Quadratic Equations

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:

The sum of the cubes of all the roots of the equation $x^4-3x^3-2x^2+3x+1=0$ is ______.

Let $p$ and $q$ be two real numbers such that $p+q=3$ and $p^4+q^4=369$. Then $\left(\dfrac1p+\dfrac1q\right)^{-2}$ is equal to ______.

The number of distinct real roots of $x^4-4x+1=0$ 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 :

If the sum of all the roots of the equation $e^{2x}-11e^x-45e^{-x}+\dfrac{81}2=0$ is $\log_ep$, then $p$ is equal to ______.

$\alpha=\sin36°$ is a root of which of the following equation?

Let $\alpha,\beta$ be the roots of the equation $x^2-4\lambda x+5=0$ and $\alpha,\gamma$ be the roots of the equation $x^2-\left(3\sqrt2+2\sqrt3\right)x+7+3\lambda\sqrt3=0$, $\lambda>0$. If $\beta+\gamma=3\sqrt2$, then $(\alpha+2\beta+\gamma)^2$ is equal to ______.

Let $[t]$ denote the greatest integer less than or equal to $t$. Then, the value of the integral $\displaystyle\int_0^1\left[-8x^2+6x-1\right]dx$ is equal to

The number of real solutions of $x^7+5x^3+3x+1=0$ is equal to ______.

The number of real solutions of the equation $e^{4x}+4e^{3x}-58e^{2x}+4e^x+1=0$ is ______.

Let $f(x)$ be a quadratic polynomial such that $f(-2)+f(3)=0$. If one of the roots of $f(x)=0$ is $-1$, then the sum of the roots of $f(x)=0$ is equal to :

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 :

Let $A=\begin{bmatrix}1&-2&\alpha\\\alpha&2&-1\end{bmatrix}$ and $B=\begin{bmatrix}2&\alpha\\-1&2\\4&-5\end{bmatrix}$, $\alpha\in\mathbf C$. Then the absolute value of the sum of all values of $\alpha$ for which $\det(AB)=0$ is :

Let $\alpha=\tan\left(\dfrac{5\pi}{16}\sin\left(2\cos^{-1}\left(\dfrac1{\sqrt5}\right)\right)\right)$ and $\beta=\cos\left(\sin^{-1}\left(\dfrac45\right)+\sec^{-1}\left(\dfrac53\right)\right)$ where the inverse trigonometric functions take principal values. Then, the equation whose roots are $\alpha$ and $\beta$ is :

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 :

Let a, b be two non-zero real numbers. If p and r are the roots of the equation $x^2-8ax+2a=0$ and q and s are the roots of the equation $x^2+12bx+6b=0$, such that $\dfrac1p,\dfrac1q,\dfrac1r,\dfrac1s$ are in A.P., then $a^{-1}-b^{-1}$ is equal to ______.

Let $f(x)$ be a quadratic polynomial with leading coefficient 1 such that $f(0)=p$, $p\ne0$, and $f(1)=\dfrac13$. If the equations $f(x)=0$ and $f\circ f\circ f\circ f(x)=0$ have a common real root, then $f(-3)$ is equal to ______.

Let $x=\sin\left(2\tan^{-1}\alpha\right)$ and $y=\sin\left(\dfrac12\tan^{-1}\dfrac43\right)$. If $S=\left\{\alpha\in\mathbb R:y^2=1-x\right\}$, then $\displaystyle\sum_{\alpha\in S}16\alpha^3$ is equal to ______.