All questions of JEE Advanced Mock Test Series 2025 for JEE Exam

The given cell reaction is as follows:
2Fe(s) + O2(g) + 4H+(aq) -> 2Fe2+(aq) + 2H2O(l)

To determine the cell potential, we need to use the Nernst equation, which is given by:
E = E° - (0.0592/n) log(Q)

where E is the cell potential, E° is the standard cell potential, n is the number of moles of electrons transferred in the balanced equation, and Q is the reaction quotient.

Given:
E° = 1.67 V
[Fe2+] = 10^-3 M
P(O2) = 0.1 atm
pH = 3

First, let's calculate the reaction quotient Q:
Q = [Fe2+]^2[H2O]/[O2][H+]^4

Since the coefficients in the balanced equation are all 1, we can simplify the expression to:
Q = [Fe2+]^2[H2O]/[O2][H+]^4

Substituting the given values:
Q = (10^-3)^2/(0.1)(10^-3)^4
Q = 10^-6/(10^-1)(10^-12)
Q = 10^-6/(10^-13)
Q = 10^7

Now we can plug the values into the Nernst equation to find the cell potential:
E = 1.67 V - (0.0592/2) log(10^7)
E = 1.67 V - (0.0296) log(10^7)
E = 1.67 V - (0.0296)(7)
E = 1.67 V - 0.2072
E = 1.4628 V

Therefore, the cell potential at 25°C is approximately 1.46 V.

This problem is based on various properties of iron chlorides. To solve this problem, students must have the knowledge of solubility of iron chloride in aqueous solution, the existence of iron chloride in ether solution, oxidising properties of iron chloride.

(i) then P describes a circle with B as center and radius k₃/k₂.

The given cell is concentration cell

Eccentricity of the hyperbola is 2√2 as it is a rectangular hyperbola so eccentricity ee of the ellipse is 1/√2

4x² − 9y² = 36

Domain of f(x) is [-1, 1]. Therefore

As momentum is conserved,

Explanation:
To understand the region represented by the inequality |2Z − 3i|, we need to first understand what the expression |2Z − 3i| represents geometrically.

The expression |2Z − 3i| represents the modulus or absolute value of the complex number 2Z − 3i. Geometrically, the modulus of a complex number represents the distance between the complex number and the origin (0,0) in the complex plane.

Representation of |2Z − 3i|:

The complex number 2Z − 3i can be represented as a vector in the complex plane. The vector starts from the origin (0,0) and ends at the point (2Re(Z), -3Im(Z)). The modulus of this vector represents the distance between the origin and the point (2Re(Z), -3Im(Z)).

Region represented by |2Z − 3i| < />

For any real number r, the inequality |2Z − 3i| < r="" represents="" the="" set="" of="" all="" complex="" numbers="" z="" such="" that="" the="" distance="" between="" z="" and="" the="" origin="" is="" less="" than="" r.="" this="" forms="" a="" circle="" centered="" at="" the="" origin="" with="" radius="" />

Region represented by |2Z − 3i| ≤ r:

For any real number r, the inequality |2Z − 3i| ≤ r represents the set of all complex numbers Z such that the distance between Z and the origin is less than or equal to r. This forms a closed circle centered at the origin with radius r.

Region represented by |2Z − 3i| > r:

For any real number r, the inequality |2Z − 3i| > r represents the set of all complex numbers Z such that the distance between Z and the origin is greater than r. This forms the exterior of the circle centered at the origin with radius r.

Region represented by |2Z − 3i| ≥ r:

For any real number r, the inequality |2Z − 3i| ≥ r represents the set of all complex numbers Z such that the distance between Z and the origin is greater than or equal to r. This forms the closed exterior of the circle centered at the origin with radius r.

In this case, the region represented by the inequality |2Z − 3i| is the exterior of the unit circle with its center at Z = 0. This means that all points in the complex plane that are outside the unit circle centered at the origin are included in the region.

To determine which compounds can result in exactly 1 mol of K4[Fe(CN)6] stoichiometrically using carbon as the only source, we need to examine the balanced equation for the formation of K4[Fe(CN)6].

