Match the statements/expressions given in Column I with the values given in Column II.

Area of the region bounded by the curve \(y=e^x\) and lines \(x=0\) and \(y=e\) is

Let \(S\) be the set of all seven-digit numbers that can be formed using the digits 0,1 and 2 . For example, 2210222 is in \(S\), but 0210222 is NOT in \(S\).
Then the number of elements \(x\) in \(S\) such that at least one the digits 0 and 1 appears exactly twice in \(x\), is equal to ______.

Let $f: \left(0, \infty \right)\to R$ be a differentiable function such that $f^{\text{'}}\left(x\right)=2-\frac{f\left(x\right)}{x}\forall x\in (0, \infty )$ and $f\left(1\right)\ne 1$. Then,

Let $T_1$ and $T_2$ be two distinct common tangents to the ellipse $E:\frac{x^2}{6}+\frac{y^2}{3}=1$ and the parabola $P:y^2=12x$. Suppose that the tangent $T_1$ touches $P$ and $E$ at the points $A_1$ and $A_2$, respectively and the tangent $T_2$ touches $P$ and $E$ at the points $A_4$ and $A_3$, respectively. Then which of the following statements is(are) true?

Let $f:R\to R$ be defined by $f\left(x\right)=\frac{x^2-3x-6}{x^2+2x+4}$ Then which of the following statements is(are) TRUE?

Let $O$ be the center of the circle $x^2+y^2=r^2,$ where $r>\frac{\sqrt{5}}{2}.$ Suppose $PQ$ is a chord of this circle and the equation of the line passing through $P$ and $Q$ is  $2x+4y=5$. If the center of the circumcircle of the triangle $OPQ$ lies on the line $x+2y=4,$ then the value of $r$ is ______

The general motion of a rigid body can be considered to be a combination of (i) a motion of its centre of mass about an axis, and (ii) its motion about an instantaneous axis passing through the centre of mass.
These axes need not be stationary. Consider, for example, a thin uniform disc welded (rigidly fixed) horizontally at its rim to a massless, stick, as shown in the figure. When the disc-stick system is rotated about the origin on a horizontal frictionless plane with angular speed \(\omega\), the motion at any instant can be taken as a combination of (i) a rotation of the centre of mass of the disc about the \(z\)-axis and (ii) a rotation of the disc through an instantaneous vertical axis passing through its centre of mass (as is seen from the changed orientation of points \(P\) and \(Q\) ). Both these motions have the same angular speed \(\omega\) in this case


Now consider two similar systems as shown in the figure: Case
(a) the disc with its face vertical and parallel to \(x-z\) plane; Case
(b) the disc with its face making an angle of
\(45^{\circ}\) with \(x-y\) plane and its horizontal diameter parallel to \(x\)-axis. In both the cascs, the disc is welded at point \(\mathrm{P}\), and the systems are rotated with constant angular speed \(\omega\) about the \(z\) axis.

Question : Which of the following statements regarding the angular speed about the instantaneous axis (passing through the centre of mass) is correct?

A soft plastic bottle, filled with water of density \(1 \mathrm{gm} / \mathrm{cc}\), carries an inverted glass test-tube with some air (ideal gas) trapped as shown in the figure. The test-tube has a mass of \(5 \mathrm{gm}\), and it is made of a thick glass of density \(2.5 \mathrm{gm} / \mathrm{cc}\). Initially the bottle is sealed at atmospheric pressure \(p_{0}=10^{5} \mathrm{~Pa}\) so that the volume of the trapped air is \(v_{0}=3.3 \mathrm{cc}\). When the bottle is squeezed from outside at constant temperature, the pressure inside rises and the volume of the trapped air reduces. It is found that the test tube begins to sink at pressure \(p_{0}+\Delta p\) without changing its orientation. At this pressure, the volume of the trapped air is \(v_{0}-\Delta v\).
Let \(\Delta v=X\) cc and \(\Delta p=Y \times 10^{3} \mathrm{~Pa}\).

The value of $Y$ is _____.

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

Two inductors $L_i$ (inductance $1 \mathrm{mH}$, internal resistance $3 \mathrm{\Omega }$ ) and $L_2$ (inductance $2 \mathrm{mH}$, internal resistance $4 \mathrm{\Omega }$ ), and a resistor $R$(resistance $12 \mathrm{\Omega }$ ) are all connected in parallel across a $5 \mathrm{V}$ battery. The circuit is switched on at time $t=0$. The ratio of the maximum current $\left(\frac{I_{max}}{I_{min}}\right)$ drawn from the battery is:

A parallel plate capacitor having plates of area $\mathrm{S}$ and plate separation $\mathrm{d}$ , has capacitance ${\mathrm{C}}_1$ in air. When two dielectrics of different relative permittivities ( ${\mathrm{\epsilon }}_1=2\mathrm{}$ and ${\mathrm{\epsilon }}_2=4$ ) are introduced between the two plates as shown in the figure, the capacitance becomes ${\mathrm{C}}_2$ . The ratio $\frac{{\mathrm{C}}_2}{{\mathrm{C}}_1}$ 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:
Iron is removed from chalcopyrite as