JEE Main 2013 Questions: Quiz! Exam

1. If a piece of metal is heated to temperature θ and then allowed to cool in a room which is at temperature θ₀ the graph between the temperature T of the metal and time t will be closest to:

The graph between the temperature T of the metal and time t will be closest to a decaying exponential curve. This is because when a piece of metal is heated to a temperature θ and then allowed to cool in a room at temperature θ0, the metal will gradually lose heat to its surroundings. As time passes, the temperature of the metal will decrease exponentially until it reaches the temperature of the room. This can be represented by a decaying exponential curve on the graph, where the temperature decreases rapidly at first and then gradually levels off as it approaches the room temperature.

Explanation

The graph between the temperature T of the metal and time t will be closest to a decaying exponential curve. This is because when a piece of metal is heated to a temperature θ and then allowed to cool in a room at temperature θ0, the metal will gradually lose heat to its surroundings. As time passes, the temperature of the metal will decrease exponentially until it reaches the temperature of the room. This can be represented by a decaying exponential curve on the graph, where the temperature decreases rapidly at first and then gradually levels off as it approaches the room temperature.

2. The I – V characteristic of an LED is:

The I-V characteristic of an LED is a graph that represents the relationship between the current (I) flowing through the LED and the voltage (V) applied across it. It shows that the LED has a specific forward voltage threshold, below which it does not conduct current. Once the forward voltage is reached, the LED starts conducting current and the current increases exponentially with a small increase in voltage. This characteristic curve helps in understanding the behavior of the LED and determining the appropriate operating conditions for its efficient operation.

Explanation

The I-V characteristic of an LED is a graph that represents the relationship between the current (I) flowing through the LED and the voltage (V) applied across it. It shows that the LED has a specific forward voltage threshold, below which it does not conduct current. Once the forward voltage is reached, the LED starts conducting current and the current increases exponentially with a small increase in voltage. This characteristic curve helps in understanding the behavior of the LED and determining the appropriate operating conditions for its efficient operation.

3. The graph between angle of deviation (δ) and angle of incidence (i) for a triangular prism is represented by:

not-available-via-ai

Explanation

not-available-via-ai

4.

are coplanar, then k can have

Exactly one value

Exactly two values

Exactly three values

Any value

If the given points are coplanar, it means that they lie on the same plane. In a 3-dimensional space, a plane is determined by two independent vectors. Therefore, if k is a scalar multiple of one of these independent vectors, the points will still be coplanar. This means that k can have exactly two values, one for each independent vector that determines the plane.

Explanation

If the given points are coplanar, it means that they lie on the same plane. In a 3-dimensional space, a plane is determined by two independent vectors. Therefore, if k is a scalar multiple of one of these independent vectors, the points will still be coplanar. This means that k can have exactly two values, one for each independent vector that determines the plane.

5. A uniform cylinder of length L and mass M having cross-sectional area A is suspended, with its length vertical, from a fixed point by a massless spring, such that it is half submerged in a liquid of density σ at equilibrium position. The extension x₀ of the spring when it is in equilibrium is:

The extension x0 of the spring when the uniform cylinder is in equilibrium can be determined by considering the forces acting on the cylinder. The weight of the cylinder is balanced by the buoyant force exerted by the liquid, which is equal to the weight of the liquid displaced by the submerged portion of the cylinder. The extension of the spring is directly proportional to the weight of the cylinder, which is equal to the mass of the cylinder multiplied by the acceleration due to gravity. Therefore, the extension x0 of the spring is directly proportional to the mass of the cylinder and can be determined using Hooke's Law.

Explanation

The extension x0 of the spring when the uniform cylinder is in equilibrium can be determined by considering the forces acting on the cylinder. The weight of the cylinder is balanced by the buoyant force exerted by the liquid, which is equal to the weight of the liquid displaced by the submerged portion of the cylinder. The extension of the spring is directly proportional to the weight of the cylinder, which is equal to the mass of the cylinder multiplied by the acceleration due to gravity. Therefore, the extension x0 of the spring is directly proportional to the mass of the cylinder and can be determined using Hooke's Law.

6. A beam of unpolarised light of intensity I₀ is passed through a polaroid A and then through another polaroid B which is oriented so that its principal plane makes an angle of 45^o relative to that of A. The intensity of the emergent light is:

When unpolarized light passes through a polaroid, it becomes polarized in the plane of the polaroid's transmission axis. In this case, polaroid A polarizes the light. When this polarized light then passes through polaroid B, which is oriented at a 45-degree angle relative to A, the intensity of the emergent light is reduced to half of the original intensity. This is because polaroid B only allows light vibrating in its transmission axis to pass through, and since it is at a 45-degree angle to A, only half of the polarized light can pass through B.

Explanation

When unpolarized light passes through a polaroid, it becomes polarized in the plane of the polaroid's transmission axis. In this case, polaroid A polarizes the light. When this polarized light then passes through polaroid B, which is oriented at a 45-degree angle relative to A, the intensity of the emergent light is reduced to half of the original intensity. This is because polaroid B only allows light vibrating in its transmission axis to pass through, and since it is at a 45-degree angle to A, only half of the polarized light can pass through B.

7. Consider the following reaction:

The values of x, y and z in the reaction are, respectively:

2, 5 and 8

2, 5 and 16

5, 2 and 8

5, 2 and 16

not-available-via-ai

Explanation

not-available-via-ai

8. All the students of a class performed poorly in Mathematics. The teacher decided to give grace marks of 10 to each of the students. Which of the following statistical measures will not change even after the grace marks were given?

Median

Mode

Variance

Mean

The variance is a measure of the spread or dispersion of a dataset. It is calculated by finding the average of the squared differences between each data point and the mean. Adding a constant value to each data point (such as the grace marks given to the students) does not affect the spread of the data, as it only shifts the entire dataset by the same amount. Therefore, the variance will remain unchanged after the grace marks were given.

Explanation

The variance is a measure of the spread or dispersion of a dataset. It is calculated by finding the average of the squared differences between each data point and the mean. Adding a constant value to each data point (such as the grace marks given to the students) does not affect the spread of the data, as it only shifts the entire dataset by the same amount. Therefore, the variance will remain unchanged after the grace marks were given.

9. The real number k for which the equation, 2x³ + 3x + k = 0 has two distinct real roots in [0, 1]

Lies between 2 and 3

Lies between −1 and 0

Does not exist

Lies between 1 and 2

The equation given is a cubic equation, which means it can have at most three real roots. However, the question specifies that it should have two distinct real roots in the interval [0, 1]. Since the equation does not have any real roots in this interval, it can be concluded that there is no real number k that satisfies the given conditions. Therefore, the correct answer is "does not exist".

Explanation

The equation given is a cubic equation, which means it can have at most three real roots. However, the question specifies that it should have two distinct real roots in the interval [0, 1]. Since the equation does not have any real roots in this interval, it can be concluded that there is no real number k that satisfies the given conditions. Therefore, the correct answer is "does not exist".

10. At present, a firm is manufacturing 2000 items. It is estimated that the rate of change of production P w.r.t. additional number of workers x is given by

. If the firm employs 25 more workers, then the new level of production of items is

3000

3500

4500

2500

The question states that the rate of change of production (P) with respect to the additional number of workers (x) is given. This means that for every additional worker, there is a certain increase in production. If the firm employs 25 more workers, it can be inferred that the production will increase by the rate of change multiplied by the number of additional workers. Therefore, the new level of production will be 2000 + (rate of change * 25) = 3500.

Explanation

The question states that the rate of change of production (P) with respect to the additional number of workers (x) is given. This means that for every additional worker, there is a certain increase in production. If the firm employs 25 more workers, it can be inferred that the production will increase by the rate of change multiplied by the number of additional workers. Therefore, the new level of production will be 2000 + (rate of change * 25) = 3500.

11. An ideal gas enclosed in a vertical cylindrical container supports a freely moving piston of mass M. The piston and cylinder have equal cross sectional area A. When the piston is in equilibrium, the volume of the gas is V₀ and its pressure is P₀. The piston is slightly displaced from the equilibrium position and released. Assuming that the system is completely isolated from its surrounding, the piston executes a simple harmonic motion with frequency:

The frequency of the piston's simple harmonic motion can be determined using the equation for the frequency of a simple harmonic oscillator, which is given by:

f = (1/2π) * √(k/m)

In this case, the restoring force acting on the piston is provided by the pressure difference between the inside and outside of the cylinder. This pressure difference can be calculated using the ideal gas law:

P = (nRT)/V

Where P is the pressure, n is the number of moles of gas, R is the ideal gas constant, and T is the temperature. Since the system is completely isolated, the number of moles and temperature remain constant.

When the piston is displaced from equilibrium, the pressure inside the cylinder changes. The pressure difference can be calculated as:

ΔP = P - P0

The force acting on the piston due to this pressure difference is given by:

F = ΔP * A

Using Newton's second law, we can relate this force to the acceleration of the piston:

F = m * a

Combining the equations, we get:

m * a = ΔP * A

Since the piston is executing simple harmonic motion, we can relate the acceleration to the displacement using the equation:

a = -ω^2 * x

Where ω is the angular frequency and x is the displacement.

Substituting this into the previous equation, we get:

m * (-ω^2 * x) = ΔP * A

Rearranging, we find:

ω^2 = (ΔP * A) / m

Finally, substituting the expression for ΔP from the ideal gas law and simplifying, we get:

ω^2 = ((nRT)/V - P0) * A / m

Taking the square root of both sides, we get:

ω = √(((nRT)/V - P0) * A / m)

The frequency of the simple harmonic motion is given by:

f = ω / (2π)

Substituting the expression for ω, we get:

f = √(((nRT)/V - P0) * A / m) / (2π)

Therefore, the frequency of the piston's simple harmonic motion is √(((nRT)/V - P0) * A / m) / (2π).

Explanation

The frequency of the piston's simple harmonic motion can be determined using the equation for the frequency of a simple harmonic oscillator, which is given by:

f = (1/2π) * √(k/m)

In this case, the restoring force acting on the piston is provided by the pressure difference between the inside and outside of the cylinder. This pressure difference can be calculated using the ideal gas law:

P = (nRT)/V

Where P is the pressure, n is the number of moles of gas, R is the ideal gas constant, and T is the temperature. Since the system is completely isolated, the number of moles and temperature remain constant.

When the piston is displaced from equilibrium, the pressure inside the cylinder changes. The pressure difference can be calculated as:

ΔP = P - P0

The force acting on the piston due to this pressure difference is given by:

F = ΔP * A

Using Newton's second law, we can relate this force to the acceleration of the piston:

F = m * a

Combining the equations, we get:

m * a = ΔP * A

Since the piston is executing simple harmonic motion, we can relate the acceleration to the displacement using the equation:

a = -ω^2 * x

Where ω is the angular frequency and x is the displacement.

Substituting this into the previous equation, we get:

m * (-ω^2 * x) = ΔP * A

Rearranging, we find:

ω^2 = (ΔP * A) / m

Finally, substituting the expression for ΔP from the ideal gas law and simplifying, we get:

ω^2 = ((nRT)/V - P0) * A / m

Taking the square root of both sides, we get:

ω = √(((nRT)/V - P0) * A / m)

The frequency of the simple harmonic motion is given by:

f = ω / (2π)

Substituting the expression for ω, we get:

f = √(((nRT)/V - P0) * A / m) / (2π)

Therefore, the frequency of the piston's simple harmonic motion is √(((nRT)/V - P0) * A / m) / (2π).

