Monotonicity

Consider the function $f:(0,\infty)\to(-\infty,\infty)$ given by
$$f(x)=\sqrt x\,\log_e(x)-x+1.$$
Then which one of the following statements is TRUE ?

Let $y=f(x)$ be the real valued function defined on the interval $(0,\infty)$, satisfying $y(1)=0$ and the differential equation
$$x\frac{dy}{dx}=y-x^3.$$
Then which of the following statements is (are) TRUE ?

Consider the system of linear equations in $x,y,z$:
$$x+2y+tz=0,\qquad 6x+y+5tz=0,\qquad 3x+t^2y+f(t)z=0,$$
where $f:\mathbb R\to\mathbb R$ is a differentiable function. If this system has infinitely many solutions for all $t\in\mathbb R$, then $f$

For the function $f(x)=e^{\sin|x|}-|x|$, $x\in\mathbf R$, consider the following statements :

Statement I : $f$ is differentiable for all $x\in\mathbf R$.

Statement II : $f$ is increasing in $\left(-\pi,-\dfrac\pi2\right)$.

In the light of the above statements, choose the correct answer from the options given below :

Let $\mathbb{R}$ denote the set of all real numbers. Define the function $f : \mathbb{R} \to \mathbb{R}$ by

$$f(x) = \begin{cases} 2 - 2x^2 - x^2\sin\dfrac{1}{x} & \text{if } x \neq 0,\\ 2 & \text{if } x = 0. \end{cases}$$

Then which one of the following statements is TRUE?

Let $\mathbb{R}$ denote the set of all real numbers. For a real number $x$, let $[x]$ denote the greatest integer less than or equal to $x$. Let $n$ denote a natural number.

Match each entry in List-I to the correct entry in List-II and choose the correct option.

Let the function $f : \mathbb{R} \to \mathbb{R}$ be defined by

$$f(x) = \frac{\sin x}{e^{\pi x}}\cdot\frac{\left(x^{2023} + 2024x + 2025\right)}{\left(x^2 - x + 3\right)} + \frac{2}{e^{\pi x}}\cdot\frac{\left(x^{2023} + 2024x + 2025\right)}{\left(x^2 - x + 3\right)}.$$

Then the number of solutions of $f(x) = 0$ in $\mathbb{R}$ is _____.

Let $S$ be the set of all twice differentiable functions $f$ from $\mathbb{R}$ to $\mathbb{R}$ such that $\dfrac{d^2f}{dx^2}(x) > 0$ for all $x \in (-1,1)$. For $f \in S$, let $X_f$ be the number of points $x \in (-1,1)$ for which $f(x) = x$. Then which of the following statements is(are) true?

For $x \in \mathbb{R}$, let the function $y(x)$ be the solution of the differential equation
$$\frac{dy}{dx} + 12y = \cos\left(\frac{\pi}{12}x\right), \qquad y(0) = 0.$$
Then, which of the following statements is/are TRUE?

For the function
$$f(x)=4\log_e(x-1)-2x^2+4x+5,\quad x>1,$$
which one of the following is NOT correct ?

The number of distinct real roots of the equation $x^7-7x-2=0$ is

Let $\lambda^*$ be the largest value of $\lambda$ for which the function $f_\lambda(x)=4\lambda x^3-36\lambda x^2+36x+48$ is increasing for all $x\in\mathbb R$. Then $f_{\lambda^*}(1)+f_{\lambda^*}(-1)$ is equal to :

Let $g:(0,\infty)\to\mathbb R$ be a differentiable function such that
$$\int\left(\frac{x(\cos x-\sin x)}{e^x+1}+\frac{g(x)\left(e^x+1-xe^x\right)}{\left(e^x+1\right)^2}\right)dx=\frac{x\,g(x)}{e^x+1}+c,$$
for all $x>0$, where $c$ is an arbitrary constant. Then :

Let $f:\mathbb R\to\mathbb R$ and $g:\mathbb R\to\mathbb R$ be two functions defined by $f(x)=\log_e\left(x^2+1\right)-e^{-x}+1$ and $g(x)=\dfrac{1-2e^{2x}}{e^x}$. Then, for which of the following range of $\alpha$, the inequality
$$f\left(g\left(\frac{(\alpha-1)^2}3\right)\right)>f\left(g\left(\alpha-\frac53\right)\right)$$
holds ?

Let $f(x)=2\cos^{-1}x+4\cot^{-1}x-3x^2-2x+10$, $x\in[-1,1]$. If $[a,b]$ is the range of the function $f$, then $4a-b$ is equal to :

The number of distinct real roots of $x^4-4x+1=0$ is :

The number of real solutions of $x^7+5x^3+3x+1=0$ is equal to ______.

The number of solutions of the equation $2\theta-\cos^2\theta+\sqrt2=0$ in $\mathbf R$ is equal to ______.

Let $f(x)=4x^3-11x^2+8x-5$, $x\in\mathbf R$. Then $f$ :

If the absolute maximum value of the function $f(x)=\left(x^2-2x+7\right)e^{\left(4x^3-12x^2-180x+31\right)}$ in the interval $[-3,0]$ is $f(\alpha)$, then :