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Edited By Ravindra Pindel | Updated on Sep 15, 2022 05:11 PM IST | #CBSE Class 12th

**NCERT exemplar Class 12 Maths solutions chapter 9** provides an understanding of equations that relate to one or more functions and their derivatives. In this continually changing world, describing how things change with respect to several factors is very important. The representation of this information is termed as a differential equation. A differential equation is a great way to express a set of information, but it can be hard to solve or formulate. One of the essential languages of Science is that of differential equations.

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NCERT exemplar Class 12 Maths chapter 9 solutions provided on this page would help you gain academic success, ensure an efficient and easy way of clearing doubts, aid in preparation for 12 board exams, and shape your perspective about how the world works. Also, read NCERT Class 12 Maths Solutions

Question:1

Answer:

Given

To find: Solution of the given differential equation

Rewrite the equation as,

Integrating on both sides,

Formula:

Here c is some arbitrary constant

d is also some arbitrary constant = c In2

Question:2

Find the differential equation of all non-vertical lines in a plane.

Answer:

To find: Differential equation of all non vertical lines

The general form of equation of line is given by y=mx+c where, m is the slope of the line

The slope of the line cannot be or for the given condition because if it is so, the line will become perpendicular wihout any necessity.

So,

Differentiate the general form of equation of line

Formula:

Differentiating it again it becomes,

Thus we get the diferential euqation of all non vertical lines.

Question:3

Given that and y = 0 when x = 5. Find the value of x when y = 3.

Answer:

Given:

(5,0) is a solution of this equation

To find: Solution of the given differential equation

Rewriting the equation.

Integrate on both the sides,

Formula:

Given (5,0) is a solution so to get c, satisfying these values

Hence the solution is

e^{2y}=2x + 9

when y=3,

e^{2(3)} =2x + 9

e^{6}=2x + 9

e^{6}+ 9=2x

Question:4

Solve the differential equation

Answer:

Given:

To find: Solution of the given differential equation

Rewriting the equations as,

It is a first order liner differential equation Compare it with,

Calculate Integrating Factor,

Hence, solution of the differential equation is given by,

Question:5

Solve the differential equation

Answer:

To find: Solution of the given differential equation

Rewriting the given equation as,

Integrate on both the sides,

Question:6

Answer:

Given:

It is a first order differential equation. Comparing it with,

P(x) =a

Q(x)=e^{xm}

Calculating Integrating Factor

Hence the solution of the given differential equation is ,

Question:7

Solve the differential equation

Answer:

To find: Solution of the given differential equation

Assume x+y=t

Differentiate on both sides with respect to x

Substitute

in the above equation

Rewriting the equation,

Integrate on both the sides,

Is the solution of the differential equation

Question:8

Answer:

Given:

To find: solution of the differential equation

Rewriting the given equation,

Question:9

Solve the differential equation when y = 0, x = 0.

Answer:

Given:

and (0,0) is solution of the equation

To find: solution of the differential equation

Rewriting the given equation as,

Integrating on both the sides

Substitute(0,0) to find câs value

0+0=c

c=0

Hence, the solution is

Question:10

Answer:

Given:

To find: Solution of the differential equation

Rewriting the equation as

It is a first order linear differential equation

Comparing it with

Calculation the integrating factor,

Therefore, the solution of the differential equation is

Question:11

If y(x) is a solution of and y(0) = 1, then find the value of

Answer:

Given:

To find: Solution of the differential equation

Rewriting the given equation as,

Integrating on both sides,

Let sinx=t and cos xdx= dt

ln(1+y)=-ln(2+t)+logc

ln(1+y)+ln(2++sinx)=logc

(1+y)(2+sinx)=c

When x=0 and y=1

c=4

Question:12

If y(t) is a solution of and y(0) = â1, then show that

Answer:

and (0,-1) is a solution

To find: Solution for the differential equation

Rewriting the given equation as,

It is a first order linear differential equation

Comparing it with,

Calculation Integrating Factor

Hence the solution for the differential equation is,

Substitution (0,-1) to find the value of c

The solution therefore y(1) is

Question:13

Answer:

Given:

To find: Solution of the differential equation

Differentiating on both the sides,

Differntiate again on both the sides

Hence the solution is

Question:14

Answer:

To find: Differential equations of all circles which pass though origin and centre lies on x axis

Assume a point (0,k) on y-axis

Radius of the circle is

General form of the equation of circle is,

Here a, c is the center and r is the radius.

Substituting the values in the above equation,

Differentiate the equation with respect to x

Substituting the value of k in (i)

Question:15

Find the equation of a curve passing through origin and satisfying the differential equation

Answer:

Given:

and (0,0) is a solution to the curve

To find: Equation of the curve satisfying the differential equation

Rewrite the given equation

Comparing with

Calculating Integrating Factor

Calculating

Assume

Hence the solution is

Satisfying (0,0) in the equation of the curve to find the value of c

0+0=c

c=0

therefore equation of the curve is

Question:16

Answer:

Given:

To find: solution for the differential equation

Rewriting the given equation as

Clearly it is a homogenous equation

Assume y=vx

Differentiate on both sides

Substituting dy/dx in the equation

Integrating on both the sides

is the solution for the differential equation

Question:17

Find the general solution of the differential equation

Answer:

Given

To find: Solution of the given differential equation

Rewrite the given equation as,

It is a first order differential equation

Comparing it with

Calculating Integrating Factor

Hence the solution of the given differential equation is

Differentiate on both the sides

Question:18

Answer:

Given:

To find: Solution for the given differential equation

Rewrite the given equation

It is a homogenous differential equation

Assume x=vy

Differentiating on both the sides

Substitute dy/dx in the given equation

Substitute v=x/y

Integrating on both the sides

Question:19

Solve: (x + y) (dx â dy) = dx + dy. [Hint: Substitute x + y = z after separating dx and dy].

Answer:

Given:

To find: Solution of the given differential equation

Rewriting the given equation

Assume x+y=z

Differentiate on both sides with respect to x

Substituting the values in the equation

Integrate on both the sides

Substitute v=xy

Question:20

Solve: given that y (1) = â2

Answer:

Given:

To Find: Solution of the differential equation

Integrating on both sides

Substitute (-2,1) to find value of c

Question:21

Solve the differential equation given that y = 2 when

Answer:

Given:

is a solution of the given differential equation

Rewriting the given equation

It is a first order differential equation

Calculate integrating factor

Therefore, the solution of the differential equation is

Substituting to find the value of c

Hence the solution is

Question:22

Form the differential equation by eliminating A and B in Ax^{2} + By^{2} = 1

Answer:

Given :

Ax^{2}+By^{2}=1

To find: Solution of the differential equation

Differentiate with respect to x

Differentiate the curve (i) again to get,

Substituting this in eq(i)

Question:23

Solve the differential equation (1+ y^{2}) tan^{-1}x dx + 2y (1 + x^{2}) dy = 0

Answer:

Given:

To find: Solution for differential equation that s given

Rewriting the given equation as.

Integrate on the both sides

For LHS

Assume tan^{-1} =t

For RHS

Assume 1+y^{2}=z

2ydy=dz

Substituting and integrating on both the sides

Substitute for t and z

Solution for the differential equation is

Question:24

Find the differential equation of system of concentric circles with centre (1, 2).

Answer:

To find: Differential equation of concentric circles whose center is (1,2)

Equation of the curve is given by

(x-a)^{2}+(y-b)^{2}=k^{2}

Where (a,b) is the center and k, radius.

Subsitute the values now,

(x-1)^{2}+(y-2)^{2}=k^{2}

Differentiate with respect to x

Question:25

Answer:

Now dx/dy (xy) refers to the differentiation of xy with respect to x

Using product rule

When we put it back originally in the differential equation given,

Divide by x

Compare

We get

The above equation is a linear differential equation with P and Q as functions of x

The first to find the solution of a linear differential equation is to find the integrating factor.