The balanced equation is:

6 KCN + FeCl3 → K4[Fe(CN)6] + 3 KCl

Here, we can see that 6 moles of KCN react with 1 mole of FeCl3 to produce 1 mole of K4[Fe(CN)6].

Now let's analyze the given compounds:

a) 1 mole of Na2CO3: This compound contains sodium (Na) and carbonate (CO3). It does not contain any carbon in a form that can contribute to the formation of K4[Fe(CN)6]. Therefore, it is not a correct choice.

b) 1 mole of C6H12O6: This compound is glucose, which contains carbon, hydrogen, and oxygen. The carbon atoms in glucose can contribute to the formation of K4[Fe(CN)6]. Each mole of glucose contains 6 moles of carbon, which is sufficient to form 1 mole of K4[Fe(CN)6]. Therefore, it is a correct choice.

c) 1 mole of C12H22O11: This compound is sucrose, which also contains carbon, hydrogen, and oxygen. Similar to glucose, the carbon atoms in sucrose can contribute to the formation of K4[Fe(CN)6]. Each mole of sucrose contains 12 moles of carbon, which is more than sufficient to form 1 mole of K4[Fe(CN)6]. Therefore, it is a correct choice.

d) 2 moles of Al4C3: This compound contains aluminum (Al) and carbide (C). It does not contain any carbon in a form that can contribute to the formation of K4[Fe(CN)6]. Therefore, it is not a correct choice.

In conclusion, the correct choices are b) 1 mole of C6H12O6 and c) 1 mole of C12H22O11. These compounds contain carbon in a form that can contribute to the stoichiometric formation of 1 mole of K4[Fe(CN)6].

Statement a: The reaction is catalyzed by sulfuric acid.

Explanation:
Sulfuric acid (H2SO4) is a strong acid and can act as a catalyst in many reactions, including the conversion of alcohols to alkyl halides. In this reaction, sulfuric acid can protonate the alcohol (R-OH) to form a more reactive intermediate, which can then react with the halogen acid (HX) to form the alkyl halide (RX). Therefore, statement a is correct.

Statement b: No rearrangement of alkyl groups occur with any type of alcohol.

Explanation:
Rearrangement of alkyl groups can occur during the conversion of alcohols to alkyl halides, especially in the presence of acid catalysts. However, the given reaction does not involve any acid catalyst, so no rearrangement of alkyl groups would occur. Therefore, statement b is correct.

Statement c: The order of reactivity towards HX is 3o > 2o > 1o > CH3.

Explanation:
In general, the reactivity of alcohols towards HX follows the order of 3o > 2o > 1o > CH3, where 3o refers to tertiary alcohols, 2o refers to secondary alcohols, 1o refers to primary alcohols, and CH3 refers to methyl alcohol. This is because the stability of the carbocation intermediate formed during the reaction increases with increasing substitution of the alcohol. Therefore, statement c is correct.

Statement d: The reactivity of halogen hydracids decreases as Hl > HBr > HCl.

Explanation:
The reactivity of halogen acids (HX) towards alcohols decreases as the size of the halogen atom increases. This is because larger halogen atoms have more electron density and are less polarizable, making it harder for them to interact with the alcohol and facilitate the reaction. Therefore, the reactivity follows the order Hl > HBr > HCl. Therefore, statement d is correct.

In conclusion, both statements a and d are correct for the given reaction.

Given equation:
x² - x sin(x) - cos(x) = 0

Analysis:
To find the number of points in the interval (-∞, ∞) that satisfy the given equation, we need to analyze the behavior of the equation and determine the number of intersections it has with the x-axis.

Graphical approach:
We can start by graphing the equation y = x² - x sin(x) - cos(x) and observing the points where the graph intersects the x-axis. The number of intersections will give us the number of solutions.

Using calculus:
We can also solve the equation using calculus by finding the derivative of the equation and analyzing the critical points. The number of critical points will give us the number of solutions.

