Let \(f(x)\) be differentiable on the interval \((0, \infty)\) such that \(f(1)=1\), and \(\lim _{t \rightarrow x} \frac{t^2 f(x)-x^2 f(t)}{t-x}=1\) for each \(x>0\). Then, \(f(x)\) is

A student appears for a quiz consisting of only true-false type questions and answers all the questions. The student knows the answers of some questions and guesses the answers for the remaining questions. Whenever the student knows the answer of a question, he gives the correct answer. Assume that the probability of the student giving the correct answer for a question, given that he has guessed it, is \(\frac{1}{2}\). Also assume that the probability of the answer for a question being guessed, given that the student's answer is correct, is \(\frac{1}{6}\). Then the probability that the student knows the answer of a randomly chosen question is

Match the following.

Let $z$ be a complex number satisfying ${\left|z\right|}^3+2z^2+4\overline{z}-8=0$, where $\overline{z}$ denotes the complex conjugate of $z$. Let the imaginary part of $z$ be nonzero.
Match each entry in List-I to the correct entries in List-II.

List - I	List - II
(P) \(|z|^2\) is equal to	(1) 12
(Q) \(|z-\bar{z}|^2\) is equal to	(2) 4
(R) \(|z|^2+|z+\bar{z}|^2\) is equal to	(3) 8
(S) \(|z+1|^2\) is equal to	(4) 10
	(5) 7

The correct option is

Let $X$ be a set with exactly $5$ elements and $Y$ be a set with exactly $7$ elements. If $\alpha$ is the number of one-one functions from $X\to Y$ and $\beta$ is the number of onto functions from $Y\to X$, then the value of $\frac{1}{5!}\left(\beta -\alpha \right)$ is _____-.

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 Correct combination?

Let m and n be two positive integers greater than 1. If $\underset{\alpha \to 0}{\lim }\left(\frac{e^{\mathrm{c}\mathrm{o}\mathrm{s}\left({\alpha }^n \right)}-e}{{\alpha }^m}\right)= -\left(\frac{e}{2}\right)$ then the value of $\frac{m}{n}$ is

One end of a horizontal thick copper wire of length $2\mathrm{L}$and radius $2\mathrm{R}$ is welded to an end of another horizontal thin copper wire of length$\mathrm{L}$ and radius$\mathrm{R}$. When the arrangement stretched by applying forces at two ends, the ratio of the elongation in the thin wire to that in the thick wire is 

If the distance between the plane \(A x-2 y+z=d\) and the plane containing the lines \(\frac{x-1}{2}=\frac{y-2}{3}=\frac{z-3}{4} \quad\) and \(\frac{x-2}{3}=\frac{y-3}{4}=\frac{z-4}{5}\) is \(\sqrt{6}\), then \(|d|\) is

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The reactions of \(\mathrm{Cl}_{2}\) gas with cold-dilute and hot-concentrated \(\mathrm{NaOH}\) in water give sodium salts of two (different) oxoacids of chlorine, \(\boldsymbol{P}\) and \(\boldsymbol{Q}\), respectively. The \(\mathrm{Cl}_{2}\) gas reacts with \(\mathrm{SO}_{2}\) gas, in presence of charcoal, to give a product \(\boldsymbol{R}\). \(\boldsymbol{R}\) reacts with white phosphorus to give a compound \(\boldsymbol{S}\). On hydrolysis, \(\boldsymbol{S}\) gives an oxoacid of phosphorus, \(\boldsymbol{T}\).
Question:
\(P\) and \(Q\), respectively, are the sodium salts of

The compound (s) that exhibit (s) geometrical isomerism is (are)

An optical arrangement consists of two concave mirrors $M_1$ and $M_2$, and a convex lens $L$ with a common principal axis, as shown in the figure. The focal length of $L$ is $10 \mathrm{cm}$. The radii of curvature of $M_1$ and $M_2$ are $20 \mathrm{cm}$ and $24 \mathrm{cm}$, respectively. The distance between $L$ and $M_2$ is $20 \mathrm{cm}$. A point object $S$ is placed at the mid-point between $L$ and $M_2$ on the axis. When the distance between $L$ and $M_1$ is $\frac{n}{7} \mathrm{cm}$, one of the images coincides with $S$. The value of $n$ is _____(Consider the case which gives minimum value).

The reaction of \(P_2\) with \(X\) leads selectively to \(\mathrm{P}_4 \mathrm{O}_6\). The \(\mathrm{X}\) is