For the reaction sequence given below, the correct statement(s) is(are):
Amines have a lone pair of electrons on the nitrogen atom, due to which they behave as Lewis bases. Greater the value of \( K_b \) or smaller the value of \( pK_b \), stronger is the base. Amines are more basic than alcohols, ethers, esters, etc. The basic character of aliphatic amines should increase with the increase of alkyl substitution. However, it does not occur in a regular manner, as a secondary aliphatic amine is unexpectedly more basic than a tertiary amine in aqueous solutions. Aromatic amines are weaker bases than ammonia and aliphatic amines. Electron-releasing groups such as \( -CH_3 \), \( -NH_2 \), etc., increase the basicity, while electron-withdrawing substituents such as \( -NO_2 \), \( -CN \), halogens, etc., decrease the basicity of amines. The effect of these substituents is more pronounced at the para-position than at the meta-position.
Arrange the following in increasing order of their basic character. Give reason:
A solid glass sphere of refractive index $ n = \sqrt{3} $ and radius $ R $ contains a spherical air cavity of radius $ \dfrac{R}{2} $, as shown in the figure. A very thin glass layer is present at the point $ O $ so that the air cavity (refractive index $ n = 1 $) remains inside the glass sphere. An unpolarized, unidirectional and monochromatic light source $ S $ emits a light ray from a point inside the glass sphere towards the periphery of the glass sphere. If the light is reflected from the point $ O $ and is fully polarized, then the angle of incidence at the inner surface of the glass sphere is $ \theta $. The value of $ \sin \theta $ is ____
The center of a disk of radius $ r $ and mass $ m $ is attached to a spring of spring constant $ k $, inside a ring of radius $ R>r $ as shown in the figure. The other end of the spring is attached on the periphery of the ring. Both the ring and the disk are in the same vertical plane. The disk can only roll along the inside periphery of the ring, without slipping. The spring can only be stretched or compressed along the periphery of the ring, following Hooke’s law. In equilibrium, the disk is at the bottom of the ring. Assuming small displacement of the disc, the time period of oscillation of center of mass of the disk is written as $ T = \frac{2\pi}{\omega} $. The correct expression for $ \omega $ is ( $ g $ is the acceleration due to gravity):
In a scattering experiment, a particle of mass $ 2m $ collides with another particle of mass $ m $, which is initially at rest. Assuming the collision to be perfectly elastic, the maximum angular deviation $ \theta $ of the heavier particle, as shown in the figure, in radians is:
A conducting square loop initially lies in the $ XZ $ plane with its lower edge hinged along the $ X $-axis. Only in the region $ y \geq 0 $, there is a time dependent magnetic field pointing along the $ Z $-direction, $ \vec{B}(t) = B_0 (\cos \omega t) \hat{k} $, where $ B_0 $ is a constant. The magnetic field is zero everywhere else. At time $ t = 0 $, the loop starts rotating with constant angular speed $ \omega $ about the $ X $ axis in the clockwise direction as viewed from the $ +X $ axis (as shown in the figure). Ignoring self-inductance of the loop and gravity, which of the following plots correctly represents the induced e.m.f. ($ V $) in the loop as a function of time:
Figure 1 shows the configuration of main scale and Vernier scale before measurement. Fig. 2 shows the configuration corresponding to the measurement of diameter $ D $ of a tube. The measured value of $ D $ is: