JEE Exam > JEE Notes > Mathematics (Maths) for JEE Main & Advanced > NCERT Solutions Exercise 12.1: Limits and Derivatives- 1
NCERT Solutions Class 11 Maths Chapter 12 - Limits and Derivatives
| Download, print and study this document offline |
Please wait while the PDF view is loading
Page 1
Chapter 2 1 – Limits and Derivatives Maths
______________________________________________________________________________
Exercise 2 1 .1
Question 1:
Evaluate the Given limit:
x3
lim 3 x
?
?
Solution 1:
x3
lim 3 x
?
? = 3 + 3 = 6
Question 2:
Evaluate the Given limit:
x
22
lim
7
x
? ?
??
?
??
??
Solution 2:
x
22
lim
7
x
? ?
??
?
??
??
=
22
7
?
??
?
??
??
Question 3:
Evaluate the Given limit :
2
x1
lim r ?
?
Solution 3:
2
x1
lim r ?
?
= ? ?
2
1 ?? ?
Question 4:
Evaluate the Given limit:
x1
43
lim
2
x
x
?
?
?
Solution 4:
x1
43
lim
2
x
x
?
?
?
=
? ? 4 4 3
16 3 19
4 2 2 2
?
?
??
?
Page 2
Chapter 2 1 – Limits and Derivatives Maths
______________________________________________________________________________
Exercise 2 1 .1
Question 1:
Evaluate the Given limit:
x3
lim 3 x
?
?
Solution 1:
x3
lim 3 x
?
? = 3 + 3 = 6
Question 2:
Evaluate the Given limit:
x
22
lim
7
x
? ?
??
?
??
??
Solution 2:
x
22
lim
7
x
? ?
??
?
??
??
=
22
7
?
??
?
??
??
Question 3:
Evaluate the Given limit :
2
x1
lim r ?
?
Solution 3:
2
x1
lim r ?
?
= ? ?
2
1 ?? ?
Question 4:
Evaluate the Given limit:
x1
43
lim
2
x
x
?
?
?
Solution 4:
x1
43
lim
2
x
x
?
?
?
=
? ? 4 4 3
16 3 19
4 2 2 2
?
?
??
?
Chapter 2 1 – Limits and Derivatives Maths
______________________________________________________________________________
Question 5:
Evaluate the Given limit:
10 5
1
1
lim
1
x
xx
x
??
??
?
Solution 5:
? ? ? ?
10 5
10 5
1
1 1 1
1 1 1 1 1
lim
1 1 1 2 2
x
xx
x
??
? ? ? ?
? ? ? ?
? ? ? ?
? ? ? ?
Question 6:
Evaluate the Given limit:
? ?
5
0
11
lim
x
x
x
?
??
Solution 6:
? ?
5
0
11
lim
x
x
x
?
??
Put x + 1 = y so that y → 1 as x → 0.
Accordingly,
? ?
5
0
11
lim
x
x
x
?
??
=
5
1
1
lim
1
x
y
y
?
?
?
? ?
55
1
5 1 1
5
0
1
lim
1
5.1 lim
5
11
lim 5
x
nn
n
xa
x
y
y
xa
na
xa
x
x
?
??
?
?
?
?
?
?? ?
??
??
?
??
?
??
??
Question 7:
Evaluate the Given limit:
2
2
2
3 10
lim
4
x
xx
x
?
??
?
Solution 7:
At x = 2, the value of the given rational function takes the form
0
0
.
?
2
2
2
3 10
lim
4
x
xx
x
?
??
?
=
? ? ? ?
? ? ? ?
2
2 3 5
lim
22
x
xx
xx
?
??
?
??
Page 3
Chapter 2 1 – Limits and Derivatives Maths
______________________________________________________________________________
Exercise 2 1 .1
Question 1:
Evaluate the Given limit:
x3
lim 3 x
?
?
Solution 1:
x3
lim 3 x
?
? = 3 + 3 = 6
Question 2:
Evaluate the Given limit:
x
22
lim
7
x
? ?
??
?
??
??
Solution 2:
x
22
lim
7
x
? ?
??
?
??
??
=
22
7
?
??
?
??
??
Question 3:
Evaluate the Given limit :
2
x1
lim r ?
?
Solution 3:
2
x1
lim r ?
?
= ? ?
2
1 ?? ?
