Calculus · Discrete Mathematics · GATE CSE

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?

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