All questions of CSIR NET Physical Science Mock Test for UGC NET Exam

In order to determine the specific heat capacity of the substance during the phase transition from liquid to solid, we need to know the values of the constants a, b, and c. Without this information, we cannot calculate the specific heat capacity accurately.

Calculation of Mean Collision Time Ratio:
- Given that the electrical conductivity of copper is approximately 90% of the electrical conductivity of silver.
- This implies that the ratio of electrical conductivities σCu/σAg is 0.9.
- Also, it is given that the electron density in silver is approximately 50% of the electron density in copper.
- This implies that the ratio of electron densities nCu/nAg is 2.
- In the Drude model, the mean collision time τ is inversely proportional to the electrical conductivity and directly proportional to the electron density.
- Therefore, the ratio of mean collision times τCu/τAg can be calculated as:
τCu/τAg = (σAg/σCu) * (nAg/nCu)
τCu/τAg = (1/0.9) * 2
τCu/τAg = 2.22

Answer: Option 'A' (0.45)
Therefore, the approximate ratio of the mean collision time in copper to the mean collision time in silver is 0.45.

Analysis:
Given letter series: __nmmn__mmnn__mnnm__

Filling the Gap:
To complete the series, we need to identify the pattern in the existing letters and then determine the missing set of letters to fill the gap.
- The first set of letters in the series is "nmmn" which is repeated after every two sets.
- The second set of letters is "mmnn" which is also repeated.
- Therefore, the missing set should also follow the same pattern.

Identifying the Correct Option:
Option 'A' is "nnmm", which fits the pattern of the series as it follows the alternating pattern of the previous sets "nmmn" and "mmnn".
Hence, the correct answer is option 'A' - nnmm.

Understanding Alpha Decay
Alpha decay involves the emission of alpha particles from a nucleus. In this case, p0210 nuclei emit two alpha particles with kinetic energies of 5.30 MeV and 4.50 MeV.
Energy Conservation in Decay
- The total energy before and after the decay must be conserved.
- The energy released during the decay is shared between the emitted alpha particles and the daughter nucleus.
Calculating the Total Energy Released
- The total kinetic energy of the emitted alpha particles is:
- Total kinetic energy = 5.30 MeV + 4.50 MeV = 9.80 MeV
Excited and Ground States of Daughter Nuclei
- Following the emission of the alpha particles, the daughter nuclei can exist in either the ground state or an excited state.
- If the daughter nucleus is in an excited state, it may release energy in the form of gamma quanta as it transitions to the ground state.
Identifying the Energy of Gamma Quanta
- The difference in energy between the total kinetic energy of the alpha particles and the energy of the excited state gives the energy of the emitted gamma quanta.
- If one daughter nucleus ends up in an excited state, the energy released as gamma radiation can be calculated as:
- Energy of gamma quanta = Total kinetic energy - Energy of excited state
- If we assume the excited state has an energy of 9.0 MeV, then:
- Energy of gamma quanta = 9.80 MeV - 9.0 MeV = 0.80 MeV
Conclusion
The energy of the gamma quanta emitted by the excited nuclei is therefore 0.80 MeV, corresponding to option 'A'. This value aligns with the principle of energy conservation during radioactive decay processes.

Problem Analysis:
We are given that the noise spectral density at any given frequency in a current amplifier is independent of frequency. We need to find the ratio of the RMS noise current before and after the bandwidth modification.

Solution:

Step 1: Understanding the Problem
To solve this problem, we need to understand the concept of noise spectral density and RMS noise current.

Step 2: Understanding Noise Spectral Density
Noise spectral density represents the amount of noise present at each frequency in a system. In this problem, we are given that the noise spectral density is independent of frequency. This means that the noise power is spread equally over all frequencies.

Step 3: Understanding RMS Noise Current
RMS noise current is a measure of the average noise current in a system. It is calculated by taking the square root of the integral of the noise spectral density over the bandwidth of the measurement.

Step 4: Calculating the Ratio of RMS Noise Current
We are given that the bandwidth of measurement is changed from 2 Hz to 30 Hz. To calculate the ratio of RMS noise current before and after the bandwidth modification, we need to compare the RMS noise current in the two cases.

Let's assume the RMS noise current before the bandwidth modification as A, and after the bandwidth modification as B.

Since the noise spectral density is independent of frequency, the noise power is spread equally over all frequencies. Therefore, the RMS noise current is directly proportional to the square root of the bandwidth.

Step 5: Applying the Formula
The ratio of the RMS noise current before and after the bandwidth modification can be calculated using the formula:

(B/A) = sqrt(B2/A2) = sqrt(30/2) = sqrt(15) ≈ 3.87

Step 6: Final Answer
Therefore, the ratio of the RMS noise current before and after the bandwidth modification is approximately 3.87.

Conclusion:
The ratio of the RMS noise current before and after the bandwidth modification is approximately 3.87.

The quantized energy levels for a particle in a 1-dimensional potential can be found by solving the time-independent Schrödinger equation:

[-(h_bar^2/2m) * d^2/dx^2 + V(x)] ψ(x) = E ψ(x)

Given that the potential is V(x) = a x^4, we can substitute this into the Schrödinger equation:

[-(h_bar^2/2m) * d^2/dx^2 + a x^4] ψ(x) = E ψ(x)

To simplify the equation, we can make a change of variables by defining y = x^2. This allows us to rewrite the equation as:

[-(h_bar^2/2m) * (1/4) d^2/dy^2 + a y^2] ψ(y) = E ψ(y)

Now, let's solve this equation. The general solution for ψ(y) can be written as:

ψ(y) = C e^(-α y^2) y^β

where α and β are constants to be determined, and C is a normalization constant.

Plugging this solution into the Schrödinger equation, we get:

[-(h_bar^2/2m) * (1/4) (2β(2β - 1) y^(2β - 2) - 4α y^2) + a y^2] C e^(-α y^2) y^β = E C e^(-α y^2) y^β

Simplifying, we find:

[-(h_bar^2/2m) * β(2β - 1) + a] y^(2β) e^(-α y^2) = E y^(2β) e^(-α y^2)

Since y = x^2, this equation can be rewritten as:

[-(h_bar^2/2m) * β(2β - 1) + a] x^(4β) e^(-α x^2) = E x^(4β) e^(-α x^2)

Dividing both sides by x^(4β) e^(-α x^2), we get:

[-(h_bar^2/2m) * β(2β - 1) + a] = E

This equation determines the quantized energy levels E in terms of the constant a and the parameter β. The value of β can be determined by considering the boundary conditions of the system. Without further information about the system, it is not possible to determine the precise dependence of E on a.

Explanation:

The grand canonical ensemble is a statistical ensemble used to describe a system in equilibrium with a reservoir of particles, energy, and volume. In this ensemble, the chemical potential (μ) is varied to control the number of particles in the system.

Statement: The number of particles in the system decreases as the chemical potential is increased.

Explanation:
When the chemical potential is increased, it means that the system is in contact with a reservoir with a higher chemical potential. This implies that there is a higher probability for particles to enter the system from the reservoir than to leave the system and enter the reservoir.

Key Points:
- The chemical potential represents the energy required to add or remove a particle from the system.
- When the chemical potential is increased, the energy required to add a particle to the system is higher than before.
- This higher energy requirement makes it less favorable for particles to enter the system.
- As a result, the number of particles in the system decreases as the chemical potential is increased.

Conclusion:
Therefore, in the grand canonical ensemble, the number of particles in the system decreases as the chemical potential is increased. This is because the higher energy requirement for particle addition makes it less likely for particles to enter the system from the reservoir.

Calculation:
- We need to first calculate the amount of Zinc and Copper in alloys X and Y.
- For alloy X: Zinc = 1/4 * 120g = 30g, Copper = 3/4 * 120g = 90g
- For alloy Y: Zinc = 2/7 * 91g = 26g, Copper = 5/7 * 91g = 65g
- Total Zinc in new alloy Z = 30g + 26g + 34g = 90g
- Total Copper in new alloy Z = 90g + 65g = 155g

Ratio Calculation:
- The ratio of Zinc to Copper in alloy Z = 90g : 155g
- Simplifying the ratio by dividing both values by 5, we get 18g : 31g
- Therefore, the ratio of Zinc to Copper in the new alloy Z is 18:31.
Therefore, option (c) 18:31 is the correct answer.