Question 4:
Evaluate the Given limit:
x1
43
lim
2
x
x
?
?
?
Solution 4:
x1
43
lim
2
x
x
?
?
?
=
? ? 4 4 3
16 3 19
4 2 2 2
?
?
??
?
Chapter 2 1 – Limits and Derivatives Maths
______________________________________________________________________________
Question 5:
Evaluate the Given limit:
10 5
1
1
lim
1
x
xx
x
??
??
?
Solution 5:
? ? ? ?
10 5
10 5
1
1 1 1
1 1 1 1 1
lim
1 1 1 2 2
x
xx
x
??
? ? ? ?
? ? ? ?
? ? ? ?
? ? ? ?
Question 6:
Evaluate the Given limit:
? ?
5
0
11
lim
x
x
x
?
??
Solution 6:
? ?
5
0
11
lim
x
x
x
?
??
Put x + 1 = y so that y → 1 as x → 0.
Accordingly,
? ?
5
0
11
lim
x
x
x
?
??
=
5
1
1
lim
1
x
y
y
?
?
?
? ?
55
1
5 1 1
5
0
1
lim
1
5.1 lim
5
11
lim 5
x
nn
n
xa
x
y
y
xa
na
xa
x
x
?
??
?
?
?
?
?
?? ?
??
??
?
??
?
??
??
Question 7:
Evaluate the Given limit:
2
2
2
3 10
lim
4
x
xx
x
?
??
?
Solution 7:
At x = 2, the value of the given rational function takes the form
0
0
.
?
2
2
2
3 10
lim
4
x
xx
x
?
??
?
=
? ? ? ?
? ? ? ?
2
2 3 5
lim
22
x
xx
xx
?
??
?
??
Chapter 2 1 – Limits and Derivatives Maths
______________________________________________________________________________
? ?
2
35
lim
2
3 2 5
22
11
4
x
x
x
?
?
?
?
?
?
?
?
Question 8:
Evaluate the Given limit:
4
2
3
81
lim
2 5 3
x
x
xx
?
?
??
Solution 8:
At x = 2, the value of the given rational function takes the form
0
0
.
? ? ? ? ? ?
? ? ? ?
? ? ? ?
? ? ? ?
? ?
2
4
2
33
2
3
2
3 3 9
81
lim lim
2 5 3 3 2 1
39
lim
21
3 3 3 9
2 3 1
6 18
7
108
7
xx
x
x x x
x
x x x x
xx
x
??
?
? ? ?
?
??
? ? ? ?
??
?
?
? ? ?
?
?
?
?
?
Question 9:
Evaluate the Given limit:
0
lim
1
x
ax b
cx
?
?
?
Solution 9:
0
lim
1
x
ax b
cx
?
?
?
=
? ?
? ?
0
01
ab
b
c
?
?
?
Page 4
Chapter 2 1 – Limits and Derivatives Maths
______________________________________________________________________________
Exercise 2 1 .1
Question 1:
Evaluate the Given limit:
x3
lim 3 x
?
?
Solution 1:
x3
lim 3 x
?
? = 3 + 3 = 6
Question 2:
Evaluate the Given limit:
x
22
lim
7
x
? ?
??
?
??
??
Solution 2:
x
22
lim
7
x
? ?
??
?
??
??
=
22
7
?
??
?
??
??
Question 3:
Evaluate the Given limit :
2
x1
lim r ?
?
Solution 3:
2
x1
lim r ?
?
= ? ?
2
1 ?? ?
Question 4:
Evaluate the Given limit:
x1
43
lim
2
x
x
?
?
?
Solution 4:
x1
43
lim
2
x
x
?
?
?
=
? ? 4 4 3
16 3 19
4 2 2 2
?
?
??
?
Chapter 2 1 – Limits and Derivatives Maths
______________________________________________________________________________
Question 5:
Evaluate the Given limit:
10 5
1
1
lim
1
x
xx
x
??
??
?
Solution 5:
? ? ? ?
10 5
10 5
1
1 1 1
1 1 1 1 1
lim
1 1 1 2 2
x
xx
x
??
? ? ? ?
? ? ? ?
? ? ? ?
? ? ? ?
Question 6:
Evaluate the Given limit:
? ?
5
0
11
lim
x
x
x
?
??
Solution 6:
? ?
5
0
11
lim
x
x
x
?
??
Put x + 1 = y so that y → 1 as x → 0.
