NCERT Solutions Class 12 Mathematics Chapter 9
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The fundamentals of calculus are used to present a new topic called Differential Equations in NCERT Solutions for class 12 mathematics chapter 9. Students were previously taught how to differentiate a function f about an independent variable, that is, how to find f ′(x) for a given function f at each x in its domain of definition. Students were also taught how to find a function f whose derivative is the function g. An equation containing the derivative of the dependent variable concerning the independent variable can be expressed as a formal definition for differential equations using these concepts. These equations can also be classified as:
- Ordinary Differential Equations – Differential Equations that involve derivatives of the dependent variable concerning only one independent variable.
- Partial Differential Equations – Differential Equations that involve derivatives concerning many independent variables.
These Equations have been explained in-depth with example problems in NCERT solutions for class 12 Mathematics chapter 9. Another aspect of calculus shown in NCERT answers for class 12 Mathematics chapter 9 is that students can effortlessly make a transition to solving the problems in this lesson once they are familiar with the various ways of differentiating and integrating a function. Differential equations in NCERT Solutions class 12 Mathematics chapter 9 have applications in determining population growth and decay, glucose absorption by the body, gauging epidemic spread, and solving various geometric questions involving various types of curves.
Key Topics Covered in NCERT Solutions Class 12 Mathematics Chapter 9
This chapter introduces students to the fundamentals of differential equations. The various topics covered in this chapter assist students in comprehending the real-world applications of differential equations. Students can learn the definition of differential equations here, which can aid in their understanding of the concepts. The order and degree of differential equations are also taught to students. The creation of ordinary differential equations and the solution of a differential equation are two additional essential concepts addressed in this chapter. These are crucial elements for students to understand as they prepare for the main exam.
Every topic in NCERT Solutions Class 12 Mathematics Chapter 9 is equally essential because it focuses on distinct elements of differential equations. Furthermore, each section is linked to the next. Thus, students must devote equal and enough time to all sections of this chapter, so as to avoid any superficial knowledge or learning gaps. It also gives a good review of the previous topic matter covered in the session.
List of NCERT Solutions Class 12 Mathematics Chapter 9 Exercises
Because of Gottfried Wilhelm Freiherr Leibnitz (1646 – 1716), who gave the calculus identity to Mathematics, the concept of differential equations was born on November 11, 1675. Leibnitz was more engaged in finding a curve with stipulated tangents. As a result, he discovered a slew of related differential equation principles. The equations took on their current shape due to various researchers working on the topic’s intricacy. Many of these facts and advice to help youngsters overcome boredom and study with excitement may be found in the NCERT solutions for class 12 Mathematics chapter 9.
Differential equations in NCERT Solutions Class 12 Mathematics Chapter 9 are challenging and have lengthy lessons. As a result, the only method to master the chapter is to review the contents frequently. It will be easier for students to progress through the chapter and build clear concepts if they have a strong foundation in the fundamentals. Extramarks has provided a full examination of all the exercise questions below to assist students in pursuing high-quality education.
Class 12 Mathematics Chapter No. 9 Ex 9.1 Solutions – 12 Questions
Class 12 Mathematics Chapter No. 9 Ex 9.2 Solutions – 12 Questions
Class 12 Mathematics Chapter No. 9 Ex 9.3 Solutions – 12 Questions
Class 12 Mathematics Chapter No. 9 Ex 9.4 Solutions – 23 Questions
Class 12 Mathematics Chapter No. 9 Ex 9.5 Solutions – 17 Questions
Class 12 Mathematics Chapter No. 9 Ex 9.6 Solutions – 19 Questions
Class 12 Mathematics Chapter No. 9 Miscellaneous Ex – 18 Questions
NCERT Solutions Class 12 Mathematics Chapter 9 Formula List
NCERT solutions class 12 Mathematics chapter 9 solves differential equations using the formulas provided in earlier calculus lessons. This chapter focuses on formulating and solving differential equations under particular conditions. As a result, to successfully combine both lectures, it is critical to review the calculus concepts taught previously. There are some terminologies in the NCERT solutions for class 12 Mathematics chapter 9 that students should be familiar with. A few of which are as under:
- The General form of a differential equation: dy/dx = g(x), where y = f(x).
- The general form of nth order derivative: dny/dxn.
