Given below are two statements: one is labelled as Assertion (A) and the other is labelled as Reason (R).
Assertion (A): Nucleus having more binding energy per nucleon is more stable.
Reason (R): Stability increase with increase in number of nucleons.
In the light of the above statements, choose the correct answer from the options given below.
1. Both (A) and (R) are true and (R) is the correct explanation of (A)
2. Both (A) and (R) are true but (R) is not the correct explanation of (A)
3. (A) is true but (R) is false
4. (A) is false but (R) is true
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Assertion is true because binding energy per nucleon is the direct measure of nuclear stability. Reason is false because stability does not simply increase with nucleon number; heavy nuclei with a very large number of nucleons become unstable.
Assertion (A): Two photons having equal wavelengths have equal linear momentum.
Reason (R): When monochromatic light shows its photon character, each photon has a linear momentum \(p = \frac{h}{\lambda}\).
1. Both (A) & (R) are true and the (R) is the correct explanation of the (A)
2. Both (A) & (R) are true but the (R) is not the correct explanation of the (A)
3. (A) is true but (R) is false
4. Both (A) and (R) are false
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The linear momentum of a photon is given by \(p = h/\lambda\). If two photons have equal wavelengths \(\lambda\), then their linear momenta \(p\) must also be equal. The Reason (R) correctly states the formula and explains Assertion (A).
Assertion (A): If the accelerating potential of a X-Ray tube is increased then the characteristic wavelength decreases.
Reason (R): The cut-off wavelength for a X-Ray tube is given by \(\lambda_{\text{min}} = \frac{hc}{eV}\), where \(V\) is accelerating potential.
1. Both (A) & (R) are true and the (R) is the correct explanation of the (A)
2. Both (A) & (R) are true but the (R) is not the correct explanation of the (A)
3. (A) is true but (R) is false
4. Both (A) and (R) are false
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Characteristic wavelength depends on target material, not accelerating potential. However, if 'characteristic wavelength' in (A) refers to 'cut-off wavelength', then \(\lambda_{\text{min}} = hc/(eV)\) implies increasing \(V\) decreases \(\lambda_{\text{min}}\). Under this interpretation, (A) is true, and (R) is true and explains (A).
Assertion (A): A photon and an electron both have energy \(50\text{ eV}\). Both have different wavelengths.
Reason (R): Wavelength depends on energy and not on mass.
1. Both (A) & (R) are true and the (R) is the correct explanation of the (A)
2. Both (A) & (R) are true but the (R) is not the correct explanation of the (A)
3. (A) is true but (R) is false
4. Both (A) and (R) are false
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For a photon, wavelength is \(\lambda_p = hc/E\). For an electron, de Broglie wavelength is \(\lambda_e = h/\sqrt{2mE}\). Since their formulas are different and \(\lambda_e\) depends on mass \(m\), their wavelengths will be different for the same energy. So (A) is true. Reason (R) is false because an electron's de Broglie wavelength depends on its mass.
Assertion (A): In photoelectric effect, cathode or emitter plate is usually coated with barium oxide, barium sulphide or strontium oxide.
Reason (R): Coating prevents cathode from erosion.
1. Both (A) & (R) are true and the (R) is the correct explanation of the (A)
2. Both (A) & (R) are true but the (R) is not the correct explanation of the (A)
3. (A) is true but (R) is false
4. Both (A) and (R) are false
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Cathodes are coated with materials like barium oxide to lower their work function, enhancing photoemission efficiency. So (A) is true. Coatings can indeed prevent erosion, so (R) is also true. However, preventing erosion is not the primary reason for choosing these specific low work function materials, so (R) is not the correct explanation for (A).
Assertion (A): A particle at rest breaks into two particles of different masses. They fly off in different directions. Their de Broglie wavelengths will be different.
Reason (R): Their speed will be different.
1. Both (A) & (R) are true and the (R) is the correct explanation of the (A)
2. Both (A) & (R) are true but the (R) is not the correct explanation of the (A)
3. (A) is true but (R) is false
4. Both (A) and (R) are false
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When a particle at rest decays, conservation of momentum dictates that the two resulting particles must have equal and opposite momenta \( |p_1| = |p_2| \). Since the de Broglie wavelength is \( \lambda = h/p \), both particles must have the same wavelength, making the assertion that they are different false.
