Question 1
Which of the following statements is correct regarding the supplementary physical quantities, plane angle and solid angle?
✅ Correct Answer: They have units but no dimensions.
Explanation: NCERT Units and Measurements ke mutabiq, plane angle (measured in radian) aur solid angle (measured in steradian) dono ke paas specific units hoti hain. Magar jab hum inke formulas dekhte hain ($\text{angle} = \text{arc}/\text{radius}$), toh ye pure ratios hote hain, isliye ye dimensionless quantities hain.
Question 2
With a rise in temperature, the viscosity of liquids and gases changes respectively as:
✅ Correct Answer: Decreases for liquids, increases for gases
Explanation: Liquids mein viscosity intermolecular cohesive forces ki wajah se hoti hai, jo temperature badhne par kamzor ho jaati hain (viscosity decreases). Gases mein viscosity molecular collisions aur random momentum transfer ki wajah se hoti hai; temperature badhne par gas molecules ki thermal velocity badh jaati hai, jisse collisions badhte hain (viscosity increases).
Question 3
A parallel-plate capacitor with plate area $A$ and separation $d$ is being charged by a steady current $I$. The displacement current ($I_d$) passing through an imaginary plane surface of area $A/2$ parallel to the plates and situated symmetrically between them is:
✅ Correct Answer: $I/2$
Explanation: Total displacement current plates ke beech mein conduction current ke barabar hota hai ($I_d = I$). Kyunki electric field uniformly distributed hoti hai, isliye displacement current density ($J_d = I_d/A = I/A$) bhi uniform hogi. Agar hum adha area ($A/2$) lenge, toh usse pass hone wala displacement current bhi half ho jayega: $I_d' = J_d \times (A/2) = (I/A) \times (A/2) = I/2$.
Question 4
The total energy of an electron revolving in the second excited state of a hydrogen atom is $-1.51\text{ eV}$. What does the negative sign of this total energy physically signify?
✅ Correct Answer: The electron is bound to the nucleus, and energy must be supplied to remove it.
Explanation: Total energy ka negative hona ($E<0$) ek bound state ko represent karta hai. Yeh dikhata hai ki electron nucleus ke attractive gravitational/electrostatic field mein trapped hai. Agar electron ko atom se completely aazad (infinitely away) karna hai, toh hume $+1.51\text{ eV}$ energy external source se supply karni padegi.
Question 5
A heavy wooden block is placed stationary on a rough horizontal conveyor belt. When the conveyor belt suddenly switches on and accelerates forward, the static friction force acting on the wooden block does:
✅ Correct Answer: Positive work on the block
Explanation: Jab belt forward accelerate karegi, toh block belt ke relative peeche slide karne ki koshish karega. Isko rokne ke liye static friction block par forward direction mein lagega. Kyunki block friction ki wajah se forward direction mein hi displace ho raha hai, force aur displacement ke beech ka angle $0^\circ$ hai. Therefore, work done $W = F_s \cos(0^\circ) = +F_s d$ positive hoga.
Question 6
In an optical fiber communication system, light signals travel through the core via multiple total internal reflections. For this setup to function perfectly, the refractive indices of the core ($\mu_1$) and the cladding ($\mu_2$) must satisfy which relationship?
✅ Correct Answer: $\mu_1 > \mu_2$
Explanation: Total Internal Reflection (TIR) hone ki sabse basic aur mandatory shart ye hai ki light ray hamesha optically denser medium se rarer medium mein travel karni chahiye. Isliye core ka refractive index ($\mu_1$) cladding ke refractive index ($\mu_2$) se hamesha bada hona chahiye.
Question 7
An ideal gas undergoes two separate processes: one isothermal and one adiabatic. The ratio of the adiabatic bulk modulus ($B_{ad}$) to the isothermal bulk modulus ($B_{iso}$) of the gas is equal to:
✅ Correct Answer: $\gamma$
Explanation: Isothermal process ke liye gas equation $PV = \text{constant}$ hoti hai, jisse isothermal bulk modulus $B_{iso} = P$ nikalta hai. Adiabatic process ke liye equation $PV^\gamma = \text{constant}$ hoti hai, jisse adiabatic bulk modulus $B_{ad} = \gamma P$ aata hai. In dono ka ratio: $\frac{B_{ad}}{B_{iso}} = \frac{\gamma P}{P} = \gamma$.
Question 8
When a permanent bar magnet is dropped vertically downwards through a long, hollow copper tube, its acceleration ($a$) during the fall inside the tube behaves as:
✅ Correct Answer: $a < g$ initially, and eventually it moves with a constant terminal velocity ($a = 0$)
Explanation: Falling magnet ki wajah se copper tube ke cross-sections mein changing magnetic flux pass hoga, jisse eddy currents induce honge. According to Lenz's Law, ye eddy currents magnet ke motion ko oppose karenge (upward magnetic force lagayenge). Isliye acceleration $a < g$ ho jata hai. Jaise-jaise speed badhegi, opposing force badhega aur ek point par upward force weight ko balance kar dega, jisse magnet constant terminal velocity se girne lagega ($a=0$).