- The general form of a linear differential equation: dy/dx + Py = Q
Class 12 NCERT Mathematics Syllabus
Term – 1
Unit Name
|
Chapter Name |
Relations and Function |
Relations and Functions
Inverse Trigonometric Functions |
Algebra
|
Matrices
Determinants |
Calculus
|
Continuity and Differentiability
Application of Derivatives |
Linear Programming | Linear Programming |
Term – 2
Unit Name | Chapter Name |
Calculus
|
Integrals
Application of Integrals Differential Equations |
Vectors and Three-Dimensional Geometry | Vector Algebra
Three Dimensional Geometry |
Probability | Probability |
Subject experts at Extramarks create NCERT Solutions to assist students in understanding concepts more quickly and correctly. NCERT Solutions provide extensive, step-by-step explanations of textbook problems. All classes can benefit from such solutions –
- NCERT Solutions class 1
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NCERT CBSE Mathematics Exam Pattern
Duration of Marks | 3 hours 15 minutes |
Marks for Internal | 20 marks |
Marks for Theory | 80 marks |
Total Number of Questions | 38 Questions |
Very short answer question | 20 Questions |
Short answer questions | 7 Questions |
Long Answer Questions (4 marks each) | 7 Questions |
Long Answer Questions (6 marks each) | 4 Questions |
Key Features of NCERT Solutions for Class 12 Mathematics Chapter 9
Using the NCERT Solutions to learn the chapter, Differential Equations, students will be able to understand the following:
- A Differential Equation’s definition, order and degree.
- General and Special Solutions.
- The formation of a differential equation with a given general solution.
- Solutions of homogeneous differential equations of the first order and first degree.
- Solutions of differential equations using the separation of variables approach.
NCERT Exemplar Class 12 CBSE Mathematics
NCERT Exemplars contain solutions and problems that help students prepare for their final exams. These example questions are a little more complex, and they cover every concept in each chapter of the Class 12 Mathematics subject.
Students will fully understand all the concepts covered in each chapter by practising these NCERT Exemplars for Mathematics Class 12. Each question in these materials is connected to topics covered in the CBSE Class 12 syllabus (2022-2023). They provide some of the best solutions to challenges that students confront. To match the ideas taught in each class and provide the greatest practising materials or worksheets for students, all of these questions reflect the question pattern found in NCERT books.
Q.1 Determine order and degree (if defined) of differential equation
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Q.5
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Q.6
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Q.7
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Q.8
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Q.9
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Q.10
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Q.11
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Q.12
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Q.13
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Q.14
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Q.15
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Q.16
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Q.17
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Q.18
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Q.19
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Q.20
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Q.21
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Q.22
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Q.23 The number of arbitrary constants in the general solution of a differential equation of fourth order are:
(A) 0 (B) 2 (C) 3 (D) 4
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Since, number of constants in a differential equation of order n is equal to its order i.e., n.
Thus, number of arbitrary constants in a differential equation of fourth order are 4.
Thus, option D is correct.
Q.24 The number of arbitrary constants in the particular solution of a differential equation of third order are:
(A) 3 (B) 2 (C) 1 (D) 0
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In a particular solution, there are no arbitrary constants.
∴ Number of arbitrary constants = 0
Thus, option D is correct.
Q.25 Form a differential equation representing the given family of curves by eliminating arbitrary constants a and b.
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Q.26 Form a differential equation representing the given family of curves by eliminating arbitrary constants a and b.
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Q.27 Form a differential equation representing the given family of curves by eliminating arbitrary constants a and b.
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Q.28 Form a differential equation representing the given family of curves by eliminating arbitrary constants a and b.
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Q.29 Form a differential equation representing the given family of curves by eliminating arbitrary constants a and b.
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Q.30 Form the differential equation of the family of circles touching the y-axis at origin.
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The centre of the circle touching y-axis at origin lies on x-axis. Let the centre of circle be (a, 0).
Radius of circle will be ‘a’ because circle touches y-axis at origin. So, the equation of circle with centre (a, 0) and radius ‘a’ is as follows
(x – a)2 + y2 = a2
x2 – 2ax + a2 + y2 = a2
x2 + y2 = 2ax … (i)
Differentiating equation (i), w.r.t. x, we get
2x + 2yy’ = 2a
x + yy’ = a
Putting value of a in equation (i), we get
x2 + y2 = 2(x + yy’)x
= 2x2 + 2xyy’
y2 = x2 + 2xyy’
Thus, the required differential equation is 2xyy’ + x2 = y2.
Q.31 Form the differential equation of the family of parabolas having vertex at origin and axis along positive y-axis.
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The equation of parabolas having vertex at origin and axis along positive y-axis is as follows
x2 = 4ay …(i)
Differentiating w.r.t. x, we get
2x = 4ay’
or (2x/y’) = 4a
Putting value of 4a in equation (i), we get
x2 = (2x/y’)y
x2y’ = 2xy
or xy’ = 2y
or xy’ – 2y = 0
Thus, the required differential equation is
xy’ – 2y = 0.