The solution of the linear differential equation is

Substituting values for Q and IF

Find the integrals individually,

Using uv for integration

Now

Use product rule

Substitute (i) and (ii) in (a)

Divide by

Question:26

Answer:

Divide throughout by dy

Divide by (1+tany)

Compare

We get

This is the linear differential equation with P and Q as functions of x

Put

Adding and subtracting siny in the numerator

Consider the integral

Let

Differentiate with respect to y

We get

The solution of the linear differential equation will be

Substitute values for Q and IF

Put and differentiate with respect to y

We get

Which means

Hence

Substitute t again

Question:27

Solve: [Hint: Substitute x + y = z]

Answer:

Using the given hint substitute x+y=z

Differentiate z- x with respect to x

Integrate

As we know

And

Differentiate with respect to z

We get

hence

Again substitute t

Similarly substitute z

Question:28

Answer:

We get, P= -3 and Q= sin2x

The equation is a linear differential equation where P and Q are functions of x

For the solution of the linear differential equation, we need to find Integrating factor,

The solution of the linear differential equation is

Substitute values for Q and IF

Let

If are two functions, then by integration by parts.

after applying the formula we get,

Again, applying the above stated rule in

Put this value in (1) to get

Question:29

Answer:

Slope of the tangent is given by

Slope of the tangent of the curve

Put y=VX

Using product rule differentiate vx

Integrate

Put

Resubstitute 1

Resubstitute v

The curve is passing through (2,1)

Hence (2,1) will satisfy the equation (a)

Put x=1 and y=2 in (a)

Use loga+logb=logab

Put c in equation (a)

Question:30

Given: Slope of the tangent is

Slope of tangent of a curve

Integrate

Use partial fraction for

Equate the numerator

Put x=0

A=1

Put x=-1

B=-1

Hence

Hence equation (a) becomes,

Now it is given that the curve is passing through (1,0)

Hence (1,0) will satisfy the equation (b)

Put x=1 and y=0 in b

When we put y=0 in equation b the result is which is undefined

hence, we must simplify equation (b) further

using loga-logb=loga/b

Constant c must be taken as log c to eliminate undefined elements in the

equation.(log cand not any other terms because taking logc completely

eliminates the log terms and we don't have to worry about undefined terms

in the equation)

Eliminate log

Substitute x=1 and y=0

c=-2

put back c=-2 in (c)

Question:31

Answer:

Abscissa refers to the x coordinate and ordinate refers to the y coordinate.

Slope of the tangent is the square of the difference of the abscissa and the ordinate.

Difference of the abscissa and ordinate is (x-y) and its square is

Hence the Slope of the tangent is

The curve passes through the (0,0)

Question:32

Answer:

Points on the y axis and x axis are namely A(0,a), B(b,0). The midpoint of AB is P(x,y).

The x coordinate of the points is given by the addition of the x coordinates of A and B divided by 2.

Therefore, the coordinates of A and B are (0,2y) and (2x,0) respectively.

AB is the tangent to curve where P is the point of contact.

Slope of the line given with two points

Here respectively.

Slope of the tangent AB is

Hence the slope of the tangent is -y/x

Slope of the tangent curve is given by,

Integrate

using logat logb=logab.

as given curve is passing through(1,a)

Hence (1,1) will satisfy the equation of the curve(a)

Putting

put c back in (a)

Hence the equation of the curve is

Question:33

Answer:

Using loga-logb =loga/b

Put y=v x

Differentiate yx with respect to x using product rule

Now Integrate

Substitute log v =t

Differentiate with respect to v.

logt= logx + logc

Resubstitute value of t

log(log v)=log x + logc.

Resubstitute v

Therefore the solution of the differential equation is

Question:34

The degree of the differential equation is:

A. 1

B. 2

C. 3

D. Not defined

Answer:

Degree of differential equation is defined as the highest integer power of the highest order derivative in the equation.

Hereâs the differential equation

Now for the degree to exit the differential equation must be a polynomial in some differentials.

Differential means

The given differential equation is not a polynomial because of the term sin dy/dx and therefore degree of such a differential equation is not defined.

Option D is correct.

Question:35

The degree of the differential equation is:

A. 4

B. 3/4

C. not defined

D. 2

Answer:

Generally, for a polynomial degree is the highest power.

Differential equation is Squaring both the sides,

Now for the degree to exit the differential equation must be a polynomial in

some differentials.

The given differential equation is polynomial in differential is

Degree of differential equation is the highest integer power of the highest order

derivative in the equation.