To understand why the correct answer is option 'D', let's analyze the concept of lattice vacancies and their relation to temperature.

Introduction to Lattice Vacancies:

In a crystal lattice, vacancies are the empty spaces where an atom is missing. These vacancies can occur due to various reasons such as thermal fluctuations or crystal defects. The creation of a lattice vacancy requires an input of energy, known as the vacancy formation energy.

Given Information:

According to the question, the energy required to create a lattice vacancy in the crystal is 1 eV. Additionally, we are provided with the temperatures at which the crystal is at equilibrium, 1200 K and 300 K.

Equilibrium at 300 K:

At equilibrium, the number of vacancies created is balanced by the number of vacancies annihilated. At 300 K, the crystal is at a lower temperature, which implies that the thermal energy available is less. Therefore, the number of vacancies created is comparatively smaller.

Calculating the Ratio of Number Densities:

To calculate the ratio of number densities of vacancies at 1200 K and 300 K, we need to consider the Boltzmann distribution. According to this distribution, the probability of finding an atom in a particular energy state is given by the exponential of the negative energy divided by the product of Boltzmann's constant and the temperature.

Let's denote the ratio of number densities as n(1200 K)/n(300 K). Using the Boltzmann distribution, we have:

n(1200 K)/n(300 K) = exp[(E_vacancy)/(k*T_1200)] / exp[(E_vacancy)/(k*T_300)]

where:
- E_vacancy is the vacancy formation energy (1 eV)
- k is Boltzmann's constant (8.617333262145 x 10^-5 eV/K)
- T_1200 is the temperature at 1200 K
- T_300 is the temperature at 300 K

Simplifying the Equation:

Substituting the given values into the equation, we get:

n(1200 K)/n(300 K) = exp[(1 eV)/(8.617333262145 x 10^-5 eV/K * 1200 K)] / exp[(1 eV)/(8.617333262145 x 10^-5 eV/K * 300 K)]

n(1200 K)/n(300 K) = exp[1/(8.617333262145 x 10^-5 * 1200)] / exp[1/(8.617333262145 x 10^-5 * 300)]

n(1200 K)/n(300 K) = exp(1/103.408) / exp(1/25.808)

n(1200 K)/n(300 K) ≈ exp(0.00967) / exp(0.03876)

n(1200 K)/n(300 K) ≈ 1.00972 / 1.0395

n(1200 K)/n(300 K) ≈ 0.9703

Approximating the Result:

The ratio of number densities is approximately 0.9703. Since the value is less than 1, it means that at 1200 K, there are fewer vacancies compared to 300 K.

Converting the Ratio to Exponential Form:

To convert the ratio

The density of states D(E) for a two-dimensional electron system with energy dispersion E = uk is given by:

D(E) = A / |dE/dk|

where A is a constant and dE/dk is the derivative of energy with respect to momentum.

In this case, E = uk, so we can differentiate both sides of the equation with respect to k:

dE/dk = u

Substituting this back into the expression for D(E), we get:

D(E) = A / |u|

Since u is a constant, the absolute value of u is also a constant. Therefore, the density of states D(E) is independent of the energy E. So the correct answer is:

a) 1/

Given Information:
- P is sitting at the center of the row.
- Between P and Q only one person sits.
- S sits second from the left corner.
- Between U and T only one person sits and T is at the right corner.
- R does not sit in any corner.

Arranging the Information:
- Since P is at the center, we can arrange the seating order as follows: T _ _ P _ _ _ _ S
- S sits second from the left corner, so S must be sitting to the right of P.
- Between P and Q only one person sits, so Q must be sitting to the right of P.
- Between U and T only one person sits, so U must be sitting to the left of T.
- R does not sit in any corner, so R must be placed in one of the remaining seats.

Final Seating Arrangement:
- T U R P Q V S

Answer:
Therefore, the person sitting between P and Q is U.