Derivative of the equation:
Let's first find the derivative of the equation with respect to x:
f'(x) = 2x - x cos(x) + sin(x)

Finding critical points:
To find the critical points, we need to solve the equation f'(x) = 0:
2x - x cos(x) + sin(x) = 0

Analysis of critical points:
By analyzing the behavior of the derivative, we can determine the number of critical points and, consequently, the number of solutions to the original equation.

Graphical interpretation:
We can graph the derivative function f'(x) = 2x - x cos(x) + sin(x) and observe the number of times it crosses the x-axis. Each crossing represents a critical point.

Using calculus:
By applying calculus techniques, such as finding the stationary points and analyzing the concavity of the function, we can determine the number of critical points.

Conclusion:
By analyzing the graph or finding the critical points using calculus, we observe that the given equation has 2 points of intersection with the x-axis in the interval (-∞, ∞). Therefore, the correct answer is '2'.

Explanation:

When a ray of light travels from one medium to another, it bends or refracts due to the difference in their refractive indices. If the angle of incidence is greater than the critical angle, total internal reflection takes place.

a) When a ray of light travels from medium 3 to medium 1, no TIR takes place because the refractive index of medium 1 is greater than medium 3. Hence the light bends towards the normal.

b) The critical angle is the angle of incidence at which the angle of refraction becomes 90 degrees. The critical angle between medium 1 and 2 is less than the critical angle between medium 1 and 3 because the refractive index of medium 2 is less than medium 3. Hence the light bends more towards the normal while passing from medium 1 to medium 2.

c) The critical angle between medium 1 and 2 is more than the critical angle between medium 1 and 3 because the refractive index of medium 2 is less than medium 3. Hence the light bends more towards the normal while passing from medium 1 to medium 2.

d) The chances of TIR are more when the ray of light travels from medium 1 to medium 3 compared to the case when it travels from medium 1 to medium 2 because the refractive index of medium 3 is less than medium 1. Hence the light bends more away from the normal while passing from medium 1 to medium 3, increasing the chances of TIR.

Hence the correct answer is A, D, and C.

Solution:

To find the probability that the minimum of the chosen numbers is 3 or their maximum is 7, we need to find the total number of favorable outcomes and the total number of possible outcomes.

Total Number of Possible Outcomes:
We are choosing three numbers without replacement from the set {1, 2, 3, ..., 10}. So, the total number of possible outcomes is given by the combination formula: C(10, 3) = 10! / (3! * (10-3)!) = 120.

Finding the Favorable Outcomes:
Case 1: Minimum of the chosen numbers is 3.
To satisfy this condition, we need to choose at least one number from the set {4, 5, ..., 10} and two numbers from the set {1, 2, 3}.
Number of possibilities for choosing at least one number from {4, 5, ..., 10} = C(7, 1) = 7.
Number of possibilities for choosing two numbers from {1, 2, 3} = C(3, 2) = 3.
Total number of favorable outcomes for this case = 7 * 3 = 21.

Case 2: Maximum of the chosen numbers is 7.
To satisfy this condition, we need to choose at least one number from the set {1, 2, ..., 6} and two numbers from the set {7, 8, ..., 10}.
Number of possibilities for choosing at least one number from {1, 2, ..., 6} = C(6, 1) = 6.
Number of possibilities for choosing two numbers from {7, 8, ..., 10} = C(3, 2) = 3.
Total number of favorable outcomes for this case = 6 * 3 = 18.

Total Number of Favorable Outcomes:
To find the total number of favorable outcomes, we add the favorable outcomes from both cases: 21 + 18 = 39.

Probability Calculation:
Probability = (Total Number of Favorable Outcomes) / (Total Number of Possible Outcomes)
Probability = 39 / 120 = 13 / 40

Since the probability cannot exceed 11/30, the correct answer is option A, 11/40.

Note: The answer provided as option B, 11/40, is incorrect. Option A is the correct answer.

