Inequalities and modulus

Let $E_1 = \left\{x \in \mathbb{R} : x \ne 1 \text{ and } \frac{x}{x-1} > 0\right\}$ and $E_2 = \left\{x \in E_1 : \sin^{-1}\left(\log_e\left(\frac{x}{x-1}\right)\right) \text{ is a real number}\right\}$.

(Here, the inverse trigonometric function $\sin^{-1} x$ assumes values in $\left[-\frac{\pi}{2}, \frac{\pi}{2}\right]$.)

Let $f: E_1 \to \mathbb{R}$ be the function defined by $f(x) = \log_e\left(\frac{x}{x-1}\right)$ and $g: E_2 \to \mathbb{R}$ be the function defined by $g(x) = \sin^{-1}\left(\log_e\left(\frac{x}{x-1}\right)\right)$.

The correct option is:

For any three positive real numbers a, b and c,
$$9(25a^2+b^2)+25(c^2-3ac)=15b(3a+c).$$
Then :

If x is a solution of the equation, $\sqrt{2x+1}-\sqrt{2x-1}=1$, $\left(x\ge\dfrac12\right)$, then $\sqrt{4x^2-1}$ is equal to :

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 $S$?

The sum of the roots of the equation, $x^2+|2x-3|-4=0$, is :

A number $x$ is chosen at random from the set {1, 2, 3, 4, ....., 100}. Define the event : A = the chosen number $x$ satisfies
$$\frac{(x-10)(x-50)}{(x-30)}\ge0$$
Then P(A) is :

The function $f(x)=2|x|+|x+2|-\big||x+2|-2|x|\big|$ has a local minimum or a local maximum at $x=$

Let $f:\mathbb R\to\mathbb R$ be defined as $f(x)=|x|+|x^2-1|$. The total number of points at which $f$ attains either a local maximum or a local minimum is

Match the statements/expressions given in Column I with the values given in Column II.

Let $f(x)=\dfrac{x^2-6x+5}{x^2-5x+6}$.

Match the expressions/statements in Column I with expressions/statements in Column II and indicate your answer by darkening the appropriate bubbles in the 4 × 4 matrix given in the ORS.