Which of the following cannot be the probability of occurrence of an event?
Question: Which of the following cannot be the probability of occurrence of an event? (i) 0 (ii) $\frac{-3}{4}$ (iii) $\frac{3}{4}$ (iv) $\frac{4}{3}$ Solution: (ii) and (iv) cant be the probability of occurrence of an event. So, (ii) and (iv) are the answers to our question. Explanation: We know that, $0 \leq$ probability $\leq 1$ i.e. probability can vary from 0 to 1 (both are inclusive) So, (i) 0 can be possible as $0 \leq$ probability $\leq 1$ (ii) $-\frac{3}{4}$ is not possible as it is les...
Read More →If 5/14 Is the probability of occurrence of an event, find
Question: If 5/14 Is the probability of occurrence of an event, find (i) the odds in favor of its occurrence (ii) the odds against its occurrence Solution: (i) We know that, If odds in favor of the occurrence an event are a:b, then the probability of an event to occur is $\frac{\mathrm{a}}{\mathrm{a}+\mathrm{b}}$ Given, probability $=\frac{5}{14}$ We know, probability $=\frac{a}{a+b} .$ So, $\frac{a}{a+b}=\frac{5}{14}$ a = 5 and a+b = 14 i.e. b = 9 odds in favor of its occurrence = a:b = 5:9 Con...
Read More →Evaluate the following integrals:
Question: Evaluate the following integrals: $\int \frac{x}{\sqrt{x^{2}+x+1}} d x$ Solution: Given $I=\int \frac{x}{\sqrt{x^{2}+x+1}} d x$ Integral is of form $\int \frac{p x+q}{\sqrt{a x^{2}+b x+c}} d x$ Writing numerator as $\mathrm{px}+\mathrm{q}=\lambda\left\{\frac{\mathrm{d}}{\mathrm{dx}}\left(\mathrm{ax}^{2}+\mathrm{bx}+\mathrm{c}\right)\right\}+\mu$ $\Rightarrow p x+q=\lambda(2 a x+b)+\mu$ $\Rightarrow x=\lambda(2 x+1)+\mu$ $\therefore \lambda=1 / 2$ and $\mu=-1 / 2$ Let $x=1 / 2(2 x+1)-1 ...
Read More →Sometimes a radioactive nucleus decays
Question: Sometimes a radioactive nucleus decays into a nucleus which itself is radioactive. An example is Sulphur $\stackrel{38}{\stackrel{\text { half-life }}{=2.48 h}} C l^{38} \stackrel{\text { half-life }}{=0.62 h} A r^{38}$ (stable) Assume that we start with 1000 38S nuclei at time t = 0. The number of 38Cl is of count zero at t = 0 and will again be zero at t = . At what value of t, would the number of counts be a maximum? Solution: Let 1 and 2 be the disintegration constants for S38 and ...
Read More →Evaluate the following integrals:
Question: Evaluate the following integrals: Solution: Given $I=\int \frac{x-1}{\sqrt{x^{2}+1}} d x$ Integral is of form $\int \frac{p x+q}{\sqrt{2 x^{2}+b x+c}} d x$ Writing numerator as $\mathrm{px}+\mathrm{q}=\lambda\left\{\frac{\mathrm{d}}{\mathrm{dx}}\left(\mathrm{ax}^{2}+\mathrm{bx}+\mathrm{c}\right)\right\}+\mu$ $\Rightarrow p x+q=\lambda(2 a x+b)+\mu$ $\Rightarrow x-1=\lambda(2 x)+\mu$ Let $x-1=1 / 2(2 x)-1$ and split, $\Rightarrow \int \frac{\mathrm{x}-1}{\sqrt{\mathrm{x}^{2}+1}} \mathrm...
Read More →A nuclide 1 is said to be the mirror isobar
Question: A nuclide 1 is said to be the mirror isobar of nuclide 2 if Z1 =N2 and Z2 =N1. (a) What nuclide is a mirror isobar ofNa2311Na1123? (b) Which nuclide out of the two mirror isobars have greater binding energy and why? Solution: (a) From the question, we know that a nuclide 1 is to be the mirror isobar of nuclide 2 if Z1 = N2 and Z2 = N1 Therefore, mirror isobar is Z2 = 12 N1 and N2 = 23 -12 = 11 = Z1 (b) Mg has greater binding energy than Na....
