Explore Exams
Login Signup
Sign in Open App
JEE Exam  >  JEE Notes  >  Mathematics (Maths) Class 12  >  RD Sharma Class 12 Solutions - Tangents and Normals

RD Sharma Class 12 Solutions - Tangents and Normals

Download, print and study this document offline
Please wait while the PDF view is loading
 Page 1


16. Tangents and Normals
Exercise 16.1
1 A. Question
Find the The Slopes of the tangent and the normal to the following curves at the indicated points :
 at x = 4
Answer
Given:
y =  at x = 4
First, we have to find  of given function, f(x),i.e, to find the derivative of f(x)
y = 
? y = (
? y = 
(x
n
) = n.x
n – 1
The Slope of the tangent is 
Since, x = 4
x = 4
 = 
x = 4
 = 
x = 4
 = 
x = 4
 = 3
The Slope of the tangent at x = 4 is 3
Page 2


16. Tangents and Normals
Exercise 16.1
1 A. Question
Find the The Slopes of the tangent and the normal to the following curves at the indicated points :
 at x = 4
Answer
Given:
y =  at x = 4
First, we have to find  of given function, f(x),i.e, to find the derivative of f(x)
y = 
? y = (
? y = 
(x
n
) = n.x
n – 1
The Slope of the tangent is 
Since, x = 4
x = 4
 = 
x = 4
 = 
x = 4
 = 
x = 4
 = 3
The Slope of the tangent at x = 4 is 3
? The Slope of the normal = 
? The Slope of the normal = 
? The Slope of the normal = 
1 B. Question
Find the The Slopes of the tangent and the normal to the following curves at the indicated points :
 at x = 9
Answer
Given:
y =  at x = 9
First, we have to find  of given function, f(x),i.e, to find the derivative of f(x)
? y = 
? y = (
(x
n
) = n.x
n – 1
The Slope of the tangent is 
? y = (
Since, x = 9
x = 9
 = 
x = 9
 = 
x = 9
 = 
x = 9
 = 
x = 9
 = 
The Slope of the tangent at x = 9 is 
? The Slope of the normal = 
? The Slope of the normal = 
Page 3


16. Tangents and Normals
Exercise 16.1
1 A. Question
Find the The Slopes of the tangent and the normal to the following curves at the indicated points :
 at x = 4
Answer
Given:
y =  at x = 4
First, we have to find  of given function, f(x),i.e, to find the derivative of f(x)
y = 
? y = (
? y = 
(x
n
) = n.x
n – 1
The Slope of the tangent is 
Since, x = 4
x = 4
 = 
x = 4
 = 
x = 4
 = 
x = 4
 = 3
The Slope of the tangent at x = 4 is 3
? The Slope of the normal = 
? The Slope of the normal = 
? The Slope of the normal = 
1 B. Question
Find the The Slopes of the tangent and the normal to the following curves at the indicated points :
 at x = 9
Answer
Given:
y =  at x = 9
First, we have to find  of given function, f(x),i.e, to find the derivative of f(x)
? y = 
? y = (
(x
n
) = n.x
n – 1
The Slope of the tangent is 
? y = (
Since, x = 9
x = 9
 = 
x = 9
 = 
x = 9
 = 
x = 9
 = 
x = 9
 = 
The Slope of the tangent at x = 9 is 
? The Slope of the normal = 
? The Slope of the normal = 
? The Slope of the normal = 
? The Slope of the normal = – 6
1 C. Question
Find the The Slopes of the tangent and the normal to the following curves at the indicated points :
y = x
3
 – x at x = 2
Answer
Given:
y = x
3
 – x at x = 2
First, we have to find  of given function, f(x),i.e, to find the derivative of f(x)
(x
n
) = n.x
n – 1
The Slope of the tangent is 
? y = x
3
 – x
(x
3
) + 3 (x)
 = 3.x
3 – 1
 – 1.x
1 – 0
 = 3x
2
 – 1
Since, x = 2
x = 2
 = 3 (2)
2
 – 1
x = 2
 = (3 4) – 1
x = 2
 = 12 – 1
x = 2
 = 11
The Slope of the tangent at x = 2 is 11
? The Slope of the normal = 
? The Slope of the normal = 
? The Slope of the normal = 
1 D. Question
Find the The Slopes of the tangent and the normal to the following curves at the indicated points :
y = 2x
2
 + 3 sin x at x = 0
Answer
Given:
y = 2x
2
 + 3sinx at x = 0
Page 4