Q.32 Form the differential equation of the family of ellipses having foci on y-axis and centre at origin.
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Q.33 Form the differential equation of the family of hyperbolas having foci on x-axis and centre at origin.
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Q.34 Form the differential equation of the family of circles having centre on y-axis and radius 3 units.
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The centre of the circle lies on y-axis. Let the centre of circle be (0, b).
So, the equation of circle with centre (0, b) and radius 3 is as follows
x2 + (y – b)2 = 32 … (i)
Q.35
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Q.36
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Q.37 For the differential equations, find the general solution:
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Q.38 For the differential equations, find the general solution:
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Q.39 For the differential equations, find the general solution:
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Q.40 For the differential equations, find the general solution:
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Q.41 For the differential equations, find the general solution:
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Q.42 For the differential equations, find the general solution:
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Q.43 For the differential equations, find the general solution:
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Q.44 For the differential equations, find the general solution:
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Q.45 For the differential equation given below, find the general solution:
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Q.46 For the differential equation given below, find the general solution:
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Q.47 For the differential equation given below, find a particular solution satisfying the given condition:
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Q.48 For the differential equation given below, find a particular solution satisfying the given condition:
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Q.49 For the differential equation given below, find a particular solution satisfying the given condition:
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Q.50 For the differential equation given below, find a particular solution satisfying the given condition:
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Q.51 Find the equation of a curve passing through the point (0, 0) and whose differential equation is y’ = ex sin x.
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Q.52
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Q.53 Find the equation of the curve passing through the point (0, –2) given that at any point (x, y) on the curve, the product of the slope of its tangent and y coordinate of the point is equal to the x coordinate of the point.
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Q.54 At any point (x, y) of a curve, the slope of the tangent is twice the slope of the line segment joining the point of contact to the point (– 4, –3). Find the equation of the curve given that it passes through (–2, 1).
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Q.55 The volume of spherical balloon being inflated changes at a constant rate. If initially its radius is 3 units and after 3 seconds it is 6 units. Find the radius of balloon after t seconds.
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Q.56 In a bank, principal increases continuously at the rate of r% per year. Find the value of r if ` 100 double itself in 10 years (loge2= 0.6931).
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Q.57 In a bank, principal increases continuously at the rate of 5% per year. An amount of ₹ 1000 is deposited with this bank, how much will it worth after 10 years (e0.5 = 1.648).
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Q.58 In a culture, the bacteria count is 1,00,000.
The number is increased by 10% in 2 hours.
In how many hours will the count reach 2,00,000, if the rate of growth of bacteria is proportional to the number present?
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Q.59
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Q.60 Show that the given differential equation is homogeneous and solve it.
(x2 + xy) dy = (x2 + y2) dx
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Q.61 Show that the given differential equation is homogeneous and solve each of them.
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Q.62 Show that the given differential equation is homogeneous and solve it.
(x – y)dy – (x + y) dx = 0
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Q.63 Show that the given differential equation is homogeneous and solve it.
(x2 – y2)dx + 2xy dy = 0
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Q.64 Show that the given differential equation is homogeneous and solve it.
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Q.65 Show that the given differential equation is homogeneous and solve it.
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Q.66 Show that the given differential equation is homogeneous and solve each of them.
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Q.67 Show that the given differential equation is homogeneous and solve each of them.
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Q.68 Show that the given differential equation is homogeneous and solve it.
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Q.69 Show that the given differential equation is homogeneous and solve each of them.
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Q.70 For the differential equation, find the particular situation satisfying the given condition:
(x + y) dy + (x – y) dx = 0; y = 1 when x = 1
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Q.71 For the differential equation, find the particular situation satisfying the given condition:
x2 dy + (xy + y2) dx = 0; y = 1 when x = 1
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Q.72 For the differential equations, find the particular situation satisfying the given condition:
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Q.73 For the differential equations, find the particular situation satisfying the given condition:
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Q.74 For the differential equation given below, find a particular situation satisfying the given condition:
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Q.75
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Q.76 Which of the following is a homogeneous differential equation?
(A) (4x + 6y + 5)dy – (3y + 2x + 4) dx = 0
(B) (xy) dx – (x3 + y3) dy = 0
(C) (x3 + 2y2) dx + 2xy dy = 0
(D) y2 dx + (x2– xy – y2) dy = 0
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A function F(x, y) is said to be homogeneous function of degree n if F(λx, λy)= λn F(x, y) for any nonzero constant λ.
Q.77 Find the general solution of the differential equation given below.
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Q.78 Find the general solution of the differential equation given below.
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Q.79 Find the general solution of the differential equation given below.
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Q.80 Find the general solution of the differential equation given below.