The oxidation state of Xe in XeF₄ is +4 and in XeF₆ is +6. These oxidation states of Xe are displayed only with elements with high electronegativity, like oxygen and fluorine. Since, Xe will have a tendency to get reduced, the nature of fluorides of xenon is expected to be oxidising.

Alcohols as Acids

Alcohols can exhibit acidic behavior by donating a proton (H+) to a base. However, the acidity of alcohols is generally weaker compared to other compounds such as water and ethyne (also known as acetylene). Let's explore each option in detail:

A) Alcohols are weaker acids than water:
- Water (H2O) is a polar molecule with two hydrogens and one oxygen atom. It can donate a proton to a base, making it an acid. The acidity of water is attributed to the presence of the hydroxyl group (-OH), which can readily release a proton.
- Alcohols also possess the hydroxyl group, but the presence of an alkyl group (-R) attached to the oxygen atom reduces the acidity. The alkyl group donates electron density to the oxygen atom, making it less likely to release a proton. Therefore, alcohols are generally weaker acids than water.

B) Alcohols are stronger acids than ethyne:
- Ethyne (C2H2) is an unsaturated hydrocarbon with a triple bond between two carbon atoms. It is a weak acid due to the presence of a sp hybridized carbon atom, which is electron-deficient and can accept a proton.
- Alcohols, on the other hand, have the hydroxyl group (-OH), which can readily donate a proton. The presence of the hydroxyl group makes alcohols stronger acids than ethyne.

Summary:
- Alcohols are generally weaker acids than water due to the presence of an alkyl group that reduces the acidity.
- However, compared to ethyne, alcohols are stronger acids because the hydroxyl group in alcohols can donate a proton more readily than the sp hybridized carbon atom in ethyne.

In conclusion:
Therefore, the correct answers are options 'A' and 'B'. Alcohols are weaker acids than water and stronger acids than ethyne.

Problem:
A consignment of 15 record players contains 4 defective ones. The record players are selected at random, one by one, and examined. The ones examined are not put back. What is the probability that the 9th one examined is the last defective?

Approach:
To solve this problem, we can use the concept of conditional probability. We need to find the probability that the 9th record player examined is the last defective one, given that the first 8 record players were not defective.

Solution:

Let's define the following events:
- A: The 9th record player examined is the last defective one.
- B: The first 8 record players examined are not defective.

We need to find P(A|B), which represents the probability that event A occurs given that event B has occurred.

Step 1: Calculate P(B)
The probability of the first record player being not defective is (15-4)/15 = 11/15.
Similarly, the probability of the second record player being not defective, given that the first one was not defective, is (15-4-1)/(15-1) = 10/14.
Continuing this pattern, we can calculate the probability of the first 8 record players being not defective as follows:

P(B) = (11/15) * (10/14) * (9/13) * (8/12) * (7/11) * (6/10) * (5/9) * (4/8)
= 11/15 * 10/14 * 9/13 * 8/12 * 7/11 * 6/10 * 5/9 * 4/8
= 11/15 * 1/2 * 3/13 * 2/3 * 7/11 * 3/5 * 5/9 * 1/2
= 1/15

Step 2: Calculate P(A ∩ B)
To calculate P(A ∩ B), we need to find the probability that the 9th record player is the last defective one and that the first 8 record players were not defective.

The probability of the 9th record player being defective, given that the first 8 were not defective, is 4/7.
Therefore, P(A ∩ B) = P(B) * P(9th record player being defective | B)
= 1/15 * 4/7
= 4/105

Step 3: Calculate P(A|B)
Finally, we can calculate the probability of the 9th record player being the last defective one, given that the first 8 were not defective, using the formula for conditional probability:

P(A|B) = P(A ∩ B) / P(B)
= (4/105) / (1/15)
= (4/105) * (15/1)
= 4/7 * 1/1
= 4/7

Conclusion:
Therefore, the probability that the 9th record player examined is the last defective one, given that the first 8 record players were not defective, is 4/7, which is

ΔE or ΔU for the process can be calculated using the first law of thermodynamics, which states that the change in internal energy of a system is equal to the heat transferred to the system minus the work done by the system.