Read More →Are the nucleons fundamental particles,
Question: Are the nucleons fundamental particles, or do they consist of still smaller parts? One way to find out is to probe a nucleon just as Rutherford probed an atom. What should be the kinetic energy of an electron for it to be able to probe a nucleon? Assume the diameter of a nucleon to be approximately 1015m. Solution: = h/p and kinetic energy = potential energy E = hc/ Kinetic energy of an electron KE = PE = hc/ = 109eV...
Read More →A piece of wood from the ruins of an ancient
Question: A piece of wood from the ruins of an ancient building was found to have a C14activity of 12 disintegrations per minute per gram of its carbon content. The 14C activity of the living wood is 16 disintegrations per minute per gram. How long ago did the tree, from which the wooden sample came, die? Given the half-life of C14is 5760 years. Solution: C14activity of a piece of wood from the ruins is R = 12 dis/min per gram C14activity of a living wood = Ro = 16 dis/min per gram Half-life of ...
Read More →Consider a radioactive nucleus
Question: Consider a radioactive nucleus A which decays to a stable nucleus C through the following sequence: A B C Here B is an intermediate nuclei which is also radioactive. Considering that there are N0atoms of A initially, plot the graph showing the variation of a number of atoms of A and B versus time. Solution: A No.of atoms B Time...
Read More →Why do stable nuclei never have
Question: Why do stable nuclei never have more protons than neutrons? Solution: A stable nuclei never have more protons than neutrons because protons are charged particles and they repel each other. The repulsion is so much that excess neutrons only produce attractive forces and this is sufficient enough to build stability....
Read More →In pair annihilation,
Question: In pair annihilation, an electron and a positron destroy each other to produce gamma radiation. How is the momentum conserved? Solution: In pair annihilation, an electron and a positron destroy each other to produce gamma radiation and their momentum is conserved as they move in opposite directions to conserve the momentum....
Read More →Which one of the following cannot
Question: Which one of the following cannot emit radiation and why? Excited nucleus, excited electron. Solution: An excited electron cannot emit radiation because the energy of the electronic energy level is in the range of eV and not in MeV....
Read More →Which sample, A or B shown in the figure
Question: Which sample, A or B shown in the figure has shorter mean-life? dN/dt A B T Solution: At t= 0, (dN/dt)A = (dN/dt)B dN/dt = -N (No)A = (No)B ANA = BNB Na NB B A...
Read More →Prove the following
Question: $H e_{2}^{3}$ and $H e_{1}^{3}$ nuclei have the same mass number. Do they have the same binding energy? Solution: He23and He13have the same mass number but the binding energy of these two nuclei is different. The binding energy of the He13is greater than the He23because the number of protons and neutrons present in both the nuclei are different. He13has one proton and two neutrons while He23has two protons and one neutron....
Read More →The variation of the decay rate of two radioactive
Question: The variation of the decay rate of two radioactive samples A and B with time is shown in the figure. Which of the following statements are true? dN/dt P B A (a) The decay constant of A is greater than that of B, hence A always decays faster than B (b) The decay constant of B is greater than that of A but its decay rate is always smaller than that of A (c) The decay constant of A is greater than that of B but it does not always decay faster than B (d) The decay constant of B is smaller ...
Read More →Samples of two radioactive nuclides A and B
Question: Samples of two radioactive nuclides A and B are taken. A and B are the disintegration constants of A and B respectively. In which of the following cases, the two samples can simultaneously have the same decay rate at any time? (a) Initial rate of decay of A is twice the initial rate of decay of B and A = B (b) Initial rate of decay of A is twice the initial rate of decay of B and A B (c) Initial rate of decay of B is twice the initial rate of decay of A and A B (d) Initial rate of deca...
Read More →Fusion processes, like combining two deuterons
Question: Fusion processes, like combining two deuterons to form a He nucleus are impossible at ordinary temperatures and pressure. The reasons for this can be traced to the fact: (a) nuclear forces have short-range (b) nuclei are positively charged (c) the original nuclei must be completely ionized before fusion can take place (d) the original nuclei must first break up before combining with each other Solution: (a) nuclear forces have short-range (b) nuclei are positively charged...
