Paragraph:
Read the following passage and answer the questions.
Let \(A B C D\) be a square of side length 2 units. \(C_2\) is the circle through vertices \(A, B, C, D\) and \(C_1\) is the circle touching all the sides of square \(A B C D\). \(L\) is the line through \(A\).Question:
A circle touches the line \(L\) and the circle \(C_1\) externally such that both the circles are on the same side of the line, then the locus of centre of the circle is

Let $S$ be the reflection of a point $Q$ with respect to the plane given by
$\overrightarrow{r}=-\left(t+p\right)\hat{i}+t\hat{j}+\left(1+p\right)\hat{k}$
where $t,p$ are real parameters and $\hat{i},\hat{j},\hat{k}$ are the unit vectors along the three positive coordinate axes. If the position vectors of $Q$ and $S$ are $10\hat{i}+15\hat{j}+20\hat{k}$ and $\alpha \hat{i}+\beta \hat{j}+\gamma \hat{k}$ respectively, then which of the following is/are TRUE?

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?

Words of length$10$ are formed using the letters A, B, C, D, E, F, G, H, I, J. Let $x$ be the number of such words where no letter is repeated; and let $y$ be the number of such words where exactly one letter is repeated twice and no other letter is repeated. The, $\frac{y}{9x}=$

Let $a, b\mathbb{ }\in \mathbb{R}$ and $a^2+b^2\ne 0$ . Suppose $S=\left\{z\in \mathbb{C}\mathbb{ }:z=\frac{1}{a+ibt}, t\mathbb{ }\in \mathbb{R},\mathbb{ }t \ne 0\right\},$ where $i= \sqrt{-1}.$ If $z=x+iy$ and $z\in S,$ then $\left(x, y\right)$ lies on

Let $F\mathbb{ }:\mathbb{ }\mathbb{R}\to \mathbb{R}$ be a thrice differentiable function. Suppose that $F\left(1\right)=0, F\left(3\right)= -4$ and $F\prime \left(x\right)<0$ all $x\in \left(\frac{1}{2}, 3\right).$ Let $f\left(x\right)=xF\left(x\right)$ for all $x\in \mathbb{ }\mathbb{R}$ .
If $\underset{1}{\overset{3}{\int }}{x}^2{F}^{\prime }\left(x\right)dx= -12$ and $\underset{1}{\overset{3}{\int }}{x}^3 F^{\prime \prime }\left(x\right)dx=40,$ then the correct expression(s) is(are)

If $y=y\left(x\right)$ satisfies the differential equation $8\sqrt{x}\left(\sqrt{9+\sqrt{x}}\right)dy={\left(\sqrt{4+\sqrt{9+\sqrt{x}}}\right)}^{-1}dx$ and $x>0, y\left(0\right)=\sqrt{7},$ then $y\left(256\right)=$

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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:
Iron is removed from chalcopyrite as

Let $a, \lambda , u\mathbb{ }\in \mathbb{R}$ . Consider the system of linear equations
$ax+2y=\lambda$
$3x-2y=\mu$
Which of the following statement(s) is (are) correct ?

Paragraph : In electromagnetic theory, the electric and magnetic phenomena are related to each other. Therefore, the dimensions of electric and magnetic quantities must also be related to each other. In the questions below, $\left[E\right] \mathrm{a}\mathrm{n}\mathrm{d} \left[B\right]$ stand for dimensions of electric and magnetic fields respectively, while $\left[{\epsilon }_0\right]$ and $\left[{\mu }_0\right]$ stand for dimensions of the permittivity and permeability of free space respectively. $\left[L\right]\mathrm{a}\mathrm{n}\mathrm{d} \left[T\right]$ are dimensions of length and time respectively. All the quantities are given in $SI$ units

Question :The relation between $\left[E\right]$ and $\left[B\right]$ is

The volume (in \(\mathrm{mL}\) ) of \(0.1 \mathrm{M} \mathrm{AgNO}_3\) required for complete precipitation of chloride ions present in \(30 \mathrm{~mL}\) of \(0.01 \mathrm{M}\) solution of \(\left[\mathrm{Cr}\left(\mathrm{H}_2 \mathrm{O}_{)_5} \mathrm{Cl}\right] \mathrm{Cl}_2\right.\), as silver chloride is close to

Paragraph :
A special metal \(S\) conducts electricity without any resistance. A closed wire loop, made of \(S\), does not allow any change in flux through itself by inducing a suitable current to generate a compensating flux. The induced current in the loop cannot decay due to its zero resistance. This current gives rise to a magnetic moment which in turn repels the source of magnetic field or flux. Consider such a loop, of radius \(a\), with its center at the origin. A magnetic dipole of moment \(m\) is brought along the axis of this loop from infinity to a point at distance \(r(\gg a)\) from the center of the loop with its north pole always facing the loop, as shown in the figure below.

The magnitude of magnetic field of a dipole \(m\), at a point on its axis at distance \(r\), is \(\frac{\mu_{0}}{2 \pi} \frac{m}{r^{3}}\), where \(\mu_{0}\) is the permeability of free space. The magnitude of the force between two magnetic dipoles with moments, \(m_{1}\) and \(m_{2}\), separated by a distance \(r\) on the common axis, with their north poles facing each other, is \(\frac{k m_{1} m_{2}}{r^{4}}\), where \(k\) is a constant of appropriate dimensions. The direction of this force is along the line joining the two dipoles.

Question :
When the dipole m is placed at a distance r from the centre of the loop (as shown in the figure), the current induced in the loop will be proportional to:

Match the compounds in LIST-$\mathrm{I}$ with the observations in LIST-$\mathrm{II}$, and choose the correct option.

	LIST-$\mathrm{I}$		LIST-$\mathrm{II}$
$\left(\mathrm{I}\right)$	Aniline	$\left(\mathrm{P}\right)$	Sodium fusion extract of the compound on boiling with ${\mathrm{FeSO}}_4$, followed by acidification with conc. ${\mathrm{H}}_2{\mathrm{SO}}_4$, gives Prussian blue color.
$\left(\mathrm{II}\right)$	$\mathrm{o}-$Cresol	$\left(\mathrm{Q}\right)$	Sodium fusion extract of the compound on treatment with sodium nitroprusside gives blood red color.
$\left(\mathrm{III}\right)$	Cysteine	$\left(\mathrm{R}\right)$	Addition of the compound to a saturated solution of ${\mathrm{NaHCO}}_3$ results in effervescence.
$\left(\mathrm{IV}\right)$	Caprolactam	$\left(\mathrm{S}\right)$	The compound reacts with bromine water to give a white precipitate.
		$\left(\mathrm{T}\right)$	Treating the compound with neutral ${\mathrm{FeCl}}_3$ solution produces violet color.