Question 9
Assertion (A): To safely complete a vertical circular loop of radius $R$ under gravity, the minimum speed required at the lowest point of the track must be $\sqrt{5gR}$.
Reason (R): At the highest point of the vertical circle, the tension in the string can completely vanish ($T=0$), where the required centripetal force is provided solely by the component of weight.
✅ Correct Answer: Both (A) and (R) are true and (R) is the correct explanation of (A).
Explanation: Critical condition par vertical circle looping complete karne ke liye highest point par minimum velocity $v_{top} = \sqrt{gR}$ honi chahiye, jahan tension $T=0$ ho jata hai aur weight pure centripetal force provide karta hai ($\frac{mv_{top}^2}{R} = mg$). Is energy condition ko conservation of energy se lowest point par apply karne se lowest velocity $v_{bottom} = \sqrt{5gR}$ aati hai. Reason perfectly explains Assertion.
Question 10
Assertion (A): A capacitor serves as a perfect block for a direct current (DC) circuit once it reaches its steady state.
Reason (R): The capacitive reactance ($X_C$) offered by a capacitor is inversely proportional to the frequency ($\nu$) of the applied voltage signal.
✅ Correct Answer: Both (A) and (R) are true and (R) is the correct explanation of (A).
Explanation: Capacitive reactance ka formula $X_C = \frac{1}{2\pi \nu C}$ hota hai. Direct Current (DC) ke liye frequency $\nu=0$ hoti hai. Isko formula mein put karne par $X_C = \frac{1}{0} = \infty$ (infinite resistance). Infinite reactance ki wajah se capacitor steady state mein DC current ko aage pass nahi hone deta. Reason correctly explains Assertion.
Question 11
Consider the following two statements regarding semiconductors:
Statement I: In an n-type extrinsic semiconductor, the concentration of conduction electrons ($n_e$) is vastly greater than the hole concentration ($n_h$), making the overall bulk material carry a net negative electrical charge.
Statement II: When an intrinsic semiconductor is doped with a trivalent impurity (like Indium or Boron), it forms a p-type semiconductor.
✅ Correct Answer: Statement I is incorrect but Statement II is correct.
Explanation: Statement I galat hai: No matter p-type ho ya n-type, bulk semiconductor material hamesha electrically neutral hota hai. Electrons ka number zyadah hota hai kyunki donor atoms extra valence electrons dete hain, par unke nuclei ke paas utne hi protons bhi hote hain. Statement II bilkul sahi hai: Trivalent impurities (acceptors) add karne par p-type semiconductor banta hai.
Question 12
Consider the following two statements regarding light interference in Wave Optics:
Statement I: Two independent, highly identical monochromatic laser sources can act as perfect coherent sources to produce a stable, steady interference pattern on a screen.
Statement II: Coherent sources are characterized by having a constant phase difference that does not fluctuate with time.
✅ Correct Answer: Statement I is incorrect but Statement II is correct.
Explanation: Statement I galat hai: Do independent sources kabhi bhi completely coherent nahi ho sakte. Unke atoms ke beech mein spontaneous photon emissions independent hote hain, jisse absolute phase constantly random tareeke se badalta rehta hai (rapid phase fluctuations). Statement II sahi hai: Coherent sources ki strict definition hi ye hai ki unka phase difference time ke sath constant rehna chahiye.
Question 13
A particle of mass $2\text{ kg}$ moves along the x-axis under the action of a variable force given by $F(x) = (3x^2 - 2x)\text{ N}$. If the particle starts from rest at $x=0$, its kinetic energy when it reaches the position $x=2\text{ m}$ will be:
✅ Correct Answer: $4\text{ J}$
Explanation: According to the Work-Energy Theorem, change in kinetic energy is equal to the total work done by the force ($\Delta K = W$).
$W = \int_{0}^{2} F(x) dx = \int_{0}^{2} (3x^2 - 2x) dx$
$W = \left[x^3 - x^2\right]_0^2 = (2^3 - 2^2) - 0 = 8 - 4 = 4\text{ J}$.
Since it started from rest ($K_{initial}=0$), $K_{final} = 4\text{ J}$.
Question 14
A uniform solid cylinder of mass $M$ and radius $R$ is mounted on a frictionless horizontal axle. A massless string is wound tightly around the cylinder, and a block of mass $M$ is suspended from the free end of the string. When the block is released from rest, the linear acceleration ($a$) of the falling block is:
✅ Correct Answer: $2g/3$
Explanation: Let $T$ be the tension in the string.
