Nature of roots

Let the eccentricity e of a hyperbola satisfy the equation $6e^2-11e+3=0$. If the foci of the hyperbola are $(3,5)$ and $(3,-4)$, then the length of its latus rectum is :

Let $\mathbb{R}$ denote the set of all real numbers. Let $a_i, b_i \in \mathbb{R}$ for $i \in \{1, 2, 3\}$. Define the functions $f : \mathbb{R} \to \mathbb{R}$, $g : \mathbb{R} \to \mathbb{R}$, and $h : \mathbb{R} \to \mathbb{R}$ by

$$\begin{aligned} f(x) &= a_1 + 10x + a_2x^2 + a_3x^3 + x^4,\\ g(x) &= b_1 + 3x + b_2x^2 + b_3x^3 + x^4,\\ h(x) &= f(x+1) - g(x+2). \end{aligned}$$

If $f(x) \neq g(x)$ for every $x \in \mathbb{R}$, then the coefficient of $x^3$ in $h(x)$ is

Let $a,b\in\mathbb R$ be such that the equation $ax^2-2bx+15=0$ has a repeated root $\alpha$. If $\alpha$ and $\beta$ are the roots of the equation $x^2-2bx+21=0$, then $\alpha^2+\beta^2$ is equal to:

If the line $y=4+kx$, $k>0$, is the tangent to the parabola $y=x-x^2$ at the point $P$ and $V$ is the vertex of the parabola, then the slope of the line through $P$ and $V$ is:

If two straight lines whose direction cosines are given by the relations $l+m-n=0$, $3l^2+m^2+cnl=0$ are parallel, then the positive value of $c$ is :

Let S be the set of all integral values of $\alpha$ for which the sum of squares of two real roots of the quadratic equation $3x^2+(\alpha-6)x+(\alpha+3)=0$ is minimum. Then S :

If the numbers appeared on the two throws of a fair six faced die are $\alpha$ and $\beta$, then the probability that $x^2+\alpha x+\beta>0$, for all $x\in\mathbf R$, is :

If for some $p,q,r\in\mathbf R$, not all have same sign, one of the roots of the equation $\left(p^2+q^2\right)x^2-2q(p+r)x+q^2+r^2=0$ is also a root of the equation $x^2+2x-8=0$, then $\dfrac{q^2+r^2}{p^2}$ is equal to ______.

Let $\alpha$, $\beta$ be the roots of the equation $x^2-\sqrt2x+\sqrt6=0$ and $\dfrac1{\alpha^2}+1$, $\dfrac1{\beta^2}+1$ be the roots of the equation $x^2+ax+b=0$. Then the roots of the equation $x^2-(a+b-2)x+(a+b+2)=0$ are :

Let $C_1$ and $C_2$ be two biased coins such that the probabilities of getting head in a single toss are $\dfrac23$ and $\dfrac13$, respectively. Suppose $\alpha$ is the number of heads that appear when $C_1$ is tossed twice, independently, and suppose $\beta$ is the number of heads that appear when $C_2$ is tossed twice, independently. Then the probability that the roots of the quadratic polynomial $x^2 - \alpha x + \beta$ are real and equal, is

Let $S$ be the set of all non-zero real numbers $\alpha$ such that the quadratic equation $\alpha x^2-x+\alpha=0$ has two distinct real roots $x_1$ and $x_2$ satisfying the inequality $|x_1-x_2|<1$. Which of the following intervals is(are) a subset(s) of $S$?

If the two roots of the equation, $(a-1)(x^4+x^2+1)+(a+1)(x^2+x+1)^2=0$ are real and distinct, then the set of all values of 'a' is :

The quadratic equation $p(x)=0$ with real coefficients has purely imaginary roots. Then the equation
$$p(p(x))=0$$
has

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 :

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 $c\ne qa$