NCERT Solutions for Exercise 9.1 Class 12 Maths Chapter 9- Differential Equations

NCERT Solutions for Exercise 9.1 Class 12 Maths Chapter 9- Differential Equations

Edited By Ramraj Saini | Updated on Dec 04, 2023 08:02 AM IST | #CBSE Class 12th

NCERT Solutions For Class 12 Maths Chapter 9 Exercise 9.1

NCERT Solutions for Exercise 9.1 Class 12 Maths Chapter 9 Differential Equations are discussed here. These NCERT solutions are created by subject matter expert at Careers360 considering the latest syllabus and pattern of CBSE 2023-24. NCERT solutions for exercise 9.1 Class 12 Maths chapter 9 introduces the questions related to differential equations. In the NCERT Class 11 Mathematics Book and also chapter 5 of Class 12 Maths, the concepts of derivatives are discussed. Exercise 9.1 Class 12 Maths gives an idea about equations involving derivatives. NCERT solutions for Class 12 Maths chapter 9 exercise 9.1 give clarity about the concept of degree and order of a differential equation. A few examples are also given in the NCERT Book to understand the same.

Here are solutions to Class 12 Maths chapter 9 exercise 9.1 prepared by expert Mathematics faculties. 12th class Maths exercise 9.1 answers are designed as per the students demand covering comprehensive, step by step solutions of every problem. Practice these questions and answers to command the concepts, boost confidence and in depth understanding of concepts. Students can find all exercise together using the link provided below.

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Access NCERT Solutions for Class 12 Maths Chapter 9 Exercise 9.1

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Differential Equations Class 12 Chapter 9 Exercise: 9.1

Question:1 Determine order and degree (if defined) of differential equation \frac{\mathrm{d} ^4y}{\mathrm{d} x^4} +\sin(y''')=0

Answer:

Given function is
\frac{\mathrm{d} ^4y}{\mathrm{d} x^4} +\sin(y''')=0
We can rewrite it as
y^{''''}+\sin(y''') =0
Now, it is clear from the above that, the highest order derivative present in differential equation is y^{''''}

Therefore, the order of the given differential equation \frac{\mathrm{d} ^4y}{\mathrm{d} x^4} +\sin(y''')=0 is 4
Now, the given differential equation is not a polynomial equation in its derivatives
Therefore, it's a degree is not defined

Question:2 Determine order and degree (if defined) of differential equation y' + 5y = 0

Answer:

Given function is
y' + 5y = 0
Now, it is clear from the above that, the highest order derivative present in differential equation is y^{'}
Therefore, the order of the given differential equation y' + 5y = 0 is 1
Now, the given differential equation is a polynomial equation in its derivatives and its highest power raised to y ' is 1
Therefore, it's a degree is 1.

Question:3 Determine order and degree (if defined) of differential equation \left(\frac{\mathrm{d} s}{\mathrm{d} t} \right )^4 + 3s \frac{\mathrm{d}^2 s}{\mathrm{d} t^2} = 0

Answer:

Given function is
\left(\frac{\mathrm{d} s}{\mathrm{d} t} \right )^4 + 3s \frac{\mathrm{d}^2 s}{\mathrm{d} t^2} = 0
We can rewrite it as
(s^{'})^4+3s.s^{''} =0
Now, it is clear from the above that, the highest order derivative present in differential equation is s^{''}

Therefore, the order of the given differential equation \left(\frac{\mathrm{d} s}{\mathrm{d} t} \right )^4 + 3s \frac{\mathrm{d}^2 s}{\mathrm{d} t^2} = 0 is 2
Now, the given differential equation is a polynomial equation in its derivatives and power raised to s '' is 1
Therefore, it's a degree is 1

Question:4 Determine order and degree (if defined) of differential equation.

