Paragraph:
Read the following passage and answer the questions.
For every function \(f(x)\) which is twice differentiable, these will be good approximation of \(\int_a^b f(x) d x \cong\left(\frac{b-a}{2}\right)\{f(a)+f(b)\}\). Now, if we take \(c=\frac{a+b}{2}\), then using above again, we get \(\int_a^b f(x) d x=\int_a^c f(x) d x+\int_c^b f(x) d x \cong \frac{b-a}{4}\{f(a)+f(b)+2 f(c)\}\) and so on.
We get approximation for value of \(\int_a^b f(x) d x\).Question:
Good approximation of \(\int_0^{\pi / 2} \sin x d x\), is

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:

Consider the region \(R=\left\{(x, y) \in \mathbb{R} \times \mathbb{R}: x \geq 0\right.\) and \(\left.y^{2} \leq 4-x\right\}\). Let \(\mathcal{F}\) be the family of all circles that are contained in \(R\) and have centers on the \(x\)-axis. Let \(C\) be the circle that has largest radius among the circles in \(\mathcal{F}\). Let \((\alpha, \beta)\) be a point where the circle \(C\) meets the curve \(y^{2}=4-x\).
The radius of the circles $C$ is____.

For a complex number $z$, let $Re\left(z\right)$ denote the real part of $z$. Let $S$ be the set of all complex numbers $z$ satisfying $z^4-|z{|}^4=4i{z}^2,$ where $i=\sqrt{-1}$. Then the minimum possible value of ${\left|z_1-z_2\right|}^2,$ where $z_1,z_2\in S$ with $Re\left(z_1\right)>0$ and $Re\left(z_2\right)<0,$ is _______

If the angles \(A, B\) and \(C\) of a triangle are in an arithmetic progression and if \(a, b\) and \(c\) denote the lengths of the sides opposite to \(A, B\) and \(C\) respectively, then the value of the expression \(\frac{a}{c} \sin 2 C+\frac{c}{a} \sin 2 A\) is

A group of 9 students, \(\mathrm{s}_1, \mathrm{~s}_2, \ldots \ldots, \mathrm{s}_9\), is to be divided to from three teams \(\mathrm{X}, \mathrm{Y}\), and \(\mathrm{Z}\) of sizes 2,3 , and 4 , respectively. Suppose that \(s_1\) cannot be selected for the team \(\mathrm{X}\), and \(\mathrm{s}_2\) cannot be selected for the team Y. Then the number of ways to from such teams, is

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 reversible cyclic process for an ideal gas is shown below. Here, $P,V,\mathrm{a}\mathrm{n}\mathrm{d}T$ are pressure, volume and temperature, respectively. The thermodynamic parameters $q,w,H\mathrm{a}\mathrm{n}\mathrm{d}U$ are heat, work, enthalpy and internal energy, respectively.

The correct option(s) is (are)

A sample \((5.6 \mathrm{~g})\) containing iron is completely dissolved in cold dilute \(\mathrm{HCl}\) to prepare a \(250 \mathrm{~mL}\) of solution. Titration of \(25.0 \mathrm{~mL}\) of this solution requires \(12.5 \mathrm{~mL}\) of \(0.03 \mathrm{M} \mathrm{KMnO}_{4}\) solution to reach the end point. Number of moles of \(\mathrm{Fe}^{2+}\) present in \(250 \mathrm{~mL}\) solution is \(\mathbf{x} \times 10^{-2}\) (consider complete dissolution of \(\mathrm{FeCl}_{2}\) ). The amount of iron present in the sample is \(\mathbf{y} \%\) by weight.
(Assume: \(\mathrm{KMnO}_{4}\) reacts only with \(\mathrm{Fe}^{2+}\) in the solution Use: Molar mass of iron as \(56 \mathrm{~g} \mathrm{~mol}^{-1}\) )
The value of $\mathrm{x}$ is________ .

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The capacitor of capacitance \(C\) can be charged (with the help of a resistance \(R\) ) by a voltage source \(V\), by closing switch \(S_1\) while keeping switch \(S_2\) open. The capacitor can be connected in series with an inductor \(L\) by closing switch \(S_2\) and opening \(S_1\).

Question :
If the total charge stored in the \(L C\) circuit is \(Q_0\), then for \(t \geq 0\)

Liquids $\mathrm{A}$ and $\mathrm{B}$ form ideal solution for all compositions of $\mathrm{A}$ and $\mathrm{B}$ at ${25}^{\circ }C$. Two such solutions with $0.25$ and $0.50$ mole fractions of $\mathrm{A}$ have the total vapor pressures of $0.3$ and $0.4$ bar, respectively. What is the vapor pressure of pure liquid $\mathrm{B}$ in bar?

The correct order of acid strength of the following carboxylic acids is -

When light of a given wavelength is incident on a metallic surface, the minimum potential needed to stop the emitted photoelectrons is $6.0 \mathrm{V}$. This potential drops to $0.6 \mathrm{V}$ if another source with wavelength four times that of the first one and intensity half of the first one is used. What are the wavelength of the first source and the work function of the metal, respectively? [Take$\frac{hc}{e}=1.24\times {10}^{-6} \mathrm{J} {\mathrm{mC}}^{-1}$ .]