Let $f\left(x\right)= \underset{n\to \infty }{\lim }{\left(\frac{n^n\left(x+n\right)\left(x+\frac{n}{2}\right)\dots ..\left(x+\frac{n}{n}\right)}{n!\left(x^2+n^2\right) \left(x^2+\frac{n^2}{4}\right)\dots ..\left(x^2+\frac{n^2}{n^2}\right)}\right)}^{\frac{x}{n}}$ , for all $x>0$ . Then

A ship is fitted with three engines \(E_{1}, E_{2}\) and \(E_{3}\). The engines function independently of each other with respective probabilities \(\frac{1}{2}, \frac{1}{4}\) and \(\frac{1}{4}\). For the ship to be operational at least two of its engines must function. Let \(X\) denote the event that the ship is operational and let \(X_{1}, X_{2}\) and \(X_{3}\) denote respectively the events that the engines \(E_{1}, E_{2}\) and \(E_{3}\) are functioning. Which of the following is(are) true?

For any $y\in \mathrm{ℝ}$, let ${\cot }^{-1}\left(y\right)\in \left(0,\pi \right)$ and ${\tan }^{-1}\left(y\right)\in \left(-\frac{\pi }{2},\frac{\pi }{2}\right)$. Then the sum of all the solutions of the equation ${\tan }^{-1}\left(\frac{6y}{9-y^2}\right)+{\cot }^{-1}\left(\frac{9-y^2}{6y}\right)=\frac{2\pi }{3}$ for $0<\left|y\right|<3$, is equal to

Let $p,q,r$ be non-zero real numbers that are, respectively, the ${10}^{\mathrm{th} },{100}^{\mathrm{th} }$ and ${1000}^{\mathrm{th} }$ terms of a harmonic progression. Consider the system of linear equations
$x+y+z=1$ $10x+100y+1000z=0$ $qrx+pry+pqz=0$

	List-$\mathrm{I}$		List-$\mathrm{II}$
$\left(\mathrm{I}\right)$	If $\frac{q}{r}=10$, then the system of linear equations has	$\left(\mathrm{P}\right)$	$x=0,y=\frac{10}{9},z=-\frac{1}{9}$ as a solution
$\left(\mathrm{II}\right)$	If $\frac{p}{r}\ne 100$, then the system of linear equations has	$\left(\mathrm{Q}\right)$	$x=\frac{10}{9},y=-\frac{1}{9},z=0$ as a solution
$\left(\mathrm{III}\right)$	If $\frac{p}{q}\ne 10$, then the system of linear equations has	$\left(\mathrm{R}\right)$	infinitely many solutions
$\left(\mathrm{IV}\right)$	If $\frac{p}{q}=10$, then the system of linear equations has	$\left(\mathrm{S}\right)$	no solution
		$\left(\mathrm{T}\right)$	at least one solution

The correct option 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

Let \(P=\left[a_{i j}\right]\) be a \(3 \times 3\) matrix and let \(Q=\left[b_{i j}\right]\), where \(b_{i j}=2^{i+j} a_{i j}\) for \(1 \leq i, j \leq 3\). If the determinant of \(P\) is 2 , then the determinant of the matrix \(Q\) is

A pack contains n cards numbered from 1 to n. Two consecutive numbered cards are removed from the pack and the sum of the numbers on the remaining cards is 1224. If the smaller of the numbers on the removed cards is k, then $\text{k}-20=$

A piece of ice (heat capacity \(=2100 \mathrm{~J} \mathrm{~kg}^{-1}{ }^{\circ} \mathrm{C}^{-1}\) and latent heat \(=3.36 \times 10^5 \mathrm{~J} \mathrm{~kg}^{-1}\) ) of mass \(\mathrm{m}\) gram is at \(-5^{\circ} \mathrm{C}\) at atmospheric pressure. It is given \(420 \mathrm{~J}\) of heat so that the ice starts melting. Finally when the ice-water mixture is in equilibrium, it is found that \(1 \mathrm{~g}\) of ice has melted. Assuming there is no other heat exchange in the process, the value of \(m\) is

List-I shows four configurations, each consisting of a pair of ideal electric dipoles. Each dipole has a dipole moment of magnitude \(p\), oriented as marked by arrows in the figures. In all the configurations the dipoles are fixed such that they are at a distance \(2 r\) apart along the \(x\) direction. The midpoint of the line joining the two dipoles is \(X\). The possible resultant electric fields \(\vec{E}\) at \(X\) are given in List-II. Choose the option that describes the correct match between the entries in List-I to those in List-II.

	List-I		List-II
\((P)\)		\((1)\)	\(\vec{E}=0\)
\((Q)\)		\((2)\)	\(\vec{E}=-\frac{p}{2 \pi \epsilon_0 \mathrm{r}^3} \hat{\mathrm{j}}\)
\((R)\)		\((3)\)	\(\vec{E}=-\frac{p}{4 \pi \epsilon_0 \mathrm{r}^3}(\hat{\mathrm{i}}-\hat{\mathrm{j}})\)
\((S)\)		\((4)\)	\(\vec{E}=\frac{p}{4 \pi \epsilon_0 \mathrm{r}^3}(2 \hat{\mathrm{i}}-\hat{\mathrm{j}})\)
		\((5)\)	\(\vec{E}=\frac{p}{\pi \epsilon_0 \mathrm{r}^3} \hat{\mathrm{i}}\)

Two large circular discs separated by a distance of $0.01\mathrm{m}$ are connected to a battery via a switch as shown in the figure. Charged oil drops of density $900\mathrm{kg}{\mathrm{m}}^{-3}$ are released through a tiny hole at the centre of the top disc. Once some oil drops achieve terminal velocity, the switch is closed to apply a voltage of $200\mathrm{V}$ across the discs. As a result, an oil drop of radius $8\times {10}^{-7}\mathrm{m}$ stops moving vertically and floats between the discs. The number of electrons present in this oil drop is __________. (neglect the buoyancy force, take acceleration due to gravity $=10\mathrm{m}{\mathrm{s}}^{-2}$ and charge on an electron (e) $=1.6\times {10}^{-19}\mathrm{C}$)

At \(25^{\circ} \mathrm{C}\), the concentration of \(\mathrm{H}^{+}\)ions in \(1.00 \times 10^{-3} \mathrm{M}\) aqueous solution of a weak monobasic acid having acid dissociation constant \(\left(K_a\right)=4.00 \times 10^{-11}\) is \(\boldsymbol{X} \times 10^{-7} \mathrm{M}\). The value of \(\boldsymbol{X}\) is _____ .
Use: Ionic product of water \(\left(K_w\right)=1.00 \times 10^{-14}\) at \(25^{\circ} \mathrm{C}\)

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 _____

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Copper is the most noble of the first row transition metals and occurs in small deposits in several countries. Ores of copper include chalcanthite \(\left(\mathrm{CuSO}_4 \cdot 5 \mathrm{H}_2 \mathrm{O}\right)\), atacamite \(\left(\mathrm{Cu}_2 \mathrm{Cl}(\mathrm{OH})_3\right)\), cuprite \(\left(\mathrm{Cu}_2 \mathrm{O}\right)\), copper glance \(\left(\mathrm{Cu}_2 \mathrm{~S}\right)\) and malachite \(\left(\mathrm{Cu}_2(\mathrm{OH})_2 \mathrm{CO}_3\right)\). However, 80\% of the world copper production comes from the ore chalcopyrite \(\left(\mathrm{CuFeS}_2\right)\). The extraction of copper from chalcopyrite involves partial roasting, removal of iron and self-reduction.
Question:
In self-reduction, the reducing species is