\left(\frac{d^2y}{dx^2} \right )^2 + \cos\left(\frac{dy}{dx} \right )= 0

Answer:

Given function is
\left(\frac{d^2y}{dx^2} \right )^2 + \cos\left(\frac{dy}{dx} \right )= 0
We can rewrite it as
(y^{''})^2+\cos y^{''} =0
Now, it is clear from the above that, the highest order derivative present in differential equation is y^{''}

Therefore, the order of the given differential equation \left(\frac{d^2y}{dx^2} \right )^2 + \cos\left(\frac{dy}{dx} \right )= 0 is 2
Now, the given differential equation is not a polynomial equation in its derivatives
Therefore, it's a degree is not defined

Question:5 Determine order and degree (if defined) of differential equation.

\frac{d^2y}{dx^2} = \cos 3x + \sin 3x

Answer:

Given function is
\frac{d^2y}{dx^2} = \cos 3x + \sin 3x
\Rightarrow \frac{d^2y}{dx^2}- \cos 3x - \sin 3x = 0

Now, it is clear from the above that, the highest order derivative present in differential equation is y^{''}\left ( \frac{d^2y}{dx^2} \right )

Therefore, order of given differential equation \frac{d^2y}{dx^2}- \cos 3x - \sin 3x = 0 is 2
Now, the given differential equation is a polynomial equation in it's dervatives \frac{d^2y}{dx^2} and power raised to \frac{d^2y}{dx^2} is 1
Therefore, it's degree is 1

Question:6 Determine order and degree (if defined) of differential equation (y''')^2 + (y'')^3 + (y')^4 + y^5= 0

Answer:

Given function is
(y''')^2 + (y'')^3 + (y')^4 + y^5= 0
Now, it is clear from the above that, the highest order derivative present in differential equation is y^{'''}

Therefore, order of given differential equation (y''')^2 + (y'')^3 + (y')^4 + y^5= 0 is 3
Now, the given differential equation is a polynomial equation in it's dervatives y^{'''} , y^{''} \ and \ y^{'} and power raised to y^{'''} is 2
Therefore, it's degree is 2

Question:7 Determine order and degree (if defined) of differential equation

y''' + 2y'' + y' =0

Answer:

Given function is
y''' + 2y'' + y' =0
Now, it is clear from the above that, the highest order derivative present in differential equation is y^{'''}

Therefore, order of given differential equation y''' + 2y'' + y' =0 is 3
Now, the given differential equation is a polynomial equation in it's dervatives y^{'''} , y^{''} \ and \ y^{'} and power raised to y^{'''} is 1
Therefore, it's degree is 1

Question:8 Determine order and degree (if defined) of differential equation

y' + y = e^x

Answer:

Given function is
y' + y = e^x
\Rightarrow y^{'}+y-e^x=0

Now, it is clear from the above that, the highest order derivative present in differential equation is y^{'}

Therefore, order of given differential equation y^{'}+y-e^x=0 is 1
Now, the given differential equation is a polynomial equation in it's dervatives y^{'} and power raised to y^{'} is 1
Therefore, it's degree is 1

Question:9 Determine order and degree (if defined) of differential equation

y'' + (y')^2 + 2y = 0

Answer:

Given function is
y'' + (y')^2 + 2y = 0
Now, it is clear from the above that, the highest order derivative present in differential equation is y^{''}

Therefore, order of given differential equation y'' + (y')^2 + 2y = 0 is 2
Now, the given differential equation is a polynomial equation in it's dervatives y^{''} \ and \ y^{'} and power raised to y^{''} is 1
Therefore, it's degree is 1

Question:10 Determine order and degree (if defined) of differential equation

y'' + 2y' + \sin y = 0

Answer:

Given function is
y'' + 2y' + \sin y = 0
Now, it is clear from the above that, the highest order derivative present in differential equation is y^{''}

Therefore, order of given differential equation y'' + 2y' + \sin y = 0 is 2
Now, the given differential equation is a polynomial equation in it's dervatives y^{''} \ and \ y^{'} and power raised to y^{''} is 1
Therefore, it's degree is 1

Question:11 The degree of the differential equation \left(\frac{d^2y}{dx^2} \right )^3 + \left(\frac{dy}{dx} \right )^2 + \sin\left(\frac{dy}{dx}\right ) + 1= 0 is

(A) 3

(B) 2

(C) 1

(D) not defined

Answer:

