Recent questions tagged goclasses-da-dpp

Let $$f(x)=\frac{(x-3)(x^2+1)}{|x-3|} \text{ for } x\ne 3$$ If $f$ is extended to be continuous at $x=3$, what should be the value of $f(3)$?$10$ $-10$ $0$ No such value ...

The function $f(x)$ is defined by $$f(x)=\begin{cases}\dfrac{x^2+ax+b}{\sqrt{x+3}},&x\le 1\\\\ \ln(2x-1)+\dfrac{5x}{x+1},&x>1\end{cases}$$ where $a$ and $b$ are constants...

Consider the piecewise function $$f(x)=\begin{cases}\dfrac{e^{Cx}-1+4x}{\sin(2x)},&x<0\\\\ \dfrac{(C^2-1)x^2+7x+3}{2x^2+\sqrt{x^4+1}},&x\ge 0\end{cases}$$where $C$ is a c...

Suppose $p(x)$ is twice differentiable for all real values of $x$. Some values of $p'(x)$ and $p''(x)$ are given in the table below:$$\begin{array}{|c|c|c|c|c|c|c|c|c|c|}...

A portion of the graph of an even function $f(x)$ is shown only on the interval $[0,6]$, although $f(x)$ is defined for all real numbers. The graph has a vertical asympto...

Evaluate the following limit and find the value of $a$:$$\lim_{x\to 0}\frac{\sqrt{1+4x+7x^2}-\sqrt[3]{1+6x+9x^2}+x-\ln(1+x)}{x^2}=\frac{a}{6}$$What is the value of $a$?

Let $f$ be a continuous function defined for all real numbers. Suppose :$$\int_0^3 f(x)dx=4, \qquad\int_0^6 f(x)dx=9, \qquad\int_6^{10} f(x)dx=-2, \qquad \text{and} \qqua...

Evaluate the definite integral:$$I=\int_0^\pi x\sin x\ln(1+\cos^2 x),dx$$Which of the following is the correct value?$\frac{\pi^2}{2}+\pi\ln 2-2\pi$ $\frac{\pi^2}{4}+\pi\...

Determine the following limit:$$L=\lim_{x\to\infty}x^2\left(\sqrt{9x^2+6x+5}-3x-1-\frac{2}{3x}\right)$$Which of the following is the correct value of the limit?$0$ $\frac...

Suppose that a function $f$ is defined by $$f(x)=\begin{cases}\frac{x^2-2x-8}{x-4}+3, & x\lt 4 \\\\ 25, & x=4 \\\\ \frac{\sqrt{x+5}-3}{x-4}, & x\gt 4\end{cases}$$Let $M=\...

Let $f(x)$ be continuous on $[1,3]$ and differentiable on $(1,3)$. If $f(1)=5$ and $2x+1\le f'(x)\le 4x+3$ for all $x\in(1,3)$, then the smallest and largest possible val...

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...