Let $f\left(x\right)=x+{\log }_{e}x-x{\log }_{\mathrm{e}}x,x\in \left(0, \infty \right)$
$•$   Column 1 contains information about zeros of $f\left(x\right)$ , $f\text{'}\left(x\right)$ and $f\text{'}\text{'}\left(x\right)$
$•$   Column 2 contains information about the limiting behaviour of $f\left(x\right)$ , $f\text{'}\left(x\right)$ and $f\text{'}\text{'}\left(x\right)$ at infinity.
$•$   Column 3 contains information about increasing-decreasing nature of $f\left(x\right)$ and $f\text{'}\left(x\right)$

Column 1	Column 2	Column 3
(I) $f\left(x\right)=0$ for some $x\in \left(1, e^2\right)$	(i) $\underset{x\to \infty }{\lim }f\left(x\right)=0$	(P) $f$ is increasing in $(0, 1)$
(II) $f^{\prime }\left(x\right)=0$ for some $x in \left(1, e\right)$	(ii) $\underset{x\to \infty }{\lim }f\left(x\right)=-\infty$	(Q) $f$ is decreasing in $\left(e, e^2\right)$
(III) $f^{\prime }\left(x\right)=0$ for some $x\in \left(0, 1\right)$	(iii) $\underset{x\to \infty }{\lim }{f}^{\prime }\left(x\right)=-\infty$	(R) $f^{\prime }$ is increasing in $(0, 1)$
(IV) $f^{\prime \prime }\left(x\right)=0$ for some $x\in \left(1, e\right)$	(iv) $\underset{x\to \infty }{\lim }{f}^{\prime \prime }\left(x\right)=0$	(S) $f^{\prime }$ is decreasing in $\left(e, e^2\right)$

The number of all possible values of \(\theta\), where \(0 < \theta < \pi\), for which the system of equations
\((y+z) \cos 3 \theta=(x y z) \sin 3 \theta \) \( x \sin 3 \theta=\frac{2 \cos 3 \theta}{y}+\frac{2 \sin 3 \theta}{z}\) and \((x y z) \sin 3 \theta=(y+2 z) \cos 3 \theta\) \(+y \sin 3 \theta\) have a solution \(\left(x_0, y_0, z_0\right)\) with \(\quad y_0 z_0 \neq 0\), is

Let two non-collinear unit vectors \(\mathbf{a}\) and \(\hat{\mathbf{b}}\) form an acute angle.
A point \(P\) moves so that at any time \(t\) the position vector \(\mathbf{O P}\) (where, \(O\) is the origin) is given by \(\mathbf{a} \cos t+\hat{\mathbf{b}} \sin t\). When \(P\) is farthest from origin \(O\), let \(M\) be the length of \(\mathbf{O P}\) and \(\hat{\mathbf{u}}\) be the unit vector along \(\mathbf{O P}\). Then,

Internal bisector of \(\angle A\) of \(\triangle A B C\) meets side \(B C\) at \(D\). A line drawn through \(D\) perpendicular to \(A D\) intersects the side \(A C\) at \(E\) and side \(A B\) at \(F\). If \(a, b\) and \(c\) represent sides of \(\triangle A B C\), then

Let \(\mathbb{R}^2\) denote \(\mathbb{R} \times \mathbb{R}\). Let \(S=\left\{(a, b, c): a, b, c \in \mathbb{R} \text { and } a x^2+2 b x y+c y^2>0 \text { for all }(x, y) \in \mathbb{R}^2-\{(0,0)\}\right\} .\)
Then which of the following statements is (are) TRUE?

