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Consider the functions defined implicitly by the equation \(y^3-3 y+x=0\) on various intervals in the real line. If \(x \in(-\infty,-2) \cup(2, \infty)\), the equation implicitly defines a unique real valued differentiable function \(y=f(x)\). If \(x \in(-2,2)\), the equation implicitly defines a unique real valued differentiable function \(y=g(x)\), satisfying \(g(0)=0\).Question:
\(\int_{-1}^1 g^{\prime}(x) d x\) is equal to

Consider the lines \(L_{1}\) and \(L_{2}\) defined by
\[
L_{1}: x \sqrt{2}+y-1=0 \text { and } L_{2}: x \sqrt{2}-y+1=0
\]
For a fixed constant \(\lambda\), let \(C\) be the locus of a point \(P\) such that the product of the distance of \(P\) from \(L_{1}\) and the distance of \(P\) from \(L_{2}\) is \(\lambda^{2}\). The line \(y=2 x+1\) meets \(C\) at two points \(R\) and \(S\), where the distance between \(R\) and \(S\) is \(\sqrt{270}\).
Let the perpendicular bisector of \(R S\) meet \(C\) at two distinct points \(R^{\prime}\) and \(S^{\prime}\). Let \(D\) be the square of the distance between \(R^{\prime}\) and \(S^{\prime}\).
The value of ${\lambda }^2$ is

Let \(\alpha\) and \(\beta\) be the real numbers such that \(\lim _{x \rightarrow 0} \frac{1}{x^3}\left(\frac{\alpha}{2} \int_0^x \frac{1}{1-t^2} d t+\beta x \cos x\right)=2\). Then the value of \(\alpha+\beta\) is _______

Let the point $\mathrm{B}$ be the reflection of the point $\mathrm{A}\left(2,\mathrm{ }3\right)$ with respect to the line $8\mathrm{x}-6\mathrm{y}-23=0.$ Let ${\mathrm{T}}_{\mathrm{A}}$ and ${\mathrm{T}}_{\mathrm{B}}$ be circle of radii $2$ and $1$ with centres $\mathrm{A}$ and $\mathrm{B}$ respectively. Let $\mathrm{T}$ be a common tangent to the circle ${\mathrm{T}}_{\mathrm{A}}$ and ${\mathrm{T}}_{\mathrm{B}}$ such that both the circle are on the same side of $\mathrm{T}.$ If $\mathrm{C}$ is the point of intersection of $\mathrm{T}$ and the line passing through $\mathrm{A}$ and $\mathrm{B},$ then the length of the line segment $\mathrm{A}\mathrm{C}$ is ___________.

Let \(\alpha, \beta\) and \(\gamma\) be real numbers such that the system of linear equations
\[
\begin{array}{c}
x+2 y+3 z=\alpha \\
4 x+5 y+6 z=\beta \\
7 x+8 y+9 z=\gamma-1
\end{array}
\]
is consistent. Let \(|M|\) represent the determinant of the matrix
\[
M=\left[\begin{array}{ccc}
\alpha & 2 & \gamma \\
\beta & 1 & 0 \\
-1 & 0 & 1
\end{array}\right]
\]
Let \(P\) be the plane containing all those \((\alpha, \beta, \gamma)\) for which the above system of linear equations is consistent, and \(D\) be the square of the distance of the point \((0,1,0)\) from the plane \(P\).
The value of $D$ is

The number of points in $\left(-\infty ,\infty \right)$, for which $x^2-x\sin x-\cos x=0$, is:

Define the collections $\left\{{\mathrm{E}}_1,{\mathrm{E}}_2,{\mathrm{E}}_3,....\right\}$ of ellipses and $\left\{{\mathrm{R}}_1,{\mathrm{R}}_2,{\mathrm{R}}_3,....\right\}$ of rectangles as follows:
${\mathrm{E}}_1:\frac{{\mathrm{x}}^2}{9}+\frac{{\mathrm{y}}^2}{4}=1;$
${\mathrm{R}}_1:$ rectangle of largest area, with sides parallel to the axes, inscribed in ${\mathrm{E}}_1;$
${\mathrm{E}}_{\mathrm{n}}:$ ellipse $\frac{{\mathrm{x}}^2}{{\mathrm{a}}_{\mathrm{n}}^2}+\frac{{\mathrm{y}}^2}{{\mathrm{b}}_{\mathrm{n}}^2}=1$ of largest area inscribed in ${\mathrm{R}}_{\mathrm{n}-1},\mathrm{n}>1;$
${\mathrm{R}}_{\mathrm{n}}:$ rectangle of largest area, with sides parallel to the axes, inscribed in ${\mathrm{E}}_{\mathrm{n}},\mathrm{n}>1.$
Then which of the following options is/are correct?

Six point charges are kept at the vertices of a regular hexagon of side \(\mathrm{L}\) and centre \(O\), as shown in the figure.
Given that \(K=\frac{1}{4 \pi \varepsilon_{0}} \frac{q}{L^{2}}\), which of the following statement(s) is (are) correct?

Suppose $a, b$ denote the distinct real roots of the quadratic polynomial $x^2+20x-2020$ and suppose $c, d$ denote the distinct complex roots of the quadratic polynomial $x^2-20x+2020$. Then the value of $ac\left(a-c\right)+ad\left(a-d\right)+bc\left(b-c\right)+bd\left(b-d\right)$

An ideal gas is expanding such that \(p T^2=\) constant. The coefficient of volume expansion of the gas is

Two lines ${\text{L}}_1\text{ : }\text{x}=5\text{,}\frac{\text{y}}{3-\alpha }=\frac{\text{z}}{-2}$ and ${\text{L}}_2\text{ : }\text{x}=\alpha \text{,}\frac{\text{y}}{-1}=\frac{\text{z}}{2-\alpha }$ are coplanar. Then $\alpha$ can take value(s)

If the straight lines \(\frac{x-1}{2}=\frac{y+1}{k}=\frac{z}{2}\) and \(\frac{x+1}{5}=\frac{y+1}{2}=\frac{z}{k}\) are coplanar, then the plane (s) containing these two lines is (are)

Let $E_1=${$x\in \mathrm{ℝ}: x\ne 1$ and $\frac{x}{x-1}>0$} and $E_2=${$x\in E_1: {\sin }^{-1}\left({\log }_e\left(\frac{x}{x-1}\right)\right)$ is a real number} (Here the inverse trigonometric function ${\sin }^{-1}\left(x\right)$ assumes values in $\left[-\frac{\mathrm{\pi }}{2}. \frac{\mathrm{\pi }}{2}\right]$.)
Let $f:E_1\to \mathrm{ℝ}$ be the function defined by $f\left(x\right)={\log }_e\left(\frac{x}{x-1}\right)$
And $g:E_2\to \mathbb{R}$ be the function defined by $g\left(x\right)={\sin }^{-1}\left({\log }_e\left(\frac{x}{x-1}\right)\right)$ .

LIST-I	LIST-II
A. The range of $f$ is	P. $\left(-\infty ,\frac{1}{1-e}\right]\cup \left[\frac{e}{e-1}, \infty \right)$
B. The range of $g$ contains	Q. (0, 1)
C. The domain of $f$ contains	R. $\left[-\frac{1}{2},\frac{1}{2}\right]$
D. The domain of $g$ is	S. $\left(-\infty ,0\right)\cup \left(0, \infty \right)$
	T. $\left(-\infty ,\frac{e}{e-1}\right]$
	U. $\left(-\infty ,0\right)\cup \left(\frac{1}{2},\frac{e}{e-1}\right]$

The correct option is: