Physics
- Two identical glass (μ_g=3/2) equiconvex lenses of focal length f each are kept in contact. The space between the two lenses is filled with water (μ_w=4/3). The focal length of the combination is Answer: 3. 3f/4 First find the radius of each lens, then the water lens, then combine powers. Glass lens: 1/f=(3/2-1)(2/R)=1/R , so R=f . Wa... 1
- A person can see clearly objects only when they lie between 50 cm and 400 cm from his eyes. In order to increase the maximum distance of distinct vision to infinity, the type and power of the correcting lens, the person has to use, will be Answer: 1. concave, -0.25 diopter The person's far point is 400 cm; to see infinity the lens must form the image of a distant object at 400... 1
- An air bubble in a glass slab with refractive index 1.5 (near normal incidence) is 5 cm deep when viewed from one surface and 3 cm deep when viewed from the opposite face. The thickness (in cm) of the slab is Answer: 2. 12 Apparent depth =real depth/μ . The bubble is at a fixed real depth x from one face and t-x from the other. x/1.5=5⇒ x=7.5 cm;... 1
- Three sound waves of equal amplitudes have frequencies (n-1), n, (n+1). They superimpose to give beats. The number of beats produced per second will be Answer: 3. 2 Add the three waves of equal amplitude and see how the intensity repeats. y=A[cos2π(n-1)t+cos2π nt+cos2π(n+1)t]=Acos2π nt (1+2c... 1
- A body of mass m is attached to the lower end of a spring whose upper end is fixed. The spring has negligible mass. When the mass m is slightly pulled down and released, it oscillates with a time period of 3 s. When the mass m is increased by 1 kg, t… Answer: 3. 9/16 Time period of a spring-mass system is T=2π√(m/k) , so T²∝ m . T₁²/T₂²=m/m+1⇒9/25=m/m+1 9m+9=25m , so 16m=9 and m=9/16 kg. O... 1
- The second overtone of an open organ pipe has the same frequency as the first overtone of a closed pipe L metre long. The length of the open pipe will be Answer: 1. 2L Compare the frequencies using v as the speed of sound. Open pipe of length l : harmonics nv/2l ; the second overtone is the 3r... 1
- A body cools from a temperature 3T to 2T in 10 minutes. The room temperature is T. Assume that Newton's law of cooling is applicable. The temperature of the body at the end of next 10 minutes will be Answer: 1. 3/2T Use Newton's law in the average form: rate of fall =k (mean temperature-T) . First 10 min: 3T-2T/10=k(3T+2T/2-T)=k·3/2T , so... 1
- Two identical bodies are made of a material for which the heat capacity increases with temperature. One of these is at 100 °C, while the other one is at 0 °C. If the two bodies are brought into contact, then, assuming no heat loss, the final common t… Answer: 1. more than 50 °C A body with larger heat capacity changes temperature less for the same heat exchanged. The 100 °C body has the la... 1
- A rectangular film of liquid is extended from (4 cm × 2 cm) to (5 cm × 4 cm). If the work done is 3×10^-4 J, the value of the surface tension of the liquid is Answer: 1. 0.125 N m ⁻¹ Work done in stretching a liquid film is W=T Δ A , where the film has two free surfaces, so Δ A=2×(change in area) .... 1
- Three liquids of densities ρ_1,ρ_2 and ρ_3 (with ρ_1>ρ_2>ρ_3), having the same value of surface tension T, rise to the same height in three identical capillaries. The angles of contact θ_1,θ_2 and θ_3 obey Answer: 1. 0≤θ₁<θ₂<θ₃<π/2 Capillary rise is h=2Tcosθ/rρ g . Here h , T and r are the same for all three liquids, so cosθ∝ρ . Since the liqui... 1
- A light rod of length l has two masses m_1 and m_2 attached to its two ends. The moment of inertia of the system about an axis perpendicular to the rod and passing through the centre of mass is Answer: 4. m₁m₂/m₁+m₂l² Take the two masses at distances x₁ and x₂ from the centre of mass, with x₁+x₂=l and m₁x₁=m₂x₂ . This gives x₁=m₂ l/... 1