Let $f:\left[0, 2\right]\to ℝ$ be the function defined by $f\left(x\right)=\left(3-\sin \left(2\pi x\right)\right)\sin \left(\pi x-\frac{\pi }{4}\right)-\sin \left(3\pi x+\frac{\pi }{4}\right)$. If $\alpha , \beta \in \left[0, 2\right]$ are such that $\left\{x\in \left[0, 2\right] :f\left(x\right)\ge 0\right\}=\left[\alpha , \beta \right]$, then the value of $\beta -\alpha$ is _____

Let \(S=\{1,2,3,4\}\). The total number of unordered pairs of disjoint subsets of \(S\) is equal to

Paragraph:
Carbon-14 is used to determine the age of organic material. The procedure is based on the formation of \({ }^{14} \mathrm{C}\) by neutron capture in the upper atmosphere.
\(
{ }_7^{14} \mathrm{~N}+{ }_0^1 n \longrightarrow{ }_6^{14} \mathrm{C}+{ }_1 n^1
\)
\({ }^{14} \mathrm{C}\) is absorbed by living organisms during photosynthesis. The \({ }^{14} \mathrm{C}\) content is constant in living organism once the plant or animal dies, the uptake of carbon dioxide by it ceases and the level of \({ }^{14} \mathrm{C}\) in the dead being, falls to the decay which \({ }^{14} \mathrm{C}\) undergoes.
\(
{ }_6^{14} \mathrm{C} \longrightarrow{ }_7^{14} \mathrm{~N}+\beta^{-}
\)
The half life period of \({ }^{14} \mathrm{C}\) is 5770 years. The decay constant \((\lambda)\) can be calculated by using the following formula \(\lambda=\frac{0.693}{t_{1 / 2}}\).
The comparison of the \(\beta^{-}\)activity of the dead matter with that of the carbon still in circulation enables measurement of the period of the isolation of the material from the living cycle. The method however, ceases to be accurate over periods longer than 30,000 years. The proportion of \({ }^{14} \mathrm{C}\) to \({ }^{12} \mathrm{C}\) in living matter is \(1: 10^{12}\).
Question:
Which of the following option is correct?

Among the following, the state function(s) is (are)

Match the temperature of a black body given in List-$\mathrm{I}$ with an appropriate statement in List-$\mathrm{II}$, and choose the correct option.
[Given: Wien’s constant as $2.9\times {10}^{-3} \mathrm{m}-\mathrm{K}$ and $\frac{hc}{e}=1.24\times {10}^{-6} \mathrm{V}-\mathrm{m}$]

	List-$\mathrm{I}$		List-$\mathrm{II}$
$\mathrm{P}$	$2000 \mathrm{K}$	$1$	The radiation at peak wavelength can lead to emission of photoelectrons from a metal of work function $4 \mathrm{eV}$
$\mathrm{Q}$	$3000 \mathrm{K}$	$2$	The radiation at peak wavelength is visible to human eye
$\mathrm{R}$	$5000 \mathrm{K}$	$3$	The radiation at peak emission wavelength will result in the widest central maximum of a single slit diffraction
$\mathrm{S}$	$10000 \mathrm{K}$	$4$	The power emitted per unit area is $\frac{1}{16}$ of that emitted by a blackbody at temperature $6000 \mathrm{K}$
		$5$	The radiation at peak emission wavelength can be used to image human bones

List-I contains various reaction sequences and List-II contains the possible products. Match each entry in List-I with the appropriate entry in List-II and choose the correct option.

Paragraph:

Let \(a, r, s, t\) be nonzero real numbers. Let \(P\left(a t^{2}, 2 a t\right), Q, R\left(a r^{2}, 2 a r\right)\) and \(S\left(a s^{2}, 2 a s\right)\) be distinct points on the parabola \(y^{2}=4 a x\). Suppose that \(P Q\) is the focal chord and lines \(Q R\) and \(P K\) are parallel, where \(K\) is the point \((2 a, 0)\).


Question:

If \(s t=1\), then the tangent at \(P\) and the normal at \(S\) to the parabola meet at a point whose ordinate is

A particle of mass $m$ is initially at rest at the origin. It is subjected to a force and starts moving along the x - axis. Its kinetic energy changes with time as $dK/dt=\gamma t,$ where $\gamma$ is a positive constant of appropriate dimensions. Which of the following statements is (are) true?