Read More →Evaluate the following integrals:
Question: Evaluate the following integrals: $\int \frac{x+2}{\sqrt{x^{2}-1}} d x$ Solution: Given $I=\int \frac{x+2}{\sqrt{x^{2}-1}} d x$ Integral is of form $\int \frac{p x+q}{\sqrt{a x^{2}+b x+c}} d x$ Writing numerator as $\mathrm{px}+\mathrm{q}=\lambda\left\{\frac{\mathrm{d}}{\mathrm{dx}}\left(\mathrm{ax}^{2}+\mathrm{bx}+\mathrm{c}\right)\right\}+\mu$ $\Rightarrow \mathrm{px}+\mathrm{q}=\lambda(2 \mathrm{ax}+\mathrm{b})+\mu$ $\Rightarrow \mathrm{x}+2=\lambda(2 \mathrm{x})+\mu$ $\therefore \l...
Read More →In a nuclear reactor, moderators slow down
Question: In a nuclear reactor, moderators slow down the neutrons which come out in a fission process. The moderator used to have light nuclei. Heavy nuclei will not serve the purpose because (a) they will break up (b) elastic collision of neutrons with heavy nuclei will not slow them down (c) the net weight of the reactor would be unbearably high (d) substances with heavy nuclei do not occur in a liquid or gaseous state at room temperature Solution: (b) elastic collision of neutrons with heavy ...
Read More →If the odds against the occurrence of an event be
Question: If the odds against the occurrence of an event be 4 : 7, find the probability of the occurrence of the event. Solution: We know that If odds in favor of the occurrence an event are a:b, then the probability of an event to occur is $\frac{a}{a+b}$ similarly, if odds are not in the favor of the occurrence an event are a:b, then the probability of not occurrence of the event is $\frac{a}{a+b}$ We also know that, Probability of occurring $=1-$ the probability of not occurring $=1-\frac{a}{...
Read More →Heavy stable nuclei have more neutrons
Question: Heavy stable nuclei have more neutrons than protons. This is because of the fact that (a) neutrons are heavier than protons (b) electrostatic force between protons are repulsive (c) neutrons decay into protons through beta decay (d) nuclear forces between neutrons are weaker than that between protons Solution: (b) electrostatic force between protons are repulsive...
Read More →Tritium is an isotope of hydrogen whose nucleus Triton contains 2
Question: Tritium is an isotope of hydrogen whose nucleus Triton contains 2 neutrons and 1 proton. Free neutrons decay intoP+e+P+e+. If one of the neutrons in Triton decays, it would transform into He3 nucleus. This does not happen. This is because (a) Triton energy is less than that of a He3nucleus (b) the electron created in the beta decay process cannot remain in the nucleus (c) both the neutrons in triton have to decay simultaneously resulting in a nucleus with 3 protons, which is not a He3n...
Read More →Mx and My denote the atomic masses
Question: Mx and My denote the atomic masses of the parent and the daughter nuclei respectively in a radioactive decay. The Q-value for a decay is Q1 and that for a + decay is Q2. If me denotes the mass of an electron, then which of the following statements is correct? (a) Q1 = (Mx My) c2and Q2 = (Mx My 2me )c2 (b) Q1 = (Mx My ) c2and Q2 = (Mx My )c2 (c) Q1 = (Mx My 2me ) c2and Q2 = (Mx My +2 me )c2 (d) Q1 = (Mx My + 2me ) c2and Q2 = (Mx My +2 me )c2 Solution: (a) Q1 = (Mx My) c2and Q2 = (Mx My ...
Read More →When a nucleus in an atom undergoes
Question: When a nucleus in an atom undergoes radioactive decay, the electronic energy levels of the atom (a) do not change for any type of radioactivity (b) change for and radioactivity but not for -radioactivity (c) change for -radioactivity but not for others (d) change for -radioactivity but not for others Solution: (b) change for and radioactivity but not for -radioactivity...
Read More →Evaluate the following integrals:
Question: Evaluate the following integrals: $\int \frac{x+2}{\sqrt{x^{2}+2 x-1}} d x$ Solution: Given $I=\int \frac{x+2}{\sqrt{x^{2}+2 x-1}} d x$ Integral is of form $\int \frac{\mathrm{px}+\mathrm{q}}{\sqrt{\mathrm{ax}^{2}+\mathrm{bx}+\mathrm{c}}} \mathrm{dx}$ Writing numerator as $\mathrm{px}+\mathrm{q}=\lambda\left\{\frac{\mathrm{d}}{\mathrm{dx}}\left(\mathrm{ax}^{2}+\mathrm{bx}+\mathrm{c}\right)\right\}+\mu$ $\Rightarrow p x+q=\lambda(2 a x+b)+\mu$ $\Rightarrow x+2=\lambda(2 x+2)+\mu$ $\ther...
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