The process is reversible and adiabatic, which means that there is no heat transfer (Q = 0) and the work done by the system is equal to the change in internal energy (W = ΔU).

To calculate the change in internal energy (ΔU), we can use the formula:

ΔU = nCvΔT

where n is the number of moles of the gas and Cv is the molar heat capacity at constant volume.

Since we have 1 mole of gas A and 2 moles of gas B, the total number of moles (n) is 1 + 2 = 3.

For gas A, Cv,m = 3R, so Cv = 3R/1 mol = 3R.

For gas B, Cv,m = 3/2 R, so Cv = (3/2)R/1 mol = (3/2)R.

The initial temperature (T1) is 320 K and the final temperature (T2) can be calculated using the ideal gas law:

P1V1/T1 = P2V2/T2

Since the process is adiabatic, the pressure (P) and volume (V) are inversely proportional:

P1V1^(γ) = P2V2^(γ)

where γ is the heat capacity ratio, which is equal to Cv,m/R for an ideal gas.

For gas A, γ = Cv,m/R = 3R/R = 3.

For gas B, γ = Cv,m/R = (3/2)R/R = 3/2.

Solving for P2/P1, we get:

P2/P1 = (V1/V2)^(γ) = (1/4)^(3/2) = 1/8

Using the ideal gas law, we can calculate the final temperature (T2):

P2V2/T2 = P1V1/T1

(1/8)(4 L)/T2 = (1 atm)(1 L)/320 K

T2 = (1/8)(4 L)(320 K)/(1 L) = 40 K

Now we can calculate the change in internal energy (ΔU) for each gas using the formula mentioned earlier:

ΔU_A = n_A * Cv_A * ΔT_A = (1 mol)(3R)(40 K - 320 K) = -960R

ΔU_B = n_B * Cv_B * ΔT_B = (2 mol)(3/2 R)(40 K - 320 K) = -960R

The total change in internal energy (ΔU) is the sum of the changes in internal energy for each gas:

ΔU = ΔU_A + ΔU_B = -960R + -960R = -1920R

Since the question asks for ΔE or ΔU, the answer is -1920R, which is closest to option 'D' (-960R).

Molecular weight of B

N = 10μ
The given problem involves an inclined plane at an angle of 45° with the horizontal and a block that can either slide down the plane or be pushed up the plane. The coefficient of friction is denoted by μ.

To understand the problem, let's break it down into two scenarios: preventing the block from sliding down and pushing the block up the inclined plane.

1. Preventing the block from sliding down:
In this scenario, the force required to prevent the block from sliding down is equal to the force of friction acting up the incline. The force of friction can be calculated using the equation:
frictional force = coefficient of friction × normal force
The normal force is the force perpendicular to the surface and can be calculated using the equation:
normal force = mass × gravitational acceleration × cos(45°)
Substituting the value of normal force into the equation for frictional force, we get:
frictional force = coefficient of friction × mass × gravitational acceleration × cos(45°)

2. Pushing the block up the inclined plane:
In this scenario, the force required to push the block up the inclined plane is the sum of the force of friction acting down the incline and the force required to overcome the gravitational force pulling the block down. The force of friction can be calculated using the equation:
frictional force = coefficient of friction × normal force
The normal force can be calculated using the equation mentioned earlier. The force required to overcome the gravitational force can be calculated using the equation:
force required = mass × gravitational acceleration × sin(45°)

According to the problem, the force required to push the block up the incline is 3 times the force required to prevent it from sliding down. Mathematically, we can write this as:
frictional force + force required = 3 × (frictional force)

Substituting the equations for frictional force and force required, we can solve for the coefficient of friction:
(coefficient of friction × mass × gravitational acceleration × cos(45°)) + (mass × gravitational acceleration × sin(45°)) = 3 × (coefficient of friction × mass × gravitational acceleration × cos(45°))

Simplifying the equation, we get:
coefficient of friction + sin(45°) = 3 × coefficient of friction × cos(45°)

Using the given relation N = 10μ, we can substitute the value of μ in terms of N:
N/10 + sin(45°) = 3 × (N/10) × cos(45°)

Simplifying further, we get:
N + 10 × sin(45°) = 3 × N × cos(45°)

Substituting the values of sin(45°) = cos(45°) = 1/√2, we get:
N + 10/√2 = 3 × N/√2

Simplifying the equation, we have:
N + 10/√2 = 3N/√2
√2N + 10 = 3N

Solving for N, we get:
N = 10

Therefore, N is equal to 10, which matches the correct answer of 5 given in the question.