Highest derivative is

There is only one term of the highest order derivative in the equation which is

Whose power is 2 hence the degree is 2

Option D is correct.

Question:36

The order and degree of the differential equation respectively, are

A. 2 and 4

B. 2 and 2

C. 2 and 3

D. 3 and 3

Answer:

The differential equation is

Order is defined as the number which represents the highest derivative in a differential equation.

Is the highest derivative in the given equation is second order hence the degree of the equation is 2 .

Integer powers on the differentials,

Now for the degree to exit the differential equation must be a polynomial in some differentials.

Here differentials means

The given differential equation is polynomial in differentials

Degree of differential equation is the highest integer power of the highest

order derivative in the equation.

Observe that

Of differential equation (a) the maximum power

Highest order is and highest power is 4

Degree of the given differential equation is 4 .

Hence order is 2 and degree is 4

Option A is correct.

Question:37

Answer:

If is a solution of differential equation, then differentiating it will give the same differential equation.

Differentiate the differential equation twice. Twice because all the options have order as 2 and also because there are two constants A and B

Differentiating using product rule

But

Differentiating again with respect to x,

But

Also,

Means,

Question:38

The differential equation for are arbitrary constants is

Answer:

Let us find the differential equation by differentiating y with respect to x twice

Twice because we have to eliminate two constants .

Differentiating,

Differentiating again

Option B is correct.

Question:39

Solution of differential equation xdy â ydx = 0 represents:

A. a rectangular hyperbola

B. parabola whose vertex is at origin

C. straight line passing through origin

D. a circle whose centre is at origin

Answer:

is constant because e is a constant and c is the integration constant let it be denoted as k hence

is the equation of straight line and (0,0) satisfies the equation.

Option C is correct.

Question:40

Integrating factor of the differential equation is:

A. cosx

B. tanx

C. sec x

D. sinx

Answer:

Differential equation is

Compare

With

The IF integrating factor is given by

Substitute hence

Resubstitute the value of t

hence IF is sec x

Option C is correct.

Question:41

Solution of the differential equation is:

A. tanx + tany = k

B. tanx â tan y = k

C.

D. tanx . tany = k

Answer:

The given differential equation is

Divide it by tanx tany

Integrate

Put tanx=t hence,

Put tany =z hence

That is

Resubstitue t and z

is constant because e is a constant and c is the integration constant let it be denoted as

Option D is correct.

Question:42

Family of curves is represented by the differential equation of degree:

A. 1

B. 2

C. 3

D. 4

Answer:

let us find the differential equation representing it so we have to eliminate

the constant A

Differentiate with respect to x

Put back value of A in y

Now for the degree to exit the differential equation must be a polynomial in some differentials.

Here the differentials mean

The given differential equation is polynomial in differentials

Degree of differential equation is the highest integer power of the highest order derivative in the equation.

Highest derivative is

And highest power to it is 3 . Hence degree is 3 .

Option C is correct.

Question:43

Integrating factor of is:

A. x

B. logx

C.

D. âx

Answer:

Given differential equation

Divide though by x

Compare

We get

The IF integrating factor is given by el $^{\mathrm{iPdx}}$

Option C is correct.

Question:44

B.

C.

D.

Answer:

Integrate

now it is given that y(0)=1 which means when x=0, y=1 hence substitute x=0 and y=0 in (a)

put back in (a)

using

Hence solution of differential equation is

Option D is correct.

Question:45

The number of solutions of when y(1) = 2 is:

A. none

B. one

C. two

D. infinite

Answer:

using

Now as given y(1)=2 which means when x=1, y=2 Substitute x=1 and y=2 in (a)

So only one solution exists.

Option B is correct.

Question:46

Which of the following is a second order differential equation?

Answer:

Order is defined as the number which defines the highest derivative in a differential equation

Second order means the order should be 2 which means the highest

derivative in the equation should be or y Let's examine each of the option given

A.

The highest order derivative is is in first order.

B.

The highest order derivative is is in second order

C.

The highest order derivative is is in third order

D.

The highest order derivative is is in first order

Option B is correct.

Question:47

Integrating factor of the differential equation is:

A. -x

B.

C.

D.

