Roots and coefficients

For a natural number n, let $\alpha_n=19^n-12^n$. Then, the value of $\dfrac{31\alpha_9-\alpha_{10}}{57\alpha_8}$ 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:

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 ______.

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 $f(x)$ 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 $\alpha=\tan\left(\dfrac{5\pi}{16}\sin\left(2\cos^{-1}\left(\dfrac1{\sqrt5}\right)\right)\right)$ 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 :

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 ______.

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 ______.

The minimum value of the sum of the squares of the roots of $x^2+(3-a)x+1=2a$ is:

Let the abscissae of the two points P and Q on a circle be the roots of $x^2-4x-6=0$ and the ordinates of P and Q be the roots of $y^2+2y-7=0$. If PQ is a diameter of the circle $x^2+y^2+2ax+2by+c=0$, then the value of $(a+b-c)$ is ______.

For $k\in\mathbf R$, let the solutions of the equation $\cos\left(\sin^{-1}\left(x\cot\left(\tan^{-1}\left(\cos\left(\sin^{-1}x\right)\right)\right)\right)\right)=k$, $0<|x|<\dfrac1{\sqrt2}$ be $\alpha$ and $\beta$, where the inverse trigonometric functions take only principal values. If the solutions of the equation $x^2-bx-5=0$ are $\dfrac1{\alpha^2}+\dfrac1{\beta^2}$ and $\dfrac\alpha\beta$, then $\dfrac b{k^2}$ is equal to ______.

If $\alpha$, $\beta$ are the roots of the equation

$$x^2-\left(5+3^{\sqrt{\log_35}}-5^{\sqrt{\log_53}}\right)x+3\left(3^{(\log_35)^{\frac13}}-5^{(\log_53)^{\frac23}}-1\right)=0,$$

then the equation, whose roots are $\alpha+\dfrac1\beta$ and $\beta+\dfrac1\alpha$, is :

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 :

If $\dfrac1{(20-a)(40-a)}+\dfrac1{(40-a)(60-a)}+\dots+\dfrac1{(180-a)(200-a)}=\dfrac1{256}$, then the maximum value of a is :

Let S = {θ ∈ (0, 2π) : $7\cos^2\theta-3\sin^2\theta-2\cos^22\theta=2$}. Then, the sum of roots of all the equations $x^2-2(\tan^2\theta+\cot^2\theta)x+6\sin^2\theta=0$, θ ∈ S, is _______.

Let α, β (α > β) be the roots of the quadratic equation $x^2-x-4=0$. If $P_n=\alpha^n-\beta^n$, n ∈ ℕ, then $\dfrac{P_{15}P_{16}-P_{14}P_{16}-P_{15}^2+P_{14}P_{15}}{P_{13}P_{14}}$ is equal to _______.

Let A be a matrix and $\begin{bmatrix}1&2\\0&1\end{bmatrix}A\begin{bmatrix}2&0\\-1&1\end{bmatrix}=\begin{bmatrix}1&0\\0&1\end{bmatrix}$. If |A| and |A$^2$| are the roots of the quadratic equation $ax^2+bx+3=0$, then a + b − ab is equal to