Basic Algebra

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 remainder when $(2021)^{2023}$ is divided by 7 is :

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 total number of 3-digit numbers, whose greatest common divisor with 36 is 2, is ______.

If the probability that a randomly chosen 6-digit number formed by using digits 1 and 8 only is a multiple of 21 is $p$, then $96p$ is equal to ______.

If the sum of all the roots of the equation $e^{2x}-11e^x-45e^{-x}+\dfrac{81}2=0$ is $\log_ep$, then $p$ is equal to ______.

If the coefficient of $x^{10}$ in the binomial expansion of $\left(\dfrac{\sqrt x}{5^{1/4}}+\dfrac{\sqrt5}{x^{1/3}}\right)^{60}$ is $5^k\cdot l$, where $l,k\in\mathbb N$ and $l$ is co-prime to 5, then $k$ 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 number of positive integers $k$ such that the constant term in the binomial expansion of $\left(2x^3+\dfrac3{x^k}\right)^{12}$, $x\ne0$ is $2^8\cdot l$, where $l$ is an odd integer, is ______.

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

The number of values of $a\in\mathbb N$ such that the variance of $3,7,12,a,43-a$ is a natural number is:

Let $f(x)$ and $g(x)$ be two real polynomials of degree 2 and 1 respectively. If $f(g(x))=8x^2-2x$, and $g(f(x))=4x^2+6x+1$, then the value of $f(2)+g(2)$ 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

The remainder when $(11)^{1011}+(1011)^{11}$ is divided by 9 is

Let $f(x)=\begin{cases}x^3-x^2+10x-7, & x\le1\\-2x+\log_2\left(b^2-4\right), & x>1\end{cases}$. Then the set of all values of b, for which $f(x)$ has maximum value at $x=1$, is :

The value of $\log_e2\,\dfrac{d}{dx}\left(\log_{\cos x}\operatorname{cosec}x\right)$ at $x=\dfrac\pi4$ is