Paragraph:
Consider the lines: \(L_1: \frac{x+1}{3}=\frac{y+2}{1}=\frac{z+1}{2}, L_2: \frac{x-2}{1}=\frac{y+2}{2}=\frac{z-3}{3}\)Question:
The shortest distance between \(L_1\) and \(L_2\) is

Let ${\mathrm{L}}_1$ and ${\mathrm{L}}_2$ denotes the lines
$\overrightarrow{\mathrm{r}}=\widehat{\mathrm{i}}+\mathrm{\lambda }\left(-\widehat{\mathrm{i}}+2\widehat{\mathrm{j}}+2\widehat{\mathrm{k}}\right),\mathrm{\lambda }\in \mathrm{R}$ and $\overrightarrow{\mathrm{r}}=\mathrm{\mu }\left(2\widehat{\mathrm{i}}-\widehat{\mathrm{j}}+2\widehat{\mathrm{k}}\right),\mathrm{\mu }\in \mathrm{R}$
respectively. If ${\mathrm{L}}_3$ is a line which is perpendicular to both ${\mathrm{L}}_1$ and ${\mathrm{L}}_2$ and cuts both of them, then which of the following options describe(s) ${\mathrm{L}}_3$ ?

Let $\mathrm{g}\mathrm{ }:\mathbb{R}\to \mathbb{R}$ be a differentiable function with $\mathrm{g}\left(0\right)=0,\mathrm{ }\mathrm{ }\mathrm{ }\mathrm{ }{\mathrm{g}}^{\mathrm{\prime }}\left(0\right)=0$ and ${\mathrm{g}}^{\mathrm{\prime }}\left(1\right)\mathrm{ }\ne 0.\mathrm{ }$ Let $\mathrm{f}\left(\mathrm{x}\right)=\mathrm{ }\left\{\begin{array}{cc}\frac{\mathrm{x}}{\left|\mathrm{x}\right|}\mathrm{ }\mathrm{g}\left(\mathrm{x}\right),& \mathrm{x}\mathrm{ }\ne 0\\ 0,& \mathrm{x}=0\end{array}\right.$ and $\mathrm{h}\left(\mathrm{x}\right)={\mathrm{e}}^{\left|\mathrm{x}\right|}$ for all $\mathrm{x}\in \mathbb{R}.$ Let $\left(\mathrm{foh}\right)\mathrm{ }\left(\mathrm{x}\right)$ denote $\mathrm{f}\left(\mathrm{h}\left(\mathrm{x}\right)\right)\mathrm{a}\mathrm{n}\mathrm{d}\mathrm{ }\left(\mathrm{hof}\right)\mathrm{ }\left(\mathrm{x}\right)\mathrm{ }$denote $\mathrm{h}\left(\mathrm{f}\left(\mathrm{x}\right)\right)$ .Then which of the following is (are) true?

Consider the ellipse $\frac{x^2}{4}+\frac{y^2}{3}=1$. Let $H\left(\alpha ,0\right),0<\alpha <2$, be a point. A straight line drawn through $H$ parallel to the $y$-axis crosses the ellipse and its auxiliary circle at points $E$ and $F$ respectively, in the first quadrant. The tangent to the ellipse at the point $E$ intersects the positive $x$-axis at a point $G$. Suppose the straight line joining $F$ and the origin makes an angle $\varphi$ with the positive $x$-axis.

	List-$\mathrm{I}$		List-$\mathrm{II}$
$\left(\mathrm{I}\right)$	If $\varphi =\frac{\pi }{4}$, then the area of the triangle $FGH$ is	$\left(\mathrm{P}\right)$	$\frac{{\left(\sqrt{3}-1\right)}^4}{8}$
$\left(\mathrm{II}\right)$	If $\varphi =\frac{\pi }{3}$, then the area of the triangle $FGH$ is	$\left(\mathrm{Q}\right)$	$1$
$\left(\mathrm{III}\right)$	If $\varphi =\frac{\pi }{6}$, then the area of the triangle $FGH$ is	$\left(\mathrm{R}\right)$	$\frac{3}{4}$
$\left(\mathrm{IV}\right)$	If $\varphi =\frac{\pi }{12}$, then the area of the triangle $FGH$ is	$\left(\mathrm{S}\right)$	$\frac{1}{2\sqrt{3}}$
		$\left(\mathrm{T}\right)$	$\frac{3\sqrt{3}}{2}$

The correct option is:

Let \(f\) be a function defined on \(R\) (the set of all real numbers) such that \(f^{\prime}(x)=2010(x-2009)\) \((x-2010)^2(x-2011)^3(x-2012)^4\), for all \(x \in R\). If \(g\) is a function defined on \(R\) with values in the interval \((0, \infty)\) such that \(f(x)=\ln (g(x))\), for all \(x \in R\), then the number of points in \(R\) at which \(g\) has a local maximum is

Consider three sets $E_1=\{1,2,3\},F_1=\{1,3,4\}$ and $G_1=\{2,3,4,5\}$. Two elements are chosen at random, without replacement, from the set $E_1$, and let $S_1$ denote the set of these chosen elements. Let $E_2=E_1-S_1$ and $F_2=F_1\cup S_1$. Now two elements are chosen at random, without replacement, from the set $F_2$ and let $S_2$ denote the set of these chosen elements. Let $G_2=G_1\cup S_2$. Finally, two elements are chosen at random, without replacement, from the set $G_2$ and let $S_3$ denote the set of these chosen elements.
Let $E_3=E_2\cup S_3$. Given that $E_1=E_3$, let $p$ be the conditional probability of the event $S_1=\{1,2\}$. Then the value of $p$ is

Let $E$ denote the parabola $y^2=8x.$ Let $P=\left(-2,4\right)$, and let $Q$ and $Q^{\text{'}}$ be two distinct points on $E$ such that the lines $PQ$ and $P{Q}^{\text{'}}$ are tangents to $E$. Let $F$ be the focus of $E$. Then which of the following statements is (are) TRUE?