16. Tangents and Normals
Exercise 16.1
1 A. Question
Find the The Slopes of the tangent and the normal to the following curves at the indicated points :
 at x = 4
Answer
Given:
y =  at x = 4
First, we have to find  of given function, f(x),i.e, to find the derivative of f(x)
y = 
? y = (
? y = 
(x
n
) = n.x
n – 1
The Slope of the tangent is 
Since, x = 4
x = 4
 = 
x = 4
 = 
x = 4
 = 
x = 4
 = 3
The Slope of the tangent at x = 4 is 3
? The Slope of the normal = 
? The Slope of the normal = 
? The Slope of the normal = 
1 B. Question
Find the The Slopes of the tangent and the normal to the following curves at the indicated points :
 at x = 9
Answer
Given:
y =  at x = 9
First, we have to find  of given function, f(x),i.e, to find the derivative of f(x)
? y = 
? y = (
(x
n
) = n.x
n – 1
The Slope of the tangent is 
? y = (
Since, x = 9
x = 9
 = 
x = 9
 = 
x = 9
 = 
x = 9
 = 
x = 9
 = 
The Slope of the tangent at x = 9 is 
? The Slope of the normal = 
? The Slope of the normal = 
? The Slope of the normal = 
? The Slope of the normal = – 6
1 C. Question
Find the The Slopes of the tangent and the normal to the following curves at the indicated points :
y = x
3
 – x at x = 2
Answer
Given:
y = x
3
 – x at x = 2
First, we have to find  of given function, f(x),i.e, to find the derivative of f(x)
(x
n
) = n.x
n – 1
The Slope of the tangent is 
? y = x
3
 – x
(x
3
) + 3 (x)
 = 3.x
3 – 1
 – 1.x
1 – 0
 = 3x
2
 – 1
Since, x = 2
x = 2
 = 3 (2)
2
 – 1
x = 2
 = (3 4) – 1
x = 2
 = 12 – 1
x = 2
 = 11
The Slope of the tangent at x = 2 is 11
? The Slope of the normal = 
? The Slope of the normal = 
? The Slope of the normal = 
1 D. Question
Find the The Slopes of the tangent and the normal to the following curves at the indicated points :
y = 2x
2
 + 3 sin x at x = 0
Answer
Given:
y = 2x
2
 + 3sinx at x = 0
First, we have to find  of given function, f(x),i.e, to find the derivative of f(x)
(x
n
) = n.x
n – 1
The Slope of the tangent is 
? y = 2x
2
 + 3sinx
 
=
 2 (x
2
) + 3 (sinx)
 = 2 2x
2 – 1
 + 3 (cosx)
 (sinx) = cosx
 = 4x + 3cosx
Since, x = 2
? 
x = 0
 = 4 0 + 3cos(0)
cos(0) = 1
? 
x = 0
 = 0 + 3 1
? 
x = 0
 = 3
The Slope of the tangent at x = 0 is 3
? The Slope of the normal = 
? The Slope of the normal = 
? The Slope of the normal = 
1 E. Question
Find the The Slopes of the tangent and the normal to the following curves at the indicated points :
x = a (? – sin ?), y = a(1 + cos ?) at
? = – p/2
Answer
Given:
x = a( ) & y = a(1 + cos ) at 
Here, To find , we have to find  &  and and divide  and we get our desired .
(x
n
) = n.x
n – 1
? x = a( )
?  = a( ( ) – (sin ))
Page 5


