Calculus · Discrete Mathematics · GATE CSE

For a real number $a$, let $I(a)=\int\limits_{-1}^1\left(3 x^2-a x+1\right) d x$. Which of the following statements is/are true?

Consider the function $f: \mathbb{R} \rightarrow \mathbb{R}$ defined as follows:

$$ f(x)=\left\{\begin{array}{cc} c_1 e^x-c_2 \log _e\left(\frac{1}{x}\right), & \text { if } x>0 \\ 3, & \text { otherwise } \end{array}\right. $$

where $c_1, c_2 \in \mathbb{R}$.

If $f$ is continuous at $x=0$, then $c_1+c_2=$ $\_\_\_\_$ . (answer in integer)

The value of $x$ such that $x>1$, satisfying the equation $\int_1^x t \ln t d t=\frac{1}{4}$ is

Consider the given function $f(x)$.

$$f(x)=\left\{\begin{array}{cc} a x+b & \text { for } x<1 \\ x^3+x^2+1 & \text { for } x \geq 1 \end{array}\right.$$

If the function is differentiable everywhere, the value of $b$ must be _________ (Rounded off to one decimal place)

Let $f(x)$ be a continuous function from $\mathbb{R}$ to $\mathbb{R}$ such that

$f(x) = 1 - f(2 - x)$

Which one of the following options is the CORRECT value of $\int_0^2 f(x) dx$?

Let $f : \mathbb{R} \rightarrow \mathbb{R}$ be a function such that $f(x) = \max \{x, x^3\}, x \in \mathbb{R}$, where $\mathbb{R}$ is the set of all real numbers. The set of all points where $f(x)$ is NOT differentiable is

Let $$f(x) = {x^3} + 15{x^2} - 33x - 36$$ be a real-valued function. Which of the following statements is/are TRUE?

The value of the definite integral

$$\int\limits_{ - 3}^3 {\int\limits_{ - 2}^2 {\int\limits_{ - 1}^1 {(4{x^2}y - {z^3})dz\,dy\,dx} } } $$

is ___________. (Rounded off to the nearest integer)

The value of the following limit is _____________.

$$\mathop {\lim }\limits_{x \to {0^ + }} {{\sqrt x } \over {1 - {e^{2\sqrt x }}}}$$

Consider the following expression

$$\mathop {\lim }\limits_{x \to -3} \frac{{\sqrt {2x + 22} - 4}}{{x + 3}}$$

The value of the above expression (rounded to 2 decimal places) is ______

Where $$\theta \in \left[ {{\pi \over 6},{\pi \over 3}} \right]$$ and $$f\left( \theta \right)$$ denote the derivative of $$f$$ with repect to $$\theta $$. Which of the following statements is/are TRUE?

$${\rm I})$$ There exists $$\theta \in \left[ {{\pi \over 6},{\pi \over 3}} \right]$$ such that $$f\left( \theta \right)$$ $$= 0$$.
$${\rm I}{\rm I})$$ There exists $$\theta \in \left[ {{\pi \over 6},{\pi \over 3}} \right]$$ such that $$f\left( \theta \right)$$ $$ \ne 0$$.

The value of $$\int\limits_0^3 {f\left( x \right)dx} $$ computed using the trapezoidal rule is

$$P.\,\,f\left( x \right)$$ is continuous for all real values of $$x$$
$$Q.\,\,f\left( x \right)$$ is differentiable for all real values of $$x$$

Which of the following is True?

Consider a function $f:(0,1) \rightarrow\{0,1\}$ defined as follows.

For a real number $r \in(0,1), f(r)=1$ if the second digit after the decimal point in $r$ is one of the four digits $2,3,6$ and 7 . Otherwise, $f(r)$ is equal to 0 .

The number of points in $(0,1)$ at which $f$ is discontinuous is $\_\_\_\_$ . (answer in integer)

Let $f: \mathbb{R} \rightarrow \mathbb{R}$ be defined as follows:

$$ f(x)=\left(\frac{|x|}{2}-x\right)\left(x-\frac{|x|}{2}\right) $$

Which of the following statements is/are true?

(b) Integrate $$\,\,\,\int\limits_{ - \pi }^\pi {x\,\cos \,x\,dx} $$