Answer:

Divide through by

Compare

We get

The IF factor is given by

Substitute hence

Which means

Resubstitute

Hence the IF integrating factor is

Option C is correct.

Question:48

is the general solution of the differential equation:

A.

B.

C.

D.

Answer:

If is a solution of differential equation then differentiating it will give the same differential equation.

To find the differential equation differentiate with respect to x.

Option C is correct..

Question:49

The differential equation represents:

A. Family of hyperbolas

B. Family of parabolas

C. Family of ellipses

D. Family of circles

Answer:

integrate

k is the integration constant

This is the equation of circle because there is no âxyâ term and and have the same coefficient.

This equation represents the family of circles because for different values of c and k we will get different circles.

Option D is correct.

Question:50

The general solution of is:

A.

B.

C.

D.

Answer:

Integrate

substitute cosy =t hence

Which means sinydy=-dt

Option A is correct.

Question:51

The degree of the differential equation is

(a) 1

(b) 2

(c) 3

(d) 5

Answer:

The answer is the option (a) 1 as the degree of a differential equation is the highest exponent of the order derivative.

Question:52

The solution of is

(a)

(b)

(c)

(d)

Answer:

The answer is the option (b)

Explanation: -

This is a linear differential equation.

On comparing it with , we get

So, the general solution is:

Given that when x=0 and y=0

Eq. (i) becomes

Question:53

Integrating factor of the differential equation

(a)

(b)

(c)

(d)

Answer:

The answer is the option (b) Sec x

Explanation: -

Question:54

The solution of the differential equation is

(a)

(b)

(c)

(d)

Answer:

The answer is the option (b) y â x = k(1 + xy)

Explanation: -

Question:55

The integrating factor of the differential equation is

(a)

(b)

(c)

(d)

Answer:

The answer is the option (b)

Explanation: -

This is a linear differential equation.

On comparing it with we get

Question:58

The solution of is

(a)

(b)

(c)

(d)

Answer:

The answer is the option (a)

Explanation: -

This is a linear differential equation. Dn comparing it with $\frac{d y}{d x}+P y=Q$, we get

So, the general solution is:

Question:59

The differential equation of the family of curves where a is arbitrary constant, is

(a)

(b)

(c)

(d)

Answer:

The answer is the option (a)

Given:

Question:60

Family y = Ax + A^{3} of curves will correspond to a differential equation of order ,

(a) 3 (b) 2 (c) 1 (d) not defined.

Answer:

The answer is the option (c) 1.

Explanation: -

Putting the value of A in Eq. (i), we gt

Question:61

The general solution of is

(a)

(b)

(c)

(d)

Answer:

The answer is the option (c)

Explanation: -

Question:62

The curve for which the slope of the tangent at any point is equal to the ratio of the abscissa to the ordinate of the point is

(a) an ellipse (b) parabola (c) circle (d) rectangular hyperbola

Answer:

The answer is the option (d) Rectangular Hyperbola

Explanation: -

According to the question,

On integrating both sides, we get

which is an equation of rectangular hyperbola.

Question:63

The general solution of differential equation is

(a)

(b)

(c)

(d)

Answer:

The answer is the option (c)

Explanation: -

This is a linear differential equation. On comparing it with we get

So, the general solution is:

Question:64

The solution of equation is

(a)

(b)

(c)

(d)

Answer:

The answer is the option (c)

Explanation: -

Question:65

The differential equation for which is a solution, is

(a)

(b)

(c)

(d)

Answer:

The answer is the option (a)

Explanation: -

On differentiating both sides w.r.t. x, we get

Again, differentiating w.r.t. x, we get

Question:66

The solution of is

(a)

(b)

(c)

(d)

Answer:

The answer is the option (d)

Explanation: -

Here,

Given, when x=0 and y=0

Eq. (i) reduces to

Question:67

The order and degree of the differential equation

(a) 1,4

(b) 3,4

(c) 2,4

(d) 3,2

Answer:

Ans: - The answer is the option (d) 3, 2

Question:68

The order and degree of the differential equation are

(a)

(b) 2,3

(c) 2,1

(d) 3,4

Answer:

Ans: -

The answer is the option (c) 2, 1.