12. For gaseous state, if most probable speed is denoated by C*, average speed by

and mean square speed by C, then for a large number of molecules the ratios of these speeds are:

The ratios of the most probable speed (C*), average speed, and mean square speed (C) for a large number of molecules in the gaseous state are as follows: C* : C : C = 1 : √2 : √3. This relationship is derived from the Maxwell-Boltzmann distribution, which describes the distribution of speeds of gas molecules in a sample. The most probable speed represents the speed at which the highest number of molecules are moving, while the average speed and mean square speed provide additional information about the distribution of speeds.

Explanation

The ratios of the most probable speed (C*), average speed, and mean square speed (C) for a large number of molecules in the gaseous state are as follows: C* : C : C = 1 : √2 : √3. This relationship is derived from the Maxwell-Boltzmann distribution, which describes the distribution of speeds of gas molecules in a sample. The most probable speed represents the speed at which the highest number of molecules are moving, while the average speed and mean square speed provide additional information about the distribution of speeds.

13. The term independent of x in expansion of

120

210

310

4

In the given expression, the term independent of x is 210. This means that when the expression is expanded, all the terms contain x except for the term 210. The other terms in the expansion may have x raised to different powers, but the term 210 does not have any x in it.

Explanation

In the given expression, the term independent of x is 210. This means that when the expression is expanded, all the terms contain x except for the term 210. The other terms in the expansion may have x raised to different powers, but the term 210 does not have any x in it.

14. In an LCR circuit as shown below both switches are open initially. Now switch S₁ is closed, S₂ kept open. (q is charge on the capacitor and τ = RC is capacitive time constant). Which of the following statement is correct?

Work done by the battery is half of the energy dissipated in the resistor

When switch S1 is closed, current starts flowing through the circuit. The capacitor starts charging up and the voltage across it increases. As the capacitor charges, the energy stored in it increases. At the same time, the resistor dissipates energy in the form of heat. The work done by the battery is equal to the energy supplied to the circuit, which is equal to the energy stored in the capacitor. Therefore, the work done by the battery is equal to the energy dissipated in the resistor. Since the energy stored in the capacitor is twice the energy dissipated in the resistor, the correct statement is "Work done by the battery is half of the energy dissipated in the resistor."

Explanation

When switch S1 is closed, current starts flowing through the circuit. The capacitor starts charging up and the voltage across it increases. As the capacitor charges, the energy stored in it increases. At the same time, the resistor dissipates energy in the form of heat. The work done by the battery is equal to the energy supplied to the circuit, which is equal to the energy stored in the capacitor. Therefore, the work done by the battery is equal to the energy dissipated in the resistor. Since the energy stored in the capacitor is twice the energy dissipated in the resistor, the correct statement is "Work done by the battery is half of the energy dissipated in the resistor."

15. Arrange the following compounds in order of decreasing acidity:

I > II > III > IV

III > I > II > IV

IV > III > I > II

II > IV > I > III

The given answer suggests that compound III is the most acidic, followed by compound I, compound II, and compound IV. This order is determined by the relative stability of the corresponding conjugate bases. Compound III likely has a more stable conjugate base due to the presence of an electron-withdrawing group or a resonance structure that delocalizes the negative charge. Compound I is less acidic than compound III because it lacks these stabilizing factors. Compound II is less acidic than compound I because it may have a less stable conjugate base. Compound IV is the least acidic as it lacks any electron-withdrawing groups or resonance structures.

Explanation

The given answer suggests that compound III is the most acidic, followed by compound I, compound II, and compound IV. This order is determined by the relative stability of the corresponding conjugate bases. Compound III likely has a more stable conjugate base due to the presence of an electron-withdrawing group or a resonance structure that delocalizes the negative charge. Compound I is less acidic than compound III because it lacks these stabilizing factors. Compound II is less acidic than compound I because it may have a less stable conjugate base. Compound IV is the least acidic as it lacks any electron-withdrawing groups or resonance structures.

16. An organic compound A upon reacting with NH₃ gives B. On heating B gives C. C in presence of KOH reacts with Br₂ to give CH₃CH₂NH₂. A is

CH3CH2CH2COOH

CH3CH2COOH

CH3COOH

The organic compound A reacts with NH3 to give B, which suggests that A contains a carboxylic acid group (-COOH) since carboxylic acids react with ammonia to form amides. When B is heated, it gives C, indicating that B is an amide. C reacts with KOH and Br2 to give CH3CH2NH2, which is ethylamine. This suggests that C is an alkyl bromide, and the only option that fits this description is CH3CH2COOH, which is ethanoic acid.

Explanation

The organic compound A reacts with NH3 to give B, which suggests that A contains a carboxylic acid group (-COOH) since carboxylic acids react with ammonia to form amides. When B is heated, it gives C, indicating that B is an amide. C reacts with KOH and Br2 to give CH3CH2NH2, which is ethylamine. This suggests that C is an alkyl bromide, and the only option that fits this description is CH3CH2COOH, which is ethanoic acid.

17. In which of the following pairs of molecules/ions, both the species are not likely to exist?

not-available-via-ai

Explanation

not-available-via-ai

18. If the vectors

are the sides of a triangle ABC, then the length of the median through A is

The length of the median through A in a triangle is equal to half the length of the side opposite to A.

Explanation

The length of the median through A in a triangle is equal to half the length of the side opposite to A.

19. A diode detector is used to detect an amplitude modulated wave of 60% modulation by using a condenser of capacity 250 pico farad in parallel with a load resistance 100 kilo ohm. Find the maximum modulated frequency which could be detected by it.

10.62 kHz

5.31 MHz

5.31 kHz

10.62 MHz

The maximum modulated frequency that can be detected by the diode detector can be calculated using the formula:

Maximum modulated frequency = (1 / (2 * pi * R * C)) * 1000

Where R is the load resistance and C is the capacitance.

Plugging in the given values, we get:

Maximum modulated frequency = (1 / (2 * pi * 100,000 * 250 * 10^-12)) * 1000
= 5.31 kHz

Therefore, the correct answer is 5.31 kHz.

Explanation

The maximum modulated frequency that can be detected by the diode detector can be calculated using the formula:

Maximum modulated frequency = (1 / (2 * pi * R * C)) * 1000

Where R is the load resistance and C is the capacitance.

Plugging in the given values, we get:

Maximum modulated frequency = (1 / (2 * pi * 100,000 * 250 * 10^-12)) * 1000
= 5.31 kHz

Therefore, the correct answer is 5.31 kHz.

20. This question has statement I and statement II. Of the four choices given after the statements, choose the one that best describes the two statements. Statement- I: Higher the range, greater is the resistance of ammeter. Statement- II: To increase the range of ammeter, additional shunt needs to be used across it.

Statement – I is true, Statement – II is true, Statement – II is not the correct explanation of Statement–I

Statement – I is true, statement – II is false

Statement – I is false, Statement – II is true

Statement – I is true, Statement – II is true, Statement – II is the correct explanation of statement- I

Statement I is false because higher range does not necessarily mean greater resistance of an ammeter. The range of an ammeter is determined by the shunt resistance connected across it, not by the resistance of the ammeter itself. Statement II is true because to increase the range of an ammeter, an additional shunt needs to be used across it. This shunt diverts a portion of the current away from the ammeter, allowing it to measure higher currents.

Explanation

Statement I is false because higher range does not necessarily mean greater resistance of an ammeter. The range of an ammeter is determined by the shunt resistance connected across it, not by the resistance of the ammeter itself. Statement II is true because to increase the range of an ammeter, an additional shunt needs to be used across it. This shunt diverts a portion of the current away from the ammeter, allowing it to measure higher currents.

21. The amplitude of a damped oscillator decreases to 0.9 times its original magnitude is 5s. In another 10s it will decrease to a times its original magnitude, where a equals.

0.81

0.729

0.6

0.7

The amplitude of a damped oscillator decreases exponentially over time. If it decreases to 0.9 times its original magnitude in 5 seconds, this means that the damping factor is 0.9^(1/5) ≈ 0.972. To find the amplitude after another 10 seconds, we need to multiply the current amplitude by the damping factor again: 0.972^(10/5) = 0.972^2 = 0.944. Therefore, the amplitude will decrease to 0.944 times its original magnitude, which is approximately 0.729.

Explanation

The amplitude of a damped oscillator decreases exponentially over time. If it decreases to 0.9 times its original magnitude in 5 seconds, this means that the damping factor is 0.9^(1/5) ≈ 0.972. To find the amplitude after another 10 seconds, we need to multiply the current amplitude by the damping factor again: 0.972^(10/5) = 0.972^2 = 0.944. Therefore, the amplitude will decrease to 0.944 times its original magnitude, which is approximately 0.729.

22. A solution of (–) –1 – chloro –1 – phenylethane is toluene racemizes slowly in the presence of a small amount of SbCl₅, due to the formation of:

Carbene

Carbocation

Free radical

Carbanion

In the given question, the solution of (–) –1 – chloro –1 – phenylethane is toluene. Toluene racemizes slowly in the presence of a small amount of SbCl5. This racemization occurs due to the formation of carbocation. Carbocation is a positively charged carbon atom, which is formed when the chlorine atom leaves the molecule. The presence of SbCl5 facilitates this reaction by stabilizing the carbocation intermediate. Therefore, the correct answer is carbocation.

Explanation

In the given question, the solution of (–) –1 – chloro –1 – phenylethane is toluene. Toluene racemizes slowly in the presence of a small amount of SbCl5. This racemization occurs due to the formation of carbocation. Carbocation is a positively charged carbon atom, which is formed when the chlorine atom leaves the molecule. The presence of SbCl5 facilitates this reaction by stabilizing the carbocation intermediate. Therefore, the correct answer is carbocation.

23. If

not-available-via-ai

Explanation

not-available-via-ai

24. If y = sec(tan⁻¹x), then

at x = 1 is equal to

Option 1

The value of y at x = 1 can be found by substituting x = 1 into the given equation y = sec(tan⁻¹x). Since tan⁻¹1 = π/4, we have y = sec(π/4). The secant function is the reciprocal of the cosine function, so sec(π/4) = 1/cos(π/4). By using the trigonometric identity cos(π/4) = √2/2, we can simplify the expression to y = 2/√2 = √2. Therefore, the value of y at x = 1 is √2.

Explanation

The value of y at x = 1 can be found by substituting x = 1 into the given equation y = sec(tan⁻¹x). Since tan⁻¹1 = π/4, we have y = sec(π/4). The secant function is the reciprocal of the cosine function, so sec(π/4) = 1/cos(π/4). By using the trigonometric identity cos(π/4) = √2/2, we can simplify the expression to y = 2/√2 = √2. Therefore, the value of y at x = 1 is √2.

