Quadratic Equations

If $a\in\mathbf R$ and the equation
$$-3(x-[x])^2+2(x-[x])+a^2=0$$
(where $[x]$ denotes the greatest integer $\le x$) has no integral solution, then all possible values of $a$ lie in the interval :

Let $\alpha$ and $\beta$ be the roots of equation $px^2+qx+r=0$, $p\ne0$. If $p$, $q$, $r$ are in A.P. and $\dfrac1\alpha+\dfrac1\beta=4$, then the value of $|\alpha-\beta|$ is :

If equations $ax^2+bx+c=0$, ($a,b,c\in\mathbf R$, $a\ne0$) and $2x^2+3x+4=0$ have a common root, then $a:b:c$ equals :

If $\dfrac1{\sqrt\alpha}$ and $\dfrac1{\sqrt\beta}$ are the roots of the equation, $ax^2+bx+1=0$ ($a\ne0$, $a,b\in\mathbf R$), then the equation, $x\left(x+b^3\right)+\left(a^3-3abx\right)=0$ has roots :

The number of values of $\alpha$ in $[0,2\pi]$ for which $2\sin^3\alpha-7\sin^2\alpha+7\sin\alpha=2$, is :

If $\alpha$ and $\beta$ are roots of the equation, $x^2-4\sqrt2\,kx+2e^{4\ln k}-1=0$ for some $k$, and $\alpha^2+\beta^2=66$, then $\alpha^3+\beta^3$ is equal to :

The sum of the roots of the equation, $x^2+|2x-3|-4=0$, is :

The equation $\sqrt{3x^2+x+5}=x-3$, where $x$ is real, has :

If non-zero real numbers b and c are such that min $f(x)$ > max $g(x)$, where
$$f(x)=x^2+2bx+2c^2\ \text{and}\ g(x)=-x^2-2cx+b^2\ (x\in\mathbb R);$$
then $\left|\dfrac cb\right|$ lies in the interval :

The value of $6+\log_{3/2}\left(\dfrac1{3\sqrt2}\sqrt{4-\dfrac1{3\sqrt2}\sqrt{4-\dfrac1{3\sqrt2}\sqrt{4-\dfrac1{3\sqrt2}\cdots}}}\right)$ is

Let $\alpha(a)$ and $\beta(a)$ be the roots of the equation
$$\left(\sqrt[3]{1+a}-1\right)x^2+\left(\sqrt{1+a}-1\right)x+\left(\sqrt[6]{1+a}-1\right)=0$$
where $a>-1$. Then $\displaystyle\lim_{a\to0^+}\alpha(a)$ and $\displaystyle\lim_{a\to0^+}\beta(a)$ are

Let $\alpha$ and $\beta$ be the roots of $x^2-6x-2=0$, with $\alpha>\beta$. If $a_n=\alpha^n-\beta^n$ for $n\ge1$, then the value of $\dfrac{a_{10}-2a_8}{2a_9}$ is

Let $b=6$, with $a$ and $c$ satisfying (E). If $\alpha$ and $\beta$ are the roots of the quadratic equation $ax^2+bx+c=0$, then
$$\sum_{n=0}^\infty\left(\frac1\alpha+\frac1\beta\right)^n$$
is

A value of $b$ for which the equations
$$x^2+bx-1=0,\qquad x^2+x+b=0,$$
have one root in common is

The number of distinct real roots of $x^4-4x^3+12x^2+x-1=0$ is

Let p and q be real numbers such that $p\ne0$, $p^3\ne q$ and $p^3\ne-q$. If $\alpha$ and $\beta$ are nonzero complex numbers satisfying $\alpha+\beta=-p$ and $\alpha^3+\beta^3=q$, then a quadratic equation having $\dfrac\alpha\beta$ and $\dfrac\beta\alpha$ as its roots is

The real number $s$ lies in the interval

The maximum value of the function $f(x)=2x^3-15x^2+36x-48$ on the set $A=\left\{x\mid x^2+20\le9x\right\}$ is

The smallest value of $k$, for which both the roots of the equation
$$x^2-8kx+16\left(k^2-k+1\right)=0$$
are real, distinct and have values at least 4, is

Let $a,b,c,p,q$ be real numbers. Suppose $\alpha,\beta$ are the roots of the equation $x^2+2px+q=0$ and $\alpha,\frac1\beta$ are the roots of the equation $ax^2+2bx+c=0$, where $\beta^2\notin\{-1,0,1\}$.

STATEMENT-1: $(p^2-q)(b^2-ac)\ge0$

and

STATEMENT-2: $b\ne pa$ or $c\ne qa$