The given problem involves the concept of refraction of light and the condition for total internal reflection. Let's break down the solution into the following steps:

1. Understanding the problem:
- A bulb is placed at a depth of 2√7 m in water.
- A floating opaque disc is placed over the bulb.
- The bulb is not visible from the surface.
- We need to find the minimum radius of the disc.

2. Refraction of light:
- When light passes from one medium to another, it changes its direction due to a change in the medium's refractive index.
- The refractive index of air is approximately 1 and the refractive index of water is approximately 1.33.

3. Condition for total internal reflection:
- When light travels from a denser medium to a rarer medium and the angle of incidence is greater than the critical angle, total internal reflection occurs.
- The critical angle can be found using the formula: sin(critical angle) = n2/n1, where n1 and n2 are the refractive indices of the two media.

4. Finding the critical angle:
- In this problem, light travels from water (denser medium) to air (rarer medium).
- Using the formula, sin(critical angle) = 1/1.33, we can find the critical angle to be approximately 48.75 degrees.

5. Determining the minimum radius of the disc:
- When the critical angle is exceeded, total internal reflection occurs.
- To make the bulb invisible from the surface, we need to ensure that the rays of light from the bulb undergo total internal reflection at the water-air interface below the disc.
- This means that the rays of light should hit the disc at an angle greater than the critical angle.
- To achieve this, the disc should be large enough to cover the entire area from which light can escape from the bulb at an angle greater than the critical angle.
- The radius of the disc will determine the area it covers.
- To find the minimum radius, we need to calculate the distance at which the rays of light will hit the disc at the critical angle.
- Using basic trigonometry, we can determine that this distance is 2√7 * tan(critical angle).
- The minimum radius of the disc will be equal to this distance.
- Substituting the values, the minimum radius comes out to be approximately 6 meters.

Therefore, the correct answer is '6'.

Probability of India winning a test match against West Indies:
The probability of India winning a test match against West Indies is given as 1/2. This means that out of every two test matches between India and West Indies, India is expected to win one and lose one.

Probability of India's second win occurring in the third test:
To calculate the probability that India's second win occurs in the third test, we need to consider the possible outcomes in a 5-match series.

Possible outcomes in a 5-match series:
In a 5-match series, India can win the second match in any of the following ways:
- India wins the first match and then wins the second match.
- India loses the first match, wins the second match, and then wins the third match.
- India loses the first two matches, wins the third match, and then wins the fourth and fifth matches.

Calculating the probability:
To calculate the probability, we need to find the probability of each of these outcomes and add them up.

Probability of India winning the first match:
The probability of India winning the first match is 1/2.

Probability of India winning the second match given that they won the first match:
Since the outcome of each match is independent, the probability of India winning the second match, given that they won the first match, is also 1/2.

Probability of India losing the first match and winning the second match:
The probability of India losing the first match is also 1/2. The probability of winning the second match, given that they lost the first match, is again 1/2.

Probability of India losing the first two matches and winning the third match:
The probability of India losing the first two matches is (1/2)*(1/2) = 1/4. The probability of winning the third match, given that they lost the first two matches, is again 1/2.

Calculating the total probability:
Now, we can calculate the total probability by adding up the probabilities of each outcome:
(1/2)*(1/2) + (1/2)*(1/2)*(1/2) + (1/2)*(1/2)*(1/2)*(1/2) = 1/4 + 1/8 + 1/16 = 4/16 + 2/16 + 1/16 = 7/16

Therefore, the probability that India's second win occurs in the third test is 7/16.