Accordingly,
? ?
5
0
11
lim
x
x
x
?
??
=
5
1
1
lim
1
x
y
y
?
?
?
? ?
55
1
5 1 1
5
0
1
lim
1
5.1 lim
5
11
lim 5
x
nn
n
xa
x
y
y
xa
na
xa
x
x
?
??
?
?
?
?
?
?? ?
??
??
?
??
?
??
??
Question 7:
Evaluate the Given limit:
2
2
2
3 10
lim
4
x
xx
x
?
??
?
Solution 7:
At x = 2, the value of the given rational function takes the form
0
0
.
?
2
2
2
3 10
lim
4
x
xx
x
?
??
?
=
? ? ? ?
? ? ? ?
2
2 3 5
lim
22
x
xx
xx
?
??
?
??
Chapter 2 1 – Limits and Derivatives Maths
______________________________________________________________________________
? ?
2
35
lim
2
3 2 5
22
11
4
x
x
x
?
?
?
?
?
?
?
?
Question 8:
Evaluate the Given limit:
4
2
3
81
lim
2 5 3
x
x
xx
?
?
??
Solution 8:
At x = 2, the value of the given rational function takes the form
0
0
.
? ? ? ? ? ?
? ? ? ?
? ? ? ?
? ? ? ?
? ?
2
4
2
33
2
3
2
3 3 9
81
lim lim
2 5 3 3 2 1
39
lim
21
3 3 3 9
2 3 1
6 18
7
108
7
xx
x
x x x
x
x x x x
xx
x
??
?
? ? ?
?
??
? ? ? ?
??
?
?
? ? ?
?
?
?
?
?
Question 9:
Evaluate the Given limit:
0
lim
1
x
ax b
cx
?
?
?
Solution 9:
0
lim
1
x
ax b
cx
?
?
?
=
? ?
? ?
0
01
ab
b
c
?
?
?
Chapter 2 1 – Limits and Derivatives Maths
______________________________________________________________________________
Question 10:
Evaluate the Given limit:
1
3
1
1
6
1
lim
1
z
z
z
?
?
?
Solution 10:
1
3
1
1
6
1
lim
1
z
z
z
?
?
?
At z = 1, the value of the given function takes the form
0
0
.
Put
1
6
z = x so that z →1 as x → 1.
Accordingly,
1
3
1
1
6
1
lim
1
z
z
z
?
?
?
=
2
1
1
lim
1
x
x
x
?
?
?
2
1
2 1 1
1
lim
1
2.1 lim
2
x
nn
n
xa
x
x
xa
na
xa
?
??
?
?
?
?
?? ?
??
??
?
??
?
1
3
1
1
6
1
lim
1
z
z
z
?
?
?
= 2
Question 11:
Evaluate the Given limit:
? ?
2
2
1
lim
1
x
ax bx c
cx b a
?
??
??
, a + b + c ? 0
Solution 11:
2
2
1
lim
x
ax bx c
cx bx a
?
??
??
=
? ? ? ?
? ? ? ?
2
2
11
11
a b c
c b a
??
??
=
abc
abc
??
??
= 1 [a + b + c ? 0]
Page 5
Chapter 2 1 – Limits and Derivatives Maths
______________________________________________________________________________
Exercise 2 1 .1
Question 1:
Evaluate the Given limit:
x3
lim 3 x
?
?
Solution 1:
x3
lim 3 x
?
? = 3 + 3 = 6
Question 2:
Evaluate the Given limit:
x
22
lim
7
x
? ?
??
?
??
??
Solution 2:
x
22
lim
7
x
? ?
??
?
??
??
=
22
7
?
??
?
??
??
Question 3:
Evaluate the Given limit :
2
x1
lim r ?
?
Solution 3:
2
x1
lim r ?
?
= ? ?
2
1 ?? ?
Question 4:
Evaluate the Given limit:
x1
43
lim
2
x
x
?
?
?
Solution 4:
x1
43
lim
2
x
x
?
?
?
=
? ? 4 4 3
16 3 19
4 2 2 2
?
?
??
?
Chapter 2 1 – Limits and Derivatives Maths
______________________________________________________________________________
Question 5:
Evaluate the Given limit:
10 5
1
1
lim
1
x
xx
x
??
??
?
Solution 5:
? ? ? ?
