Mark against the correct answer in each of the following:
Question: Mark $(\sqrt{)}$ against the correct answer in each of the following: $\int \sin ^{-1}\left(\frac{2 \tan x}{1+\tan ^{2} x}\right) d x=?$ A. $-x^{2}+C$ B. $\mathrm{x}^{2}+\mathrm{C}$ C. $\frac{x^{2}}{2}+C$ D. $2 \mathrm{x}^{2}+\mathrm{C}$ Solution:...
Read More →According to the station clocks, two babies are born at the same instant,
Question: According to the station clocks, two babies are born at the same instant, one in Howrah and other in Delhi. (a) Who is elder in the frame of 2301 Up Rajdhani Express going from Howrah to Delhi? (b) Who is elder in the frame of 2302 Dn Rajdhani Express going from Delhi to Howrah? Solution:...
Read More →A person travels on a spaceship moving at a speed of
Question: A person travels on a spaceship moving at a speed of $5 \mathrm{c} / 13$ (a) Find the time interval calculated by him between the consecutive birthday celebrations of his friend on the earth. (b) Find the time interval calculated by the friend on the earth between the consecutive birthday celebrations of the traveller. Solution:...
Read More →Mark against the correct answer in each of the following:
Question: Mark $(\sqrt{)}$ against the correct answer in each of the following: $\int \cot ^{-1}\left(\frac{\sin 2 x}{1-\cos 2 x}\right) d x=?$ A. $\frac{-1}{\left(1+x^{2}\right)}+C$ B. $\frac{-1}{\left(1-x^{2}\right)}+C$ C. $\frac{x^{2}}{2}+C$ D. $2 \mathrm{x}^{2}+\mathrm{C}$ Solution:...
Read More →A person travels by a car at a speed of
Question: A person travels by a car at a speed of $180 \mathrm{~km} / \mathrm{h}$. It takes exactly 10 hours by wristwatch to go from the station A to the station B. (a) What is the rest distance between the two stations? (b) How much time is taken in the road frame by the car to go from the station A to the station B? Solution:...
Read More →Mark against the correct answer in each of the following:
Question: Mark $(\sqrt{)}$ against the correct answer in each of the following: $\int \tan ^{-1}\left\{\sqrt{\frac{1-\cos 2 x}{1+\cos 2 x}}\right\} d x=?$ A. $\frac{-1}{\left(1+x^{2}\right)}+C$ B. $\frac{1}{\sqrt{1+x^{2}}}+C$ C. $\frac{1}{\sqrt{1-x^{2}}}+C$ D. $\frac{x^{2}}{2}+C$ Solution:...
Read More →The rest distance between Patna and Delhi is
Question: The rest distance between Patna and Delhi is $1000 \mathrm{~km}$. A nonstop train travels at $360 \mathrm{~km} / \mathrm{h}$. (a) What is the distance between Patna and Delhi in the time frame? (b) How much time elapses in the time frame between Patna and Delhi? Solution:...
Read More →An aeroplane travels a rectangular field 100m × 50m,
Question: An aeroplane travels a rectangular field $100 \mathrm{~m} \times 50 \mathrm{~m}$, parallel to its length. What should be the speed of the plane so that field becomes square in the plane frame? Solution:...
Read More →A person standing on a platform finds that a train moving with velocity
Question: A person standing on a platform finds that a train moving with velocity $0.6 \mathrm{c}$ takes one second to pass by him. Find (a) the length of the train as seen by the person and (b) the rest length of the train. Solution:...
Read More →Mark against the correct answer in each of the following:
Question: Mark $(\sqrt{)}$ against the correct answer in each of the following: $\int \sin ^{-1}(\cos x) d x=?$ A. $\operatorname{cosec} x+C$ B. $\frac{\pi x}{2}+\frac{x^{2}}{2}+C$ C. $\frac{\pi x}{2}-\frac{x^{2}}{2}+C$ D. $\frac{x^{2}}{2}-\frac{\pi x}{2}+C$ Solution:...
Read More →The length of a rod is exactly 1m when measured at rest.
Question: The length of a rod is exactly $1 \mathrm{~m}$ when measured at rest. What will be its length when it moves at a speed of (a) $3 \times 10^{5} \mathrm{~m} / \mathrm{s}$, (b) $3 \times 10^{6} \mathrm{~m} / \mathrm{s}$ and (c) $3 \times 10^{7} \mathrm{~m} / \mathrm{s}$ ? Solution:...
Read More →A suitcase kept on a shop's rack is measured 50cm×25cm×10cm by shop's owner.
Question: A suitcase kept on a shop's rack is measured $50 \mathrm{~cm} \times 25 \mathrm{~cm} \times 10 \mathrm{~cm}$ by shop's owner. A traveller takes this suitcase in a train moving with a velocity $0.6 \mathrm{c}$. If the suitcase is placed with its length along the train's velocity, find the dimensions measured by (a) the traveler and (b) a ground observer. Solution:...
Read More →The guru of a yogi lives in a Himalayan cave,
Question: The guru of a yogi lives in a Himalayan cave, $1000 \mathrm{~km}$ away from the house of the yogi. The yogi claims that whenever he thinks about his guru, the guru immediately knows about it. Calculate the minimum possible time interval between the yogi thinking about the guru and the guru knowing about it. Solution:...
