Quadratic Equations

Match each entry in List-I to the correct entry in List-II and choose the correct option.

Let $a,b,c$ be positive integers in arithmetic progression such that the equation
$$ax^2+bx+c=0$$
has only integer solutions.

Then which of the following statements is (are) TRUE ?

Let $\alpha,\ \alpha+2$, $\alpha\in\mathbb Z$, be the roots of the quadratic equation
$$x(x+2)+(x+1)(x+3)+(x+2)(x+4)+\dots+(x+n-1)(x+n+1)=4n$$
for some $n\in\mathbb N$. Then $n+\alpha$ is equal to :

Let $a,b,c\in\{1,2,3,4\}$. If the probability, that $ax^2+2\sqrt2\,bx+c>0$ for all $x\in\mathbb R$, is $\dfrac mn$, $\gcd(m,n)=1$, then $m+n$ is equal to ________.

Let $\alpha,\beta$ be the roots of the equation $x^2-3x+r=0$, and $\dfrac\alpha2,2\beta$ be the roots of the equation $x^2+3x+r=0$. If the roots of the equation $x^2+6x=m$ are $2\alpha+\beta+2r$ and $\alpha-2\beta-\dfrac r2$, then $m$ is equal to :

Let the parabola $y=x^2+px+q$ passing through the point $(1,-1)$ be such that the distance between its vertex and the $x$-axis is minimum. Then the value of $p^2+q^2$ is :

Let $S=\{z\in\mathbb C:z^2+4z+16=0\}$. Then $\displaystyle\sum_{z\in S}\left|z+\sqrt3i\right|^2$ is equal to:

If $\alpha=1$ and $\beta=1+i\sqrt2$, where $i=\sqrt{-1}$ are two roots of the equation $x^3+ax^2+bx+c=0$, $a,b,c\in\mathbb R$, then $\displaystyle\int_{-1}^1\left(x^3+ax^2+bx+c\right)dx$ is equal to:

If the quadratic equation $(\lambda+2)x^2-3\lambda x+4\lambda=0$, $\lambda\ne-2$, has two positive roots, then the number of possible integral values of $\lambda$ is:

Let $a,b\in\mathbb C$. Let $\alpha,\beta$ be the roots of the equation $x^2+ax+b=0$. If $\beta-\alpha=\sqrt{11}$ and $\beta^2-\alpha^2=3i\sqrt{11}$, then $\left(\beta^3-\alpha^3\right)^2$ is equal to:

Let $\tan A$, $\tan B$, where $A,B\in\left(-\dfrac\pi2,\dfrac\pi2\right)$, be the roots of the quadratic equation $x^2-2x-5=0$. Then $20\sin^2\left(\dfrac{A+B}2\right)$ is equal to:

The sum of all the integral values of $p$ such that the equation $3\sin^2x+12\cos x-3=p$, $x\in\mathbb R$, has at least one solution, is:

Let $\alpha,\beta$ be the roots of the equation $x^2-x+p=0$ and $\gamma,\delta$ be the roots the equation $x^2-4x+q=0$; $p,q\in\mathbb Z$. If $\alpha,\beta,\gamma,\delta$ are in G.P., then $|p+q|$ equals :

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 one root of the quadratic equation in $x$:
$$(k^2-15k+27)x^2+9(k-1)x+18=0$$
be twice the other. Then the length of the latus rectum of the parabola $y^2=6kx$ is equal to:

Let $e_1$ and $e_2$ be two distinct roots of the equation $x^2-ax+2=0$. Let the sets
$\{a\in\mathbb R:e_1\text{ and }e_2\text{ are the eccentricities of hyperbolas}\}=(\alpha,\beta)$, and
$\{a\in\mathbb R:e_1\text{ and }e_2\text{ are the eccentricities of an ellipse and a hyperbola, respectively}\}=(\gamma,\infty)$.
Then $\alpha^2+\beta^2+\gamma^2$ is equal to:

The sum of the first ten terms of an A.P. is 160 and the sum of the first two terms of a G.P. is 8. If the first term of the A.P. is equal to the common ratio of the G.P. and the first term of the G.P. is equal to common difference of the A.P., then the sum of all possible values of the first term of the G.P. is:

Consider the quadratic equation $(n^2-2n+2)x^2-3x+(n^2-2n+2)^2=0$, $n\in\mathbf R$. Let $\alpha$ be the minimum value of the product of its roots and $\beta$ be the maximum value of the sum of its roots. Then the sum of the first six terms of the G.P., whose first term is $\alpha$ and the common ratio is $\dfrac\alpha\beta$, is :

Let $S=\left\{z\in\mathbb C:z^2+\sqrt6\,iz-3=0\right\}$. Then $\displaystyle\sum_{z\in S}z^8$ is equal to :

Let the mean and the variance of seven observations $2,4,\alpha,8,\beta,12,14$, $\alpha<\beta$, be 8 and 16 respectively. Then the quadratic equation whose roots are $3\alpha+2$ and $2\beta+1$ is :