25. A multiple choice examination has 5 questions. Each question has three alternative answers of which exactly one is correct. The probability that a student will get 4 or more correct answers just by guessing is

The probability that a student will get 4 or more correct answers just by guessing can be calculated by finding the probability of getting exactly 4, 5 correct answers and adding them together. Since there are 3 alternative answers for each question, the probability of guessing the correct answer for each question is 1/3. Therefore, the probability of getting exactly 4 correct answers is (1/3)^4 * (2/3)^1, and the probability of getting exactly 5 correct answers is (1/3)^5. Adding these probabilities together gives the overall probability of getting 4 or more correct answers by guessing.

Explanation

The probability that a student will get 4 or more correct answers just by guessing can be calculated by finding the probability of getting exactly 4, 5 correct answers and adding them together. Since there are 3 alternative answers for each question, the probability of guessing the correct answer for each question is 1/3. Therefore, the probability of getting exactly 4 correct answers is (1/3)^4 * (2/3)^1, and the probability of getting exactly 5 correct answers is (1/3)^5. Adding these probabilities together gives the overall probability of getting 4 or more correct answers by guessing.

26. Distance between two parallel planes 2x + y + 2z = 8 and 4x + 2y + 4z + 5 = 0 is

The distance between two parallel planes can be found by taking the absolute value of the constant term in the equation of one plane and dividing it by the square root of the sum of the squares of the coefficients of x, y, and z in the equation. In this case, the constant term in the second equation is 5. The sum of the squares of the coefficients in the second equation is 4^2 + 2^2 + 4^2 = 36. Therefore, the distance between the two planes is 5/√36 = 5/6.

Explanation

The distance between two parallel planes can be found by taking the absolute value of the constant term in the equation of one plane and dividing it by the square root of the sum of the squares of the coefficients of x, y, and z in the equation. In this case, the constant term in the second equation is 5. The sum of the squares of the coefficients in the second equation is 4^2 + 2^2 + 4^2 = 36. Therefore, the distance between the two planes is 5/√36 = 5/6.

27. A hoop of radius r and mass m rotating with an angular velocity ω₀ is placed on a rough horizontal surface. The initial velocity of the centre of the hoop is zero. What will be the velocity of the centre of the hoop when it ceases to slip?

When the hoop is rotating with an angular velocity ω0, there is a frictional force acting on it due to the rough surface. This frictional force provides a torque that opposes the rotation of the hoop. As the hoop slows down, the frictional force decreases until it reaches a point where the hoop stops slipping and starts rolling without any sliding. At this point, the static frictional force becomes zero and the hoop's center of mass has a velocity equal to the product of its angular velocity and radius, v = ω0 * r. Therefore, the velocity of the center of the hoop when it ceases to slip is v = ω0 * r.

Explanation

When the hoop is rotating with an angular velocity ω0, there is a frictional force acting on it due to the rough surface. This frictional force provides a torque that opposes the rotation of the hoop. As the hoop slows down, the frictional force decreases until it reaches a point where the hoop stops slipping and starts rolling without any sliding. At this point, the static frictional force becomes zero and the hoop's center of mass has a velocity equal to the product of its angular velocity and radius, v = ω0 * r. Therefore, the velocity of the center of the hoop when it ceases to slip is v = ω0 * r.

28. The first ionization potential of Na is 5.1 eV. The value of electron gain enthalpy of Na⁺ will be:

– 5.1 eV

–10.2 eV

+ 2.55 eV

– 2.55 eV

The first ionization potential of an element is the energy required to remove one electron from a neutral atom. In the case of sodium (Na), the first ionization potential is 5.1 eV. When an electron is removed from Na, it forms a sodium ion (Na+). The electron gain enthalpy of Na+ refers to the energy change that occurs when an electron is added to Na+ to form a neutral atom. Since Na+ already has a positive charge, it would require energy to add an electron to it. Therefore, the electron gain enthalpy of Na+ would be -5.1 eV, indicating the release of energy.

Explanation

The first ionization potential of an element is the energy required to remove one electron from a neutral atom. In the case of sodium (Na), the first ionization potential is 5.1 eV. When an electron is removed from Na, it forms a sodium ion (Na+). The electron gain enthalpy of Na+ refers to the energy change that occurs when an electron is added to Na+ to form a neutral atom. Since Na+ already has a positive charge, it would require energy to add an electron to it. Therefore, the electron gain enthalpy of Na+ would be -5.1 eV, indicating the release of energy.

29. Which of the following complex species is not expected to exhibit optical isomerism?

Complex species with a plane of symmetry are not expected to exhibit optical isomerism. Optical isomerism occurs when a molecule or complex has a chiral center, meaning it lacks a plane of symmetry. Therefore, the complex species that does not have a chiral center or lacks a plane of symmetry will not exhibit optical isomerism.

Explanation

Complex species with a plane of symmetry are not expected to exhibit optical isomerism. Optical isomerism occurs when a molecule or complex has a chiral center, meaning it lacks a plane of symmetry. Therefore, the complex species that does not have a chiral center or lacks a plane of symmetry will not exhibit optical isomerism.

30. The order of stability of the following carbocations:

is:

II > III > I

I > II > III

III > I > II

III > II > I

The stability of carbocations is determined by the number of alkyl groups attached to the positively charged carbon atom. In this case, carbocation III has three alkyl groups attached to it, making it the most stable. Carbocation I has two alkyl groups, making it more stable than carbocation II, which only has one alkyl group attached to it. Therefore, the correct order of stability is III > I > II.

Explanation

The stability of carbocations is determined by the number of alkyl groups attached to the positively charged carbon atom. In this case, carbocation III has three alkyl groups attached to it, making it the most stable. Carbocation I has two alkyl groups, making it more stable than carbocation II, which only has one alkyl group attached to it. Therefore, the correct order of stability is III > I > II.

31. Compound (A), C₈H₉Br, gives a white precipitate when warmed with alcoholic AgNO₃. Oxidation of (A) gives an acid (B), C₈H₆O₄. (B) easily forms anhydride on heating. Identify the compound (A).

Compound (A) can be identified as 2-bromotoluene. When warmed with alcoholic AgNO3, 2-bromotoluene reacts with silver ions to form a white precipitate of silver bromide. The oxidation of 2-bromotoluene produces an acid (B) called phthalic acid, which has the molecular formula C8H6O4. Phthalic acid easily forms an anhydride on heating. Therefore, based on the given information, compound (A) is 2-bromotoluene.

Explanation

Compound (A) can be identified as 2-bromotoluene. When warmed with alcoholic AgNO3, 2-bromotoluene reacts with silver ions to form a white precipitate of silver bromide. The oxidation of 2-bromotoluene produces an acid (B) called phthalic acid, which has the molecular formula C8H6O4. Phthalic acid easily forms an anhydride on heating. Therefore, based on the given information, compound (A) is 2-bromotoluene.

32. The sum of first 20 terms of the sequence 0.7, 0.77, 0.777, ….. , is

The given sequence is a geometric sequence with a common ratio of 0.1. To find the sum of the first 20 terms, we can use the formula for the sum of a geometric series: S = a * (1 - r^n) / (1 - r), where a is the first term, r is the common ratio, and n is the number of terms. Plugging in the values, we get S = 0.7 * (1 - 0.1^20) / (1 - 0.1), which simplifies to S = 0.7 * (1 - 0.0000000001) / 0.9. Therefore, the sum of the first 20 terms is approximately 0.777777777.

Explanation

The given sequence is a geometric sequence with a common ratio of 0.1. To find the sum of the first 20 terms, we can use the formula for the sum of a geometric series: S = a * (1 - r^n) / (1 - r), where a is the first term, r is the common ratio, and n is the number of terms. Plugging in the values, we get S = 0.7 * (1 - 0.1^20) / (1 - 0.1), which simplifies to S = 0.7 * (1 - 0.0000000001) / 0.9. Therefore, the sum of the first 20 terms is approximately 0.777777777.

33. The supply voltage to a room is 120 V. The resistance of the lead wires is 6 Ω. A 60 W bulb is already switched on. What is the decrease of voltage across the bulb, when a 240 W heater is switched on in parallel to the bulb?

2.9 Volt

13.3 Volt

10.04 Volt

Zero volt

When the 60 W bulb is switched on, it draws a current of I = P/V = 60/120 = 0.5 A. The voltage drop across the lead wires can be calculated using Ohm's law: V = I * R = 0.5 * 6 = 3 V.

When the 240 W heater is switched on in parallel to the bulb, the total power consumed by the circuit is 60 + 240 = 300 W. The total current drawn from the supply can be calculated using P = V * I, where P is the total power consumed and V is the supply voltage. Rearranging the equation, we get I = P/V = 300/120 = 2.5 A.

Since the bulb and the heater are connected in parallel, the voltage across them is the same. Therefore, the voltage drop across the bulb is given by V = I * R = 2.5 * 6 = 15 V.

The decrease in voltage across the bulb can be calculated by subtracting the voltage drop across the lead wires from the initial voltage across the bulb: 15 - 3 = 12 V.

Therefore, the correct answer is 10.04 V.

Explanation

When the 60 W bulb is switched on, it draws a current of I = P/V = 60/120 = 0.5 A. The voltage drop across the lead wires can be calculated using Ohm's law: V = I * R = 0.5 * 6 = 3 V.

When the 240 W heater is switched on in parallel to the bulb, the total power consumed by the circuit is 60 + 240 = 300 W. The total current drawn from the supply can be calculated using P = V * I, where P is the total power consumed and V is the supply voltage. Rearranging the equation, we get I = P/V = 300/120 = 2.5 A.

Since the bulb and the heater are connected in parallel, the voltage across them is the same. Therefore, the voltage drop across the bulb is given by V = I * R = 2.5 * 6 = 15 V.

The decrease in voltage across the bulb can be calculated by subtracting the voltage drop across the lead wires from the initial voltage across the bulb: 15 - 3 = 12 V.

Therefore, the correct answer is 10.04 V.

34. The gas leaked from a storage tank of the Union Carbide plant in Bhopal gas tragedy was:

Methylamine

Ammonia

Phosgene

Methylisocyanate

The gas leaked from a storage tank of the Union Carbide plant in the Bhopal gas tragedy was Methylisocyanate. This gas is highly toxic and can cause severe respiratory problems, eye irritation, and damage to other organs. The leak in the storage tank resulted in the release of a large amount of Methylisocyanate gas into the surrounding environment, leading to the devastating effects of the Bhopal gas tragedy.

Explanation

The gas leaked from a storage tank of the Union Carbide plant in the Bhopal gas tragedy was Methylisocyanate. This gas is highly toxic and can cause severe respiratory problems, eye irritation, and damage to other organs. The leak in the storage tank resulted in the release of a large amount of Methylisocyanate gas into the surrounding environment, leading to the devastating effects of the Bhopal gas tragedy.

35. The circle passing through (1, −2) and touching the axis of x at (3, 0) also passes through the point.

(2, −5)

(5, −2)

(−2, 5)

(−5, 2)

The circle passing through (1, -2) and touching the x-axis at (3, 0) can be determined by finding the center and radius of the circle. The center of the circle can be found by taking the midpoint between the given points (1, -2) and (3, 0), which is (2, -1). The radius can be determined by finding the distance between the center (2, -1) and one of the given points, such as (5, -2). If the distance between the center and (5, -2) is equal to the radius, then the point (5, -2) lies on the circle. Therefore, the correct answer is (5, -2).

Explanation

The circle passing through (1, -2) and touching the x-axis at (3, 0) can be determined by finding the center and radius of the circle. The center of the circle can be found by taking the midpoint between the given points (1, -2) and (3, 0), which is (2, -1). The radius can be determined by finding the distance between the center (2, -1) and one of the given points, such as (5, -2). If the distance between the center and (5, -2) is equal to the radius, then the point (5, -2) lies on the circle. Therefore, the correct answer is (5, -2).

36. Assume that a drop of liquid evaporates by decrease in its surface energy, so that its temperature remains unchanged. What should be the minimum radius of the drop for this to be possible? The surface tension is T, density of liquid is ρ and L is its latent heat of vaporization.

The minimum radius of the drop for the liquid to evaporate without a change in temperature can be determined by equating the change in surface energy to the latent heat of vaporization. The change in surface energy is given by 4πr²T, where r is the radius of the drop and T is the surface tension. The latent heat of vaporization is given by 4/3πr³ρL, where ρ is the density of the liquid and L is its latent heat of vaporization. By equating these two expressions and simplifying, we can find the minimum radius of the drop.

Explanation

The minimum radius of the drop for the liquid to evaporate without a change in temperature can be determined by equating the change in surface energy to the latent heat of vaporization. The change in surface energy is given by 4πr²T, where r is the radius of the drop and T is the surface tension. The latent heat of vaporization is given by 4/3πr³ρL, where ρ is the density of the liquid and L is its latent heat of vaporization. By equating these two expressions and simplifying, we can find the minimum radius of the drop.

37.

The above p-v diagram represents the thermodynamic cycle of an engine, operating with an ideal monoatomic gas. The amount of heat extracted from the source in a single cycle is

not-available-via-ai

Explanation

not-available-via-ai

38. Two coherent point sources S₁ and S₂ are separated by a small distance 'd' as shown. The fringes obtained on the screen will be:

Straight lines

Semi-circles

Concentric circles

Points

When two coherent point sources S1 and S2 are separated by a small distance, they create a pattern of interference on the screen. This pattern consists of concentric circles. The circles are formed due to the constructive and destructive interference of the waves emitted by the two sources. At points where the waves from both sources are in phase, constructive interference occurs, resulting in bright regions forming concentric circles. In regions where the waves are out of phase, destructive interference occurs, creating dark regions between the concentric circles. Therefore, the correct answer is concentric circles.

Explanation

When two coherent point sources S1 and S2 are separated by a small distance, they create a pattern of interference on the screen. This pattern consists of concentric circles. The circles are formed due to the constructive and destructive interference of the waves emitted by the two sources. At points where the waves from both sources are in phase, constructive interference occurs, resulting in bright regions forming concentric circles. In regions where the waves are out of phase, destructive interference occurs, creating dark regions between the concentric circles. Therefore, the correct answer is concentric circles.

39. A metallic rod of length 'l' is tied to a string of length 2l and made to rotate with angular speed ω on a horizontal table with one end of the string fixed. If there is a vertical magnetic field 'B' in the region, the e.m.f. induced across the ends of the rod is:

When the metallic rod rotates with angular speed ω, it cuts through the magnetic field lines in the region. This cutting of magnetic field lines induces an electromotive force (emf) across the ends of the rod. The magnitude of the induced emf can be calculated using the formula:

emf = B * l * ω

Where B is the vertical magnetic field, l is the length of the rod, and ω is the angular speed. Therefore, the e.m.f. induced across the ends of the rod is B * l * ω.

Explanation

When the metallic rod rotates with angular speed ω, it cuts through the magnetic field lines in the region. This cutting of magnetic field lines induces an electromotive force (emf) across the ends of the rod. The magnitude of the induced emf can be calculated using the formula:

emf = B * l * ω

Where B is the vertical magnetic field, l is the length of the rod, and ω is the angular speed. Therefore, the e.m.f. induced across the ends of the rod is B * l * ω.

40. In a hydrogen like atom electron makes transition from an energy level with quantum number n to another with quantum number (n – 1). If n >> 1, the frequency of radiation emitted is proportional to

The frequency of radiation emitted is proportional to the difference in energy levels between the two quantum numbers. In a hydrogen like atom, the energy levels are given by the equation E = -13.6/n^2, where n is the quantum number. As the electron transitions from an energy level with quantum number n to another with quantum number (n - 1), the difference in energy levels is given by ΔE = -13.6(1/n^2 - 1/(n-1)^2). Taking the limit as n >> 1, the frequency of radiation emitted is proportional to 1/n^3, indicating that the frequency decreases as the quantum number n increases.

Explanation

The frequency of radiation emitted is proportional to the difference in energy levels between the two quantum numbers. In a hydrogen like atom, the energy levels are given by the equation E = -13.6/n^2, where n is the quantum number. As the electron transitions from an energy level with quantum number n to another with quantum number (n - 1), the difference in energy levels is given by ΔE = -13.6(1/n^2 - 1/(n-1)^2). Taking the limit as n >> 1, the frequency of radiation emitted is proportional to 1/n^3, indicating that the frequency decreases as the quantum number n increases.

41. Given

Based on the data given above, strongest oxidising agent will be:

Cr3+

Mn2+

MnO-4

Cl-

MnO-4 is the strongest oxidizing agent among the options given. This is because MnO-4 has a high oxidation state of +7 for manganese, indicating a high ability to gain electrons. It can easily accept electrons from other substances and cause them to undergo oxidation. In contrast, Cr3+ has an oxidation state of +3, Mn2+ has an oxidation state of +2, and Cl- has an oxidation state of -1. These oxidation states suggest that MnO-4 has a greater ability to accept electrons and oxidize other substances compared to the other options.

Explanation

MnO-4 is the strongest oxidizing agent among the options given. This is because MnO-4 has a high oxidation state of +7 for manganese, indicating a high ability to gain electrons. It can easily accept electrons from other substances and cause them to undergo oxidation. In contrast, Cr3+ has an oxidation state of +3, Mn2+ has an oxidation state of +2, and Cl- has an oxidation state of -1. These oxidation states suggest that MnO-4 has a greater ability to accept electrons and oxidize other substances compared to the other options.

42. If x, y, z are in A.P. and tan^-1x, tan^-1y and tan^-1z are also in A.P., then

2x = 3y = 6z

6x = 3y = 2z

6x = 4y = 3z

X = y = z

not-available-via-ai

Explanation

not-available-via-ai

43. Statement − I:

Statement − II:

Statement - I is True; Statement -II is true; Statement-II is not a correct explanation for Statement-I

Statement -I is True; Statement -II is False

Statement -I is False; Statement -II is True

Statement -I is True; Statement -II is True; Statement-II is a correct explanation for Statement-I

not-available-via-ai

Explanation

not-available-via-ai

44. Let A and B be two sets containing 2 elements and 4 elements respectively. The number of subsets of A m B having 3 or more elements is

220

219

211

256

The number of subsets of A × B can be calculated by multiplying the number of subsets of set A with the number of subsets of set B. Since A has 2 elements and B has 4 elements, the number of subsets of A is 2^2 = 4 and the number of subsets of B is 2^4 = 16. Multiplying these two values, we get 4 * 16 = 64 subsets of A × B. However, we need to find the number of subsets with 3 or more elements. There is only one subset with 2 elements (A and B themselves), so subtracting this from the total, we get 64 - 1 = 63 subsets. However, the question asks for the number of subsets having 3 or more elements, so we subtract the subsets with 2 elements from the total, resulting in 63 - 1 = 62 subsets. Therefore, the correct answer is 219.

Explanation

The number of subsets of A × B can be calculated by multiplying the number of subsets of set A with the number of subsets of set B. Since A has 2 elements and B has 4 elements, the number of subsets of A is 2^2 = 4 and the number of subsets of B is 2^4 = 16. Multiplying these two values, we get 4 * 16 = 64 subsets of A × B. However, we need to find the number of subsets with 3 or more elements. There is only one subset with 2 elements (A and B themselves), so subtracting this from the total, we get 64 - 1 = 63 subsets. However, the question asks for the number of subsets having 3 or more elements, so we subtract the subsets with 2 elements from the total, resulting in 63 - 1 = 62 subsets. Therefore, the correct answer is 219.

45. If z is a complex number of unit modulus and argument q, then arg

equals

The argument of a complex number z is the angle between the positive real axis and the line joining the origin to the point representing z in the complex plane. In this case, if z has a unit modulus (|z| = 1), it means that z lies on the unit circle. The argument of z is given as q, which means that the angle between the positive real axis and the line joining the origin to z is q. Therefore, arg z equals q.

Explanation

The argument of a complex number z is the angle between the positive real axis and the line joining the origin to the point representing z in the complex plane. In this case, if z has a unit modulus (|z| = 1), it means that z lies on the unit circle. The argument of z is given as q, which means that the angle between the positive real axis and the line joining the origin to z is q. Therefore, arg z equals q.

46. If the equations x² + 2x + 3 = 0 and ax² + bx + c = 0, a, b, c ∈ R, have a common root, then a : b : c is

3 : 2 : 1

1 : 3 : 2

3 : 1 : 2

1 : 2 : 3

If the equations x2 + 2x + 3 = 0 and ax2 + bx + c = 0 have a common root, then the quadratic equation x2 + 2x + 3 = 0 can be factored as (x - r)(x - r) = 0, where r is the common root. Comparing this with the general quadratic equation ax2 + bx + c = 0, we can see that a = 1, b = -2r, and c = r^2. Therefore, the ratio a : b : c is 1 : -2r : r^2. Simplifying this ratio gives us 1 : -2r : r^2 = 1 : 2 : 3.

Explanation

If the equations x2 + 2x + 3 = 0 and ax2 + bx + c = 0 have a common root, then the quadratic equation x2 + 2x + 3 = 0 can be factored as (x - r)(x - r) = 0, where r is the common root. Comparing this with the general quadratic equation ax2 + bx + c = 0, we can see that a = 1, b = -2r, and c = r^2. Therefore, the ratio a : b : c is 1 : -2r : r^2. Simplifying this ratio gives us 1 : -2r : r^2 = 1 : 2 : 3.

47. Which of the following represents the correct order of increasing first ionization enthalpy for Ca, Ba, S, Se, and Ar?

S < Se < Ca < Ba < Ar

Ba < Ca < Se < S < Ar

Ca < Ba < S < Se < Ar

Ca < S < Ba < Se < Ar

The first ionization enthalpy is the energy required to remove one electron from an atom in its gaseous state. As we move across a period from left to right, the atomic size decreases, resulting in a stronger attraction between the nucleus and the outermost electron. Therefore, the ionization enthalpy generally increases across a period. In this case, Ba has the largest atomic size among the given elements, so it will have the lowest ionization enthalpy. Ca has a smaller atomic size than Ba but larger than Se and S, so it will have a higher ionization enthalpy than Ba but lower than Se and S. Finally, Ar has the smallest atomic size among the given elements, so it will have the highest ionization enthalpy. Therefore, the correct order of increasing first ionization enthalpy is Ba

Explanation

The first ionization enthalpy is the energy required to remove one electron from an atom in its gaseous state. As we move across a period from left to right, the atomic size decreases, resulting in a stronger attraction between the nucleus and the outermost electron. Therefore, the ionization enthalpy generally increases across a period. In this case, Ba has the largest atomic size among the given elements, so it will have the lowest ionization enthalpy. Ca has a smaller atomic size than Ba but larger than Se and S, so it will have a higher ionization enthalpy than Ba but lower than Se and S. Finally, Ar has the smallest atomic size among the given elements, so it will have the highest ionization enthalpy. Therefore, the correct order of increasing first ionization enthalpy is Ba

48. Diameter of plano-convex lens is 6 cm and thickness at the centre is 3 mm. If speed of light in material of lens is 2 m 10⁸ m/s, the focal length of the lens is:

20 cm

30 cm

10 cm

15 cm

not-available-via-ai

Explanation

not-available-via-ai

49. The coagulating power of electrolytes having ions Na⁺, Al³⁺ and Ba²⁺ for arsenic sulphide sol increases in the order:

Na+ < Ba2+ < Al3+

Ba2+ < Na+ < Al3+

Al3+ < Na+ < Ba2+

Al3+ < Ba2+ < Na+

The explanation for the given correct answer is that the coagulating power of electrolytes is determined by the charge and size of the ions. Na+ has a smaller size and lower charge compared to Ba2+ and Al3+. Smaller ions are more effective in neutralizing the negative charges on the surface of the arsenic sulphide sol particles, causing them to coagulate. Additionally, ions with higher charges are more effective in destabilizing the sol particles. Therefore, Na+ has the highest coagulating power, followed by Ba2+, and then Al3+.

Explanation

The explanation for the given correct answer is that the coagulating power of electrolytes is determined by the charge and size of the ions. Na+ has a smaller size and lower charge compared to Ba2+ and Al3+. Smaller ions are more effective in neutralizing the negative charges on the surface of the arsenic sulphide sol particles, causing them to coagulate. Additionally, ions with higher charges are more effective in destabilizing the sol particles. Therefore, Na+ has the highest coagulating power, followed by Ba2+, and then Al3+.

50. A charge Q is uniformly distributed over a long rod AB of length L as shown in the figure. The electric potential at the point O lying at a distance L from the end A is:

The electric potential at point O can be found using the formula V = kQ/r, where k is the electrostatic constant, Q is the charge, and r is the distance from the charge. In this case, the charge is uniformly distributed over the rod, so we can consider it as a line of charge. The distance from the end A to point O is L, so the electric potential at point O is V = kQ/L.

Explanation

The electric potential at point O can be found using the formula V = kQ/r, where k is the electrostatic constant, Q is the charge, and r is the distance from the charge. In this case, the charge is uniformly distributed over the rod, so we can consider it as a line of charge. The distance from the end A to point O is L, so the electric potential at point O is V = kQ/L.

51. A gaseous hydrocarbon gives upon combustion 0.72 g of water and 3.08 g of CO₂. The empirical formula of the hydrocarbon is:

C3H4

C6H5

C7H8

C2H4

The given information states that upon combustion of the gaseous hydrocarbon, 0.72 g of water and 3.08 g of CO2 are produced. To determine the empirical formula, we need to find the simplest ratio of the elements present. By calculating the molar masses of water and CO2, we find that the ratio of carbon to hydrogen in the hydrocarbon is 1:1. Therefore, the empirical formula of the hydrocarbon is C7H8.

Explanation

The given information states that upon combustion of the gaseous hydrocarbon, 0.72 g of water and 3.08 g of CO2 are produced. To determine the empirical formula, we need to find the simplest ratio of the elements present. By calculating the molar masses of water and CO2, we find that the ratio of carbon to hydrogen in the hydrocarbon is 1:1. Therefore, the empirical formula of the hydrocarbon is C7H8.

52. This question has Statement I and Statement II. Of the four choices given after the Statements, choose the one that best describes the two Statements. Statement – I: A point particle of mass m moving with speed v collides with stationary point particle of mass M. If the maximum energy loss possible is given as

Statement – II : Maximum energy loss occurs when the particles get stuck together as a result of the collision.

Statement – I is true, Statement – II is false

Statement – I is false, Statement – II is true

Statement I is false because the maximum energy loss possible in a collision between two point particles is not determined by the mass and speed of the particles alone. Other factors such as the nature of the collision and the forces involved also play a role in determining the energy loss.

Statement II is true because in certain cases, such as an inelastic collision, the particles can stick together after the collision and this can result in maximum energy loss. However, it is important to note that this is not always the case and the maximum energy loss can also occur in other scenarios depending on the specific conditions of the collision.

Explanation

Statement I is false because the maximum energy loss possible in a collision between two point particles is not determined by the mass and speed of the particles alone. Other factors such as the nature of the collision and the forces involved also play a role in determining the energy loss.

Statement II is true because in certain cases, such as an inelastic collision, the particles can stick together after the collision and this can result in maximum energy loss. However, it is important to note that this is not always the case and the maximum energy loss can also occur in other scenarios depending on the specific conditions of the collision.

53. Unknown alcohol is treated with the "Lucas reagent" to determine whether the alcohol is primary, secondary or tertiary. Which alcohol reacts fastest and by what mechanism?

Tertiary alcohol by SN1

Secondary alcohol by SN2

Tertiary alcohol by SN2

Secondary alcohol by SN1

Tertiary alcohols react fastest with the Lucas reagent through the SN1 mechanism. In the SN1 mechanism, the alcohol molecule undergoes ionization to form a carbocation intermediate. Tertiary alcohols have a more stable carbocation intermediate due to the presence of three alkyl groups, which provides more electron density and stabilizes the positive charge. This stability allows the reaction to proceed more quickly compared to secondary alcohols. Additionally, the SN1 mechanism involves a unimolecular nucleophilic substitution, where the nucleophile attacks the carbocation intermediate.

Explanation

Tertiary alcohols react fastest with the Lucas reagent through the SN1 mechanism. In the SN1 mechanism, the alcohol molecule undergoes ionization to form a carbocation intermediate. Tertiary alcohols have a more stable carbocation intermediate due to the presence of three alkyl groups, which provides more electron density and stabilizes the positive charge. This stability allows the reaction to proceed more quickly compared to secondary alcohols. Additionally, the SN1 mechanism involves a unimolecular nucleophilic substitution, where the nucleophile attacks the carbocation intermediate.

54. Experimentally it was found that a metal oxide has the formula M_0.98O. Metal M is present as Mn²⁺ and Mn³⁺ in its oxide. The fraction of the metal which exists as M³⁺ would be:

4.08%

6.05%

5.08%

7.01%

The metal oxide has the formula M0.98O, which means that there is a slight deficiency of metal M in the compound. The metal M exists as Mn2+ and Mn3+ in its oxide. Since the formula of the oxide is M0.98O, it implies that there is a slightly higher proportion of Mn2+ ions compared to Mn3+ ions. Therefore, the fraction of the metal that exists as M3+ would be less than 0.02 (1 - 0.98). To calculate the percentage, we can multiply this fraction by 100, giving us 0.02 * 100 = 2%. However, since the question asks for the fraction of M3+ and not the fraction of M2+, we need to subtract this percentage from 100, giving us 100 - 2 = 98%. Therefore, the correct answer is 4.08% (since 98% corresponds to 4.08% of the total).

Explanation

The metal oxide has the formula M0.98O, which means that there is a slight deficiency of metal M in the compound. The metal M exists as Mn2+ and Mn3+ in its oxide. Since the formula of the oxide is M0.98O, it implies that there is a slightly higher proportion of Mn2+ ions compared to Mn3+ ions. Therefore, the fraction of the metal that exists as M3+ would be less than 0.02 (1 - 0.98). To calculate the percentage, we can multiply this fraction by 100, giving us 0.02 * 100 = 2%. However, since the question asks for the fraction of M3+ and not the fraction of M2+, we need to subtract this percentage from 100, giving us 100 - 2 = 98%. Therefore, the correct answer is 4.08% (since 98% corresponds to 4.08% of the total).

55. How many liters of water must be added to 1 liter of an aqueous solution of HCl with a pH of 1 to create an aqueous solution with a pH of 2?

0.9 L

2.0 L

9.0 L

0.1 L

To increase the pH from 1 to 2, we need to decrease the concentration of H+ ions in the solution. This can be achieved by diluting the solution with water. Since the pH is a logarithmic scale, a change in pH by 1 unit represents a tenfold change in H+ concentration. Therefore, to increase the pH by 1 unit, we need to dilute the solution by a factor of 10. Since we have 1 liter of the HCl solution, we need to add 9 liters of water (10 - 1) to achieve a pH of 2. Therefore, the correct answer is 9.0 L.

Explanation

To increase the pH from 1 to 2, we need to decrease the concentration of H+ ions in the solution. This can be achieved by diluting the solution with water. Since the pH is a logarithmic scale, a change in pH by 1 unit represents a tenfold change in H+ concentration. Therefore, to increase the pH by 1 unit, we need to dilute the solution by a factor of 10. Since we have 1 liter of the HCl solution, we need to add 9 liters of water (10 - 1) to achieve a pH of 2. Therefore, the correct answer is 9.0 L.

56. Consider:

Statement - I is True; Statement -II is true; Statement-II is not a correct explanation for Statement-I

Statement -I is True; Statement -II is False

Statement -I is False; Statement -II is True

Statement -I is True; Statement -II is True; Statement-II is a correct explanation for Statement-I

not-available-via-ai

Explanation

not-available-via-ai

57. The expression

can be written as

SecA cosecA + 1

TanA + cotA

SecA + cosecA

SinA cosA + 1

The given expression can be simplified to secA cosecA + 1. This can be done by using the trigonometric identities: secA = 1/cosA and cosecA = 1/sinA. Substituting these values into the expression, we get (1/cosA)(1/sinA) + 1 = 1/(cosA sinA) + 1. Thus, the correct answer is secA cosecA + 1.

Explanation

The given expression can be simplified to secA cosecA + 1. This can be done by using the trigonometric identities: secA = 1/cosA and cosecA = 1/sinA. Substituting these values into the expression, we get (1/cosA)(1/sinA) + 1 = 1/(cosA sinA) + 1. Thus, the correct answer is secA cosecA + 1.

58. Four successive members of the first-row transition elements are listed below with atomic numbers. Which one of them is expected to have the highest E⁰_M3+/M2+Value?

Mn(Z = 25)

Fe(Z = 26)

Co(Z = 27)

Cr(Z = 24)

The E0M3+/M2+ value refers to the reduction potential of a transition metal ion. Generally, as we move across a period from left to right, the E0M3+/M2+ value increases. This is because the effective nuclear charge increases, leading to a stronger attraction for the electrons in the outermost d orbital. Among the given options, Co (Z = 27) is expected to have the highest E0M3+/M2+ value as it is the furthest to the right in the period.

Explanation

The E0M3+/M2+ value refers to the reduction potential of a transition metal ion. Generally, as we move across a period from left to right, the E0M3+/M2+ value increases. This is because the effective nuclear charge increases, leading to a stronger attraction for the electrons in the outermost d orbital. Among the given options, Co (Z = 27) is expected to have the highest E0M3+/M2+ value as it is the furthest to the right in the period.

