The carboxyl functional group \((-\mathrm{COOH})\) is present in

Which of the following statement(s) (are) correct regarding the root mean square speed $\left({\mathrm{U}}_{\mathrm{r}\mathrm{m}\mathrm{s}}\right)$ and average translational kinetic energy $\left({\mathrm{\epsilon }}_{\mathrm{a}\mathrm{v}}\right)$ of a molecule in a gas at equilibrium?

The correct acidity order of the following is

For the reaction sequence given below, the correct statement(s) is(are)

(In the options, \(X\) is any atom other than carbon and hydrogen, and it is different in \(\mathbf{P}, \mathbf{Q}\) and \(\mathbf{R}\) )

The desired product X can be prepared by reacting the major product of the reactions in LIST-I with one or more appropriate reagents in LIST-II.
(given, order of migratory aptitude: aryl > alkyl hydrogen)

	LIST-I		LIST -II
A)		P)	${\mathrm{I}}_2,\mathrm{NaOH}$
B)		Q)	$\left[\mathrm{Ag}{\left({\mathrm{NH}}_3\right)}_2\mathrm{OH}\right]$
C)		R)	Fehling solution
D)		S)	$\mathrm{H}\mathrm{C}\mathrm{H}\mathrm{O},\mathrm{N}\mathrm{a}\mathrm{O}\mathrm{H}$
		T)	$\mathrm{NaOBr}$

In a bimolecular reaction, the steric factor $\mathrm{P}$ was experimentally determined to be $4.5$. The correct option(s) among the following is(are)

An aqueous solution of hydrazine \(\left(\mathrm{N}_2 \mathrm{H}_4\right)\) is electrochemically oxidized by \(\mathrm{O}_2\), thereby releasing chemical energy in the form of electrical energy. One of the products generated from the electrochemical reaction is \(\mathrm{N}_2(\mathrm{~g})\).
Choose the correct statement(s) about the above process :

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The amount of energy required to break a bond is same as the amount of energy released when the same bond is formed. In gaseous state, the energy required for homolytic cleavage of a bond is called Bond Dissociation Energy (BDE) or Bond Strength. BDE is affected by \(s\)-character of the bond and the stability of the radicals formed. Shorter bonds are typically stronger bonds. BDEs for some bonds are given below:





Question:
\(\mathrm{CH}_{4}(\mathrm{~g})+\mathrm{Cl}_{2}(\mathrm{~g}) \stackrel{\text { light }}{\longrightarrow} \mathrm{CH}_{3} \mathrm{Cl}(\mathrm{g})+\mathrm{HCl}(\mathrm{g})\) the correct statement is

Which of the following is (are) NOT the square of a $3\times 3$ matrix with real entries?

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Let \(A, B, C\) be three sets of complex numbers as defined below.
\(A=\{z ; \operatorname{lm} z \geq 1\} ; B=\{z:|z-2-i|=3\} ;\) \(C=\{z: \operatorname{Re}((1-i) z)=\sqrt{2}\}\).
Question:
Let \(z\) be any point in \(A \cap B \cap C\) and let \(w\) be any point satisfying \(|w-2-i| < 3\). Then, \(|z|-|w|+3\) lies between

Consider a thin square sheet of side \(L\) and thickness \(t\), made of a material of resistivity \(\rho\). The resistance between two opposite faces, shown by the shaded areas in the figure is

Two fair dice, each with faces numbered $1,2,3,4,5$ and $6$, are rolled together and the sum of the numbers on the faces is observed. This process is repeated till the sum is either a prime number or a perfect square. Suppose the sum turns out to be a perfect square before it turns out to be a prime number. If $p$ is the probability that this perfect square is an odd number, then the value of $14p$ is_____

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A small block of mass \(1 \mathrm{~kg}\) is released from rest at the top of a rough track. The track is a circular arc of radius \(40 \mathrm{~m}\). The block slides along the track without toppling and a frictional force acts on it in the direction opposite to the instantaneous velocity. The work done in overcoming the friction up to the point \(Q\), as shown in the figure below, is \(150 \mathrm{~J}\). (Take the acceleration due to gravity, \(g=10 \mathrm{~m} \mathrm{~s}^{-2}\) ).


Question :
The magnitude of the normal reaction that acts on the block at the point \(Q\) is