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Edited By Ramraj Saini | Updated on Dec 04, 2023 01:24 PM IST | #CBSE Class 12th

**NCERT Solutions for miscellaneous exercise chapter 8 class 12 Application of Integrals **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 Class 12 Maths chapter 8 Miscellaneous exercise deals with the area finding of the various curves and intersection of two curves. Class 12 Maths chapter 8 miscellaneous exercise is a good source to practice good questions before the exam. NCERT book Class 12 Maths chapter 8 miscellaneous exercise is relatively easier than the chapter 7 exercise as it provides questions based on the application part. Class 12 Maths chapter 8 miscellaneous exercise solutions with other two exercises are sufficient to score good marks.

This Story also Contains

- NCERT Solutions For Class 12 Chapter 8 Miscellaneous Exercise
- Application of Integrals Class 12 Chapter 8 Miscellaneous: Exercise
- More About NCERT Solutions for Class 12 Maths Chapter 8 Miscellaneous Exercise
- Also Read| Application of Integrals Class 12 Notes
- Benefits of Ncert Solutions for Class 12 Maths Chapter 8 Miscellaneous Exercises
- Key Features Of NCERT Solutions For Class 12 Chapter 8 Miscellaneous Exercise
- Also see-
- NCERT Solutions Subject Wise
- Subject Wise NCERT Exemplar Solutions

**Miscellaneous exercise class 12 chapter 8**** **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 enumerated in NCERT Book together using the link provided below.

** Question:1 ** Find the area under the given curves and given lines:

(i) and -axis

** Answer: **

The area bounded by the curve and -axis

The area of the required region = area of ABCD

Hence the area of shaded region is 7/3 units

** Question:1 ** Find the area under the given curves and given lines:

(ii) and -axis

** Answer: **

The area bounded by the curev and -axis

The area of the required region = area of ABCD

Hence the area of the shaded region is 624.8 units

** Question:2 ** Find the area between the curves and .

** Answer: **

the area between the curves and .

The curves intersect at A(1,1)

Draw a normal to AC to OC(x-axis)

therefore, the required area (OBAO)= area of (OCAO) - area of (OCABO)

Thus the area of shaded region is 1/6 units

** Question:3 ** Find the area of the region lying in the first quadrant and bounded by and .

** Answer: **

the area of the region lying in the first quadrant and bounded by and .

The required area (ABCD) =

The area of the shaded region is 7/3 units

** Question:4 ** Sketch the graph of and evaluate

** Answer: **

y=|x+3|

the given modulus function can be written as

x+3>0

x>-3

for x>-3

y=|x+3|=x+3

x+3<0

x<-3

For x<-3

y=|x+3|=-(x+3)

Integral to be evaluated is

** Question:5 ** Find the area bounded by the curve between and .

** Answer: **

The graph of y=sinx is as follows

We need to find the area of the shaded region

ar(OAB)+ar(BCD)

=2ar(OAB)

The bounded area is 4 units.

** Question:6 ** Find the area enclosed between the parabola and the line .

** Answer: **

We have to find the area of the shaded region OBA

The curves y=mx and y ^{ 2 } =4ax intersect at the following points

The required area is

** Question:7 ** Find the area enclosed by the parabola and the line .

** Answer: **

We have to find the area of the shaded region COB

The two curves intersect at points (2,3) and (4,12)

Required area is

** Question:8 ** Find the area of the smaller region bounded by the ellipse and the line .

** Answer: **

We have to find the area of the shaded region

The given ellipse and the given line intersect at following points

Since the shaded region lies above x axis we take y to be positive

The required area is

** Question:9 ** Find the area of the smaller region bounded by the ellipse and the line .

** Answer: **

The area of the shaded region ACB is to be found

The given ellipse and the line intersect at following points

Y will always be positive since the shaded region lies above x axis

The required area is

** Question:10 ** Find the area of the region enclosed by the parabola the line and the -axis.

** Answer: **

We have to find the area of the shaded region BAOB

O is(0,0)

The line and the parabola intersect in the second quadrant at (-1,1)

The line y=x+2 intersects the x axis at (-2,0)

The area of the region enclosed by the parabola the line and the -axis is 5/6 units.

** Question:11 ** Using the method of integration find the area bounded by the curve

[ ** Hint: ** The required region is bounded by lines and ]

** Answer: **

We need to find the area of the shaded region ABCD

ar(ABCD)=4ar(AOB)

Coordinates of points A and B are (0,1) and (1,0)

Equation of line through A and B is y=1-x

The area bounded by the curve is 2 units.

** Question:12 ** Find the area bounded by curves .

** Answer: **

We have to find the area of the shaded region

In the first quadrant

y=|x|=x

Area of the shaded region=2ar(OADO)

The area bounded by the curves is 1/3 units.

** Question:13 ** Using the method of integration find the area of the triangle ABC, coordinates of whose vertices are

** Answer: **

Equation of line joining A and B is

Equation of line joining B and C is

Equation of line joining A and C is

ar(ABC)=ar(ABL)+ar(LBCM)-ar(ACM)

ar(ABC)=8+5-6=7

Therefore the area of the triangle ABC is 7 units.