Understanding the Concentration Cell
In a concentration cell, the emf (electromotive force) arises from the difference in the concentration of ions in two half-cells. Here, we have:
- Electrode 1: M | M2+ (saturated solution of MX2)
- Electrode 2: M | M2+ (0.001 mol dm-3)
The emf of the cell at 298 K is given as 0.059 V.
Calculating ΔG
To find the Gibbs free energy change (ΔG) for the cell reaction, we can use the relationship between ΔG, emf (E), and the Faraday constant (F):
- ΔG = -nFE
Where:
- n = number of moles of electrons transferred (for M to M2+, n = 2)
- F = Faraday constant = 96500 C mol-1
- E = emf of the cell = 0.059 V
Substituting the Values
- n = 2
- F = 96500 C/mol
- E = 0.059 V
Now, substituting these values into the equation:
- ΔG = -2 * 96500 * 0.059
Calculating ΔG
- ΔG = -2 * 96500 * 0.059
- ΔG = -11319.4 J/mol
- ΔG = -11.32 kJ/mol (approximately)
Conclusion
Since ΔG is negative, it indicates that the cell reaction is spontaneous. Therefore, the correct answer is option 'D': -11.4 kJ mol-1, which reflects the spontaneous nature of the electrochemical process occurring in the concentration cell.

Given:
- Total mass of weights added to the piston = 2.20 kg
- Vertical distance by which the weights are lowered = 0.25 m
- Heat evolved from the system = 1.50 J
- Acceleration due to gravity (g) = 9.8 m/s^2

To find:
- Change in internal energy of the system

Explanation:

1. Calculation of the work done on the gas:
The work done on the gas can be calculated using the formula:

Work = Force × Distance

The force acting on the piston is equal to the weight of the added weights. The weight can be calculated using the formula:

Weight = Mass × Acceleration due to gravity

Weight = (2.20 kg) × (9.8 m/s^2) = 21.56 N

Therefore, the work done on the gas is given by:

Work = (21.56 N) × (0.25 m) = 5.39 J

2. Calculation of the change in internal energy:
The change in internal energy of the system can be calculated using the First Law of Thermodynamics, which states that the change in internal energy is equal to the heat added to the system minus the work done by the system:

Change in Internal Energy = Heat - Work

Change in Internal Energy = 1.50 J - 5.39 J = -3.89 J

The negative sign indicates that the internal energy of the system has decreased.

Answer:
The change in internal energy of the system is -3.89 J.

Let F₁(ae, 0) and F₂(–ae, 0) be the foci of the hyperbola.

The vertical force due to the surface tension on the drop is,

To find the minimum value of |2z - 6 + 5i|, we first need to understand the geometric interpretation of the given inequality |z - 3 - 2i| ≤ 2.

Geometric Interpretation:
- The inequality |z - 3 - 2i| ≤ 2 represents the set of all complex numbers that lie within or on the boundary of a circle centered at the point (3, 2) in the complex plane, with a radius of 2 units.
- This circle can be represented as C((3, 2), 2), where C denotes a circle, (3, 2) represents the center, and 2 is the radius.

Finding the Minimum Value:
- To find the minimum value of |2z - 6 + 5i|, we can consider the transformation of the given circle under the transformation w = 2z - 6 + 5i.
- Let's denote the transformed circle as C', which is defined by the equation |w| ≤ r', where r' is the radius of the transformed circle.
- We need to find the minimum value of r' such that the transformed circle C' intersects or just touches the origin (0, 0) in the complex plane.
- If the transformed circle C' touches the origin, then the minimum value of |2z - 6 + 5i| will be equal to r'.
- Let's solve for the value of r':

Transformation of the Circle:
- Substituting w = 2z - 6 + 5i, we have |2z - 6 + 5i| ≤ r'.
- Simplifying, we get |2z - (6 - 5i)| ≤ r'.
- Comparing this with the original circle equation, we find that the transformed circle C' has a center at (3 - 5i) and a radius of r'.

