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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
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Nov 23, 2024 Last updated
Document Description: RD Sharma Class 12 Solutions - Tangents and Normals for JEE 2024 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 2024 Exam. Find important definitions, questions, notes, meanings, examples, exercises and tests below for RD Sharma Class 12 Solutions - Tangents and Normals.
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