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 $f\left(x\right), f^{\prime }\left(x\right)$ and $f^{\prime \prime }\left(x\right).$
$•$   Column 2 contains information about the limiting behaviour of $f\left(x\right), f^{\prime }\left(x\right)$ and $f^{\prime \prime }\left(x\right)$ at infinity.
$•$   Column 3 contains information about increasing-decreasing nature of $f\left(x\right)$ and $f^{\prime }\left(x\right).$
 

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 $\left(e, e^2\right)$

Which of the following options is the only Incorrect combination?

The point \(P\) is the intersection of the straight line joining the points \(Q(2,3,5)\) and \(R(1,-1,4)\) with the plane \(5 x-4 y-\) \(z=1\). If \(S\) is the foot of the perpendicular drawn from the point \(T(2,1,4)\) to \(Q R\), then the length of the line segment \(P S\) is

The quadratic equation $\mathrm{p}\left(\mathrm{x}\right)\mathrm{ }=\mathrm{ }0$ with real coefficients has purely imaginary roots. Then the equation $\mathrm{p}\left(\mathrm{p}\right(\mathrm{x}\left)\right)\mathrm{ }=\mathrm{ }0$ has

Let \(g_{i}:\left[\frac{\pi}{8}, \frac{3 \pi}{8}\right] \rightarrow \mathbb{R}, i=1,2\), and \(f:\left[\frac{\pi}{8}, \frac{3 \pi}{8}\right] \rightarrow \mathbb{R}\) be functions such that \(g_{1}(x)=1, g_{2}(x)=|4 x-\pi|\) and \(f(x)=\sin ^{2} x\), for all \(x \in\left[\frac{\pi}{8}, \frac{3 \pi}{8}\right]\)

Define

\[

S_{i}=\int_{\frac{\pi}{8}}^{\frac{3 \pi}{8}} f(x) \cdot g_{i}(x) d x, \quad i=1,2

\]

The value of $\frac{16S_1}{\pi }$ is ___.

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A circle \(C\) of radius 1 is inscribed in an equilateral \(\triangle P Q R\). The points of contact of \(C\) with the sides \(P Q, Q R, R P\) are \(D, E, F\) respectively. The line \(P Q\) is given by the equation
\(\sqrt{3} x+y-6=0\) and the point \(D\) is \(\left(\frac{3 \sqrt{3}}{2}, \frac{3}{2}\right)\). Further, it is given that the origin and the centre of \(C\) are on the same side of the line \(P Q\).Question:
The equation of circle \(C\) is

For $x\in \mathrm{ℝ}$, let ${\tan }^{-1}\left(x\right)\in \left(-\frac{\pi }{2},\frac{\pi }{2}\right)$. Then the minimum value of the function $f:\mathrm{ℝ}\to \mathrm{ℝ}$ defined by $f\left(x\right)={\int }_0^{x{\tan }^{-1}x}\frac{e^{\left(t-\cos t\right)}}{1+t^{2023}}dt$ is

Let \(\mathbb{R}\) denote the set of all real numbers. Let \(f: \mathbb{R} \rightarrow \mathbb{R}\) be a function such that \(f(x)>0\) for all \(x \in \mathbb{R}\), and \(f(x+y)=f(x) f(y)\) for all \(x, y \in \mathbb{R}\).
Let the real numbers \(\mathrm{a}_1, \mathrm{a}_2 \ldots, \mathrm{a}_{50}\) be in an arithmetic progression. If \(f\left(\mathrm{a}_{31}\right)=64 f\left(\mathrm{a}_{25}\right)\), and \(\sum_{i=1}^{50} f\left(a_i\right)=3\left(2^{25}+1\right)\) then the value of \(\sum_{i=6}^{30} f\left(a_i\right)\) is ________

Paragraph:

One twirls a circular ring (of mass \(M\) and radius \(R\) ) near the tip of one's finger as shown in Figure 1 . In the process the finger never loses contact with the inner rim of the ring. The finger traces out the surface of a cone, shown by the dotted line. The radius of the path traced out by the point where the ring and the finger is in contact is \(r\). The finger rotates with an angular velocity \(\omega_{0}\). The rotating ring rolls without slipping on the outside of a smaller circle described by the point where the ring and the finger is in contact (Figure 2). The coefficient of friction between the ring and the finger is \(\mu\) and the acceleration due to gravity is \(g\).








Question:

The minimum value of \(\omega_{0}\) below which the ring will drop down is

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A thermal power plant produces electric power of \(600 \mathrm{~kW}\) at \(4000 \mathrm{~V}\), which is to be transported to a place \(20 \mathrm{~km}\) away from the power plant for consumers' usage. It can be transported either directly with a cable of large current carrying capacity or by using a combination of step-up and step-down transformers at the two ends. The drawback of the direct transmission is the large energy dissipation. In the method using transformers, the dissipation is much smaller. In this method, a step-up transformer is used at the plant side so that the current is reduced to a smaller value. At the consumers' end, a step-down transformer is used to supply power to the consumers at the specified lower voltage. It is reasonable to assume that the power cable is purely resistive and the transformers are ideal with a power factor unity. All the currents and voltages mentioned are rms values.

Question :

In the method using the transformers, assume that the ratio of the number of turns in the primary to that in the secondary in the step-up transformer is \(1: 10\). If the power to the consumers has to be supplied at \(200 \mathrm{~V}\), the ratio of the number of turns in the primary to that in the secondary in the step-down transformer is

A current carrying infinitely long wire is kept along the diameter of a circular wire loop, without touching it, the correct statement(s) is(are)

The wavelength of the first spectral line in the Balmer series of hydrogen atom is \(6561 Å\). The wavelength of the second spectral line in the Balmer series of singly ionized helium atom is

A symmetric star shaped conducting wire loop is carrying a steady state current $I$ as shown in the figure. The distance between the diametrically opposite vertices of the star is $4a$ .

Three students \(S_1, S_2\) and \(S_1\) are given a problem to solve. Consider the following events :
\(U:\) At least one of \(S_1, S_2\) and \(S_3\) can solve the problem,
\(V: S_1\) can solve the problem, given that neither \(\mathrm{S}_2\) nor \(\mathrm{S}_3\) can solve the problem,
\(W: S_2\) can solve the problem and \(S_3\) cannot solve the problem,
\(T: S_3\) can solve the problem.
For any event \(E\), let \(P(E)\) denote the probability of \(E\).
If \(P(U)=\frac{1}{2}, P(V)=\frac{1}{10}\) and \(P(W)=\frac{1}{12}\), then \(P(T)\) is equal to