Understanding the Statements
The provided statements are:
1. Some insects are cockroaches.
2. Some lizards are insects.
Based on these statements, we need to analyze the conclusions.
Analyzing Conclusion I
- Conclusion I: Some cockroaches are insects.
- This conclusion directly follows from the first statement. Since it states that "some insects are cockroaches," we can infer that there is an intersection between the sets of insects and cockroaches.
- Therefore, this conclusion is valid.
Analyzing Conclusion II
- Conclusion II: Some lizards are cockroaches.
- This conclusion does not logically follow from the given statements. The second statement only indicates that some lizards belong to the category of insects but does not provide any information linking lizards directly to cockroaches.
- Hence, this conclusion cannot be established based on the provided information.
Final Evaluation of Conclusions
- Conclusion I is valid and follows from the statements.
- Conclusion II does not follow.
Correct Answer
The correct answer is option 'D' — Only conclusion I follows.

Understanding Noise in Current Amplifiers
In a current amplifier, the noise spectral density is constant across frequencies. The RMS noise current is influenced by the bandwidth of the measurement.
RMS Noise Current Calculation
The RMS noise current (I_n) can be calculated using the formula:
I_n = √(4kBT * B)
where:
- kB is the Boltzmann constant
- T is the temperature in Kelvin
- B is the bandwidth
When the bandwidth changes, the RMS noise current also changes proportionally to the square root of the bandwidth.
Initial and Final Bandwidths
- Initial Bandwidth (A): 2 Hz
- Final Bandwidth (B): 30 Hz
Calculating RMS Noise Currents
1. Initial RMS Noise Current (I_nA):
I_nA = √(4kBT * 2)
2. Final RMS Noise Current (I_nB):
I_nB = √(4kBT * 30)
Calculating the Ratio (B/A)
To find the ratio of the RMS noise currents:
- Ratio (B/A) = I_nB / I_nA
- This simplifies to:
Ratio = √(30/2) = √15
Numerical Value of the Ratio
Calculating √15 yields approximately 3.87, which corresponds to option 'C'.
Conclusion
The correct ratio of the RMS noise current before and after the bandwidth modification is 3.87. This demonstrates how noise increases with bandwidth, highlighting the importance of bandwidth selection in noise-sensitive applications.

Explanation:

To determine which of the given functions cannot be the real part of a complex analytic function, we need to analyze the Cauchy-Riemann equations.

The Cauchy-Riemann equations state that if f(z) = u(x, y) + iv(x, y) is a complex analytic function, then the partial derivatives of u and v with respect to x and y must satisfy the following conditions:

1. ∂u/∂x = ∂v/∂y
2. ∂u/∂y = -∂v/∂x

We will examine each option one by one to see if the given function satisfies the Cauchy-Riemann equations.

Option A: 3x^2y - y - y^3
To check if this function satisfies the Cauchy-Riemann equations, we need to calculate the partial derivatives:

∂u/∂x = 6xy
∂u/∂y = 3x^2 - 1 - 3y^2

∂v/∂x = 0 (since there is no variable 'v' in the function)
∂v/∂y = 0 (since there is no variable 'v' in the function)

Comparing the partial derivatives, we can see that ∂u/∂x ≠ ∂v/∂y and ∂u/∂y ≠ -∂v/∂x. Therefore, this function does not satisfy the Cauchy-Riemann equations and cannot be the real part of a complex analytic function.

Option B: x^2y
To check if this function satisfies the Cauchy-Riemann equations, we need to calculate the partial derivatives:

∂u/∂x = 2xy
∂u/∂y = x^2

∂v/∂x = 0 (since there is no variable 'v' in the function)
∂v/∂y = 0 (since there is no variable 'v' in the function)

Comparing the partial derivatives, we can see that ∂u/∂x ≠ ∂v/∂y and ∂u/∂y ≠ -∂v/∂x. Therefore, this function does not satisfy the Cauchy-Riemann equations and cannot be the real part of a complex analytic function.

