Recent questions tagged tifr2012

The minimum value of $f(x)=\ln \left(1+\exp \left(x^{2}-3 x+2\right)\right)$ for $x \geq 0$, where $\ln (\cdot)$ denotes the natural logarithm, is$\ln \left(1+e^{-1 / 4}\...

Let $\alpha_{1}, \alpha_{2}, \cdots, \alpha_{k}$ be complex numbers. Then\[\lim _{n \rightarrow \infty}\left|\sum_{i=1}^{k} \alpha_{i}^{n}\right|^{1 / n}\]is$0$$\infty$$\...

A sequence of numbers $\left(x_{n}: n=1,2,3, \ldots\right)$ is said to have a limit $x$, if given any number $\epsilon>0$, there exists an integer $n_{\epsilon}$ such tha...

The signal $x_{n}=0$ for $n<0$ and $x_{n}=a^{n} / n$ ! for $n \geq 0$. Its $z$-transform $X(z)=\sum_{n=-\infty}^{\infty} x_{n} z^{-n}$ is$1 /\left(z^{-1}-a\right)$, regio...

Consider a periodic square wave $f(t)$ with a period of $1$ second such that $f(t)=1$ for $t \in[0,1 / 2)$ and $f(t)=-1$ for $t \in[1 / 2,1)$. It is passed through an ide...

Let $u(t)$ be the unit step function that takes value $1$ for $t \geq 0$ and is zero otherwise. Let $f(t)=e^{-t} u(t)$ and $g(t)=u(t) u(1-t)$. Then the convolution of $f(...

A linear time-invariant system has a transfer function $H(s)=1 /(1+s)$. If the input to the system is $\cos (t)$, the output is$\left(e^{j t}+e^{-j t}\right) / 2$ where $...

The input to a series $\text{RLC}$ circuit is a sinusoidal voltage source and the output is the current in the circuit. Which of the following is true about the magnitude...

$x(t)$ is a signal of bandwidth $4 \mathrm{~kHz}$. It was sampled at a rate of $16 \mathrm{~kHz}$.\[x_{n}=x(n T), \quad n \text { integer, } \quad T=\frac{1}{16} \mathrm{...

Suppose three dice are rolled independently. Each dice can take values $1$ to $6$ with equal probability. Find the probability that the second highest outcome equals the ...

A Poisson random variable $X$ is given by $\operatorname{Pr}\{X=k\}=\mathrm{e}^{-\lambda} \lambda^{k} / k !, k=0,1,2, \ldots$ for $\lambda>0$. The variance of $X$ scales ...

In modeling the number of health insurance claims filed by an individual during a three year period, an analyst makes a simplifying assumption that for all non-negative i...

Consider a single amoeba that at each time slot splits into two with probability $p$ or dies otherwise with probability $1-p$. This process is repeated independently infi...

Let $X$ and $Y$ be indepedent, identically distributed standard normal random variables, i.e., the probability density function of $X$ is\[f_{X}(x)=\frac{1}{\sqrt{2 \pi}}...

Consider a string of length $1 \mathrm{~m}$. Two points are chosen independently and uniformly random on it thereby dividing the string into three parts. What is the prob...

Let $P$ be a $n \times n$ matrix such that $P^{k}=\mathbf{0}$, for some $k \in \mathbb{N}$ and where $\mathbf{0}$ is an all zeros matrix. Then at least how many eigenvalu...

Let $A=U \Lambda U^{\dagger}$ be a $n \times n$ matrix, where $U U^{\dagger}=I$. Which of the following statements is TRUE.The matrix $I+A$ has non-negative eigen valuesT...

Under a certain coordinate transformation from $(x, y)$ to $(u, v)$ the circle $x^{2}+y^{2}=1$ shown below on the left side was transformed into the ellipse shown on the ...

$X$ and $Y$ are two $3$ by $3$ matrices. If\[X Y=\left(\begin{array}{rrr}1 & 3 & -2 \\-4 & 2 & 5 \\2 & -8 & -1\end{array}\right)\]then$X$ has rank $2$at least one of $X, ...

Let $A$ be a $2 \times 2$ matrix with all entries equal to $1.$ Define $B=\sum_{n=0}^{\infty} A^{n} / n !$. Then$B=e^{2} A / 2$$B=\left(\begin{array}{cc}1+e & e \\e & 1+e...