Question:69

The differential equation of family of curves is

(a)

(b)

(c)

(d)

Answer:

Ans: - The answer is the option (d)

Explanation: -

On differentiating both sides w.r.t. x, we get

On putting the value of a in Eq. (i), we get

Question:70

Which of the following is the general solution of ?

(a)

(b)

(c)

(d)

Answer:

Ans: -

The answer is the option (a)

Explanation: -

Question:71

General solution of is

(a)

(b)

(c)

(d)

Answer:

Ans: - The answer is the option (a) y sec x = tan x + C

Explanation: -

Here,

Question:72

Solution of the differential equation is

(a)

(b)

(c)

(d)

Answer:

Ans: - The answer is the option (a) x(y + cos x) = sin x + C

Explanation: -

Here,

Question:73

The general solution of differential equation is

(a)

(b)

(c)

(d)

Answer:

Ans: - The answer is the option (c)

Explanation: -

Question:74

The solution of the differential equation is

(a)

(b)

(c)

(d)

Answer:

The answer is the option (b)

Explaination:

Question:75

The solution of the differential equation

(a)

(b)

(c)

(d)

Answer:

Ans: - The answer is the option (a)

Explanation: -

Here,

Question:76

Answer:

(i) Given differential equation is

Degree of this equation is not defined as it cannot be expresses as polynomial of derivatives.

(ii) We have

So, degree of this equation is two.

(iii) Given that the general solution of a differential equation has three arbitrary constants. So we require three more equations to eliminate these three constants. We can get three more equations by differentiating given equation three times. So, the order of the differential equation is three.

(iv) We have

The equation is of the type

Hence it is linear differential equation.

(v) We have

For solving such equation we multiply both sides by

So we get

This is the required solution of the given differential equation.

(vi) We have,

This equation of the form

The general solution is

(vii) We have

This equation is of the form

So, the general solution is:

(viii) We have, $

(ix) We have,

Which is of the form

So, the general solution is:

(x) Given differential equation is

(xi) Given differential equation is

Which is linear differential equation.

Question:77

Answer:

i) Integrating factor of the differential of the form is given by . Hence given statement is true.

(ii) Solution of the differential equation of the type is given by .

Hence given statement is true.

iii) Correct substitution for the solution of the differential equation of the type is a homogeneous function of zero degree is y=v x.

Hence given statement is true.

(iv) Correct substitution for the solution of the differential equation of the type where g(x, y) is a homogeneous function of the degree zero is x=v y.

Hence given statement is true.

(V) There is no arbitrary constants in the particular solution of a differential equation. Hence given statement is Flase.

(vi) In thegiven equation the number of arbitrary constant is one. So the order order will be one.

Hence given statement is False.

(vii)

Hence the given statement is true.

(viii)

.

Hence the given statement is true.

ix) Given:

Compare with

Here ,

General solution

Hence the given statement is true.

x) Given:

Let y =vx

Hence the given statement is true.

xi) Assume equation of a non-horizontal line in the plane

y = mx +c

Hence the given statement is true.

Question:56

satisfies which of the following differential equation.

Answer:

given

upon differentiation, we get

after differentiation again we get

Option c is correct

Below is the list of topics which are covered in Class 12 Maths NCERT exemplar solutions chapter 9

- Introductory Concepts
- Ordinary differential equations
- Order of a differential equation
- Degree of a differential equation
- General and particular solutions of a differential equation
- Formation of a differential equation
- Formation of a differential equation whose general solution is given
- Formation of a differential equation that will represent a given family of curves
- Methods of solving First order, First degree differential equations
- Differential equations with separate variables
- Homogeneous Differential equations
- Linear differential equations

- These equations have a variety of uses in academic subjects like Physics, Chemistry, Biology, Geology, etc., which makes it important to acquire detailed learning of these equations.
- A thorough understanding of differential equations is needed to solve Newton's laws of motion and cooling, the rate of spread of a pandemic or an epidemic, and also help measure market competition.
- If understood well, NCERT exemplar solutions for Class 12 Maths chapter 9 would help you answer real-world questions like - How do you model an antibiotic-resistant bacteria's growth? How do you study ever-changing online purchasing trends? At what rate, a radioactive material decays? What is the trajectory of a biological cell motion? How the suspension system of a car works to give you a smooth ride? These equations are a way to describe many things in the universe and model nearly anything around us. Scientists and geniuses understand the world through differential equations; you can too.

