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If $\alpha ,\beta$are the roots of the equation $a{x}^2+bx+c=0$ then the equation whose roots are $\alpha +\frac{1}{\beta }$ and $\beta +\frac{1}{\alpha }$, is

If the expression $(n-2)x^2+8x+(n+4)$ is negative $\forall x\in R$, then $n$ lies in

The range of $f\left(x\right)=\frac{1}{-x^2+4x+5}$,$x\in R-\{-1,5\}$ is

The correct statement about the roots of the equation $x^2-4\sqrt{2}+8=0$

The zeroes of the quadratic polynomial $f\left(x\right)=x^2+7x+10$ are

Let $p,q$ be integers and $\alpha ,\beta$ be the roots of the equation $x^2-2x+3=0$ where $\alpha \ne \beta$.
$\text{If}{a}_n=p{\alpha }^n+q{\beta }^n\text{where}n\in \{0,1,2,.....\}$, then the value of $a_9$ is

Let $\alpha ,\beta$ be the values of m for which the equation $(1+m)x^2-2(1+3m)x+(1+8m)$ has equal roots. Find the equation whose roots are $\alpha +2$ and $\beta +2$.

The number of real roots of the equation, $e^{4x}+e^{3x}-4e^{2x}+e^x+1=0$ is :

The value(s) of $a$ for which the roots of $2x^2+(a^2-1)x+a^2+3a+4=0$ are reciprocal to each other is/are

Find the equation whose roots are the cubes of the roots of $x^3+3x^2+2=0$

The sum of values of $x$ satisfying the equation $\sqrt{\frac{x}{1-x}}+\sqrt{\frac{1-x}{x}}=\frac{13}{6}$ is

If equations $x^2-3x+4=0$ and $4x^2-2[3a+b]x+b=0(a,b\in R)$ have a common root, then the complete set of values of $a$ is
$($Here, $\left[K\right]$ denotes the greatest integer less than or equal to $K.)$

Let $\alpha ,\beta$ be the roots of $a{x}^2+bx+c=0$. The roots of $a(x-2{)}^2-b(x-2\left)\right(x-3)+c(x-3{)}^2=0$, where $a\ne 0$ are

If $m$ is choosen in the quadratic equation $(m^2+1)x^2-3x+(m^2+1{)}^2=0$ such that the sum of its roots is greatest, then the absolute difference of the cubes of its roots is :

Consider the quadratic equation $(c-5)x^2-2cx+(c-4)=0$. Let $S$ be the set of all integral values of $c$ for which one root of the equation lies in the interval $(0,2)$ and another root lies in the interval $(2,3)$. The number of elements in $S$ is

If the three equations $x^2+ax+12=0,$ $x^2+bx+15=0,$ $x^2+(a+b)x+36=0$ have a common possible root. Then, the sum of roots is

$\sqrt{5}{x}^2+x+\sqrt{5}=0$

Let $f\left(x\right)=({\lambda }^2+\lambda -2)x^2+(\lambda +2)x$ be a quadratic polynomial. The sum of all integral values of $\lambda$ for which $f\left(x\right)<1\forall x\in R$, is

If the difference of the roots of the equation $(k-2)x^2-(k-4)x-2=0,k\ne 2$ is $3$, then the sum of all the values of $k$ is

The zeroes of the polynomial $f\left(x\right)=6x^2-x-2$ is/are

Let $\alpha ,\beta$ be the roots of $x^2-x-1=0$ $(\alpha >\beta )$ and $m,n\in Z,k\in W$ such that $a_k=m{\alpha }^k+n{\beta }^k.$ If $a_4=35,$ then the value of $3m+2n$ is equal to