For the falling block: $Mg - T = Ma \implies T = M(g - a)$
For the rotating cylinder: $T \cdot R = I\alpha \implies T \cdot R = \left(\frac{1}{2}MR^2\right)\left(\frac{a}{R}\right) \implies T = \frac{1}{2}Ma$
Equating the two expressions for tension $T$:
$M(g - a) = \frac{1}{2}Ma \implies g - a = \frac{a}{2} \implies g = \frac{3a}{2} \implies a = \frac{2g}{3}$.
Question 15
A police car sounding its siren at a frequency of $800\text{ Hz}$ drops down a straight road at a constant speed of $30\text{ m/s}$ towards a stationary observer. If the velocity of sound in air is $330\text{ m/s}$, the apparent frequency ($n'$) heard by the stationary observer is:
✅ Correct Answer: $880\text{ Hz}$
Explanation: Using the Doppler Effect formula when the source is moving towards a stationary observer ($v_o=0$):
$n' = n \left(\frac{v}{v - v_s}\right)$
Given $n = 800\text{ Hz}$, $v = 330\text{ m/s}$, and $v_s = 30\text{ m/s}$.
$n' = 800 \left(\frac{330}{330 - 30}\right) = 800 \left(\frac{330}{300}\right) = 800 \times 1.1 = 880\text{ Hz}$.
Question 16
The temperature dependence of electrical resistivity ($\rho$) for a standard alloy like Nichrome is best described by which graphical or physical parameter profile?
✅ Correct Answer: Resistivity is very high and exhibits a weak, linear dependence with temperature, not dropping to zero even at 0 K.
Explanation: NCERT Current Electricity ke graphs ke mutabiq, alloys (jaise Nichrome, Manganin, Constantan) ki resistivity bohot high hoti hai aur inka temperature coefficient of resistance ($\alpha$) bohot chhota hota hai. Isliye temperature badhne par inki resistivity bohot minor tarike se linearly badhti hai, aur absolute zero (0 K) par bhi inke paas significant residual resistivity hoti hai.
Question 17
A particle executing Simple Harmonic Motion (SHM) has an amplitude $A$. At what displacement $x$ from its mean position will its kinetic energy be exactly equal to its potential energy?
✅ Correct Answer: $x = \pm A/\sqrt{2}$
Explanation: For a particle in SHM:
Kinetic Energy ($K$) = $\frac{1}{2}m\omega^2(A^2 - x^2)$
Potential Energy ($U$) = $\frac{1}{2}m\omega^2 x^2$
Given the condition $K = U$:
$\frac{1}{2}m\omega^2(A^2 - x^2) = \frac{1}{2}m\omega^2 x^2 \implies A^2 - x^2 = x^2 \implies A^2 = 2x^2 \implies x = \pm \frac{A}{\sqrt{2}}$.
Question 18
A moving coil galvanometer of resistance $G$ is converted into an ammeter using a small shunt resistance $S$, and into a voltmeter using a large series resistance $R$. Let $R_A$ be the effective total resistance of the ammeter and $R_V$ be the effective total resistance of the voltmeter. The correct relation comparing these resistances is:
✅ Correct Answer: $R_V > G > R_A$
Explanation: Ammeter banane ke liye galvanometer ke parallel mein chota resistance ($S$) lagaya jata hai. Parallel combination ka equivalent resistance humesha smallest individual component se bhi chota hota hai, so $R_A < G$. Voltmeter banane ke liye series mein badha resistance ($R$) lagaya jata hai, jisse equivalent resistance scale up ho jata hai, so $R_V = G + R \implies R_V > G$. Combining both gives: $R_V > G > R_A$.
Question 19
Inside a uniform solid sphere of mass $M$ and radius $R$, the gravitational potential ($V$) at a distance $r (r
✅ Correct Answer: It varies quadratically with $r$, reaching its minimum value at the absolute center ($r=0$).
Explanation: The formula for gravitational potential inside a solid sphere is: $V_{inside} = -\frac{GM}{2R^3}(3R^2 - r^2)$. At the center ($r=0$), $V_{center} = -\frac{1.5GM}{R}$, which represents the maximum depth or absolute minimum numerical value on the curve. As $r$ increases, the term $-r^2$ inside makes it vary parabolically/quadratically up to the surface value ($-\frac{GM}{R}$).
Question 20
At a constant temperature, if the pressure of an ideal gas inside an acoustic chamber is completely doubled, the speed of sound waves propagating through the gas will:
✅ Correct Answer: Remain completely unchanged
Explanation: According to the Laplace-corrected formula, the speed of sound in a gas is $v = \sqrt{\frac{\gamma P}{\rho}}$. Ideal gas law ke mutabiq, $P = \frac{\rho R T}{M} \implies \frac{P}{\rho} = \frac{RT}{M}$. Substituting this gives $v = \sqrt{\frac{\gamma R T}{M}}$. Kyunki temperature constant hai, pressure badhane se density bhi proportional rate par badh jayegi, jisse ratio $\frac{P}{\rho}$ bilkul constant rahega. Isliye speed of sound pressure changes par depend nahi karegi.
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