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:

The value of \(\lim_{x \to +\infty}\sqrt[3]{\frac{3x+5}{6x-8}}\) is given by:

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 :

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

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:

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:

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

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

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

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

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

In actual machines ____________.

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:

The average variable cost is the result of ____________.

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:

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.

One dimensional flow is :

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