Physics

Step: Find the value of angle \(\theta\).
The force and the tension in the string are shown in the figure below;  
                       
From the above figure, by using the property of a linear pair we get;
\(60^{\circ}+ 90^{\circ}+\theta^{\circ} = 180^{\circ} \)
\(\Rightarrow \theta^{\circ} = 180^{\circ} – 60^{\circ}- 90^{\circ}\)
\(\Rightarrow \theta^{\circ} = 30^{\circ} \)
Therefore, the value of  angle \(\theta\)  is \(30^{\circ}\).
Hence, option (1) is the correct answer.
 

Explanation: The magnetic dipoles experience a torque due to the magnetic field produced by the current-carrying wire, resulting in a perpendicular orientation. Therefore, (A) is true.
The magnetic field lines circle around the wire, and the field direction at any point is tangent to those circles, which can be considered perpendicular in relation to the wire at a given point.
Therefore, (R) is true.
As the magnetic field influences the dipole’s orientation, therefore, both (A) and (R) are True and (R) is the correct explanation of (A).
Hence, option (1) is the correct answer. 
 

Explanation: The magnetic field within the iron is given by; \(B=\mu_0(H+M)\). Since the iron is already saturated, there’s no more room for additional alignment of magnetic domains, and \(M\) (the magnetization of the material) remains constant. The magnetization of the given material is much larger than the magnetic intensity. Therefore, if the current is increased further, the magnetic field within the iron will increase slightly.
Hence, option (2) is the correct answer.
 

Step: Find the relation between the energy levels corresponding to the hydrogen atom and \(He^+\)– ion.
\(E_{n,H}=-\frac{13.6}{n^2}~\text{eV}\)
\(E_{n,He^+}=-\frac{13.6\times 2^2}{n^2}~\text{eV}\\ E_{2n,He^+}=-\frac{13.6\times 2^2}{(2n)^2}~\text{eV}=-\frac{13.6}{n^2}~\text{eV}=E_{n,H}\)
The energy level corresponding to the  \(n^{\text{th}}\) quantum number of the  \(H\)-atom has the same value as the  \(2n^{\text{th}}\) level of the  \(He^{+}\)-ion. Therefore, the energy of the photon causing the \(n=2\rightarrow4\) transition in the  \(He^+\)-ion is equal to the one causing the  \(n=1\rightarrow2\) transition in the  \(H\) -atom.
 Hence, option (1) is the correct answer.
 

Step: Find the magnetic force per unit length.
The force per unit length between two long parallel current-carrying wires is: 
\(\dfrac{F}{L}=\dfrac{\mu_0 i_1 i_2}{2\pi r}\)
\(\Rightarrow \dfrac{F}{L}\propto \dfrac{i_2 i_1}{r}\)
Hence, option (1) is the correct answer.
 

Step: Find the change in the voltage sensitivity.
The voltage sensitivity \((S_V)\) of a moving coil galvanometer is given by;
\(S_V=\frac{NAB}{KR}\)
where, \(N = \) Number of turns, \(B=\) Magnetic field, \(A=\) Area of the coil, \(K=\) Torsional constant, and \(R=\) Resistance of the coil.
The current sensitivity of the coil is given by;
\(S_I = \frac{\theta}{I}= \frac{NBA}{K}\)
The voltage sensitivity is related to current sensitivity by the expression as;
\(S_V = \frac{S_I}{R}\)
If the number of turns \(N\) increases by \(25\%\), it would increase the current sensitivity \((S_I)\) proportionally. However, the resistance \(R\) of the coil also increases proportionally to \(N\) (since more turns mean longer wire).
Since \(S_V = \frac{S_I}{R}\) and both \(S_I\)​ and \(R\) increase proportionally, the voltage sensitivity remains unchanged.
Therefore, the change in the voltage sensitivity is zero.
Hence, option (1) is the correct answer.
 

Step: Find the time period of the combination of the springs.
As both springs are connected in parallel then the effective spring constant is given by; 
\({k}_{eq}{=}{k}_{1}{+}{k}_{2}\)
The time period of the spring combination is given by;  
\({T}{=}{2}\mathit{\pi}\sqrt{\frac{m}{k_{eq}}}\)
\(\Rightarrow {T}{=}{2}\mathit{\pi}\sqrt{\frac{m}{{k}_{1}{+}{k}_{2}}}\)
Hence, option (3) is the correct answer.
 

Step: Convert given quantities in list-I into appropriate units.
(a) \(\left [ \frac{\text{H}}{\text{s}}\right ]=\left [ \frac{L}{R}\times \frac{R}{s} \right ]=\left [ R \right ]=\Omega \)

(b) \(\left [ \text{H}\times \text{A} \right ]=\left [ LI \right ]=\left [ \phi \right ]=\text{Wb}\)

(c) \(\left [\text{H}\times \text{F}\right]=\left [\frac{L}{R}\times RC\right ]=\text{s}^2\)

(d) \(\left [\Omega \times \text{F}\right ]=\left [ RC \right ]=\text{s}\)
Here, H is Henry (SI unit of electrical inductance), F is Farad (SI unit of electrical capacitance), A is Ampere (SI unit of electric current)
Therefore, the correct match is (a) – (iv), (b) – (ii), (c) – (i), (d) – (iii).
Hence, option (2) is the correct answer. 
 

Explanation: Average velocity is the total displacement divided by time and average speed is the total distance travelled divided by time.
Since displacement is the shortest distance between the initial and final positions, it can never exceed the total distance travelled. therefore, average velocity is always less than or equal to the average speed.
Therefore, both (A) and (R) are true and (R) is the correct explanation of (A).
Hence, option (1) is the correct answer. 
 

Step: Find the heat absorbed by the system in the given cyclic process.
The work done by the gas during the cyclic process is given by; 
\(W = \pi ab\)
\(\Rightarrow W = \pi[10\times 10^{3}\times 10\times 10^{-3}]\)
\(\Rightarrow W = 100\pi ~\text{J}\)
The heat absorbed by the system in the given cyclic process is given by;
\(\Delta Q ={W}=100\pi~\text{J}= 10^{2}\pi ~\text{J}~~~~~[\Delta U =0]\)
Hence, option (2) is the correct answer.