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Test: Differential Equations- Case Based Type Questions - JEE MCQ


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20 Questions MCQ Test Mathematics (Maths) Class 12 - Test: Differential Equations- Case Based Type Questions

Test: Differential Equations- Case Based Type Questions for JEE 2024 is part of Mathematics (Maths) Class 12 preparation. The Test: Differential Equations- Case Based Type Questions questions and answers have been prepared according to the JEE exam syllabus.The Test: Differential Equations- Case Based Type Questions MCQs are made for JEE 2024 Exam. Find important definitions, questions, notes, meanings, examples, exercises, MCQs and online tests for Test: Differential Equations- Case Based Type Questions below.
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Test: Differential Equations- Case Based Type Questions - Question 1

Direction: Read the following text and answer the following questions on the basis of the same:

A differential equation of the form where P and Q are functions of x alone, is called a first order linear differential equation. It has many applications in physics including RL circuits.

Consider the linear differential equation

Q. The general solution is _______.

Detailed Solution for Test: Differential Equations- Case Based Type Questions - Question 1
The general solution is

Test: Differential Equations- Case Based Type Questions - Question 2

Direction: Read the following text and answer the following questions on the basis of the same:

A differential equation of the form where P and Q are functions of x alone, is called a first order linear differential equation. It has many applications in physics including RL circuits.

Consider the linear differential equation

Q. The value of ∫ P dx = ______.

Detailed Solution for Test: Differential Equations- Case Based Type Questions - Question 2

= 2log |x| + C

= log x2 + C

Test: Differential Equations- Case Based Type Questions - Question 3

Direction: Read the following text and answer the following questions on the basis of the same:

A differential equation of the form where P and Q are functions of x alone, is called a first order linear differential equation. It has many applications in physics including RL circuits.

Consider the linear differential equation

Q. ∫ Qdx = _______.

Detailed Solution for Test: Differential Equations- Case Based Type Questions - Question 3
The given differential equation can be expressed as

P = 2/x , Q = x

Qdx = xdx

Test: Differential Equations- Case Based Type Questions - Question 4

Direction: Read the following text and answer the following questions on the basis of the same:

A differential equation of the form where P and Q are functions of x alone, is called a first order linear differential equation. It has many applications in physics including RL circuits.

Consider the linear differential equation

Q. The integrating factor is _______.

Detailed Solution for Test: Differential Equations- Case Based Type Questions - Question 4

Test: Differential Equations- Case Based Type Questions - Question 5

Direction: Read the following text and answer the following questions on the basis of the same:

A differential equation of the form where P and Q are functions of x alone, is called a first order linear differential equation. It has many applications in physics including RL circuits.

Consider the linear differential equation

Q. If y(1) = 0, then y(2) = _______.

Detailed Solution for Test: Differential Equations- Case Based Type Questions - Question 5
Given y(1) = 0

⇒ C = -1/4

The particular solution is

When x = 2, 4y = 15/4

⇒ y = 15/16

Test: Differential Equations- Case Based Type Questions - Question 6

Direction: Read the following text and answer the following questions on the basis of the same:

Polio drops are delivered to 50K children in a district. The rate at which polio drops are given is directly proportional to the number of children who have not been administered the drops. By the end of 2nd week half the children have been given the polio drops. How many will have been given the drops by the end of 3rd week can be estimated using the solution to the differential equation dy/dx = k(50 - y) where x denotes the number of weeks and y the number of children who have been given the drops.

Q. The value of C in the particular solution given that y(0) = 0 and k = 0.049 is

Detailed Solution for Test: Differential Equations- Case Based Type Questions - Question 6
Given,

y(0) = and k = 0.049

We have, -log|50 - y| = Kx + C

log |50 - y| = -Kx - C

log |50 - 0| = 0 - C

[∵ x = 0, K = 0.049, y(0) = 0]

log 50 = - C

C = log 1/50

Test: Differential Equations- Case Based Type Questions - Question 7

Direction: Read the following text and answer the following questions on the basis of the same:

Polio drops are delivered to 50K children in a district. The rate at which polio drops are given is directly proportional to the number of children who have not been administered the drops. By the end of 2nd week half the children have been given the polio drops. How many will have been given the drops by the end of 3rd week can be estimated using the solution to the differential equation dy/dx = k(50 - y) where x denotes the number of weeks and y the number of children who have been given the drops.

Q. Which method of solving a differential equation can be used to solve dy/dx = k(50 - y)?

Detailed Solution for Test: Differential Equations- Case Based Type Questions - Question 7
‘Variable separable method’ is used to solve such an equation in which variables

can be separated completely, i.e., terms containing x should remain with dx and terms containing y should remain with dy.

