Roots and coefficients

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

Let p and q be two positive numbers such that p + q = 2 and $p^4+q^4=272$. Then p and q are roots of the equation :

Suppose $a, b$ denote the distinct real roots of the quadratic polynomial $x^2 + 20x - 2020$ and suppose $c, d$ denote the distinct complex roots of the quadratic polynomial $x^2 - 20x + 2020$. Then the value of

$$ac(a-c) + ad(a-d) + bc(b-c) + bd(b-d)$$

is

The trace of a square matrix is defined to be the sum of its diagonal entries. If $A$ is a $2\times2$ matrix such that the trace of $A$ is 3 and the trace of $A^3$ is $-18$, then the value of the determinant of $A$ is ______

Let $\alpha$ and $\beta$ be the roots of $x^2 - x - 1 = 0$, with $\alpha > \beta$. For all positive integers $n$, define

$$a_n = \frac{\alpha^n - \beta^n}{\alpha - \beta},\quad n \ge 1,$$

$$b_1 = 1\quad\text{and}\quad b_n = a_{n-1} + a_{n+1},\quad n \ge 2.$$

Then which of the following options is/are correct?

$a_{12} =$

If $a_4 = 28$, then $p + 2q =$

If, for a positive integer n, the quadratic equation,
$$x(x+1)+(x+1)(x+2)+\dots+(x+\overline{n-1})(x+n)=10n$$
has two consecutive integral solutions, then n is equal to :

The sum of all the real values of $x$ satisfying the equation $2^{(x-1)(x^2+5x-50)}=1$ is :

Let $-\dfrac{\pi}{6}<\theta<-\dfrac{\pi}{12}$. Suppose $\alpha_1$ and $\beta_1$ are the roots of the equation $x^2-2x\sec\theta+1=0$ and $\alpha_2$ and $\beta_2$ are the roots of the equation $x^2+2x\tan\theta-1=0$. If $\alpha_1>\beta_1$ and $\alpha_2>\beta_2$, then $\alpha_1+\beta_2$ equals

If the equations $x^2+bx-1=0$ and $x^2+x+b=0$ have a common root different from −1, then |b| is equal to :

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

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 :

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 :

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