Option C: x^3 - 3xy^2
To check if this function satisfies the Cauchy-Riemann equations, we need to calculate the partial derivatives:

∂u/∂x = 3x^2 - 3y^2
∂u/∂y = -6xy

∂v/∂x = 0 (since there is no variable 'v' in the function)
∂v/∂y = 0 (since there is no variable 'v' in the function)

Comparing the partial derivatives, we can see that ∂u/∂x ≠ ∂v/∂y and ∂u/∂y ≠ -∂v/∂x. Therefore, this function does not satisfy the Cauchy-Riemann equations and cannot be the real

Escape velocity is the minimum velocity required for an object to escape the gravitational pull of a celestial body and move away indefinitely. It can be calculated using the formula:

v = √(2gR)

Where:
- v is the escape velocity
- g is the acceleration due to gravity
- R is the radius of the celestial body

We are given that the acceleration due to gravity on Earth (gE) is approximately 2.6 times that on Mars (gM). We are also given that the radius of Mars (RM) is about one half the radius of Earth (RE).

So, we need to find the ratio of the escape velocity on Mars (vM) to that on Earth (vE).

Let's calculate the escape velocity on Earth and Mars separately.

Escape velocity on Earth (vE):
vE = √(2gERE)

Escape velocity on Mars (vM):
vM = √(2gMRM)

Now, let's substitute the given values into the equations and calculate the ratio.

Calculation:
vE = √(2gERE)
vM = √(2gMRM)

Since gM = gE/2.6 and RM = RE/2, we can substitute these values into the equations.

vE = √(2gERE)
= √(2(gE/2.6)(RE))
= √((2gERE)/2.6)
= √((2gE/2.6)(RE))
= (1/√2.6) √(2gERE)

vM = √(2gMRM)
= √(2(gE/2.6)(RE/2))
= √((2gERE)/2.6)
= (1/√2.6) √(2gERE)

We can see that the expressions for vE and vM are the same. Therefore, the ratio of vM to vE is 1.

So, the ratio of the escape velocity on Mars to that on Earth is approximately 1 or 100%.

Therefore, the correct answer is option A) 0.44.

Given:
Rohan sells two commodities for Rs. 19,800 each.
Profit on one commodity: 10%
Loss on the other commodity: 10%

To find:
Overall profit or loss percent
Amount of profit or loss

Solution:

Calculating Profit and Loss:
Let the cost price of the first commodity be Rs. x.
Selling price of the first commodity = Rs. 19,800
Profit on the first commodity = 10%
Profit = (10/100) * x = 0.1x
Selling price of the second commodity = Rs. 19,800
Loss on the second commodity = 10%
Loss = (10/100) * x = 0.1x

Calculating Overall Cost Price and Selling Price:
Total cost price = Cost price of first commodity + Cost price of second commodity
Total cost price = x + x = 2x
Total selling price = Selling price of first commodity + Selling price of second commodity
Total selling price = 19,800 + 19,800 = Rs. 39,600

Calculating Profit or Loss:
Total profit = Profit on first commodity - Loss on second commodity
Total profit = 0.1x - 0.1x = 0
As there is no profit, total loss = Rs. 0

Calculating Overall Profit or Loss Percent:
Overall profit or loss percent = (Total loss / Total cost price) * 100
Overall profit or loss percent = (0 / 2x) * 100
Overall profit or loss percent = 0%
Since there is no profit, the answer is a 0% loss, which is not provided in the options.

Explanation:

Rayleigh Scattering:
- Rayleigh scattering (IR) is the scattering of light by particles that are much smaller than the wavelength of the light.
- It is the least intense component in Raman scattering.

Stokes and Anti-Stokes Lines:
- In Raman scattering, when a photon is scattered by a molecule, the scattered photon can have a lower energy (Stokes line) or higher energy (anti-Stokes line) compared to the incident photon.
- Stokes lines (IS) are more intense than anti-Stokes lines (IAS) because the population of molecules in the lower energy state is higher than in the higher energy state.

Order of Intensities:
- The correct order of intensities in Raman scattering is IR > IS > IAS.
- This means that Rayleigh scattering (IR) is the least intense, followed by the most intense Stokes line (IS), and then the most intense anti-Stokes line (IAS).
Therefore, the correct order of intensities is IR > IS > IAS as given in option 'B'.

This transformation is a simple substitution where the variable Q is equal to the variable q. In other words, Q and q are the same variable, just represented by different symbols.