Test: Differential Equations- Case Based Type Questions - Question 8

Direction: Read the following text and answer the following questions on the basis of the same:

Polio drops are delivered to 50K children in a district. The rate at which polio drops are given is directly proportional to the number of children who have not been administered the drops. By the end of 2nd week half the children have been given the polio drops. How many will have been given the drops by the end of 3rd week can be estimated using the solution to the differential equation dy/dx = k(50 - y) where x denotes the number of weeks and y the number of children who have been given the drops.

Q. State the order of the above given differential equation.

Test: Differential Equations- Case Based Type Questions - Question 9

Direction: Read the following text and answer the following questions on the basis of the same:

Polio drops are delivered to 50K children in a district. The rate at which polio drops are given is directly proportional to the number of children who have not been administered the drops. By the end of 2nd week half the children have been given the polio drops. How many will have been given the drops by the end of 3rd week can be estimated using the solution to the differential equation dy/dx = k(50 - y) where x denotes the number of weeks and y the number of children who have been given the drops.

Q. The solution of the differential equation dy/dx = k(50 - y) is given by,

Detailed Solution for Test: Differential Equations- Case Based Type Questions - Question 9
dy/dx = k(50 - y)

Test: Differential Equations- Case Based Type Questions - Question 10

Direction: Read the following text and answer the following questions on the basis of the same:

Polio drops are delivered to 50K children in a district. The rate at which polio drops are given is directly proportional to the number of children who have not been administered the drops. By the end of 2nd week half the children have been given the polio drops. How many will have been given the drops by the end of 3rd week can be estimated using the solution to the differential equation dy/dx = k(50 - y) where x denotes the number of weeks and y the number of children who have been given the drops.

Q. Which of the following solutions may be used to find the number of children who have been given the polio drops?

Detailed Solution for Test: Differential Equations- Case Based Type Questions - Question 10
We have

Test: Differential Equations- Case Based Type Questions - Question 11

Direction: Read the following text and answer the following questions on the basis of the same:

A Veterinary doctor was examining a sick cat brought by a pet lover. When it was brought to the hospital, it was already dead. The pet lover wanted to find its time of death. He took the temperature of the cat at 11.30 pm which was 94.6°F. He took the temperature again after one hour; the temperature was lower than the first observation. It was 93.4°F. The room in which the cat was put is always at 70°F. The normal temperature of the cat is taken as 98.6°F when it was alive. The doctor estimated the time of death using Newton law of cooling which is governed by the differential equation: dT/dt ∞ (T−70), where 70°F is the room temperature and T is the temperature of the object at time t.

Substituting the two different observations of T and t made, in the solution of the differential equation dT/dt = k(T - 70) where k is a constant of proportion, time of death is calculated.

Q. The solution of the differential equation dT/dt = k(T - 70) is given by,

Test: Differential Equations- Case Based Type Questions - Question 12

Direction: Read the following text and answer the following questions on the basis of the same:

A Veterinary doctor was examining a sick cat brought by a pet lover. When it was brought to the hospital, it was already dead. The pet lover wanted to find its time of death. He took the temperature of the cat at 11.30 pm which was 94.6°F. He took the temperature again after one hour; the temperature was lower than the first observation. It was 93.4°F. The room in which the cat was put is always at 70°F. The normal temperature of the cat is taken as 98.6°F when it was alive. The doctor estimated the time of death using Newton law of cooling which is governed by the differential equation: dT/dt ∞ (T−70), where 70°F is the room temperature and T is the temperature of the object at time t.

Substituting the two different observations of T and t made, in the solution of the differential equation dT/dt = k(T - 70) where k is a constant of proportion, time of death is calculated.

Q. Which method of solving a differential equation helped in calculation of the time of death?

Test: Differential Equations- Case Based Type Questions - Question 13

Direction: Read the following text and answer the following questions on the basis of the same:

A Veterinary doctor was examining a sick cat brought by a pet lover. When it was brought to the hospital, it was already dead. The pet lover wanted to find its time of death. He took the temperature of the cat at 11.30 pm which was 94.6°F. He took the temperature again after one hour; the temperature was lower than the first observation. It was 93.4°F. The room in which the cat was put is always at 70°F. The normal temperature of the cat is taken as 98.6°F when it was alive. The doctor estimated the time of death using Newton law of cooling which is governed by the differential equation: dT/dt ∞ (T−70), where 70°F is the room temperature and T is the temperature of the object at time t.