16. Tangents and Normals
Exercise 16.1
1 A. Question
Find the The Slopes of the tangent and the normal to the following curves at the indicated points :
 at x = 4
Answer
Given:
y =  at x = 4
First, we have to find  of given function, f(x),i.e, to find the derivative of f(x)
y = 
? y = (
? y = 
(x
n
) = n.x
n – 1
The Slope of the tangent is 
Since, x = 4
x = 4
 = 
x = 4
 = 
x = 4
 = 
x = 4
 = 3
The Slope of the tangent at x = 4 is 3
? The Slope of the normal = 
? The Slope of the normal = 
? The Slope of the normal = 
1 B. Question
Find the The Slopes of the tangent and the normal to the following curves at the indicated points :
 at x = 9
Answer
Given:
y =  at x = 9
First, we have to find  of given function, f(x),i.e, to find the derivative of f(x)
? y = 
? y = (
(x
n
) = n.x
n – 1
The Slope of the tangent is 
? y = (
Since, x = 9
x = 9
 = 
x = 9
 = 
x = 9
 = 
x = 9
 = 
x = 9
 = 
The Slope of the tangent at x = 9 is 
? The Slope of the normal = 
? The Slope of the normal = 
? The Slope of the normal = 
? The Slope of the normal = – 6
1 C. Question
Find the The Slopes of the tangent and the normal to the following curves at the indicated points :
y = x
3
 – x at x = 2
Answer
Given:
y = x
3
 – x at x = 2
First, we have to find  of given function, f(x),i.e, to find the derivative of f(x)
(x
n
) = n.x
n – 1
The Slope of the tangent is 
? y = x
3
 – x
(x
3
) + 3 (x)
 = 3.x
3 – 1
 – 1.x
1 – 0
 = 3x
2
 – 1
Since, x = 2
x = 2
 = 3 (2)
2
 – 1
x = 2
 = (3 4) – 1
x = 2
 = 12 – 1
x = 2
 = 11
The Slope of the tangent at x = 2 is 11
? The Slope of the normal = 
? The Slope of the normal = 
? The Slope of the normal = 
1 D. Question
Find the The Slopes of the tangent and the normal to the following curves at the indicated points :
y = 2x
2
 + 3 sin x at x = 0
Answer
Given:
y = 2x
2
 + 3sinx at x = 0
First, we have to find  of given function, f(x),i.e, to find the derivative of f(x)
(x
n
) = n.x
n – 1
The Slope of the tangent is 
? y = 2x
2
 + 3sinx
 
