List of top Statistical Science Questions on Calculus asked in ICAR AIEEA (PG)

A line $y = L$ is called a horizontal asymptote of the graph of a function $f(x)$ if

If $z = x^4 \sin(xy^3)$, then the value of $\frac{\partial z}{\partial x}$ at the point (0, 0) is given by}

If the equation $f(x, y) = C$ defines $y$ implicitly as a differentiable function of $x$, and if $\frac{\partial f}{\partial y} \neq 0$, then $\frac{dy}{dx}$ is equal to}

For the equation $yz - \ln(z) = x + y$, where $z$ is a function of two independent variables $x$ and $y$ and the partial derivatives exist. The value of $\frac{\partial z}{\partial x}$ is given by}

If probability of type II error of a statistical test is 0.2 then power of the test is

In a simple bar diagram, the heights of the bars are proportional to

Standard deviation of the numbers 2,2,2,2,2,2 is

Which of the following is always true?

If $Y$ follows Normal distribution with mean 4 and variance 16, then the standard Normal deviate is

To test $H_0: \mu = 0$ vs $H_1: \mu \neq 0$ when the population variance is known, which of the following test is used?

Which of following random variable is always non-negative?

For the function $f(x) = \begin{cases} \frac{\sin x}{x}, & x \neq 0 \\ h, & x = 0 \end{cases}$, the value of $h$ for which the function is continuous at $x = 0$ is:

For the function $f(x) = x^{2/3}(2x + 5)$, if:

a) $x = 0$ is a horizontal tangent
b) The curve has no asymptote
c) The curve has a cusp at $(0,0)$
d) The curve has no vertical tangent

Then:

The value of $\lim_{(x,y) \to (0,0)} \frac{x^2y}{x^2 + y^2}$ is:

The total differential $df$ for the function $f(x, y, z) = x^2 y^3 z^4$ is:

For the function $f(x, y) = x^3 + x^2 y^2$, the values of $f_x(-1, -1)$ and $f_y(-1, -1)$ are respectively:

The value of $\int \frac{2x - 9}{\sqrt{x^2 - 9x + 1}} \, dx$ is:

The value of $\int_{0}^{\frac{\pi}{2}} \sin^5 x \, dx$ is:

The value of $\int_{0}^{1} \frac{x^2}{\sqrt{1 - x^2}} \, dx$ is:

The value of $\iint_R x^2 y^5 \, dx \, dy$, where R is the rectangle with vertices (1, 0), (1, 1), (2, 1) and (2, 0) is:

The value of $\int_{0}^{4} \int_{0}^{4-x} x y \, dx \, dy$ is:

For a continuous function $f(r, \theta)$ defined on a region D in $(r, \theta)$ plane, the area of the domain D is given by:

The value of the integral $\int \frac{dx}{1 + e^x}$ is equal to:

The infinite series $3 + \frac{1}{6}x - \frac{1}{216}x^2 - \frac{1}{3888}x^3 + \cdots$ represents the Maclaurin series for the function:

First four terms of the Taylor series of the function $f(x) = x^3 + 2x^2 + x - 1$ at $c = 3$ are: