List of top Questions asked in ICAR AIEEA (PG)- 2023

If the function \(f(x,y)\) has mixed second order partial derivatives \(f_{xy}\) and \(f_{yx}\) that are continuous in an open disk containing \((x_0,y_0)\), then:

Let f be a function such that f and all its derivatives exist in an open interval I containing c, then the Taylor’s series of f at x = c is given by :

If f is continuous on an open interval containing a value $x_0$, and if f changes the direction of its concavity at the point ($x_0, f(x_0)$), then f has :

The value of \(\int_{0}^{6} f(x)\,dx\), if \(f(x)=\begin{cases}x^2, & x < 2 \\ 3x-2, & x \ge 2\end{cases}\), is given by:

If $\Delta$ and $\nabla$ be the first forward difference and first backward difference operator respectively, then the expression $(1+\Delta)(1-\nabla)$ is equivalent to :

Suppose the function f is continuous at the point P(c, f(c)). Then, the graph of f has a vertical tangent at P if :

The solution of the differential equation \(\frac{d^2x}{dt^2}+6\frac{dx}{dt}+9x=0\) is given by:

If \(A\) is any real number and \(r\) is a positive rational number, then the value of \(\lim_{x\to +\infty}\frac{A}{x^r}\) is:

The process of evaluating a definite integral from a set of tabulated values of the integrand is called:

The formula to solve simple non-linear equation f(x) = 0 by Newton’s Raphson Method is given by :

The area under the curve y = cos x on $\left[-\frac{\pi}{2}, \frac{\pi}{2}\right]$ is given by :

If \(z=f(x,y)\), \(x=e^u+e^{-v}\), and \(y=e^{-u}-e^v\), then:

The solution of the differential equation \(xy^{\prime\prime}+y^{\prime}=4x\) is given by:

The value of \(\int \cos^{-1}(2x)\,dx\) is given by:

The value of \(\iint_R y^2x\,dy\,dx\) over the rectangle \(R=\{(x,y):-3\le x\le 2,\;0\le y\le 1\}\) is given by:

The solution of the differential equation $\cos(x+y)dx + [3y^2 + 2y + \cos(x+y)]dy = 0$ is given by :

The average variable cost is the result of ____________.

One dimensional flow is :

Given below are two statements, one is labelled as Assertion (A) and other one labelled as Reason (R).
Assertion (A) : A good measure of dispersion needs to be least affected by change in the sampling.
Reason (R) : Standard deviation is the best measure of dispersion.
In light of the above statements, choose the most appropriate answer from the options given below:

If \(\theta=t^n e^{-\frac{r^2}{4t}}\), then the value of \(n\) that will make \(\frac{1}{r^2}\frac{\partial}{\partial r}\left(r^2\frac{\partial \theta}{\partial r}\right)=\frac{\partial \theta}{\partial t}\) is given by:

Given below are two statements :
Statement (I) : The commercial unit of energy is 1 kWh.
Statement (II) : 1 kWh = \(2.6 \times 10^6\) J
In light of the above statements, choose the most appropriate answer from the options given below.

In actual machines ____________.

Given below are two statements:
Statement (I) : Differentiation and Integration are just opposite of each other.
Statement (II) : Differentiation is the slope of any curve at any point, while integration is the area under that same curve specified between two points.
In light of the above statements, choose the most appropriate answer from the options given below.

If \(2x^3 + 3x^2 + 5x + 6 = 0\) has roots \(\alpha\), \(\beta\), \(\gamma\) then, find \(\alpha + \beta + \gamma\)