10 5
10 5
1
1 1 1
1 1 1 1 1
lim
1 1 1 2 2
x
xx
x
??
? ? ? ?
? ? ? ?
? ? ? ?
? ? ? ?
Question 6:
Evaluate the Given limit:
? ?
5
0
11
lim
x
x
x
?
??
Solution 6:
? ?
5
0
11
lim
x
x
x
?
??
Put x + 1 = y so that y → 1 as x → 0.
Accordingly,
? ?
5
0
11
lim
x
x
x
?
??
=
5
1
1
lim
1
x
y
y
?
?
?
? ?
55
1
5 1 1
5
0
1
lim
1
5.1 lim
5
11
lim 5
x
nn
n
xa
x
y
y
xa
na
xa
x
x
?
??
?
?
?
?
?
?? ?
??
??
?
??
?
??
??
Question 7:
Evaluate the Given limit:
2
2
2
3 10
lim
4
x
xx
x
?
??
?
Solution 7:
At x = 2, the value of the given rational function takes the form
0
0
.
?
2
2
2
3 10
lim
4
x
xx
x
?
??
?
=
? ? ? ?
? ? ? ?
2
2 3 5
lim
22
x
xx
xx
?
??
?
??
Chapter 2 1 – Limits and Derivatives Maths
______________________________________________________________________________
? ?
2
35
lim
2
3 2 5
22
11
4
x
x
x
?
?
?
?
?
?
?
?
Question 8:
Evaluate the Given limit:
4
2
3
81
lim
2 5 3
x
x
xx
?
?
??
Solution 8:
At x = 2, the value of the given rational function takes the form
0
0
.
? ? ? ? ? ?
? ? ? ?
? ? ? ?
? ? ? ?
? ?
2
4
2
33
2
3
2
3 3 9
81
lim lim
2 5 3 3 2 1
39
lim
21
3 3 3 9
2 3 1
6 18
7
108
7
xx
x
x x x
x
x x x x
xx
x
??
?
? ? ?
?
??
? ? ? ?
??
?
?
? ? ?
?
?
?
?
?
Question 9:
Evaluate the Given limit:
0
lim
1
x
ax b
cx
?
?
?
Solution 9:
0
lim
1
x
ax b
cx
?
?
?
=
? ?
? ?
0
01
ab
b
c
?
?
?
Chapter 2 1 – Limits and Derivatives Maths
______________________________________________________________________________
Question 10:
Evaluate the Given limit:
1
3
1
1
6
1
lim
1
z
z
z
?
?
?
Solution 10:
1
3
1
1
6
1
lim
1
z
z
z
?
?
?
At z = 1, the value of the given function takes the form
0
0
.
Put
1
6
z = x so that z →1 as x → 1.
Accordingly,
1
3
1
1
6
1
lim
1
z
z
z
?
?
?
=
2
1
1
lim
1
x
x
x
?
?
?
2
1
2 1 1
1
lim
1
2.1 lim
2
x
nn
n
xa
x
x
xa
na
xa
?
??
?
?
?
?
?? ?
??
??
?
??
?
1
3
1
1
6
1
lim
1
z
z
z
?
?
?
= 2
Question 11:
Evaluate the Given limit:
? ?
2
2
1
lim
1
x
ax bx c
cx b a
?
??
??
, a + b + c ? 0
Solution 11:
2
2
1
lim
x
ax bx c
cx bx a
?
??
??
=
? ? ? ?
? ? ? ?
2
2
11
11
a b c
c b a
??
??
=
abc
abc
??
??
= 1 [a + b + c ? 0]
Chapter 2 1 – Limits and Derivatives Maths
______________________________________________________________________________
Question 12:
Evaluate the Given limit:
2
11
2
lim
2
x
x
x
??
?
?
Solution 12:
2
11
2
lim
2
x
x
x
??
?
?
At x = –2, the value of the given function takes the form
0
0
Now,
2
11
2
lim
2
x
x
x
??
?
?
=
2
2
2
lim
2
x
x
x
x
??
? ??
??
??
?
=
2
1
lim
2
x
x
??
? ?
11
2 2 4
?
??
?
Question 13:
Evaluate the Given limit:
0
sin
lim
x
ax
bx
?
Solution 13:
0
sin
lim
x
ax
bx
?
At x = 0, the value of the given function takes the form
0
0
.
Now,
0
sin
lim
x
ax
bx
?
