#mht-cet pcm
- In a series LCR circuit, \(L=\frac{1}{\pi}\) H and \(C=\frac{1}{\pi}\) μF. The resonant frequency is Answer: A. 500 Hz f=1/2π√(LC)=frac12π·frac10⁻³π=500 Hz. 1
- Eight identical rain drops, each falling with terminal velocity \(v\), coalesce to form a single drop. The terminal velocity of the big drop is Answer: A. 4v Volume conserved: R³=8r³⇒ R=2r . Since v_t∝ r² , new terminal velocity =4v . 1
- Water flows through a horizontal pipe whose cross-sectional area at a wide section is twice that at a narrow section. If the speed at the wide section is 2 m/s, the pressure difference between the wide and narrow sections is (density of water = 1000… Answer: C. 6000 Pa Continuity: v₂=2×2=4 m/s. Bernoulli: Δ P=1/2ρ(v₂²-v₁²)=500(16-4)=6000 Pa. 1
- The r.m.s. speed of the molecules of a gas at 27 °C is \(v\). The temperature at which the r.m.s. speed becomes \(2v\) is Answer: D. 927 °C v_rms∝√(T) : T₂=4×300=1200 K =927 °C. 1
- If the absolute temperature of a perfect black body is doubled, the energy radiated by it per second becomes Answer: B. 16 times Stefan's law: E∝ T⁴ , so 2⁴=16 times. 1
- The wavelength of maximum emission from the sun is 480 nm. Taking Wien's constant \(b=2.88\times10^{-3}\) m K, the surface temperature of the sun is Answer: C. 6000 K T=fracbλ_max=frac2.88×10⁻³480×10⁻⁹=6000 K. 1
- In a series RL circuit, R = 30 Ω and \(X_L\) = 40 Ω. The power factor of the circuit is Answer: D. 0.6 Z=√(30²+40²)=50\ Ω , cosφ=R/Z=0.6 . 1
- The critical angle for light going from glass (\(\mu=1.5\)) to air is Answer: B. sin⁻¹(2/3) sin C=frac1μ=2/3 . 1
- A wire of resistance R is stretched uniformly until its length is doubled. Its new resistance is Answer: C. 4R Volume constant: A→ A/2 . R=ρl/A→ρ2l/A/2=4R . 1
- An ideal gas has density 1.2 kg/m3 at a pressure of \(1\times10^5\) Pa. The r.m.s. speed of its molecules is Answer: B. 500 m/s P=1/3ρ v_rms²⇒ v_rms=√(3P/ρ)=√(3×10⁵/1.2)=500 m/s. 1
- One mole of an ideal gas expands isothermally at 300 K to twice its initial volume. The work done by the gas is approximately (R = 8.314 J mol-1 K-1, ln 2 = 0.693) Answer: B. 1729 J W=nRTlnV₂/V₁=8.314×300×0.693≈1729 J. 1
- A Carnot engine has efficiency 40% with its sink at 300 K. Keeping the sink temperature constant, by how much should the source temperature be increased to raise the efficiency to 50%? Answer: C. 100 K 0.4=1-300/T₁⇒ T₁=500 K. For 50%: T₁'=600 K. Increase =100 K. 1
- A mass attached to a spring oscillates with period \(T\). The spring is cut into two equal halves and the same mass is attached to one half. The new period is Answer: D. T/sqrt2 Half spring has k'=2k . T∝1/sqrt k⇒ T'=T/sqrt2 . 1
- The fundamental frequency of a pipe closed at one end is 250 Hz. The frequency of its first overtone is Answer: D. 750 Hz A closed pipe has only odd harmonics; first overtone =3n=750 Hz. 1
- The packing efficiency of a body-centred cubic (bcc) unit cell is Answer: C. 68% In bcc, 4r=sqrt3a ; two atoms occupy 2·4/3π r³/a³≈0.68 of the cell. 1
- Which of the following defects does NOT change the density of an ionic crystal? Answer: D. Frenkel defect In a Frenkel defect an ion only shifts to an interstitial site, so mass and volume stay the same. 1
- The pH of 0.1 M acetic acid (\(K_a=1.8\times10^{-5}\)) is approximately Answer: B. 2.87 [H^+]=√(K_aC)=sqrt1.8×10⁻⁶=1.34×10⁻³ M; pH ≈2.87 . 1
- For the rate law \(r=k[\mathrm A]^2[\mathrm B]\), if [A] is doubled and [B] is halved, the rate becomes Answer: C. 2 times r'=k(2[mathrm A])²[mathrm B]/2=2r . 1
- A tuning fork of frequency 256 Hz produces 4 beats/s with an unknown fork. When the unknown fork is loaded with a little wax (which lowers its frequency), the number of beats becomes 6 per second. The original frequency of the unknown fork is Answer: A. 252 Hz Unknown is 252 or 260 Hz. Loading lowers frequency; beats increase only if it was already below 256, so it is 252 Hz. 1
- A string 1 m long, fixed at both ends, carries transverse waves with speed 200 m/s. The frequency of its second overtone is Answer: A. 300 Hz Fundamental =v/2L=100 Hz; second overtone = third harmonic =300 Hz. 1
- In Young's double slit experiment, slit separation is 0.5 mm, the screen is 1 m away and \(\lambda=600\) nm. The fringe width is Answer: A. 1.2 mm β=λ D/d=frac600×10⁻⁹×10.5×10⁻³=1.2×10⁻³ m. 1
- Light of wavelength 500 nm falls normally on a single slit of width 0.2 mm. The width of the central maximum on a screen 2 m away is Answer: D. 10 mm W=2λ D/a=frac2×5×10⁻⁷×22×10⁻⁴=10⁻² m =10 mm. 1
- A parallel plate capacitor has capacitance \(C\) with air between the plates. A dielectric slab of dielectric constant 4 and thickness equal to half the plate separation is inserted between the plates. The new capacitance is Answer: B. 1.6 C C'=ε₀A/d-t+t/K=ε₀A/d/2+d/8=8/5·ε₀A/d=1.6C . 1
- The shape of the XeF4 molecule is Answer: A. square planar Xe has 4 bond pairs and 2 lone pairs (sp³d²). The lone pairs sit trans to each other, giving a square planar shape. 1