Given function is
\left(\frac{d^2y}{dx^2} \right )^3 + \left(\frac{dy}{dx} \right )^2 + \sin\left(\frac{dy}{dx}\right ) + 1= 0
We can rewrite it as
(y^{''})^3+(y^{'})^2+\sin y^{'}+1=0
Now, it is clear from the above that, the highest order derivative present in differential equation is y^{''}

Therefore, order of given differential equation \left(\frac{d^2y}{dx^2} \right )^3 + \left(\frac{dy}{dx} \right )^2 + \sin\left(\frac{dy}{dx}\right ) + 1= 0 is 2
Now, the given differential equation is a not polynomial equation in it's dervatives
Therefore, it's degree is not defined

Therefore, answer is (D)

Question:12 The order of the differential equation 2x^2\frac{d^2y}{dx^2} - 3\frac{dy}{dx} + y = 0 is

(A) 2

(B) 1

(C) 0

(D) Not Defined

Answer:

Given function is
2x^2\frac{d^2y}{dx^2} - 3\frac{dy}{dx} + y = 0
We can rewrite it as
2x.y^{''}-3y^{'}+y=0
Now, it is clear from the above that, the highest order derivative present in differential equation is y^{''}

Therefore, order of given differential equation 2x^2\frac{d^2y}{dx^2} - 3\frac{dy}{dx} + y = 0 is 2

Therefore, answer is (A)

More About NCERT Solutions for Class 12 Maths Chapter 9 Exercise 9.1

There is one example prior to exercise 9.1 Class 12 Maths and 12 questions in the Class 12 Maths chapter 9 exercise 9.1. Two questions of Class 12th Maths chapter 6 exercise 9.1 are multiple objective type questions. All the questions in NCERT solutions for Class 12 Maths chapter 9 exercise 9.1 are to find the order and degree of the given differential equations.

Also Read| Differential Equations Class 12th Notes

Benefits of NCERT Solutions for Class 12 Maths Chapter 9 Exercise 9.1

  • One fill in the blank or multiple choice type or very short answer can be expected from exercise 9.1 Class 12 Maths for CBSE Class 12 Maths Board Exams

  • Not only CBSE, but certain state boards also follow the NCERT Syllabus. Therefore the NCERT solutions for Class 12 Maths chapter 9 exercise 9.1 can be used to prepare for state boards that follow NCERT.

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Key Features Of NCERT Solutions for Exercise 9.1 Class 12 Maths Chapter 9

  • Comprehensive Coverage: The solutions encompass all the topics covered in ex 9.1 class 12, ensuring a thorough understanding of the concepts.
  • Step-by-Step Solutions: In this class 12 maths ex 9.1, each problem is solved systematically, providing a stepwise approach to aid in better comprehension for students.
  • Accuracy and Clarity: Solutions for class 12 ex 9.1 are presented accurately and concisely, using simple language to help students grasp the concepts easily.
  • Conceptual Clarity: In this 12th class maths exercise 9.1 answers, emphasis is placed on conceptual clarity, providing explanations that assist students in understanding the underlying principles behind each problem.
  • Inclusive Approach: Solutions for ex 9.1 class 12 cater to different learning styles and abilities, ensuring that students of various levels can grasp the concepts effectively.
  • Relevance to Curriculum: The solutions for class 12 maths ex 9.1 align closely with the NCERT curriculum, ensuring that students are prepared in line with the prescribed syllabus.

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Frequently Asked Questions (FAQs)

1. How many questions are solved in Exercise 9.1 Class 12 Maths?

12 questions and their answers are given in the NCERT solutions for Class 12 Maths chapter 9 exercise 9.1

2. What number of exercises are present in the chapter differential equations?

7 exercises. In which one is miscellaneous exercises.

3. What is the difference between miscellaneous exercises and other exercises given in the chapter?

Miscellaneous exercise covers question from whole chapter and exercise questions covers topics discussed in that particular area.

4. What are the concepts covered in the Class 12th Maths chapter 6 exercise 9.1?

The concepts of order and degree of differential equations are covered in the Class 12 Maths chapter 9 exercise 9.1.

5. How many examples are solved before Class 12th Maths chapter 6 exercise 9.1?

One solved example is given in the NCERT book before exercise 9.1 Class 12 Maths.

6. Is the topic of degree and order important for JEE Main exam?

Yes, Students can expect questions from this part for JEE Mains.

7. Are the NCERT Solutions for Class 12 Maths chapter 9 exercise 9.1 reliable?

Yes, these solutions of exercise 9.1 are prepared by expert faculty and are reviewed.

8. Why do we solve Exercise 9.1 Class 12 Maths?

It is necessary to get clarity over the topics degree and order of a differential equation. NCERT Solutions for Class 12 Maths chapter 9 exercise 9.1 helps for the same. 

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A block of mass 0.50 kg is moving with a speed of 2.00 ms-1 on a smooth surface. It strikes another mass of 1.00 kg and then they move together as a single body. The energy loss during the collision is

Option 1)

0.34\; J

Option 2)

0.16\; J

Option 3)

1.00\; J

Option 4)

0.67\; J

A person trying to lose weight by burning fat lifts a mass of 10 kg upto a height of 1 m 1000 times.  Assume that the potential energy lost each time he lowers the mass is dissipated.  How much fat will he use up considering the work done only when the weight is lifted up ?  Fat supplies 3.8×107 J of energy per kg which is converted to mechanical energy with a 20% efficiency rate.  Take g = 9.8 ms−2 :

Option 1)

2.45×10−3 kg

Option 2)

 6.45×10−3 kg

Option 3)

 9.89×10−3 kg

Option 4)

12.89×10−3 kg

 

An athlete in the olympic games covers a distance of 100 m in 10 s. His kinetic energy can be estimated to be in the range

Option 1)

2,000 \; J - 5,000\; J

Option 2)

200 \, \, J - 500 \, \, J

Option 3)

2\times 10^{5}J-3\times 10^{5}J

Option 4)

20,000 \, \, J - 50,000 \, \, J

A particle is projected at 600   to the horizontal with a kinetic energy K. The kinetic energy at the highest point

Option 1)

K/2\,

Option 2)

\; K\;

Option 3)

zero\;

Option 4)

K/4

In the reaction,

2Al_{(s)}+6HCL_{(aq)}\rightarrow 2Al^{3+}\, _{(aq)}+6Cl^{-}\, _{(aq)}+3H_{2(g)}

Option 1)

11.2\, L\, H_{2(g)}  at STP  is produced for every mole HCL_{(aq)}  consumed

Option 2)

6L\, HCl_{(aq)}  is consumed for ever 3L\, H_{2(g)}      produced

Option 3)

33.6 L\, H_{2(g)} is produced regardless of temperature and pressure for every mole Al that reacts

Option 4)

67.2\, L\, H_{2(g)} at STP is produced for every mole Al that reacts .

How many moles of magnesium phosphate, Mg_{3}(PO_{4})_{2} will contain 0.25 mole of oxygen atoms?

Option 1)

0.02

Option 2)

3.125 × 10-2

Option 3)

1.25 × 10-2

Option 4)

2.5 × 10-2

If we consider that 1/6, in place of 1/12, mass of carbon atom is taken to be the relative atomic mass unit, the mass of one mole of a substance will

Option 1)

decrease twice

Option 2)

increase two fold

Option 3)

remain unchanged

Option 4)

be a function of the molecular mass of the substance.

With increase of temperature, which of these changes?

Option 1)

Molality

Option 2)

Weight fraction of solute

Option 3)

Fraction of solute present in water

Option 4)

Mole fraction.

Number of atoms in 558.5 gram Fe (at. wt.of Fe = 55.85 g mol-1) is

Option 1)

twice that in 60 g carbon

Option 2)

6.023 × 1022

Option 3)

half that in 8 g He

Option 4)

558.5 × 6.023 × 1023

A pulley of radius 2 m is rotated about its axis by a force F = (20t - 5t2) newton (where t is measured in seconds) applied tangentially. If the moment of inertia of the pulley about its axis of rotation is 10 kg m2 , the number of rotations made by the pulley before its direction of motion if reversed, is

Option 1)

less than 3

Option 2)

more than 3 but less than 6

Option 3)

more than 6 but less than 9

Option 4)

more than 9

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