Algebraic Equations of Higher Degree

The value of $x$, which satisfies the equation $\left(x-1\right)\left|x^2-4x+3\right|+2x^2+3x-5=0$, is

The number of points, where the curve $f\left(x\right)=e^{8x}-e^{6x}-3e^{4x}-e^{2x}+1,x\in \mathrm{ℝ}$ cuts $x$-axis, is equal to............
 

If $e^{8x}-e^{6x}-3e^{4x}-e^{2x}+1=0$, then number of solution to the given equation will be,

If $\alpha , \beta$ and $\gamma$ are the roots of the equation $x^3-12x^2+44x-48=0$, then the centroid of the triangle whose coordinates are $\left(\alpha ,\frac{1}{\alpha }\right), \left(\beta ,\frac{1}{\beta }\right)$ and $\left(\gamma ,\frac{1}{\gamma }\right)$ is

Let $f\left(x\right)=x^4+a{x}^3+b{x}^2+cx+d$ be a polynomial whose roots are all negative integer, if $a+b+c+d=2009$, then $d$ is 

If radii of three concentric circles are related as $r_1\left(r_2+r_3\right)+r_2\left(r_3+r_1\right)+r_3\left(r_1+r_2\right)=118$, $r_1{r}_2\left(r_1+r_2\right)+r_2{r}_3\left(r_2+r_3\right)+r_1{r}_3\left(r_1+r_3\right)=616$ and $\Sigma \left(\frac{{r_1}^2+{r_2}^2}{r_1{r}_2}\right)=\frac{44}{5}$, then the area of enclosed region between largest and smallest circle is 

If $\left|x^2-5\left|x\right|+6\right|=k$ has distinct solutions, then $k$ lies in

If the roots of the equation $x^4-3x^2+2=0$ are sides of the rectangle and if this rectangle is inscribed in a circle of radius $\lambda$ then $4{\lambda }^2$ is 

If $\alpha ,\beta ,\gamma ,\delta$ are the roots of the equation $x^4+x^3+x^2+x+1=0$, then ${\alpha }^{2021}+{\beta }^{2021}+{\gamma }^{2021}+{\delta }^{2021}$ is equal to

Let $c>0$ and $d<0$. For the equation $x^4+\left(b+c\right)x^3+\left(c+d+bc\right)x^2+\left(c^2+bd\right)x+cd=0$

The set of values of $k\left(k\in R\right)$ for which the equation $x^2-4\left|x\right|+3-\left|k-1\right|=0$ will have exactly four real roots, is:

Let $S$ denote the set of all real values of '$x$' such that $\left(x^{2010}+1\right)\left(1+x^2+x^4+\cdots \dots +x^{2008}\right)=2010x^{2009}$ then

Solve the equation $18x^3+81x^2+\lambda x+60=0$, one root being half the sum of the other two. Hence find the value of $\lambda$.

${\left(x-\frac{1}{x}\right)}^2+p\left|x+\frac{{\displaystyle 1}}{{\displaystyle x}}\right|+29=0, p\in R$ exactly four distinct real solutions, then the true set of values of $p$ is

The difference of the maximum real root and the minimum real root of the equation ${\left(x^2-5\right)}^4+{\left(x^2-7\right)}^4=16$ is

$x^7-2x+3=P\left(x\right)$. Find number of real roots.

$p$ is non-zero real number. If the equation whose roots are the squares of the roots of the equation $x^3-p{x}^2+px-1=0$ is identical with the given equation, then $p=$

If $\alpha ,\beta ,\gamma$ are the roots of $x^3+x^2+2x+3=0$ then the equation whose roots $\beta +\gamma ,\gamma +\alpha ,\alpha +\beta$ is

If $-3,1,8$ are the roots of $p{x}^3+q{x}^2+rx+s=0$ then the roots of $p{\left(x-3\right)}^3+q{\left(x-3\right)}^2+r\left(x-3\right)+s=0$ are

If $\alpha ,\beta ,\gamma$ are the roots of $x^3-x^2+ax+b=0$ and $\beta ,\gamma ,\delta$ are the roots of $x^3-4x^2+mx+n=0$. If $\alpha ,\beta ,\gamma \& \delta$ are in $A.P.$ with common difference $d$, then