Recent questions tagged goclasses-da-dpp

The value of the given integral is : $$I=\int_0^{\pi/2}\left(\frac{4\sqrt{\sin x}+7\sqrt{\cos x}}{\sqrt{\sin x}+\sqrt{\cos x}}+\sin x\cos x\right)dx$$$\frac{11\pi}{4}+\fr...

Evaluate the limit $\lim_{x\to-\infty}(\sqrt{9x^2+6x+5}+3x)$.$-1$ $0$ $1$ Limit does not exist

Consider a differentiable function $f$ whose derivative is given by $f'(x)=e^{-x}(x+2)^2(x-1)(x-3)^3$. Which of the following statements are true?There is a local maximum...

Let $f$ be not defined at $x=2$. For $x\ne 2$, suppose $f(x)=\frac{x^2+ax+b}{x-2}$. Which condition on $a$ and $b$ makes $\lim_{x\to 2}f(x)$ exist as a finite real number...

Let $f:\mathbb{R}\to\mathbb{R}$ be differentiable at $x=x^*$. If $f'(x^*)=0$, which of the following statements must be true?$f$ is continuous at $x=x^*$ The tangent line...

Find the minimum value of $f(x,y)=xy+x+y$ subject to $x^2+y^2=1$.$-1$ $-\sqrt{2}$ $-1/2$ $-2$

Choose the largest interval of $x$ on which the function $f(x)=x^2\ln x$, where $x>0$, is convex.$(0,e^{-3/2})$ $[e^{-3/2},\infty)$ $(0,\infty)$ $[e^{-1},\infty)$

A company splits $12$ research hours between two tasks. If $x$ hours are spent on task A, then $12-x$ hours are spent on task B. The total difficulty score is $D(x)=x^2+(...

A research team has $60$ hours to divide between technical validation and market validation. If $x$ hours are spent on technical validation, then $60-x$ hours are spent o...

A company models its total production cost by $C(x)=x^2+\frac{16}{x}$, where $x>0$ is the number of units produced in hundreds. Find the minimum possible value of $C(x)$.

A machine produces a random score $T$ between $0$ and $2$ with probability density function $$f(t)= \begin{cases}\frac{3t^2}{8}, & \text { if } 0<t<2 \\\\ 0, & \text { ot...

Let $Y$ be a continuous random variable with density,$$f_Y(y)= \begin{cases}2y, & \text { if } 0<y<1 \\\\ 0, & \text { otherwise }\end{cases}$$Given $Y=y$, the conditiona...

Let $X$ be a continuous random variable on $[0,1]$ with probability density function $f(x)=3x^2$ for $0\le x\le 1$ and $0$ otherwise. Let $Y=\log_e(1+X^3)$. Find the expe...

Let $X$ and $Y$ be independent standard normal random variables. Define $A=X+Y$ and $B=X-Y$. Find $P(|A|<B)$.$1/4$ $1/2$ $3/4$ $1/3$

Let $X$ be an exponentially distributed random variable with mean $1/\ln 3$. Define $Y=\lfloor X \rfloor$, where $\lfloor X \rfloor$ is the greatest integer less than or ...

A biased coin is flipped $6$ times, and the probability of getting a head on a single flip is $1/3$. Given that the total number of heads is even, what is the probability...

In the anime Jujutsu Kaisen, Kinji Hakari has a Domain Expansion called Idle Death Gamble. His domain works like a gambling machine. It does not directly attack the oppon...

Suppose $X,Y,Z,W$ are independent random variables taking values $0$ or $1$ with equal probability $1/2$. Find $P(X+Y+Z+W=2 \mid X=Y \text{ or } Z=W)$.$1/2$ $1/3$ $1/6$ $...

A game is played with a weighted coin that lands heads with probability $3/5$ and tails with probability $2/5$. The game has normal rounds and bonus rounds. The first rou...

Let $X_1,X_2,X_3$ be independent Bernoulli random variables with parameter $p$. Find $E[X_1^2X_2(2X_1+X_3)]$.$2p^2+p^3$ $3p^3$ $2p^3+p^2$ $p^2+p^3$

A website assigns each new visitor one of $6$ equally likely security icons, independently of all previous visitors. The administrator watches the sequence of assigned ic...

A random variable $X$ has probability mass function given by :$$p_X(x)=\begin{cases} c, & \text{if } x=-1 \\\\ 2c, & \text{if } x=1 \\\\ \frac{1}{6}, & \text{if } x=4 \\\...

A delivery worker is assigned one of three types of delivery routes by drawing a card from a box containing $6$ cards numbered $1$ through $6$.If the card is $1$ or $2$, ...

Suppose that a person plays $12$ independent games. In each game, he wins with probability $\frac{1}{3}$ and loses otherwise. If he wins a game, he earns $9$ dollars, and...

Suppose you keep tossing a fair coin until you get the first head. Let $Y$ be the number of tosses until the first head appears. Find the probability that $Y$ is odd, giv...

Suppose that $X$ and $Y$ are independent random variables, such that :$P(X=x)=\frac{1}{4}\left(\frac{3}{4}\right)^{x-1}$ for integers $x\ge 1$, and $P(Y=y)=\frac{1}{2}\le...

Let $X_1,X_2,\ldots,X_n$ be independent discrete uniform random variables on ${1,2,\ldots,n}$. That is, $P(X_i=j)=\frac{1}{n}$ for each $i$ and each $j\in{1,2,\ldots,n}$....

Suppose that $X$, $Y$, and $Z$ are independent random variables such that : $P(X=x)=\frac{1}{2}\left(\frac{1}{2}\right)^{x-1}$ for integers $x\ge 1$,  $P(Y=y)=\frac{1}{3}...

Let $X_1,X_2,X_3$ be independent continuous uniform random variables on $[0,1]$. Let $k\in[0,1]$ be a constant. Find $P(\max(X_1,X_2,X_3)>k)$.$1-k$ $1-k^2$ $1-k^3$ $(1-k)...

Let $X$ be a random variable with PDF :$$f(x)= \begin{cases}C x^2(1-x)^3 & 0 <x<1\\\\ 0 & \text{otherwise} \end{cases}$$What is $C$?