- One mole of an ideal monatomic gas undergoes a process described by the equation PV^3 = constant. The heat capacity of the gas during this process is Answer: 3. R For a polytropic process PV^x= const, molar heat capacity is C=C_v+R/1-x . Monatomic gas: C_v=3/2R ; here x=3 . C=3/2R+R/1-3=3/... 1
- The temperature inside a refrigerator is t_2°C and the room temperature is t_1°C. The amount of heat delivered to the room for each joule of electrical energy consumed ideally will be Answer: 1. t₁+273/t₁-t₂ The refrigerator works as a heat pump. Ideally its coefficient of performance for the heat delivered to the room is... 1
- A solid sphere of mass m and radius R is rotating about its diameter. A solid cylinder of the same mass and same radius is also rotating about its geometrical axis with an angular speed twice that of the sphere. The ratio of their kinetic energies of… Answer: 1. 1 : 5 Rotational kinetic energy is E=1/2 Iω² , so find each moment of inertia first. Sphere about a diameter: I_s=2/5mR² , so E_s... 1
- Given the value of Rydberg constant is 10^7\ m^-1, the wave number of the last line of the Balmer series in hydrogen spectrum will be : Answer: 3. 0.25×10⁷\ m⁻¹ Key idea: barν=R(dfrac12²-dfrac1n²) ; the last Balmer line has n→∞ . barν=R/4=10⁷/4=0.25×10⁷\ m⁻¹ . 1
- A potentiometer wire is 100 cm long and a constant potential difference is maintained across it. Two cells are connected in series first to support one another and then in opposite direction. The balance points are obtained at 50 cm and 10 cm from th… Answer: 4. 3 : 2 Key idea: balance length is proportional to the emf in the circuit. Supporting: ε₁+ε₂∝50 ; opposing: ε₁-ε₂∝10 . So ε₁∝30 an... 1
- When a metallic surface is illuminated with radiation of wavelength λ, the stopping potential is V. If the same surface is illuminated with radiation of wavelength 2λ, the stopping potential is V/4. The threshold wavelength for the metallic surface i… Answer: 4. 3λ Key idea: eV=hc/λ-hc/λ₀ . eV=hc/λ-φ and eV/4=hc/2λ-φ . Subtracting: 3/4eV=hc/2λ⇒ eV=2hc/3λ . φ=hc/λ-2hc/3λ=hc/3λ , so λ₀=3λ . 1
- A small signal voltage V(t) = V_0 ω t is applied across an ideal capacitor C : Answer: 2. Over a full cycle the capacitor C does not consume any energy from the voltage source. Key idea: in an ideal capacitor, current l... 1
- An inductor 20 mH, a capacitor 50 μF and a resistor 40 Ω are connected in series across a source of emf V = 10 340 t. The power loss in A.C. circuit is : Answer: 1. 0.51 W Key idea: average power P=I_rms²R , with I_rms=V_rms/Z . X_L=ω L=340×0.02=6.8\ Ω ; X_C=dfrac1ω C≈58.8\ Ω . Z=√(40²+52²)=√(... 1
- A long solenoid has 1000 turns. When a current of 4A flows through it, the magnetic flux linked with each turn of the solenoid is 4×10^-3 Wb. The self-inductance of the solenoid is : Answer: 4. 1 H Key idea: L=Nφ/I , where φ is the flux per turn. L=dfrac1000×4×10⁻³4=1 H. 1
- To get output 1 for the following circuit, the correct choice for the input is : Answer: 4. A = 1, B = 0, C = 1 Key idea: from the circuit, Y=(A+B)· C : A and B go to an OR gate, and its output goes to an AND gate with C.... 1
- The ratio of escape velocity at earth (v_e) to the escape velocity at a planet (v_p) whose radius and mean density are twice as that of earth is : Answer: 2. 1 : 2√(2) Key idea: v_e=√(2GM/R)=R√(8π Gρ/3) , so v_e∝ Rsqrtρ . Planet: radius 2R , density 2ρ , so v_p=v_e×2×sqrt2=2sqrt2 v_e .... 1
- If the velocity of a particle is v = At + Bt^2, where A and B are constants, then the distance travelled by it between 1s and 2s is : Answer: 3. 3/2A + 7/3B Key idea: distance is the integral of speed over time (here v≥0 ). s=∫₁²(At+Bt²)dt=A4-1/2+B8-1/3=3/2A+7/3B . 1
- If the magnitude of sum of two vectors is equal to the magnitude of difference of the two vectors, the angle between these vectors is : Answer: 2. 90^° Key idea: |vec A+vec B|²=A²+B²+2ABcosθ and |vec A-vec B|²=A²+B²-2ABcosθ . Setting them equal gives 4ABcosθ=0 , so cosθ=0 and... 1