- NCERT exemplar Class 12 Maths chapter 9 solutions revolve around using the mathematical tool of Differentiation, analyses properties like the intervals of increment and decrement, study the local maximum and minimum of functions that are quadratic.
- In Class 12 Maths NCERT exemplar solutions chapter 9, you will be introduced to the concepts related to differential equations, types of these equations, general and specific solutions to solve them, the formation and production of these equations, different forms of the equations, and a wide range of applications of modelling real-life situations by applying these equations
In NCERT exemplar Class 12 Maths solutions chapter 9 pdf download, we would also look at the graphical aspects of differential equations, including a family of straight lines and curves, and have a look at the devised solutions and mathematical tools to solve the most complex equations over time.

Chapter 1 | |

Chapter 2 | |

Chapter 3 | |

Chapter 4 | |

Chapter 5 | |

Chapter 6 | |

Chapter 7 | |

Chapter 8 | |

Chapter 9 | Differential Equations |

Chapter 10 | |

Chapter 11 | |

Chapter 12 | |

Chapter 13 |

- NCERT Solution for Class 12 Physics
- NCERT Solution for Class 12 Chemistry
- NCERT Solution for Class 12 Maths
- NCERT Solution for Class 12 Biology

- NCERT Notes for Class 12 Physics
- NCERT Notes for Class 12 Chemistry
- NCERT Notes for Class 12 Maths
- NCERT Notes for Class 12 Biology

1. Are these solutions helpful in competitive exams?

Yes, these NCERT exemplar Class 12 Maths solutions chapter 9 can be highly useful in understanding the way the questions should be solved in entrance exams.

2. How to make use of these NCERT exemplar Class 12 Maths chapter 9 solutions?

These solutions can be used for both getting used to the chapter and its topics and to also get an idea about how to solve questions in exams.

3. What are the basic take away from the Class 12 Maths NCERT exemplar solutions chapter 9?

One can understand how to stepwise solve these questions through NCERT exemplar Class 12 Maths solutions chapter 9 and how the CBSE expects a student to solve in their final paper.

4. Who prepare these solutions to the maths chapter?

We have the best maths teachers onboard to solve the questions as per the students understanding and also CBSE standards. These teachers prepare the NCERT exemplar solutions for Class 12 Maths chapter 9.

Application Date:20 November,2023 - 19 December,2023

Application Date:20 November,2023 - 19 December,2023

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9 minHave a question related to CBSE Class 12th ?

hello,

Yes you can appear for the compartment paper again since CBSE gives three chances to a candidate to clear his/her exams so you still have two more attempts. However, you can appear for your improvement paper for all subjects but you cannot appear for the ones in which you have failed.

I hope this was helpful!

Good Luck

Hello dear,

If you was not able to clear 1st compartment and now you giving second compartment so YES, you can go for your improvement exam next year but if a student receives an improvement, they are given the opportunity to retake the boards as a private candidate the following year, but there are some requirements. First, the student must pass all of their subjects; if they received a compartment in any subject, they must then pass the compartment exam before being eligible for the improvement.

As you can registered yourself as private candidate for giving your improvement exam of 12 standard CBSE(Central Board of Secondary Education).For that you have to wait for a whole year which is bit difficult for you.

Positive side of waiting for whole year is you have a whole year to preparing yourself for your examination. You have no distraction or something which may causes your failure in the exams. In whole year you have to stay focused on your 12 standard examination for doing well in it. By this you get a highest marks as a comparison of others.

Believe in Yourself! You can make anything happen

All the very best.

Hello Student,

I appreciate your Interest in education. See the improvement is not restricted to one subject or multiple subjects Â and Â we cannot say if improvement in one subject in one year leads to improvement in more subjects in coming year.

You just need to have a revision of all subjects what you have completed in the school. have a revision and practice of subjects and concepts helps you better.

All the best.

If you'll do hard work then by hard work of 6 months you can achieve your goal but you have to start studying for it dont waste your time its a very important year so please dont waste it otherwise you'll regret.