Assertion (A): In photon-particle collision the total energy and total momentum are conserved.
Reason (R): The number of photons are conserved in a collision.
1. Both (A) & (R) are true and the (R) is the correct explanation of the (A)
2. Both (A) & (R) are true but the (R) is not the correct explanation of the (A)
3. (A) is true but (R) is false
4. Both (A) and (R) are false
View Answer
Concept: Conservation laws in collisions.
Formula: Total energy \(E_{\text{total}}\) and total momentum \(p_{\text{total}}\) are conserved in all collisions.
Solution: In photon-particle collisions, total energy and momentum are conserved. However, photons can be absorbed or emitted, so their number is not necessarily conserved. Hence, Assertion (A) is true, but Reason (R) is false.
Assertion (A): Cut-off wavelength of x-ray is independent of type of target metal
Reason (R): Wavelength of \(K_{alpha}\) x-ray depends upon type of target metal.
1. Both (A) & (R) are true and the (R) is the correct explanation of the (A)
2. Both (A) & (R) are true but the (R) is not the correct explanation of the (A)
3. (A) is true but (R) is false
4. Both (A) and (R) are false
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Concept: X-ray production mechanisms.
Formula: Cut-off wavelength \(\lambda_{\text{min}} = \frac{hc}{eV}\) (depends on voltage). Characteristic X-ray energy \(E = h\nu\) (depends on atomic transitions).
Solution: The cut-off (minimum) wavelength of continuous X-rays depends only on the accelerating voltage, not the target material. So (A) is true. \(K_{\alpha}\) X-rays are characteristic X-rays, whose wavelengths are specific to the target material. So (R) is true. However, (R) explains characteristic X-rays, not the cut-off wavelength; thus, (R) is not a correct explanation for (A).
Assertion (A): The stopping potential increases, when frequency of incident rays are increased.
Reason (R): Stopping potential is directly proportional to the frequency of incident radiation.
1. Both (A) & (R) are true and the (R) is the correct explanation of the (A)
2. Both (A) & (R) are true but the (R) is not the correct explanation of the (A)
3. (A) is true but (R) is false
4. Both (A) and (R) are false
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Concept: Einstein's photoelectric equation.
Formula: \(eV_s = hf - \phi_0\) or \(V_s = \frac{h}{e}f - \frac{\phi_0}{e}\) where \(f\) is frequency and \(\phi_0\) is work function.
Solution: From the formula, as frequency \(f\) increases, the stopping potential \(V_s\) also increases. So (A) is true. However, \(V_s\) is linearly dependent on \(f\) with an intercept of \(-\frac{\phi_0}{e}\) (unless \(\phi_0 = 0\)), not directly proportional. So (R) is false.
Assertion (A): A metallic surface is irradiated by monochromatic light of frequency \(nu > nu_0\) (the threshold frequency). The maximum kinetic energy and stopping potential are \(K_{\text{max}}\) and \(V_s\) respectively. If the frequency of incident on the surface is doubled, both \(K_{\text{max}}\) and \(V_s\) are more than doubled.
Reason (R): The maximum kinetic energy and the stopping potential of photoelectrons emitted from a surface are linearly dependent on the frequency of incident light.
1. Both (A) & (R) are true and the (R) is the correct explanation of the (A)
2. Both (A) & (R) are true but the (R) is not the correct explanation of the (A)
3. (A) is true but (R) is false
4. Both (A) and (R) are false
View Answer
Concept: Photoelectric effect and linear dependence.
Formula: \(K_{\text{max}} = h\nu - h\nu_0\) and \(V_s = \frac{h\nu}{e} - \frac{h\nu_0}{e}\) where \(h\nu_0\) is the work function.
Solution: Both \(K_{\text{max}}\) and \(V_s\) are linearly dependent on frequency \(\nu\) with a positive slope and a negative intercept (work function term). Due to this negative intercept, if \(\nu\) is doubled, \(K_{\text{max}}\) and \(V_s\) will increase by more than double. Thus, both (A) and (R) are true, and (R) correctly explains (A).