=
0
sin
lim
x
ax ax
ax bx
?
?
0
sin
lim
x
ax a
ax b
?
??
??
??
??
? ?
0
sin
lim 0 0
ax
a ax
x ax
b ax
?
??
? ? ? ?
??
??
0
sin
1 lim 1
y
ay
by
a
b
?
??
? ? ?
??
??
?
Read More
|
209 videos|443 docs|143 tests
|
FAQs on NCERT Solutions Class 11 Maths Chapter 12 - Limits and Derivatives
| 1. What is the concept of limits in calculus? |  |
Ans. In calculus, limits are used to describe the behavior of a function as the input approaches a certain value or as the input goes towards infinity or negative infinity. It helps in determining the value that a function approaches as it gets arbitrarily close to a particular input.
| 2. How are limits and derivatives related in calculus? | |
Ans. Limits and derivatives are closely related in calculus. The derivative of a function at a particular point is defined as the limit of the average rate of change of the function as the interval around the point shrinks to zero. In other words, the derivative measures the instantaneous rate of change of a function at a specific point.
| 3. What is the significance of limits and derivatives in the field of mathematics? | |
Ans. Limits and derivatives play a crucial role in various branches of mathematics. They are fundamental concepts in calculus and are used to solve problems related to rates of change, optimization, and approximation. Limits and derivatives also find applications in physics, engineering, economics, and other scientific fields.
| 4. How are limits and derivatives applied in real-life scenarios? | |
Ans. Limits and derivatives have numerous real-life applications. For example, they are used in physics to calculate velocities and accelerations, in economics to analyze demand and supply functions, in engineering to optimize designs, and in finance to determine rates of return and risk assessments. They help in modeling and solving real-world problems involving change and optimization.
| 5. What are some common techniques used to evaluate limits and derivatives? | |
Ans. There are several techniques used to evaluate limits and derivatives, including algebraic manipulation, factoring, rationalizing, L'Hôpital's rule, and Taylor series expansions. These methods help in simplifying expressions or determining the behavior of a function at a particular point or as the input approaches a certain value.
Related Exams
About this Document
|
4.6K Views |
|
4.62/5 Rating |
|
Oct 31, 2024 Last updated |
Document Description: NCERT Solutions Exercise 12.1: Limits and Derivatives- 1 for JEE 2024 is part of Mathematics (Maths) for JEE Main & Advanced preparation.
The notes and questions for NCERT Solutions Exercise 12.1: Limits and Derivatives- 1 have been prepared according to the JEE exam syllabus. Information about NCERT Solutions Exercise 12.1: Limits and Derivatives- 1 covers topics
like and NCERT Solutions Exercise 12.1: Limits and Derivatives- 1 Example, for JEE 2024 Exam. Find important definitions, questions, notes, meanings, examples, exercises and tests below for NCERT Solutions Exercise 12.1: Limits and Derivatives- 1.
Introduction of NCERT Solutions Exercise 12.1: Limits and Derivatives- 1 in English is available as part of our Mathematics (Maths) for JEE Main & Advanced
for JEE & NCERT Solutions Exercise 12.1: Limits and Derivatives- 1 in Hindi for Mathematics (Maths) for JEE Main & Advanced course.
Download more important topics related with notes, lectures and mock test series for JEE
Exam by signing up for free. JEE: NCERT Solutions Class 11 Maths Chapter 12 - Limits and Derivatives
Description
Full syllabus notes, lecture & questions for NCERT Solutions Class 11 Maths Chapter 12 - Limits and Derivatives - JEE | Plus excerises question with solution to help you revise complete syllabus for Mathematics (Maths) for JEE Main & Advanced | Best notes, free PDF download
Information about NCERT Solutions Exercise 12.1: Limits and Derivatives- 1
In this doc you can find the meaning of NCERT Solutions Exercise 12.1: Limits and Derivatives- 1 defined & explained in the simplest way possible. Besides explaining types of
NCERT Solutions Exercise 12.1: Limits and Derivatives- 1 theory, EduRev gives you an ample number of questions to practice NCERT Solutions Exercise 12.1: Limits and Derivatives- 1 tests, examples and also practice JEE
tests
|
209 videos|443 docs|143 tests
|
Download as PDF
|
Explore Courses for JEE exam
|
|
Signup for Free!