Yes, you can take admission in class 12th privately there are many colleges in which you can give 12th privately.

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Content writing is meant to speak directly with a particular audience, such as customers, potential customers, investors, employees, or other stakeholders. The main aim of professional content writers is to speak to their targeted audience and if it is not then it is not doing its job. There are numerous kinds of the content present on the website and each is different based on the service or the product it is used for.

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Individuals who opt for a career as a reporter may often be at work on national holidays and festivities. He or she pitches various story ideas and covers news stories in risky situations. Students can pursue a BMC (Bachelor of Mass Communication), B.M.M. (Bachelor of Mass Media), or MAJMC (MA in Journalism and Mass Communication) to become a reporter. While we sit at home reporters travel to locations to collect information that carries a news value.

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Linguistic meaning is related to language or Linguistics which is the study of languages. A career as a linguistic meaning, a profession that is based on the scientific study of language, and it's a very broad field with many specialities. Famous linguists work in academia, researching and teaching different areas of language, such as phonetics (sounds), syntax (word order) and semantics (meaning).

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A quality controller plays a crucial role in an organisation. He or she is responsible for performing quality checks on manufactured products. He or she identifies the defects in a product and rejects the product.

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A career as a merchandiser requires one to promote specific products and services of one or different brands, to increase the in-house sales of the store. Merchandising job focuses on enticing the customers to enter the store and hence increasing their chances of buying a product. Although the buyer is the one who selects the lines, it all depends on the merchandiser on how much money a buyer will spend, how many lines will be purchased, and what will be the quantity of those lines. In a career as merchandiser, one is required to closely work with the display staff in order to decide in what way a product would be displayed so that sales can be maximised. In small brands or local retail stores, a merchandiser is responsible for both merchandising and buying.

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The procurement Manager is also known as Purchasing Manager. The role of the Procurement Manager is to source products and services for a company. A Procurement Manager is involved in developing a purchasing strategy, including the company's budget and the supplies as well as the vendors who can provide goods and services to the company. His or her ultimate goal is to bring the right products or services at the right time with cost-effectiveness.

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Individuals who opt for a career as a production planner are professionals who are responsible for ensuring goods manufactured by the employing company are cost-effective and meets quality specifications including ensuring the availability of ready to distribute stock in a timely fashion manner.

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ITSM Manager is a professional responsible for heading the ITSM (Information Technology Service Management) or (Information Technology Infrastructure Library) processes. He or she ensures that operation management provides appropriate resource levels for problem resolutions. The ITSM Manager oversees the level of prioritisation for the problems, critical incidents, planned as well as proactive tasks.

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Individuals in the information security manager career path involves in overseeing and controlling all aspects of computer security. The IT security manager job description includes planning and carrying out security measures to protect the business data and information from corruption, theft, unauthorised access, and deliberate attack

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Careers in computer programming primarily refer to the systematic act of writing code and moreover include wider computer science areas. The word 'programmer' or 'coder' has entered into practice with the growing number of newly self-taught tech enthusiasts. Computer programming careers involve the use of designs created by software developers and engineers and transforming them into commands that can be implemented by computers. These commands result in regular usage of social media sites, word-processing applications and browsers.

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Individuals in the computer systems analyst career path study the hardware and applications that are part of an organization's computer systems, as well as how they are used. They collaborate closely with managers and end-users to identify system specifications and business priorities, as well as to assess the efficiency of computer systems and create techniques to boost IT efficiency. Individuals who opt for a career as a computer system analyst support the implementation, modification, and debugging of new systems after they have been installed.

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A Test Manager is a professional responsible for planning, coordinating and controlling test activities. He or she develops test processes and strategies to analyse and determine test methods and tools for test activities. The test manager jobs involve documenting tests that have been carried out, analysing and evaluating software quality to determine further recommended procedures.

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A career as Azure Developer comes with the responsibility of designing and developing cloud-based applications and maintaining software components. He or she possesses an in-depth knowledge of cloud computing and Azure app service.

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A Deep Learning Engineer is an IT professional who is responsible for developing and managing data pipelines. He or she is knowledgeable about analyzing and storing data collected from various sources. A Career as a Deep Learning Engineer needs to help the data scientists and analysts to create effective data sets.

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