Substituting the two different observations of T and t made, in the solution of the differential equation dT/dt = k(T - 70) where k is a constant of proportion, time of death is calculated.

Q. What will be the degree of the above given differential equation.

Test: Differential Equations- Case Based Type Questions - Question 14

Direction: Read the following text and answer the following questions on the basis of the same:

A Veterinary doctor was examining a sick cat brought by a pet lover. When it was brought to the hospital, it was already dead. The pet lover wanted to find its time of death. He took the temperature of the cat at 11.30 pm which was 94.6°F. He took the temperature again after one hour; the temperature was lower than the first observation. It was 93.4°F. The room in which the cat was put is always at 70°F. The normal temperature of the cat is taken as 98.6°F when it was alive. The doctor estimated the time of death using Newton law of cooling which is governed by the differential equation: dT/dt ∞ (T−70), where 70°F is the room temperature and T is the temperature of the object at time t.

Substituting the two different observations of T and t made, in the solution of the differential equation dT/dt = k(T - 70) where k is a constant of proportion, time of death is calculated.

Q. If the temperature was measured 2 hours after 11.30 pm, what will be the change in time of death?

Test: Differential Equations- Case Based Type Questions - Question 15

Direction: Read the following text and answer the following questions on the basis of the same:

A Veterinary doctor was examining a sick cat brought by a pet lover. When it was brought to the hospital, it was already dead. The pet lover wanted to find its time of death. He took the temperature of the cat at 11.30 pm which was 94.6°F. He took the temperature again after one hour; the temperature was lower than the first observation. It was 93.4°F. The room in which the cat was put is always at 70°F. The normal temperature of the cat is taken as 98.6°F when it was alive. The doctor estimated the time of death using Newton law of cooling which is governed by the differential equation: dT/dt ∞ (T−70), where 70°F is the room temperature and T is the temperature of the object at time t.

Substituting the two different observations of T and t made, in the solution of the differential equation dT/dt = k(T - 70) where k is a constant of proportion, time of death is calculated.

Q. If t = 0 when T is 72, then the value of C is

Test: Differential Equations- Case Based Type Questions - Question 16

Direction: Read the following text and answer the following questions on the basis of the same:

The rate of increase in the number of bacteria in a certain bacteria culture is proportional to the number present. Given that the number triples in 5 hours.

Q. If N0 is the initial count of bacteria, after 10 hours the count is _______.

Detailed Solution for Test: Differential Equations- Case Based Type Questions - Question 16
Given when t = 0, N = N0.

From (i), log|N0| = C

∴ (i) ⇒ log|N| = Kt + log|N0|

….(ii)

Given when t = 5, N = 3N0.

From (ii), log|3| = 5K

⇒ K = 1/5 log3

∴ The particular solution is

When t = 10,

N/N0 = 9

⇒ N = 9N0

Test: Differential Equations- Case Based Type Questions - Question 17

Direction: Read the following text and answer the following questions on the basis of the same:

The rate of increase in the number of bacteria in a certain bacteria culture is proportional to the number present. Given that the number triples in 5 hours.

Q. If ‘N’ is the number of bacteria, the corresponding differential equation is _______.

Detailed Solution for Test: Differential Equations- Case Based Type Questions - Question 17
Given that N is the number of bacteria.

⇒ dN/dt = KN

Test: Differential Equations- Case Based Type Questions - Question 18

Direction: Read the following text and answer the following questions on the basis of the same:

The rate of increase in the number of bacteria in a certain bacteria culture is proportional to the number present. Given that the number triples in 5 hours.

Q.

Detailed Solution for Test: Differential Equations- Case Based Type Questions - Question 18

Test: Differential Equations- Case Based Type Questions - Question 19

Direction: Read the following text and answer the following questions on the basis of the same:

The rate of increase in the number of bacteria in a certain bacteria culture is proportional to the number present. Given that the number triples in 5 hours.

Q. The general solution is _______.

Detailed Solution for Test: Differential Equations- Case Based Type Questions - Question 19

log|N| = Kt + C ...(i)

Test: Differential Equations- Case Based Type Questions - Question 20

Direction: Read the following text and answer the following questions on the basis of the same:

The rate of increase in the number of bacteria in a certain bacteria culture is proportional to the number present. Given that the number triples in 5 hours.

Q. The bacteria becomes 10 times in _______ hours.

Detailed Solution for Test: Differential Equations- Case Based Type Questions - Question 20
Given N = 10N0,

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