** Question:14 ** Using the method of integration find the area of the region bounded by lines:

** Answer: **

We have to find the area of the shaded region ABC

ar(ABC)=ar(ACLM)-ar(ALB)-ar(BMC)

The lines intersect at points (1,2), (4,3) and (2,0)

Area of the region bounded by the lines is 3.5 units

** Question:15 ** Find the area of the region .

** Answer: **

We have to find the area of the shaded region OCBAO

Ar(OCBAO)=2ar(OCBO)

For the fist quadrant

In the first quadrant, the curves intersect at a point

Area of the unshaded region in the first quadrant is

The total area of the shaded region is-

= Area of half circle - area of the shaded region in the first quadrant

** Question:16 ** Choose the correct answer.

** ** Area bounded by the curve , the -axis and the ordinates and is

(A) (B) (C) (D)

** Answer: **

Hence the required area

** Therefore the correct answer is B. **

** Question:17 ** Choose the correct answer.

** T ** he area bounded by the curve , -axis and the ordinates and is given by

(A) (B) (C) (D)

[ ** Hint : ** if and if . ]

** Answer: **

The required area is

** Question:18 ** Choose the correct answer.

The area of the circle exterior to the parabola is

(A) (B) (C) (D)

** Answer: **

The area of the shaded region is to be found.

Required area =ar(DOC)+ar(DOA)

The region to the left of the y-axis is half of the circle with radius 4 units and centre origin.

Area of the shaded region to the left of y axis is ar(1) =

For the region to the right of y-axis and above x axis

The parabola and the circle in the first quadrant intersect at point

Remaining area is 2ar(2) is

Total area of shaded region is

** Question:19 ** Choose the correct answer ** ** The area bounded by the -axis, and when is

(A) (B) (C) (D)

** Answer: **

Given : and

Area of shaded region = area of BCDB + are of ADCA

Hence, the correct answer is B.

Miscellaneous exercise chapter 8 Class 12 chapter application of integrals is quite interesting and important from exam perspective. Class 12 Maths chapter 8 miscellaneous solutions mainly deals with application based questions like area finding between two curves etc. NCERT solutions for Class 12 Maths chapter 8 miscellaneous exercise will not take much time to solve provided chapter 7 integrals are done well before chapter 8.

- The Class 12th Maths chapter 8 exercise has some good questions which must be done before examination.
- NCERT syllabus Class 12 Maths chapter 8 miscellaneous exercises can be done along with chapter 7 as basic concepts used are the same.
- These Class 12 Maths chapter 8 miscellaneous exercises are also useful in Physics and chemistry problems.

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Download EBook**Comprehensive Coverage:**The solutions encompass all the topics covered in miscellaneous exercise class 12 chapter 8, ensuring a thorough understanding of the concepts.**Step-by-Step Solutions:**In this class 12 chapter 8 maths miscellaneous solutions, each problem is solved systematically, providing a stepwise approach to aid in better comprehension for students.**Accuracy and Clarity:**Solutions for class 12 maths miscellaneous exercise chapter 8 are presented accurately and concisely, using simple language to help students grasp the concepts easily.**Conceptual Clarity:**In this class 12 maths ch 8 miscellaneous exercise solutions, emphasis is placed on conceptual clarity, providing explanations that assist students in understanding the underlying principles behind each problem.**Inclusive Approach:**Solutions for class 12 chapter 8 miscellaneous exercise cater to different learning styles and abilities, ensuring that students of various levels can grasp the concepts effectively.**Relevance to Curriculum:**The solutions for miscellaneous exercise class 12 chapter 8 align closely with the NCERT curriculum, ensuring that students are prepared in line with the prescribed syllabus.

As per latest 2024 syllabus. Maths formulas, equations, & theorems of class 11 & 12th chapters

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Happy learning!!!

1. How many questions are there in Miscellaneous exercise Chapter 8 ?

There are 19 questions total in Miscellaneous exercise Chapter 8.

2. Can we find an area without using integrals ?

Simple figures like triangle, circle etc. can be tackled without integration but not the complex ones.

3. Are questions repeated in the examination from this Chapter ?

Yes, in the Board exam the questions are repeated every year.

4. What is the level of questions asked from this Chapter ?

Moderate level questions are asked from this Chapter.

5. Can one skip Miscellaneous exercise ?

No, as it has some good questions, miscellaneous exercise must be done.

6. What is the time it will take to complete for the first time ?

It will take around 5-6 hours to complete for the first time.

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hello mahima,

If you have uploaded screenshot of your 12th board result taken from CBSE official website,there won,t be a problem with that.If the screenshot that you have uploaded is clear and legible. It should display your name, roll number, marks obtained, and any other relevant details in a readable forma.ALSO, the screenshot clearly show it is from the official CBSE results portal.

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Hello Akash,

If you are looking for important questions of class 12th then I would like to suggest you to go with previous year questions of that particular board. You can go with last 5-10 years of PYQs so and after going through all the questions you will have a clear idea about the type and level of questions that are being asked and it will help you to boost your class 12th board preparation.

You can get the Previous Year Questions (PYQs) on the official website of the respective board.

I hope this answer helps you. If you have more queries then feel free to share your questions with us we will be happy to assist you.

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Thank you and wishing you all the best for your bright future.
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Hello student,

**
If you are planning to appear again for class 12th board exam with PCMB as a private candidate here is the right information you need:
**

- No school admission needed! Register directly with CBSE. (But if you want to attend the school then you can take admission in any private school of your choice but it will be waste of money)
- You have to appear for the 2025 12th board exams.
- Registration for class 12th board exam starts around September 2024 (check CBSE website for exact dates).
- Aim to register before late October to avoid extra fees.
- Schools might not offer classes for private students, so focus on self-study or coaching.

**
Remember
**
, these are tentative dates based on last year. Keep an eye on the CBSE website ( https://www.cbse.gov.in/ ) for the accurate and official announcement.

I hope this answer helps you. If you have more queries then feel free to share your questions with us, we will be happy to help you.

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Good luck with your studies!
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Hello Aspirant , Hope your doing great . As per your query , your eligible for JEE mains in the year of 2025 , Every candidate can appear for the JEE Main exam
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6 times over three consecutive years
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