Explanation:

Differentiating the Group of Letters:
- The group of letters in option 'a' (SENO) has letters arranged in alphabetical order.
- The group of letters in option 'b' (MLAG) has no specific pattern or arrangement.
- The group of letters in option 'c' (XUBU) has alternating consonants and vowels.
- The group of letters in option 'd' (CZHK) has letters arranged in reverse alphabetical order.
Therefore, the group of letters in option 'c' (XUBU) is different from the others as it follows a specific pattern of alternating consonants and vowels while the rest do not have a clear pattern.

To find the other number, we need to use the relationship between the LCM (Least Common Multiple) and HCF (Highest Common Factor) of two numbers.

Given:
LCM = 126
HCF = 9
One of the numbers = 18

We can write the relationship between the LCM and HCF as:
LCM × HCF = Product of the two numbers

Let the other number be x.

So, we have:
126 × 9 = 18 × x

Simplifying the equation:
1134 = 18x

Dividing both sides by 18:
x = 1134 / 18
x = 63

Therefore, the other number is 63.

Explanation:
The LCM of two numbers is the smallest multiple that is divisible by both numbers. In this case, the LCM is 126.

The HCF of two numbers is the largest factor that divides both numbers. In this case, the HCF is 9.

We are given that one of the numbers is 18.

Using the relationship between the LCM and HCF, we can write the equation: LCM × HCF = Product of the two numbers.

Substituting the given values, we get: 126 × 9 = 18 × x.

Simplifying the equation, we find: 1134 = 18x.

Dividing both sides by 18, we find: x = 1134 / 18 = 63.

Therefore, the other number is 63.

Probability of returning to the origin after certain steps
- After 10 steps:
- The walker has to take 5 steps to the left and 5 steps to the right to return to the origin.
- The number of ways this can happen is given by the binomial coefficient C(10,5).
- The total number of possible steps is 2^10 = 1024.
- Therefore, the probability of returning to the origin after 10 steps is C(10,5)/2^10 = 252/1024 = 0.25.
- After 7 steps:
- The walker has to take 3 steps to the left and 3 steps to the right to return to the origin.
- The number of ways this can happen is given by the binomial coefficient C(7,3).
- The total number of possible steps is 2^7 = 128.
- Therefore, the probability of returning to the origin after 7 steps is C(7,3)/2^7 = 35/128 ≈ 0.2734 ≈ 0.25.
Therefore, the correct answer is option 'C' with probabilities of 0.25 for both cases.

Numbers Divisible by 6 or 9 between 6 and 1300
To find the numbers between 6 and 1300 that are completely divisible by 6 or 9 or both, we need to consider the multiples of 6 and 9 within this range.

Numbers completely divisible by 6:
- We know that numbers divisible by 6 are multiples of 6.
- The first multiple of 6 after 6 is 12, and the last multiple of 6 before 1300 is 1296.
- To find the total numbers divisible by 6, we need to calculate (last multiple - first multiple)/6 + 1 = (1296-12)/6 + 1 = 216.

Numbers completely divisible by 9:
- Similarly, numbers divisible by 9 are multiples of 9.
- The first multiple of 9 after 6 is 9, and the last multiple of 9 before 1300 is 1296.
- To find the total numbers divisible by 9, we calculate (last multiple - first multiple)/9 + 1 = (1296-9)/9 + 1 = 144.

Numbers divisible by both 6 and 9 (i.e., multiples of 9*6 = 54):
- To find the numbers divisible by both 6 and 9, we consider the multiples of their least common multiple which is 54.
- The first multiple of 54 after 6 is 54, and the last multiple of 54 before 1300 is 1296.
- To find the total numbers divisible by 54, we calculate (last multiple - first multiple)/54 + 1 = (1296-54)/54 + 1 = 24.

Total numbers divisible by 6 or 9 or both:
- By adding the numbers divisible by 6, 9, and both, we get 216 + 144 - 24 = 336.
- However, we need to consider that the multiples of 54 are also counted in the multiples of 6 and 9. So, we need to subtract the numbers divisible by 54 once.
- Therefore, the total numbers between 6 and 1300 that are completely divisible by 6 or 9 or both are 336 - 24 = 312.
Therefore, the correct answer is option 'D' which is 288.