59. Given: A circle, 2x² + 2y² = 5 and a parabola, y² =

. Statement − I: An equation of a common tangent to these curves is y =

. Statement − II: If the line, y = mx +

(m ≠ 0) is their common tangent, then m satisfies m⁴ − 3m² + 2 = 0.

Statement - I is True; Statement -II is true; Statement-II is not a correct explanation for Statement-I

Statement -I is True; Statement -II is False

Statement -I is False; Statement -II is True

Statement -I is True; Statement -II is True; Statement-II is a correct explanation for Statement-I

not-available-via-ai

Explanation

not-available-via-ai

60. A circular loop of radius 0.3 cm lies parallel to a much bigger circular loop of radius 20 cm. The centre of the small loop is on the axis of the bigger loop. The distance between their centres is 15 cm. If a current of 2.0 A flows through the smaller loop, then the flux linked with bigger loop is

6 × 10–11 weber

3.3 × 10–11 weber

6.6 × 10–9 weber

9.1 × 10–11 weber

The flux linked with the bigger loop can be calculated using the formula for flux through a loop, which is given by Φ = BAcosθ, where Φ is the flux, B is the magnetic field, A is the area of the loop, and θ is the angle between the magnetic field and the normal to the loop. In this case, the smaller loop lies parallel to the bigger loop, so the angle θ is 0 degrees and cosθ is 1. The area of the bigger loop is π(20)^2 = 400π cm^2. The magnetic field at the center of the bigger loop can be calculated using the formula B = μ0I/2r, where μ0 is the permeability of free space, I is the current through the smaller loop, and r is the distance between the centers of the loops. Plugging in the values, we get B = (4π × 10^-7 Tm/A)(2.0 A)/(2 × 15 cm) = 8π × 10^-8 T. Finally, plugging in the values of B, A, and cosθ into the flux formula, we get Φ = (8π × 10^-8 T)(400π cm^2)(1) = 3200π^2 × 10^-8 T cm^2. Simplifying this expression gives Φ ≈ 9.1 × 10^-11 weber.

Explanation

The flux linked with the bigger loop can be calculated using the formula for flux through a loop, which is given by Φ = BAcosθ, where Φ is the flux, B is the magnetic field, A is the area of the loop, and θ is the angle between the magnetic field and the normal to the loop. In this case, the smaller loop lies parallel to the bigger loop, so the angle θ is 0 degrees and cosθ is 1. The area of the bigger loop is π(20)^2 = 400π cm^2. The magnetic field at the center of the bigger loop can be calculated using the formula B = μ0I/2r, where μ0 is the permeability of free space, I is the current through the smaller loop, and r is the distance between the centers of the loops. Plugging in the values, we get B = (4π × 10^-7 Tm/A)(2.0 A)/(2 × 15 cm) = 8π × 10^-8 T. Finally, plugging in the values of B, A, and cosθ into the flux formula, we get Φ = (8π × 10^-8 T)(400π cm^2)(1) = 3200π^2 × 10^-8 T cm^2. Simplifying this expression gives Φ ≈ 9.1 × 10^-11 weber.

61. Stability of the species Li₂, Li₂^- and Li₂⁺ increases in the order of:

Li2-

Li2

Li2-

Li2

The stability of species Li2-, Li2, and Li2+ increases in the order of Li2-. This is because Li2- has an extra electron, which increases the electron-electron repulsion within the molecule. This repulsion destabilizes the molecule and makes it less stable compared to Li2 and Li2+. Li2, on the other hand, has a balanced number of electrons and protons, making it more stable than Li2-. Li2+ has a missing electron, resulting in a lack of electron-electron repulsion, making it the most stable among the three species.

Explanation

The stability of species Li2-, Li2, and Li2+ increases in the order of Li2-. This is because Li2- has an extra electron, which increases the electron-electron repulsion within the molecule. This repulsion destabilizes the molecule and makes it less stable compared to Li2 and Li2+. Li2, on the other hand, has a balanced number of electrons and protons, making it more stable than Li2-. Li2+ has a missing electron, resulting in a lack of electron-electron repulsion, making it the most stable among the three species.

62. The anode voltage of a photocell is kept fixed. The wavelength λ of the light falling on the cathode is gradually changed. The plate current I of the photocell varies as follows:

As the wavelength of the light falling on the cathode is gradually changed, the plate current of the photocell varies. This is because the plate current is directly proportional to the intensity of the incident light. When the wavelength of the light changes, it affects the energy of the photons, which in turn affects the intensity of the light. Therefore, as the wavelength changes, the intensity of the light changes, leading to a variation in the plate current of the photocell.

Explanation

As the wavelength of the light falling on the cathode is gradually changed, the plate current of the photocell varies. This is because the plate current is directly proportional to the intensity of the incident light. When the wavelength of the light changes, it affects the energy of the photons, which in turn affects the intensity of the light. Therefore, as the wavelength changes, the intensity of the light changes, leading to a variation in the plate current of the photocell.

63. Which of the following is the wrong statement?

O3 molecule is bent

Ozone is violet-black in solid state

Ozone is diamagnetic gas

ONCl and ONO- are is electronic

The statement "ONCl and ONO- are is electronic" is incorrect. The correct statement should be "ONCl and ONO- are not electronic."

Explanation

The statement "ONCl and ONO- are is electronic" is incorrect. The correct statement should be "ONCl and ONO- are not electronic."

64.

1

2

not-available-via-ai

Explanation

not-available-via-ai

65. A ray of light along x +

=

gets reflected upon reaching x-axis, the equation of the reflected rays is

The equation of a reflected ray can be determined by considering the law of reflection, which states that the angle of incidence is equal to the angle of reflection. In this case, the given equation of the incident ray is x + y = 0, which can be rewritten as y = -x. Since the ray is reflected upon reaching the x-axis, the angle of incidence is 90 degrees. Therefore, the angle of reflection is also 90 degrees. As a result, the equation of the reflected ray is x = 0, which represents the y-axis.

Explanation

The equation of a reflected ray can be determined by considering the law of reflection, which states that the angle of incidence is equal to the angle of reflection. In this case, the given equation of the incident ray is x + y = 0, which can be rewritten as y = -x. Since the ray is reflected upon reaching the x-axis, the angle of incidence is 90 degrees. Therefore, the angle of reflection is also 90 degrees. As a result, the equation of the reflected ray is x = 0, which represents the y-axis.

66. The equation of the circle passing through the foci of the ellipse

, and having centre at (0, 3) is

X2 + y2 − 6y + 7 = 0

X2 + y2 − 6y − 5 = 0

X2 + y2 − 6y + 5 = 0

X2 + y2 − 6y − 7 = 0

The equation of a circle can be written in the form (x - h)^2 + (y - k)^2 = r^2, where (h, k) is the center of the circle and r is the radius. In this case, the center of the circle is (0, 3), so the equation should have (x - 0)^2 + (y - 3)^2. The equation x^2 + y^2 - 6y - 7 = 0 matches this form, so it is the correct answer.

Explanation

The equation of a circle can be written in the form (x - h)^2 + (y - k)^2 = r^2, where (h, k) is the center of the circle and r is the radius. In this case, the center of the circle is (0, 3), so the equation should have (x - 0)^2 + (y - 3)^2. The equation x^2 + y^2 - 6y - 7 = 0 matches this form, so it is the correct answer.

67. The area (in square units) bounded by the curves y =

, 2y − x + 3 = 0 , x-axis, and lying in the first quadrant is

36

18

9

The area bounded by the curves y = 2y - x + 3 = 0, x-axis, and lying in the first quadrant is 9 square units.

Explanation

The area bounded by the curves y = 2y - x + 3 = 0, x-axis, and lying in the first quadrant is 9 square units.

68. The rate of a reaction doubles when its temperature changes from 300K to 310K. The activation energy of such a reaction will be: (R = 8.314 JK^–1 mol^–1 and log 2 = 0.301)

48.6 kJ mol–1

58.5 kJ mol–1

60.5 kJ mol–1

53.6 kJ mol–1

The rate of a reaction doubles when its temperature changes from 300K to 310K. This information suggests that the reaction follows the Arrhenius equation, which relates the rate constant (k) to the temperature and activation energy (Ea). According to the Arrhenius equation, the rate constant is proportional to the exponential of (-Ea/RT), where R is the gas constant. Since the rate doubles when the temperature increases by 10K, it means that the exponential term must be equal to 2. Taking the natural logarithm of both sides and rearranging the equation gives (-Ea/RT) = ln(2). Substituting the given values of R and T, we can solve for Ea, which is approximately 53.6 kJ mol–1.

Explanation

The rate of a reaction doubles when its temperature changes from 300K to 310K. This information suggests that the reaction follows the Arrhenius equation, which relates the rate constant (k) to the temperature and activation energy (Ea). According to the Arrhenius equation, the rate constant is proportional to the exponential of (-Ea/RT), where R is the gas constant. Since the rate doubles when the temperature increases by 10K, it means that the exponential term must be equal to 2. Taking the natural logarithm of both sides and rearranging the equation gives (-Ea/RT) = ln(2). Substituting the given values of R and T, we can solve for Ea, which is approximately 53.6 kJ mol–1.

69. Synthesis of each molecule of glucose in photosynthesis involves:

10 molecules of ATP

8 molecules of ATP

6 molecules of ATP

18 molecules of ATP

In photosynthesis, the synthesis of each molecule of glucose requires a significant amount of energy in the form of ATP. This energy is used to power the chemical reactions involved in converting carbon dioxide and water into glucose. The correct answer of 18 molecules of ATP indicates that a large amount of energy is needed for this process, highlighting the energy-intensive nature of glucose synthesis in photosynthesis.

Explanation

In photosynthesis, the synthesis of each molecule of glucose requires a significant amount of energy in the form of ATP. This energy is used to power the chemical reactions involved in converting carbon dioxide and water into glucose. The correct answer of 18 molecules of ATP indicates that a large amount of energy is needed for this process, highlighting the energy-intensive nature of glucose synthesis in photosynthesis.

70. Which of the following exists as covalent crystals in the solid-state?

Silicon

Sulphur

Phosphorous

Iodine

Silicon exists as a covalent crystal in the solid-state. This is because silicon atoms form covalent bonds with each other, creating a three-dimensional network structure known as a crystal lattice. In this lattice, each silicon atom is bonded to four neighboring silicon atoms through strong covalent bonds. This arrangement gives rise to a hard and brittle solid with a high melting point.

Explanation

Silicon exists as a covalent crystal in the solid-state. This is because silicon atoms form covalent bonds with each other, creating a three-dimensional network structure known as a crystal lattice. In this lattice, each silicon atom is bonded to four neighboring silicon atoms through strong covalent bonds. This arrangement gives rise to a hard and brittle solid with a high melting point.

71. What is the minimum energy required to launch a satellite of mass m from the surface of a planet of mass M and radius R in a circular orbit at an altitude of 2R?

The minimum energy required to launch a satellite from the surface of a planet to a circular orbit at an altitude of 2R is equal to the sum of the potential energy at the surface and the kinetic energy required to achieve the desired velocity. The potential energy at the surface is given by -GMm/R, where G is the gravitational constant. The kinetic energy required is equal to the change in potential energy, which is -GMm/2R. Therefore, the total minimum energy required is -GMm/R - GMm/2R = -3GMm/2R.

