Roots and coefficients

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

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:

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 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:

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 :

Let $\alpha$ and $\beta$ be the distinct roots of the equation $x^2 + x - 1 = 0$. Consider the set $T = \{1, \alpha, \beta\}$. For a $3\times 3$ matrix $M = (a_{ij})_{3\times 3}$, define $R_i = a_{i1} + a_{i2} + a_{i3}$ and $C_j = a_{1j} + a_{2j} + a_{3j}$ for $i = 1, 2, 3$ and $j = 1, 2, 3$.

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

The correct option is

For any $y \in \mathbb{R}$, let $\cot^{-1}(y) \in (0,\pi)$ and $\tan^{-1}(y) \in \left(-\dfrac\pi2, \dfrac\pi2\right)$. Then the sum of all the solutions of the equation $\tan^{-1}\left(\dfrac{6y}{9-y^2}\right) + \cot^{-1}\left(\dfrac{9-y^2}{6y}\right) = \dfrac{2\pi}{3}$ for $0 < |y| < 3$, is equal to

Consider the equation
$$\int_1^e \frac{(\log_e x)^{1/2}}{x\left(a - (\log_e x)^{3/2}\right)^2}\,dx = 1, \qquad a \in (-\infty, 0)\cup(1,\infty).$$
Which of the following statements is/are TRUE?

The product of all positive real values of $x$ satisfying the equation
$$x^{\left(16(\log_5 x)^3 - 68\log_5 x\right)} = 5^{-16}$$
is _____.

Let $\bar z$ denote the complex conjugate of a complex number $z$. If $z$ is a non-zero complex number for which both real and imaginary parts of
$$(\bar z)^2 + \frac{1}{z^2}$$
are integers, then which of the following is/are possible value(s) of $|z|$?

If the sum of the squares of the reciprocals of the roots $\alpha$ and $\beta$ of the equation $3x^2+\lambda x-1=0$ is 15, then $6(\alpha^3+\beta^3)^2$ is equal to :

Let the abscissae of the two points P and Q be the roots of $2x^2-rx+p=0$ and the ordinates of P and Q be the roots of $x^2-sx-q=0$. If the equation of the circle described on PQ as diameter is $2\left(x^2+y^2\right)-11x-14y-22=0$, then $2r+s-2q+p$ is equal to ______.