=
 2 (x
2
) + 3 (sinx)
 = 2 2x
2 – 1
 + 3 (cosx)
 (sinx) = cosx
 = 4x + 3cosx
Since, x = 2
? 
x = 0
 = 4 0 + 3cos(0)
cos(0) = 1
? 
x = 0
 = 0 + 3 1
? 
x = 0
 = 3
The Slope of the tangent at x = 0 is 3
? The Slope of the normal = 
? The Slope of the normal = 
? The Slope of the normal = 
1 E. Question
Find the The Slopes of the tangent and the normal to the following curves at the indicated points :
x = a (? – sin ?), y = a(1 + cos ?) at
? = – p/2
Answer
Given:
x = a( ) & y = a(1 + cos ) at 
Here, To find , we have to find  &  and and divide  and we get our desired .
(x
n
) = n.x
n – 1
? x = a( )
?  = a( ( ) – (sin ))
?  = a(1 – ) ...(1)
 (sinx) = cosx
? y = a(1 + cos )
?  = a( ( ) + (cos ))
 (cosx) = – sinx
 (Constant) = 0
?  = a( + ( – sin ))
?  = a( – sin )
?  = – asin ...(2)
? 
? 
The Slope of the tangent is 
Since, 
? 
sin( ) = 1
cos( ) = 0
? 
? 
?  = 1
The Slope of the tangent at x = is 1
? The Slope of the normal = 
? The Slope of the normal = 
? The Slope of the normal = 
? The Slope of the normal = – 1
1 F. Question
Read More
Join Course for Free
About this Document
4.64/5 Rating
Apr 20, 2026 Last updated
Related Exams
Document Description: RD Sharma Class 12 Solutions - Tangents and Normals for JEE 2026 is part of Mathematics (Maths) Class 12 preparation. The notes and questions for RD Sharma Class 12 Solutions - Tangents and Normals have been prepared according to the JEE exam syllabus. Information about RD Sharma Class 12 Solutions - Tangents and Normals covers topics like and RD Sharma Class 12 Solutions - Tangents and Normals Example, for JEE 2026 Exam. Find important definitions, questions, notes, meanings, examples, exercises and tests below for RD Sharma Class 12 Solutions - Tangents and Normals.
Introduction of RD Sharma Class 12 Solutions - Tangents and Normals in English is available as part of our Mathematics (Maths) Class 12 for JEE & RD Sharma Class 12 Solutions - Tangents and Normals in Hindi for Mathematics (Maths) Class 12 course. Download more important topics related with notes, lectures and mock test series for JEE Exam by signing up for free. JEE: RD Sharma Class 12 Solutions - Tangents and Normals
Description
RD Sharma Class 12 Solutions of Maths Class 12 on EduRev - covers important questions, MCQs with answers. Helps you prepare for exam.
Information about RD Sharma Class 12 Solutions - Tangents and Normals
In this doc you can find the meaning of RD Sharma Class 12 Solutions - Tangents and Normals defined & explained in the simplest way possible. Besides explaining types of RD Sharma Class 12 Solutions - Tangents and Normals theory, EduRev gives you an ample number of questions to practice RD Sharma Class 12 Solutions - Tangents and Normals tests, examples and also practice JEE tests
Explore Courses for JEE exam
Related Searches
Sample Paper, Semester Notes, MCQs, Exam, study material, practice quizzes, Important questions, Extra Questions, Free, ppt, RD Sharma Class 12 Solutions - Tangents and Normals, mock tests for examination, past year papers, shortcuts and tricks, pdf , Viva Questions, Objective type Questions, Summary, Previous Year Questions with Solutions, video lectures, RD Sharma Class 12 Solutions - Tangents and Normals, RD Sharma Class 12 Solutions - Tangents and Normals;
;
Additional Information about RD Sharma Class 12 Solutions - Tangents and Normals for JEE Preparation

RD Sharma Class 12 Solutions - Tangents and Normals Free PDF Download

The RD Sharma Class 12 Solutions - Tangents and Normals 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 RD Sharma Class 12 Solutions - Tangents and Normals now and kickstart your journey towards success in the JEE exam.

Importance of RD Sharma Class 12 Solutions - Tangents and Normals

The importance of RD Sharma Class 12 Solutions - Tangents and Normals 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.

RD Sharma Class 12 Solutions - Tangents and Normals Notes

RD Sharma Class 12 Solutions - Tangents and Normals Notes offer in-depth insights into the specific topic to help you master it with ease. This comprehensive document covers all aspects related to RD Sharma Class 12 Solutions - Tangents and Normals. 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, RD Sharma Class 12 Solutions - Tangents and Normals Notes on EduRev are your ultimate resource for success.

RD Sharma Class 12 Solutions - Tangents and Normals JEE Questions

The "RD Sharma Class 12 Solutions - Tangents and Normals 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 RD Sharma Class 12 Solutions - Tangents and Normals on the App

Students of JEE can study RD Sharma Class 12 Solutions - Tangents and Normals alongwith tests & analysis from the EduRev app, which will help them while preparing for their exam. Apart from the RD Sharma Class 12 Solutions - Tangents and Normals, 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 RD Sharma Class 12 Solutions - Tangents and Normals is prepared as per the latest JEE syllabus.
Signup to see your scores go up within 7 days!
Access 1000+ FREE Docs, Videos and Tests
Takes less than 10 seconds to signup