Read More →Mark against the correct answer in each of the following:
Question: Mark $(\sqrt{)}$ against the correct answer in each of the following: $\int \frac{1}{(1+\cos x)} d x=?$ A. $-\cot x+\operatorname{cosec} x+C$ B. $\cot x-\operatorname{cosec} x+C$ C. $\cot x+\operatorname{cosec} x+C$ D. None of these Solution:...
Read More →Calculate the energy that can be obtained from 1 kg of water through the fusion reaction
Question: Calculate the energy that can be obtained from $1 \mathrm{~kg}$ of water through the fusion reaction ${ }^{2} \mathrm{H}+{ }^{2} \mathrm{H}^{3} \mathrm{H}+\mathrm{p}$ Assume that $1.5 \times 10^{-2} \%$ of natural water is heavy water $\mathrm{D}_{2} \mathrm{O}$ (by number of molecules) and all the deuterium is used for fusion. Solution:...
Read More →Calculate the Q-value of the fusion reaction
Question: Calculate the Q-value of the fusion reaction ${ }^{4} \mathrm{He}+{ }^{4} \mathrm{He}={ }^{8} \mathrm{Be}$ Is such a fusion energetically favorable? Atomic mass of ${ }^{8} \mathrm{Be}$ is $8.0053 \mathrm{u}$ and that of $4 \mathrm{He}$ is $4.0026 \mathrm{u}$. Solution:...
Read More →Consider the fusion in helium plasma.
Question: Consider the fusion in helium plasma. Find the temperature at which the average thermal energy $1.5 \mathrm{kT}$ equals the coulomb potential energy at $2 \mathrm{fm}$. Solution:...
Read More →Calculate the Q-values of the following fusion reactions:
Question: Calculate the Q-values of the following fusion reactions: (a) ${ }_{1}^{2} H+{ }_{1}^{2} H \rightarrow{ }_{1}^{3} H+{ }_{1}^{1} H$ (b) ${ }_{1}^{2} \mathrm{H}+{ }_{1}^{2} \mathrm{H} \rightarrow{ }_{2}^{3} \mathrm{He}+n$ (c) ${ }_{1}^{2} \mathrm{H}+{ }_{1}^{3} \mathrm{H} \rightarrow{ }_{2}^{4} \mathrm{He}+n$ Atomic masses are m Solution:...
Read More →A town has a population of 1 million.
Question: A town has a population of 1 million. The average electric power needed per person is 300 W.A reactor is to be designed to supply power to this town. The efficiency with which thermal power is converted into electric power is aimed at $25 \%$. (a) Assuming $200 \mathrm{MeV}$ of thermal energy to come from each fission event on an average, find the number of events that should take place every day. (b) Assuming the fission to take place largely through ${ }^{235} \mathrm{U}$, at what So...
Read More →Mark against the correct answer in each of the following:
Question: Mark $(\sqrt{)}$ against the correct answer in each of the following: $\int \frac{(1+\sin x)}{(1-\sin x)} d x=?$ A. $2 \tan x+2 \sec x+x+C$ B. $2 \tan x+2 \sec x-x+C$ C. $\tan x+\sec x-x+C$ D. None of these Solution:...
Read More →A uranium reactor develops thermal energy at a rate of
Question: A uranium reactor develops thermal energy at a rate of $300 \mathrm{MW}$. Calculate the amount of $235 \mathrm{U}$ being consumed every second. Average energy released per fission is $200 \mathrm{MeV}$. Solution:...
Read More →Calculate the energy released by
Question: Calculate the energy released by $1 \mathrm{~g}$ of natural uranium assuming $200 \mathrm{MeV}$ is released in each fission event and that the fissionable isotope ${ }^{235} \mathrm{U}$ has an abundance of $0.7 \%$ by weight in natural uranium. Solution:...
Read More →Radioactive isotopes are produced in a nuclear physics experiment at a constant rate
Question: Radioactive isotopes are produced in a nuclear physics experiment at a constant rate $\mathrm{dN} / \mathrm{dt}=\mathrm{R}$. An inductor of inductance $100 \mathrm{mH}$, a resistance $100 \Omega$ and a battery are connected to form a series circuit. The circuit is switched on at instant the production of radioactive isotope starts. It is found that $\mathrm{i} / \mathrm{N}$ remains constant in time where $i$ is the current in the circuit at time $t$ and $N$ is the number of active nucl...
Read More →Mark against the correct answer in each of the following:
Question: Mark $(\sqrt{)}$ against the correct answer in each of the following: $\int \frac{\sin x}{(1+\sin x)} d x=?$ A. $\sec x+\tan x+x+C$ B. $\sec x-\tan x+x+C$ C. $-\sec x+\tan x+x+C$ D. None of these Solution:...
Read More →Mark against the correct answer in each of the following:
Question: Mark $(\sqrt{)}$ against the correct answer in each of the following: $\int \frac{\cot x}{(\operatorname{cosec} x-\cot x)} d x=?$ A. $=\operatorname{cosec} x-\cot x-x+C$ B. $\operatorname{cosec} x-\cot x-x+C$ C. $-\operatorname{cosec} x+\cot x-x+C$ D. $\operatorname{cosec} x+\cot x-x+C$ Solution:...
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