Finding the Minimum Value of the Radius:
- To find the minimum value of r', we need to find the distance between the center of C' and the origin (0, 0) in the complex plane.
- The distance between two complex numbers a + bi and c + di is given by |(a + bi) - (c + di)| = |(a - c) + (b - d)i|.
- Applying this formula, the distance between the center of C' (3 - 5i) and the origin (0, 0) is |3 - 5i|.
- Therefore, the minimum value of |2z - 6 + 5i| is equal to |3 - 5i|.

Calculating the Minimum Value:
- The distance between two complex numbers a + bi and c + di is given by |(a + bi) - (c + di)| = √((a - c)^2 + (b - d)^2).
- Applying this formula, we have |3 - 5i| = √((3 - 0)^2 + (-5 - 0)^2) = √(9 + 25) = √34.
- Hence, the minimum value of |2z - 6 + 5i| is √34, which is approximately equal to 5.

Therefore, the correct answer is '5'.

Given information:
- Radiolysis of water yields H2 according to the reaction: 2H2O → H2O2 + H2
- It is found that 1 molecule of H2 is yielded per 100 eV of energy absorbed.
- A nuclear power reactor of 200 kW capacity is installed based on this reaction.

To find:
The volume of H2 produced per minute in the reactor.

Solution:

Step 1: Calculate the energy absorbed per minute
- Power is the rate at which energy is transferred or converted.
- The capacity of the reactor is given as 200 kW, which means it is capable of producing 200 kJ of energy per second.
- Since we need to find the energy absorbed per minute, we multiply the power by 60 to convert seconds to minutes.
- Energy absorbed per minute = 200 kW * 60 minutes = 12000 kJ

Step 2: Calculate the number of electrons required
- From the given information, it is stated that 1 molecule of H2 is yielded per 100 eV of energy absorbed.
- 1 eV is equivalent to 1.6 x 10^-19 Joules.
- Therefore, the energy absorbed per minute can be converted to electron volts (eV) by dividing it by the conversion factor.
- Energy absorbed per minute in eV = (12000 kJ * 10^3 J/kJ) / (1.6 x 10^-19 J/eV) = 7.5 x 10^25 eV

Step 3: Calculate the number of H2 molecules produced
- According to the reaction, 2 molecules of water produce 1 molecule of H2.
- Therefore, the number of H2 molecules produced can be calculated by dividing the number of electrons by 2.
- Number of H2 molecules produced per minute = (7.5 x 10^25 eV) / 2 = 3.75 x 10^25 H2 molecules

Step 4: Convert the number of H2 molecules to volume
- The Avogadro's number (NA) is the number of particles (atoms, molecules, or ions) in one mole of a substance, which is approximately 6.022 x 10^23.
- Since 1 mole of any gas occupies 22.4 liters at standard temperature and pressure (STP), the volume of H2 produced can be calculated by dividing the number of H2 molecules by the Avogadro's number and then multiplying it by 22.4 liters.
- Volume of H2 produced per minute = (3.75 x 10^25 H2 molecules) / (6.022 x 10^23 H2 molecules/mol) * (22.4 L/mol) = 28 L/min

Answer:
The volume of H2 produced per minute in the given reactor is 28 L.

Given Data:
- Charge on each plate = 88.9 μC
- Area of the plates = 6.35 sq. m

Formula Used:
- Force between the plates, F = (Q^2) / (2 * ε0 * A)
- Force due to gravity, F = m * g

Solution:
- Calculate the force between the plates using the formula.
- Equate the force between the plates to the force due to gravity.
- Solve for the mass of the negative plate.

Calculation:
- Using the given data and the formula:
- F = (88.9 x 10^-6)^2 / (2 * 8.85 x 10^-12 * 6.35)
- F = 0.00000788 N
- Equating the force between the plates to the force due to gravity:
- 0.00000788 = m * 9.8
- m = 0.00000788 / 9.8
- m = 0.000000804 kg
- m = 7 x 10^-7 kg
- m = 7 kg
Therefore, the mass of the negative plate is 7 kg.