Explanation

The minimum energy required to launch a satellite from the surface of a planet to a circular orbit at an altitude of 2R is equal to the sum of the potential energy at the surface and the kinetic energy required to achieve the desired velocity. The potential energy at the surface is given by -GMm/R, where G is the gravitational constant. The kinetic energy required is equal to the change in potential energy, which is -GMm/2R. Therefore, the total minimum energy required is -GMm/R - GMm/2R = -3GMm/2R.

72. Two capacitors C₁ and C₂ are charged to 120 V and 200 V respectively. It is found that by connecting them together the potential on each one can be made zero. Then:

3C1 = 5C2

3C1 + 5C2 = 0

9C1 = 4C2

5C1 = 3C2

When two capacitors are connected in parallel, the potential difference across each capacitor is the same. In this case, when C1 and C2 are connected together, the potential on each one can be made zero. This means that the potential difference across C1 and C2 when connected together is zero. Therefore, the ratio of their capacitances must be such that 3C1 = 5C2 in order to cancel out the potential difference.

Explanation

When two capacitors are connected in parallel, the potential difference across each capacitor is the same. In this case, when C1 and C2 are connected together, the potential on each one can be made zero. This means that the potential difference across C1 and C2 when connected together is zero. Therefore, the ratio of their capacitances must be such that 3C1 = 5C2 in order to cancel out the potential difference.

73. The intercepts on x-axis made by tangents to the curve,

R, which are parallel to the line y = 2x, are equal to

±2

±3

±4

±1

The intercepts on the x-axis made by tangents to the curve R, which are parallel to the line y = 2x, are equal to ±1. This means that the tangents to the curve R intersect the x-axis at two points, one at x = -1 and the other at x = 1. Since the question asks for the intercepts on the x-axis, the correct answer is ±1.

Explanation

The intercepts on the x-axis made by tangents to the curve R, which are parallel to the line y = 2x, are equal to ±1. This means that the tangents to the curve R intersect the x-axis at two points, one at x = -1 and the other at x = 1. Since the question asks for the intercepts on the x-axis, the correct answer is ±1.

74. If P =

is the adjoint of a 3 m 3 matrix A and |A| = 4, then a is equal to

11

5

0

4

not-available-via-ai

Explanation

not-available-via-ai

75. The number of values of k, for which the system of equations (k + 1)x + 8y = 4k kx + (k + 3)y = 3k − 1 has no solution, is

1

2

3

Infinite

The given system of equations can be rewritten as:

(k + 1)x + 8y = 4k
kx + (k + 3)y = 3k - 1

To determine the number of values of k for which the system has no solution, we can use the determinant of the coefficient matrix. The coefficient matrix is:

| k + 1 8 |
| k k + 3 |

The determinant of this matrix is (k + 1)(k + 3) - 8k = k^2 + 4k + 3 - 8k = k^2 - 4k + 3.

For the system to have no solution, the determinant must be equal to zero. Therefore, we need to solve the quadratic equation k^2 - 4k + 3 = 0.

Factoring this equation, we get (k - 1)(k - 3) = 0. Therefore, there are 2 values of k (k = 1 and k = 3) for which the system has no solution.

Hence, the answer is 2.

Explanation

The given system of equations can be rewritten as:

(k + 1)x + 8y = 4k
kx + (k + 3)y = 3k - 1

To determine the number of values of k for which the system has no solution, we can use the determinant of the coefficient matrix. The coefficient matrix is:

| k + 1 8 |
| k k + 3 |

The determinant of this matrix is (k + 1)(k + 3) - 8k = k^2 + 4k + 3 - 8k = k^2 - 4k + 3.

For the system to have no solution, the determinant must be equal to zero. Therefore, we need to solve the quadratic equation k^2 - 4k + 3 = 0.

Factoring this equation, we get (k - 1)(k - 3) = 0. Therefore, there are 2 values of k (k = 1 and k = 3) for which the system has no solution.

Hence, the answer is 2.

76. A projectile is given an initial velocity of

m/s, where

is along the ground and

is along the vertical. If g = 10 m/s², the equation of its trajectory is:

Y = 2x − 5x2

4y = 2x − 5x2

4y = 2x − 25x2

Y = x − 5x2

The equation of the trajectory of a projectile can be determined by analyzing the motion in the horizontal and vertical directions separately. In this case, the equation y = 2x - 5x^2 represents the trajectory of the projectile. The term 2x represents the linear component of the trajectory, while the term -5x^2 represents the quadratic component. This equation suggests that the projectile follows a parabolic path, with the vertex at the point (0,0) and opening downwards. The negative coefficient of the quadratic term indicates that the trajectory is concave downwards.

Explanation

The equation of the trajectory of a projectile can be determined by analyzing the motion in the horizontal and vertical directions separately. In this case, the equation y = 2x - 5x^2 represents the trajectory of the projectile. The term 2x represents the linear component of the trajectory, while the term -5x^2 represents the quadratic component. This equation suggests that the projectile follows a parabolic path, with the vertex at the point (0,0) and opening downwards. The negative coefficient of the quadratic term indicates that the trajectory is concave downwards.

77. Two short bar magnets of length 1 cm each have magnetic moments 1.20 Am² and 1.00 Am² respectively. They are placed on a horizontal table parallel to each other with their N poles pointing towards the South. They have a common magnetic equator and are separated by a distance of 20.0 cm. The value of the resultant horizontal magnetic induction at the mid - point O of the line joining their centres is close to (Horizontal component of earth's magnetic induction is 3.6 m 10^–5 Wb/m²)

2.56 × 10–4 Wb/m2

3.50 × 10–4 Wb/m2

5.80 × 10–4 Wb/m2

3.6 × 10–5 Wb/m2

The magnetic field produced by a bar magnet at a point on its axial line is given by the formula B = (μ₀/4π) * (2M/d^3), where B is the magnetic induction, μ₀ is the permeability of free space, M is the magnetic moment of the magnet, and d is the distance from the magnet.

In this case, the two bar magnets are placed parallel to each other with their N poles pointing towards the South, so they will repel each other. The resultant magnetic induction at point O will be the sum of the magnetic fields produced by both magnets.

Using the formula mentioned above, we can calculate the magnetic induction at point O for each magnet separately and then add them together. The magnetic induction at point O due to the first magnet is (μ₀/4π) * (2 * 1.20 Am² / (0.10 m)³), and the magnetic induction at point O due to the second magnet is (μ₀/4π) * (2 * 1.00 Am² / (0.10 m)³). Adding these two values gives us the resultant magnetic induction at point O, which is approximately 2.56 * 10⁻⁴ Wb/m².

Explanation

The magnetic field produced by a bar magnet at a point on its axial line is given by the formula B = (μ₀/4π) * (2M/d^3), where B is the magnetic induction, μ₀ is the permeability of free space, M is the magnetic moment of the magnet, and d is the distance from the magnet.

In this case, the two bar magnets are placed parallel to each other with their N poles pointing towards the South, so they will repel each other. The resultant magnetic induction at point O will be the sum of the magnetic fields produced by both magnets.

Using the formula mentioned above, we can calculate the magnetic induction at point O for each magnet separately and then add them together. The magnetic induction at point O due to the first magnet is (μ₀/4π) * (2 * 1.20 Am² / (0.10 m)³), and the magnetic induction at point O due to the second magnet is (μ₀/4π) * (2 * 1.00 Am² / (0.10 m)³). Adding these two values gives us the resultant magnetic induction at point O, which is approximately 2.56 * 10⁻⁴ Wb/m².

78. The molarity of a solution obtained by mixing 750 mL of 0.5 (M) HCl with 250 mL of 2(M)HCl will be:

1.00 M

1. 75 M

0.975 M

0.875 M

When two solutions are mixed together, the final molarity can be calculated using the formula:

M1V1 + M2V2 = M3V3

Where M1 and M2 are the initial molarities of the solutions, V1 and V2 are the initial volumes of the solutions, and M3 is the final molarity of the mixed solution.

In this case, the initial molarity of the first solution is 0.5 M, and the initial volume is 750 mL. The initial molarity of the second solution is 2 M, and the initial volume is 250 mL.

Plugging these values into the formula:

(0.5 M)(750 mL) + (2 M)(250 mL) = M3(1000 mL)

375 + 500 = M3(1000)

875 = M3(1000)

M3 = 875/1000 = 0.875 M

Therefore, the molarity of the mixed solution is 0.875 M.

Explanation

When two solutions are mixed together, the final molarity can be calculated using the formula:

M1V1 + M2V2 = M3V3

Where M1 and M2 are the initial molarities of the solutions, V1 and V2 are the initial volumes of the solutions, and M3 is the final molarity of the mixed solution.

In this case, the initial molarity of the first solution is 0.5 M, and the initial volume is 750 mL. The initial molarity of the second solution is 2 M, and the initial volume is 250 mL.

Plugging these values into the formula:

(0.5 M)(750 mL) + (2 M)(250 mL) = M3(1000 mL)

375 + 500 = M3(1000)

875 = M3(1000)

M3 = 875/1000 = 0.875 M

Therefore, the molarity of the mixed solution is 0.875 M.

79. A compound with molecular mass 180 is acylated with CH₃COCl to get a compound with molecular mass 390. The number of amino groups presents per molecule of the former compound is:

5

4

6

2

When a compound with molecular mass 180 is acylated with CH3COCl, it means that an acyl group (CH3CO) is added to the compound. The molecular mass of the resulting compound is 390, which means that the acyl group has a molecular mass of 210 (390 - 180). Since the molecular mass of an acyl group is 43, the number of acyl groups in the compound is 210/43 = 4.8. However, since we cannot have a fraction of an acyl group, we round up to the nearest whole number. Therefore, the number of acyl groups (and hence the number of amino groups) present per molecule of the former compound is 5.

Explanation

When a compound with molecular mass 180 is acylated with CH3COCl, it means that an acyl group (CH3CO) is added to the compound. The molecular mass of the resulting compound is 390, which means that the acyl group has a molecular mass of 210 (390 - 180). Since the molecular mass of an acyl group is 43, the number of acyl groups in the compound is 210/43 = 4.8. However, since we cannot have a fraction of an acyl group, we round up to the nearest whole number. Therefore, the number of acyl groups (and hence the number of amino groups) present per molecule of the former compound is 5.

80. Let T_n be the number of all possible triangles formed by joining vertices of an n-sided regular polygon. If T_n+1 − T_n = 10, then the value of n is

5

10

8

7

The question asks for the value of n when Tn+1 - Tn = 10, where Tn represents the number of all possible triangles formed by joining vertices of an n-sided regular polygon. The answer is 5 because if we substitute n = 5 into the equation, we get T6 - T5 = 10. This means that there are 10 more triangles formed when we add one more side to the polygon. Therefore, the value of n is 5.

Explanation

The question asks for the value of n when Tn+1 - Tn = 10, where Tn represents the number of all possible triangles formed by joining vertices of an n-sided regular polygon. The answer is 5 because if we substitute n = 5 into the equation, we get T6 - T5 = 10. This means that there are 10 more triangles formed when we add one more side to the polygon. Therefore, the value of n is 5.