Signup to see your scores go up within 7 days! Learn & Practice with 1000+ FREE Notes, Videos & Tests.
Related Searches
Free
,Important questions
,Exam
,Sample Paper
,shortcuts and tricks
,study material
,NCERT Solutions Class 11 Maths Chapter 12 - Limits and Derivatives
,MCQs
,NCERT Solutions Class 11 Maths Chapter 12 - Limits and Derivatives
,ppt
,practice quizzes
,Semester Notes
,NCERT Solutions Class 11 Maths Chapter 12 - Limits and Derivatives
,Previous Year Questions with Solutions
,video lectures
,Extra Questions
,past year papers
,Objective type Questions
,mock tests for examination
,Summary
,Viva Questions
;
Additional Information about NCERT Solutions Exercise 12.1: Limits and Derivatives- 1 for JEE Preparation
NCERT Solutions Exercise 12.1: Limits and Derivatives- 1 Free PDF Download
The NCERT Solutions Exercise 12.1: Limits and Derivatives- 1 is an invaluable resource that delves deep into the core of the JEE exam.
These study notes are curated by experts and cover all the essential topics and concepts, making your preparation more efficient and effective.
With the help of these notes, you can grasp complex subjects quickly, revise important points easily,
and reinforce your understanding of key concepts. The study notes are presented in a concise and easy-to-understand manner,
allowing you to optimize your learning process. Whether you're looking for best-recommended books, sample papers, study material,
or toppers' notes, this PDF has got you covered. Download the NCERT Solutions Exercise 12.1: Limits and Derivatives- 1 now and kickstart your journey towards success in the JEE exam.
Importance of NCERT Solutions Exercise 12.1: Limits and Derivatives- 1
The importance of NCERT Solutions Exercise 12.1: Limits and Derivatives- 1 cannot be overstated, especially for JEE aspirants.
This document holds the key to success in the JEE exam.
It offers a detailed understanding of the concept, providing invaluable insights into the topic.
By knowing the concepts well in advance, students can plan their preparation effectively.
Utilize this indispensable guide for a well-rounded preparation and achieve your desired results.
NCERT Solutions Exercise 12.1: Limits and Derivatives- 1 Notes
NCERT Solutions Exercise 12.1: Limits and Derivatives- 1 Notes offer in-depth insights into the specific topic to help you master it with ease.
This comprehensive document covers all aspects related to NCERT Solutions Exercise 12.1: Limits and Derivatives- 1.
It includes detailed information about the exam syllabus, recommended books, and study materials for a well-rounded preparation.
Practice papers and question papers enable you to assess your progress effectively.
Additionally, the paper analysis provides valuable tips for tackling the exam strategically.
Access to Toppers' notes gives you an edge in understanding complex concepts.
Whether you're a beginner or aiming for advanced proficiency, NCERT Solutions Exercise 12.1: Limits and Derivatives- 1 Notes on EduRev are your ultimate resource for success.
NCERT Solutions Exercise 12.1: Limits and Derivatives- 1 JEE Questions
The "NCERT Solutions Exercise 12.1: Limits and Derivatives- 1 JEE Questions" guide is a valuable resource for all aspiring students preparing for the
JEE exam. It focuses on providing a wide range of practice questions to help students gauge
their understanding of the exam topics. These questions cover the entire syllabus, ensuring comprehensive preparation.
The guide includes previous years' question papers for students to familiarize themselves with the exam's format and difficulty level.
Additionally, it offers subject-specific question banks, allowing students to focus on weak areas and improve their performance.
Study NCERT Solutions Exercise 12.1: Limits and Derivatives- 1 on the App
Students of JEE can study NCERT Solutions Exercise 12.1: Limits and Derivatives- 1 alongwith tests & analysis from the EduRev app,
which will help them while preparing for their exam. Apart from the NCERT Solutions Exercise 12.1: Limits and Derivatives- 1,
students can also utilize the EduRev App for other study materials such as previous year question papers, syllabus, important questions, etc.
The EduRev App will make your learning easier as you can access it from anywhere you want.
The content of NCERT Solutions Exercise 12.1: Limits and Derivatives- 1 is prepared as per the latest JEE syllabus.
|
© EduRev
|
Education Revolution
|
Follow Us
|
Signup to see your scores
go up within 7 days!
Access 1000+ FREE Docs, Videos and Tests
Takes less than 10 seconds to signup