Inequalities and modulus

If the domain of the function $f(x)=\sqrt{\log_{(0.6)}\left(\left|\dfrac{2x-5}{x^2-4}\right|\right)}$ is $(-\infty,a]\cup\{b\}\cup[c,d)\cup(e,\infty)$, then the value of $a+b+c+d+e$ is ________.

If the set of all solutions of $|x^2+x-9|=|x|+|x^2-9|$ is $[\alpha,\beta]\cup[\gamma,\infty)$, then $(\alpha^2+\beta^2+\gamma^2)$ is equal to:

The sum of absolute maximum and absolute minimum values of the function $f(x)=|2x^2+3x-2|+\sin x\cos x$ in the interval $[0,1]$ is :

Let $A=\{x\in\mathbb R:|x+1|<2\}$ and $B=\{x\in\mathbb R:|x-1|\ge2\}$. Then which one of the following statements is NOT true ?

Let $f(x)=\left|(x-1)\left(x^2-2x-3\right)\right|+x-3$, $x\in\mathbb R$. If $m$ and $M$ are respectively the number of points of local minimum and local maximum of $f$ in the interval $(0,4)$, then $m+M$ is equal to ______.

The sum of the absolute minimum and the absolute maximum values of the function $f(x)=\left|3x-x^2+2\right|-x$ in the interval $[-1,2]$ is :

Let $f(x)=\max\{|x+1|,|x+2|,\dots,|x+5|\}$. Then $\displaystyle\int_{-6}^0f(x)\,dx$ is equal to ______.

The set of values of $k$, for which the circle $C:4x^2+4y^2-12x+8y+k=0$ lies inside the fourth quadrant and the point $\left(1,-\dfrac13\right)$ lies on or inside the circle C, is

If the system of linear equations
$$2x+3y-z=-2$$
$$x+y+z=4$$
$$x-y+|\lambda|z=4\lambda-4$$
where $\lambda\in\mathbb R$, has no solution, then

The area of the bounded region enclosed by the curve $y=3-\left|x-\dfrac12\right|-|x+1|$ and the $x$-axis is :

Let $S_1=\left\{x\in\mathbf R-\{1,2\}:\dfrac{(x+2)\left(x^2+3x+5\right)}{-2+3x-x^2}\ge0\right\}$ and $S_2=\left\{x\in\mathbf R:3^{2x}-3^{x+1}-3^{x+2}+27\le0\right\}$. Then, $S_1\cup S_2$ is equal to :

If for some $\alpha>0$, the area of the region $\{(x,y):|x+\alpha|\le y\le2-|x|\}$ is equal to $\dfrac32$, then the area of the region $\{(x,y):0\le y\le x+2\alpha,\ |x|\le1\}$ is equal to ______.

The number of real values of $\lambda$, such that the system of linear equations
$$2x-3y+5z=9$$
$$x+3y-z=-18$$
$$3x-y+\left(\lambda^2-|\lambda|\right)z=16$$
has no solutions, is

Let $f(x)=2x^2-x-1$ and $S=\{n\in\mathbf Z:|f(n)|\le800\}$. Then, the value of $\displaystyle\sum_{n\in S}f(n)$ is equal to ______.

Let $S=\left\{x\in[-6,3]-\{-2,2\}:\dfrac{|x+3|-1}{|x|-2}\ge0\right\}$ and $T=\left\{x\in\mathbf Z:x^2-7|x|+9\le0\right\}$. Then the number of elements in $S\cap T$ is :

Let S be the set of all $a\in\mathbf R$ for which the angle between the vectors $\vec u=a(\log_eb)\hat i-6\hat j+3\hat k$ and $\vec v=(\log_eb)\hat i+2\hat j+2a(\log_eb)\hat k$, $(b>1)$ is acute. Then S is equal to :

If [t] denotes the greatest integer ≤ t, then the value of $\displaystyle\int_0^1\left[2x-\left|3x^2-5x+2\right|+1\right]dx$ is :

The number of solutions of the equation $|x^2+3x+2|+|x+5|-2=0$, x ∈ ℝ, is _______.

For $x \in \mathbb R$, the number of real roots of the equation
$$3x^2 - 4|x^2 - 1| + x - 1 = 0$$
is ____.

For any real number $x$, let $[x]$ denote the largest integer less than or equal to $x$. If
$$I = \int_0^{10}\left[\sqrt{\frac{10x}{x + 1}}\right]dx,$$
then the value of $9I$ is ____.