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

Find the vertical asymptotes of \[ f(x)=\frac{-8}{x^2-4}. \] 

A liquid form of penicillin manufactured by a pharmaceutical company is sold in bulk at a price of \(\$200\) per unit. If the total production cost (in dollars) for \(x\) units is \[ C(x)=500{,}000+80x+0.003x^2, \] and the production capacity of the firm is at most \(30{,}000\) units in a specified time, then how many units of penicillin must be manufactured and sold during that specified time to maximize the profit? 

Let \[ S=\{(2,5),\ (3,7),\ (4,-9),\ (-8,3)\}. \] Then:

Let \[ \sum_{n=1}^{\infty} a_n = \sum_{n=1}^{\infty} (-1)^n \frac{\sin(nx)}{n^2} \] be a series. Then: 

Find the vertical asymptotes of \[ f(x)=\frac{-8}{x^2-4}. \]

The following table gives the distribution of 100 families according to their daily expenditure: 

Expenditure (in Rs)0–1010–2020–3030–4040–50
Number of families14\(x\)27\(y\)15

If the mode of the distribution is \(24\), find the missing frequencies \(x\) and \(y\).

If \(A\) and \(B\) are two events such that \[ P(A \cup B)=\frac{5}{6}, \qquad P(A \cap B)=\frac{1}{3}, \qquad P(B')=\frac{1}{2}, \] then the events \(A\) and \(B\) are:
If \(X\) is a random variable with probability density function \[ f(x)=\frac{1}{\sqrt{12\pi}}\,e^{-\frac{(x-1)^2}{12}}, \qquad 0<x<\infty, \] then the value of \[ \int_{1}^{\infty} f(x)\,dx \] is:
The line \(y = L\) is a horizontal asymptote of the graph of the function \(f\) if
Find the interval in which the graph of the function given by \(f(x) = x^4 - 4x^3 + 10\) is concave up.
Find the value of \(\int_{-1}^2 |x| dx\)
If the interval of differencing being unity and \(\Delta\) be the forward difference operator, then the value of \(\Delta\left(\frac{x^2}{x^2}\right)\) is given by
Evaluate the integral \(I = \int_0^1 \frac{1}{1+x} dx\) by using Simpson's one-third rule with \(h = 0.5\)
Find constants a and b so that \(\lim_{x \to 0} \left( \frac{a}{2x} + \frac{a}{2} + b \right) = 1\)
If \(A\) is any real number and \(r\) is positive rational number, then the value of \(\lim_{x \to +\infty} \frac{x^r}{e^{Ax}}\) is given by
The formula to solve simple nonlinear equation \(f(x) = 0\) by Newton's Raphson Method is given by
Find the interval in which the graph of the function given by \(f(x) = x^4 - 4x^3 + 10\) is concave up.
The value of \(\int_0^\infty x^7 e^{-x^2} dx\) is
The line \(y = L\) is a horizontal asymptote of the graph of the function \(f\) if
The point at which the tangent to the curve \(y = x^3 - 3x^2 - 9x + 7\) is parallel to the x-axis is
If the interval of differencing being unity and \(\Delta\) be the forward difference operator, then the value of \(\Delta\left(\frac{x^2}{x^2}\right)\) is given by
The formula to solve simple nonlinear equation \(f(x) = 0\) by Newton's Raphson Method is given by
Find the value of \(\int_{-1}^2 |x| dx\)
Evaluate the integral \(I = \int_0^1 \frac{1}{1+x} dx\) by using Simpson's one-third rule with \(h = 0.5\)
Find constants a and b so that \(\lim_{x \to 0} \left( \frac{a}{2x} + \frac{a}{2} + b \right) = 1\)