matrix \[ \begin{bmatrix} 3 - x & 2 & 2 \\ 2 & 4 - x & 1 \\ -2 & -4 & -1 - x \end{bmatrix} \] is singular for the following values of \(x\).

Work done by a moving particle in the force field \( \mathbf{F} = 6x^2 \hat{i} + (3xz + y) \hat{j} + 4z \hat{k} \), moving along the straight line from (0,0,0) to (1,2,3) is _____. \(\textit{[Answer in integer]}\)

The power consumption readings (in watt) by an instrument at fixed intervals of time (in seconds) are tabulated below:  

\[ \begin{array}{|c|c|} \hline \text{Time (s)} & \text{Power consumption (W)} \\ \hline 0.0 & 8.6 \\ 0.6 & 9.2 \\ 1.2 & 7.8 \\ 1.8 & 6.4 \\ 2.4 & 7.2 \\ 3.0 & 8.6 \\ 3.6 & 11.2 \\ \hline \end{array} \] 
Using Simpson's 1/3rd rule, the energy expenditure of the instrument in joules is _____.[Round off to two decimal places]

The function \( (x - 2)^2 (x + 2)^2 \) has
Function \( f(x) \) by Maclaurin's series (as an infinite series) can be expressed as:
Determinant of a matrix remains unaltered if __________.