38. Taking the electronic charge as ' $e$ ' and the permittivity as ' $\varepsilon _ { 0 }$ ', use dimensional analysis to determine the correct expression for $\omega _ { p }$.
(A) $\sqrt { \frac { N e } { m \varepsilon _ { 0 } } }$
(B) $\sqrt { \frac { m \varepsilon _ { 0 } } { N e } }$
(C) $\sqrt { \frac { N e ^ { 2 } } { m \varepsilon _ { 0 } } }$
(D) $\sqrt { \frac { m \varepsilon _ { 0 } } { N e ^ { 2 } } }$

ANSWER: C

1. Estimate the wavelength at which plasma reflection will occur for a metal having the density of electrons $\mathrm { N } \approx 4 \times 10 ^ { 27 } \mathrm {~m} ^ { - 3 }$. Take $\varepsilon _ { 0 } \approx 10 ^ { - 11 }$ and $\mathrm { m } \approx 10 ^ { - 30 }$, where these quantities are in proper SI units.
   (A) 800 nm
   (B) 600 nm
   (C) 300 nm
   (D) 200 nm

ANSWER: B

SECTION - IV (Total Marks : 28)

(Integer Answer Type)

This section contains $\mathbf { 7 }$ questions. The answer to each of the questions is a single-digit integer, ranging from 0 to 9 . The bubble corresponding to the correct answer is to be darkened in the ORS.

Q1. Applying the principle of homogeneity of dimensions, determine which one is correct, where $T$ is time period, $G$ is gravitational constant, $M$ is mass, $r$ is radius of orbit.
(1) $T ^ { 2 } = \frac { 4 \pi ^ { 2 } r ^ { 2 } } { G M }$
(2) $T ^ { 2 } = \frac { 4 \pi ^ { 2 } r } { G M ^ { 2 } }$
(3) $T ^ { 2 } = \frac { 4 \pi ^ { 2 } r ^ { 3 } } { G M }$
(4) $T ^ { 2 } = 4 \pi ^ { 2 } r ^ { 3 }$

Q1. To find the spring constant $( k )$ of a spring experimentally, a student commits $2 \%$ positive error in the measurement of time and $1 \%$ negative error in measurement of mass. The percentage error in determining value of $k$ is :
(1) $5 \%$
(2) $1 \%$
(3) $3 \%$
(4) $4 \%$

Q2. What is the dimensional formula of $a b ^ { - 1 }$ in the equation $\left( \mathrm { P } + \frac { \mathrm { a } } { \mathrm { V } ^ { 2 } } \right) ( \mathrm { V } - \mathrm { b } ) = \mathrm { RT }$, where letters have their usual meaning.
(1) $\left[ \mathrm { M } ^ { - 1 } \mathrm {~L} ^ { 5 } \mathrm {~T} ^ { 3 } \right]$
(2) $\left[ \mathrm { M } ^ { 6 } \mathrm {~L} ^ { 7 } \mathrm {~T} ^ { 4 } \right]$
(3) $\left[ \mathrm { ML } ^ { 2 } \mathrm {~T} ^ { - 2 } \right]$
(4) $\left[ \mathrm { M } ^ { 0 } \mathrm {~L} ^ { 3 } \mathrm {~T} ^ { - 2 } \right]$

Q2.

LIST I	LIST II
A.	Torque	I.	$\left[ M ^ { 1 } L ^ { 1 } T ^ { - 2 } A ^ { - 2 } \right]$
B.	Magnetic field	II.	$\left[ L ^ { 2 } A ^ { 1 } \right]$
C.	Magnetic moment	III.	$\left[ M ^ { 1 } T ^ { - 2 } A ^ { - 1 } \right]$
D.	Permeability of free space	IV.	$\left[ M ^ { 1 } L ^ { 2 } T ^ { - 2 } \right]$

Choose the correct answer from the options given below:
(1) A-III, B-I, C-II, D-IV
(2) A-IV, B-II, C-III, D-I
(3) A-IV, B-III, C-II, D-I
(4) A-I, B-III, C-II, D-IV