To check the principle of multiple proportions, a series of pure binary compounds $\left({\mathrm{P}}_{\mathrm{m}}{\mathrm{Q}}_{\mathrm{n}}\right)$ were analysed and their composition is tabulated below. The correct option(s) is(are)

Compound	Weight $\%$ of $\mathrm{P}$	Weight $\%$ of $\mathrm{Q}$
$1$	$50$	$50$
$2$	$44.4$	$55.6$
$3$	$40$	$60$

In dilute aqueous ${\mathrm{H}}_2\mathrm{S}{\mathrm{O}}_4$ , the complex diaquodioxalatoferrate (II) is oxidized by ${\mathrm{M}\mathrm{n}\mathrm{O}}_4^{-}$ . For this reaction, the ratio of the rate of change of $\left[{\mathrm{H}}^{+}\right]$ to the rate of change of $\left[{\mathrm{M}\mathrm{n}\mathrm{O}}_4^{-}\right]$ is

The crystalline form of borax has

Based on VSEPR theory, the number of 90 degree \(\mathrm{F}-\mathrm{Br}-\mathrm{F}\) angles in \(\mathrm{BrF}_5\) is

Paragraph: When potassium iodide is added to an aqueous solution of potassium ferricyanide, a reversible reaction is observed in which a complex \(\mathbf{P}\) is formed. In a strong acidic medium, the equilibrium shifts completely towards \(\mathbf{P}\). Addition of zinc chloride to \(\mathbf{P}\) in a slightly acidic medium results in a sparingly soluble complex Q.
Question: The number of zinc ions present in the molecular formula of \(\mathbf{Q}\) is

Which ordering of compounds is according to the decreasing order of the oxidation state of nitrogen?

An ideal gas is expanded from $({\mathrm{p}}_1\mathrm{ },\mathrm{ }{\mathrm{V}}_1\mathrm{ },\mathrm{ }{\mathrm{T}}_1)\mathrm{ }\mathrm{to}\mathrm{ }({\mathrm{p}}_2\mathrm{ },\mathrm{ }{\mathrm{V}}_2\mathrm{ },\mathrm{ }{\mathrm{T}}_2)$ under different conditions. The correct statement(s) among the following is(are)

In a conductometric titration, small volume of titrant of higher concentration is added stepwise to a larger volume of titrate of much lower concentration, and the conductance is measured after each addition.
The limiting ionic conductivity \(\left(\Lambda_0\right)\) values (in \(\mathrm{mS} \mathrm{m}^2 \mathrm{~mol}^{-1}\)) for different ions in aqueous solutions are given below:

For different combinations of titrates and titrants given in List-I, the graphs of 'conductance' versus 'volume of titrant' are given in List-II.
Match each entry in List-I with the appropriate entry in List-II and choose the correct option.

Paragraph:

Consider a simple \(R C\) circuit as shown in Figure \(1\).

Process 1: In the circuit the switch \(S\) is closed at \(t=0\) and the capacitor is fully charged to voltage \(V_{0}\) (i.e., charging continues for time \(T>>R C\) ). In the process some dissipation \(\left(E_{D}\right)\) occurs across the resistance \(R\). The amount of energy finally stored in the fully charged capacitor is \(E_{C}\).

Process 2: In a different process the voltage is first set to \(\frac{V_{0}}{3}\) and maintained for a charging time \(T>>R C\). Then the voltage is raised to \(\frac{2 V_{0}}{3}\) without discharging the capacitor and again maintained for a time \(T>>R C\). The process is repeated one more time by raising the voltage to \(V_{0}\) and the capacitor is charged to the same final voltage \(V_{0}\) as in Process 1.

These two processes are depicted in Figure \(2 .\)







Question:

In Process 2, total energy dissipated across the resistance \(E_{D}\) is:

Let $f\left(\text{x}\right)=\text{x }\text{sin }\pi \text{x}$, $\text{x}>0$. Then for all natural numbers $\text{n}\text{, }{f}^{\prime }\left(\text{x}\right)$ vanishes at

A hydrogen atom, initially at rest in its ground state, absorbs a photon of frequency \(v_1\) and ejects the electron with a kinetic energy of 10 eV . The electron then combines with a positron at rest to form a positronium atom in its ground state and simultaneously emits a photon of frequency \(v_2\). The center of mass of the resulting positronium atom moves with a kinetic energy of 5 eV . It is given that positron has the same mass as that of electron and the positronium atom can be considered as a Bohr atom, in which the electron and the positron orbit around their center of mass. Considering no other energy loss during the whole process, the difference between the two photon energies (in eV ) is \(\qquad\)

A person sitting inside an elevator performs a weighing experiment with an object of mass 50 kg . Suppose that the variation of the height \(y\) (in m ) of the elevator, from the ground, with time \(t\) (in s ) is given by \(y=8\left[1+\sin \left(\frac{2 \pi t}{T}\right)\right]\), where \(T=40 \pi \mathrm{~s}\). Taking acceleration due to gravity, \(g=10 \mathrm{~m} / \mathrm{s}^2\), the maximum variation of the object's weight (in N ) as observed in the experiment is _______ .

Let $f:\left[a, b\right]\to \left[1, \infty \right)$ be a continuous function and let $g :R\to R$ be defined as
$g\left(x\right)= \left\{\begin{array}{ccc}0& if& x<a,\\ \underset{a}{\overset{x}{\int }}f\left(t\right)dt& if& a \le x \le b,\\ \underset{a}{\overset{b}{\int }}f\left(t\right)dt& if& x>b\end{array}\right.$ . Then