A positive, singly ionized atom of mass number \(A_{\mathrm{M}}\) is accelerated from rest by the voltage \(192 \mathrm{~V}\). Thereafter, it enters a rectangular region of width \(w\) with magnetic field \(\vec{B}_0=0.1 \hat{k}\) Tesla, as shown in the figure. The ion finally hits a detector at the distance \(x\) below its starting trajectory.
[Given: Mass of neutron/proton \(=(5 / 3) \times 10^{-27} \mathrm{~kg}\), charge of the electron \(=1.6 \times 10^{-19} \mathrm{C}\).]

Which of the following option(s) is(are) correct?

Consider a neutral conducting sphere. A positive point charge is placed outside the sphere. The net charge on the sphere is then

Answer the following by appropriately matching the lists based on the information given in the paragraph.
In a thermodynamics process on an ideal monatomic gas, the infinitesimal heat absorbed by the gas is given by $\mathrm{T}\mathrm{\Delta }\mathrm{X},$ where $\mathrm{T}$ is temperature of the system and $\mathrm{\Delta }\mathrm{X}$ is the infinitesimal change in a thermodynamic quantity $\mathrm{X}$ of the system. For a mole of monatomic ideal gas $\mathrm{X}=\frac{3}{2}\mathrm{R}\mathrm{l}\mathrm{n}\left(\frac{\mathrm{T}}{{\mathrm{T}}_{\mathrm{A}}}\right)+\mathrm{R}\mathrm{l}\mathrm{n}\left(\frac{\mathrm{V}}{{\mathrm{V}}_{\mathrm{A}}}\right).$ Here, $\mathrm{R}$ is gas constant, $\mathrm{V}$ is volume of gas, ${\mathrm{T}}_{\mathrm{A}}$ and ${\mathrm{V}}_{\mathrm{A}}$ are constants.
The List-I below gives some quantities involved in a process and List-II gives some possible values of these quantities.

	List I		List II
$\left(\mathrm{a}\right)$	Work done by the system in process $1\to 2\to 3$	$\left(\mathrm{p}\right)$	$\frac{1}{3}\mathrm{R}{\mathrm{T}}_0\mathrm{l}\mathrm{n}2$
$\left(\mathrm{b}\right)$	Change in internal energy in process $1\to 2\to 3$	$\left(\mathrm{q}\right)$	$\frac{1}{3}\mathrm{R}{\mathrm{T}}_0$
$\left(\mathrm{c}\right)$	Heat absorbed by the system in process $1\to 2\to 3$	$\left(\mathrm{r}\right)$	$\mathrm{R}{\mathrm{T}}_0$
$\left(\mathrm{d}\right)$	Heat absorbed by the system in process $1\to 2$	$\left(\mathrm{s}\right)$	$\frac{4}{3}\mathrm{R}{\mathrm{T}}_0$
		$\left(\mathrm{t}\right)$	$\frac{1}{3}\mathrm{R}{\mathrm{T}}_0\left(3+\mathrm{l}\mathrm{n}2\right)$
		$\left(\mathrm{u}\right)$	$\frac{5}{6}\mathrm{R}{\mathrm{T}}_0$

If the process carried out on one mole of monatomic ideal gas is as shown in figure in the $\mathrm{T}$ - $\mathrm{V}$ -diagram with ${\mathrm{P}}_0{\mathrm{V}}_0=\frac{1}{3}\mathrm{R}{\mathrm{T}}_0,$ the correct match is,

The electrochemical extraction of aluminium from bauxite ore involves

Four point charges, each of \(+q\), are rigidly fixed at the four corners of a square planar soap film of side \(a\). The surface tension of the soap film is \(\gamma\). The system of charges and planar film are in equilibrium, and \(a=k\left[\frac{q^2}{\gamma}\right]^{1 / N}\), where \(k\) is a constant. Then, \(N\) is

The value of \(\log _{10} \mathrm{~K}\) for a reaction \(A \rightleftharpoons B\) is (Given: \(\Delta_{\mathrm{r}} H_{298 \mathrm{~K}}^{\circ}=-54.07 \mathrm{~kJ} \mathrm{~mol}^{-1}, \Delta_{\mathrm{r}} S_{298 \mathrm{~K}}^{\circ}\)\(=10 \mathrm{JK}^{-1} \mathrm{~mol}^{-1}\) and \(R=8.314 \mathrm{JK}^{-1} \mathrm{~mol}^{-1} ; 2.303 \times 8.314 \times 298\)\(=5705\) )