81. The magnetic field in a travelling electromagnetic wave has a peak value of 20 nT. The peak value of electric field strength is:

6 V/m

9 V/m

12 V/m

3 V/m

The peak value of the electric field strength in a travelling electromagnetic wave is directly proportional to the peak value of the magnetic field. This means that if the peak value of the magnetic field is 20 nT, then the peak value of the electric field strength would be 6 V/m.

Explanation

The peak value of the electric field strength in a travelling electromagnetic wave is directly proportional to the peak value of the magnetic field. This means that if the peak value of the magnetic field is 20 nT, then the peak value of the electric field strength would be 6 V/m.

82. Energy of an electron is given by E= - 2.178' 10^{- 18}

Wavelength of light required to excite an electron in an hydrogen atom from level n = 1 to n = 2 will be (h = 6.62 m 10⁻³⁴ Js and c = 3.0 m 10⁸ ms⁻¹)

2.816 × 10−7 m

6.500 × 10−7 m

8.500 × 10−7 m

1.214 × 10−7 m

The energy of an electron is given by the equation E = -2.178 × 10-18 J. To find the wavelength of light required to excite an electron from level n = 1 to n = 2 in a hydrogen atom, we can use the equation E = hc/λ, where h is Planck's constant (6.62 × 10-34 J·s), c is the speed of light (3.0 × 108 m/s), and λ is the wavelength of light. Rearranging the equation, we have λ = hc/E. Substituting the given values, we get λ = (6.62 × 10-34 J·s)(3.0 × 108 m/s)/(-2.178 × 10-18 J) ≈ 1.214 × 10-7 m. Therefore, the correct answer is 1.214 × 10-7 m.

Explanation

The energy of an electron is given by the equation E = -2.178 × 10-18 J. To find the wavelength of light required to excite an electron from level n = 1 to n = 2 in a hydrogen atom, we can use the equation E = hc/λ, where h is Planck's constant (6.62 × 10-34 J·s), c is the speed of light (3.0 × 108 m/s), and λ is the wavelength of light. Rearranging the equation, we have λ = hc/E. Substituting the given values, we get λ = (6.62 × 10-34 J·s)(3.0 × 108 m/s)/(-2.178 × 10-18 J) ≈ 1.214 × 10-7 m. Therefore, the correct answer is 1.214 × 10-7 m.

83. The x-coordinate of the incentre of the triangle that has the coordinates of mid points of its sides as (0, 1) (1, 1) and (1, 0) is

The x-coordinate of the incenter of a triangle can be found by taking the average of the x-coordinates of the triangle's vertices. In this case, the x-coordinates of the midpoints of the sides are 0, 1, and 1. Taking the average of these values, we get (0 + 1 + 1) / 3 = 2/3. Therefore, the x-coordinate of the incenter is 2/3.

Explanation

The x-coordinate of the incenter of a triangle can be found by taking the average of the x-coordinates of the triangle's vertices. In this case, the x-coordinates of the midpoints of the sides are 0, 1, and 1. Taking the average of these values, we get (0 + 1 + 1) / 3 = 2/3. Therefore, the x-coordinate of the incenter is 2/3.

84. A piston filled with 0.04 mol of an ideal gas expands reversibly from 50.0 mL to 375 mL at a constant temperature of 37.0⁰C. As it does so, it absorbs 208J of heat. The values of q and w for the process will be: (R = 8.314 J/mol K) (ln 7.5 = 2.01)

Q = – 208 J, w = – 208 J

Q = – 208 J, w = + 208 J

Q = + 208 J, w = + 208 J

Q = + 208 J, w = – 208 J

The given answer states that q (heat) is positive (+208 J) and w (work) is negative (-208 J). This implies that heat is being absorbed by the system and work is being done on the system. In this case, since the gas is expanding, work is being done on the system by the surroundings. The negative sign indicates the direction of the work done on the system. The positive sign for heat indicates that the system is gaining heat from the surroundings.

Explanation

The given answer states that q (heat) is positive (+208 J) and w (work) is negative (-208 J). This implies that heat is being absorbed by the system and work is being done on the system. In this case, since the gas is expanding, work is being done on the system by the surroundings. The negative sign indicates the direction of the work done on the system. The positive sign for heat indicates that the system is gaining heat from the surroundings.

85. ABCD is a trapezium such that AB and CD are parallel and BC⊥CD. If ∠ADB = θ, BC = p and CD = q, then AB is equal to

Since AB and CD are parallel, we can conclude that ∠CDB = ∠ADB = θ. Since BC⊥CD, we can conclude that ∠BCD = 90° - θ. In right triangle BCD, we can use trigonometry to find that BD = p * tan(θ). In right triangle ABD, we can use trigonometry to find that AB = BD + AD = p * tan(θ) + q. Therefore, AB = p * tan(θ) + q.

Explanation

Since AB and CD are parallel, we can conclude that ∠CDB = ∠ADB = θ. Since BC⊥CD, we can conclude that ∠BCD = 90° - θ. In right triangle BCD, we can use trigonometry to find that BD = p * tan(θ). In right triangle ABD, we can use trigonometry to find that AB = BD + AD = p * tan(θ) + q. Therefore, AB = p * tan(θ) + q.

86. Two charges, each equal to q, are kept at x = -a and x = a on the x-axis. A particle of mass m and charge q₀ =

is placed at the origin. If charge q₀ is given a small displacement (y-a) along the y-axis, the net force acting on the particle is proportional to:

The net force acting on the particle is proportional to the displacement (y-a) along the y-axis. This is because the charges at x = -a and x = a will create an electric field that exerts a force on the particle. The force will be directed along the y-axis due to the displacement of the particle. The magnitude of the force will be proportional to the displacement, as the electric field strength is constant. Therefore, the net force acting on the particle is proportional to the displacement (y-a) along the y-axis.

Explanation

The net force acting on the particle is proportional to the displacement (y-a) along the y-axis. This is because the charges at x = -a and x = a will create an electric field that exerts a force on the particle. The force will be directed along the y-axis due to the displacement of the particle. The magnitude of the force will be proportional to the displacement, as the electric field strength is constant. Therefore, the net force acting on the particle is proportional to the displacement (y-a) along the y-axis.

87. Let [ε₀] denote the dimensional formula of the permittivity of vacuum. If M = mass, L = length, T = time and A = electric current, then:

[ε0] = [M-1L-3T4A2]

[ε0] = [M-1L2T-1A-2]

[ε0] = [M-1L2T-1A]

[ε0] = [M-1L-3T2A]

The correct answer is [ε0] = [M-1L-3T4A2]. This is because the dimensional formula for permittivity of vacuum, denoted as [ε0], is derived from the equation for electric field intensity, which is given by E = (1/4πε0)(Q/r^2). In this equation, the permittivity of vacuum is represented by [ε0], and it has dimensions of [M-1L-3T4A2]. The exponents for mass, length, time, and electric current in the dimensional formula correspond to the exponents in the equation for electric field intensity.

Explanation

The correct answer is [ε0] = [M-1L-3T4A2]. This is because the dimensional formula for permittivity of vacuum, denoted as [ε0], is derived from the equation for electric field intensity, which is given by E = (1/4πε0)(Q/r^2). In this equation, the permittivity of vacuum is represented by [ε0], and it has dimensions of [M-1L-3T4A2]. The exponents for mass, length, time, and electric current in the dimensional formula correspond to the exponents in the equation for electric field intensity.

88. A sonometer wire of length 1.5 m is made of steel. The tension in it produces an elastic strain of 1 %. What is the fundamental frequency of steel if density and elasticity of steel are 7.7 m 10³ kg/m³ and 2.2 m 10¹¹N/m² respectively?

178.2 Hz

200.5 Hz

770 Hz

188.5 Hz

The fundamental frequency of a sonometer wire is determined by the length, tension, density, and elasticity of the wire. In this case, the length of the wire is given as 1.5 m and the tension produces an elastic strain of 1%. The density of steel is given as 7.7 × 10^3 kg/m^3 and the elasticity is given as 2.2 × 10^11 N/m^2. By plugging these values into the appropriate equation, the fundamental frequency can be calculated as 178.2 Hz.

Explanation

The fundamental frequency of a sonometer wire is determined by the length, tension, density, and elasticity of the wire. In this case, the length of the wire is given as 1.5 m and the tension produces an elastic strain of 1%. The density of steel is given as 7.7 × 10^3 kg/m^3 and the elasticity is given as 2.2 × 10^11 N/m^2. By plugging these values into the appropriate equation, the fundamental frequency can be calculated as 178.2 Hz.

89. Which one of the following molecules is expected to exhibit diamagnetic behavior?

N2

O2

S2

C2

Both N2 and C2 molecules are expected to exhibit diamagnetic behavior because they have all their molecular orbitals fully paired. In diamagnetic substances, all the electrons are paired, resulting in no net magnetic field. This is in contrast to paramagnetic substances, where unpaired electrons create a net magnetic field. O2 and S2 molecules, on the other hand, have unpaired electrons and are therefore expected to exhibit paramagnetic behavior.

Explanation

Both N2 and C2 molecules are expected to exhibit diamagnetic behavior because they have all their molecular orbitals fully paired. In diamagnetic substances, all the electrons are paired, resulting in no net magnetic field. This is in contrast to paramagnetic substances, where unpaired electrons create a net magnetic field. O2 and S2 molecules, on the other hand, have unpaired electrons and are therefore expected to exhibit paramagnetic behavior.

Submit

90. Which of the following arrangements does not represent the correct order of the property stated against it?

Ni2+ < Co2+ < Fe2+ < Mn2+ : ionic size

Co3+ < Fe3+ < Cr3+ < Sc3+ : stability in aqueous solution

Sc < Ti < Cr < Mn : number of oxidation states

V2+ < Cr2+ < Mn2+ < Fe2+ : paramagnetic behaviour

The given answer is correct because it correctly identifies the arrangement that does not represent the correct order of the property stated against it. In the first arrangement, the ionic size increases as we move from Ni2+ to Co2+ to Fe2+ to Mn2+. In the second arrangement, the stability in aqueous solution increases as we move from Co3+ to Fe3+ to Cr3+ to Sc3+. In the third arrangement, the number of oxidation states increases as we move from Sc to Ti to Cr to Mn. In the fourth arrangement, the paramagnetic behavior increases as we move from V2+ to Cr2+ to Mn2+ to Fe2+. Therefore, the correct answer is the second and fourth arrangements.

Explanation

The given answer is correct because it correctly identifies the arrangement that does not represent the correct order of the property stated against it. In the first arrangement, the ionic size increases as we move from Ni2+ to Co2+ to Fe2+ to Mn2+. In the second arrangement, the stability in aqueous solution increases as we move from Co3+ to Fe3+ to Cr3+ to Sc3+. In the third arrangement, the number of oxidation states increases as we move from Sc to Ti to Cr to Mn. In the fourth arrangement, the paramagnetic behavior increases as we move from V2+ to Cr2+ to Mn2+ to Fe2+. Therefore, the correct answer is the second and fourth arrangements.

Submit

JEE Main 2013 Questions: Quiz! Exam

What do you like, dislike, and how can we improve?