Choose a course and topic, attempt each question and open its complete worked solution when you are ready.
Name two metals that can occur in nature in the free state.
Very unreactive metals may remain uncombined in nature. Gold and platinum are standard examples; silver and copper may also occur free.
Gold and platinum.
Solution A has pH 6 and solution B has pH 8. Compare
A lower pH means a higher hydrogen-ion concentration. Because pH differs by 2 units, solution A has \(10^2=100\) times the hydrogen-ion concentration of solution B.
Since pH 6 is below 7, A is acidic. Since pH 8 is above 7, B is basic.
For y=0.06sin(2πx/3)cos(120πt), decide whether all string points share frequency, phase, and amplitude. Find the amplitude at \(x=0.375\,\mathrm{m}\).
Principle. At fixed x, the time factor is cos(120πt), while the signed spatial factor sets local amplitude and phase.
Every non-node point oscillates at 60 Hz. Points within one loop are in phase; points in adjacent loops differ by π. Amplitude varies from zero at nodes to \(0.06\,\mathrm{m}\) at antinodes.
For anticlockwise reference circles, give radius, angular speed, and initial angular position for: (a) x=−2sin(3t+π/3)(b) x=cos(π/6−t)(c) x=3sin(2πt+π/4)(d) x=2cosπt.
Principle and reasoning. Work from the governing physical relation and apply it to every stated part.
Write every motion as x=Acosθ with θ increasing anticlockwise. (a) −2sin(3t+π/3)=2cos(3t+5π/6): radius \(2\,\mathrm{cm}\), ω=3 rad s⁻¹, θ₀=5π/6.
(b) cos(π/6−t)=cos(t−π/6): radius \(1\,\mathrm{cm}\), ω=1 rad s⁻¹, θ₀=−π/6. (c) 3sin(2πt+π/4)=3cos(2πt−π/4): radius \(3\,\mathrm{cm}\), ω=2π rad s⁻¹, θ₀=−π/4. (d) already has cosine form: radius \(2\,\mathrm{cm}\), ω=π rad s⁻¹, θ₀=0.
May gauge pressures replace absolute pressures in Bernoulli's equation? Give the mathematical reason.
Principle. Subtracting the same atmospheric pressure from both pressure terms leaves every pressure difference unchanged.
Yes, provided the same reference pressure is used at all points.
A pendulum string is cut when its bob is (a) at an extreme(b) at the mean position. Describe the subsequent path.
Principle. After cutting, tension vanishes and gravity alone determines motion; the initial velocity is the pendulum velocity at the cut.
A \(40\,\mathrm{m\,s^{-1}}\) projectile is fired inside a hall whose ceiling is \(25\,\mathrm{m}\) high. Find the greatest horizontal range possible without striking the ceiling.
Principle. For launch angle theta, ceiling clearance requires u^2 sin^2(theta)/(2g) at most H. The maximum permissible angle gives the maximum allowed range on the relevant branch.
Four proposed graphs represent: position versus time, velocity versus time, speed versus time, and path length versus time. Decide which are physically impossible and state the violated condition.
Principle. A particle has one position and velocity at each instant; speed cannot be negative; accumulated path length cannot decrease.
A box reads \(2.30\,\mathrm{kg}\) on a coarse balance. Gold pieces of \(20.15\,\mathrm{g}\) and \(20.17\,\mathrm{g}\) are added. Report (a) the combined mass(b) their mass difference with proper precision.
Principle. For addition or subtraction, retain only the decimal place supported by the least precise measurement.
A fair two-faced token marked 1 and 6 and a fair die are tossed. Find the probability that their sum is (i) 3(ii) 12.
Define the sample space and count each favourable outcome once.
The final value lies in [0,1]; complement pairs also sum to one, providing an independent check.