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RD Sharma Solutions are the highly recommended reference material by most of the CBSE schools. In the absence of a teacher, this book will guide the students in the right direction by clarifying their doubts and providing accurate answers. Mathematics is a subject where doubt arises quite often. And when it comes to chapter 15, things get even more complicated. RD Sharma Solutions The students with such issues can use the RD Sharma Class 12th Exercise 15.2 to clarify their doubts.

Chapter 15 - Tangents and Normals Ex 15.1

Chapter 15 - Tangents and Normals Ex 15.3

Chapter 15 -Tangents and Normals Ex-FBQ

Chapter 15 -Tangents and Normals Ex-MCQ

Chapter 15 -Tangents and Normals Ex-VSA

Tangents and Normals Exercise 15.2 Question 1

HINTS:

We find the slope and differentiate slope

Tangents and Normals Exercise 15.2 Question 2

Hint:

We find the slope of tangent and using relation,

Equation of normal will be

Tangents and Normals Exercise 15.2 Question 3 Sub Question 1

Equation of normal ,

HINTS:

Differentiating the given equation and find the slope of the tangent.

Equation of tangent is,

Equation of Normal is,

Tangents and Normals Exercise 15.2 Question 3 Sub Question 3

Equation of normal,

HINTS:

Differentiating the given equation .

Slope of tangent ,

Equation of tangent is,

Equation of Normal is,

Tangents and Normals Exercise 15.2 Question 3 Sub Question 4

Answer: Equation of tangent,Equation of normal,

HINTS:

Differentiating the given equation with respect to x.

Slope of tangent,

Equation of tangent is,

Equation of Normal is,

Tangents and Normals Exercise 15.2 Question 3 Sub Question 5

Equation of normal ,

HINTS:

Differentiating the given equation with respect to x.

Equation of tangent is,

Equation of Normal is,

Tangents and Normals Exercise 15.2 Question 3 Sub Question 6

Equation of normal,

HINTS:

Differentiating the given curve and find the slope.

Slope of tangent ,

Equation of tangent is,

Equation of Normal is,

Tangents and Normals Exercise 15.2 Question 3 Sub Question 7

Equation of normal ,

HINTS:

Differentiating the given equation with respect to x.

Slope of tangent ,

Equation of tangent is,

Dividing by ab

Equation of Normal is,

Dividing by

Tangents and Normals Exercise 15.2 Question 3 Sub Question 8

Equation of normal ,

HINTS:

Differentiating the given curve with respect to x and find its slope.

Slope of tangent ,

Equation of tangent is,

Divinding by

Equation of Normal is,

Dividing by

Tangents and Normals Exercise 15.2 Question 3 Sub Question 9

Equation of normal ,

HINTS:

Differentiating the given curve and find its slope.

Equation of tangent is,

Equation of Normal is,

Tangents and Normals Exercise 15.2 Question 3 Sub Question 10

Equation of normal ,

HINTS:

Differentiating the given curve with respect to x and find its slope.

Slope of tangent,

Given,

Equation of tangent is,

Equation of Normal is,

Tangents and Normals Exercise 15.2 Question 3 Sub Question 11

Equation of normal ,

HINTS:

Differentiating the given curve with respect to x and find its slope.

Slope of tangent,

Equation of tangent is,

Equation of Normal is,

Tangents and Normals Exercise 15.2 Question 3 Sub Question 12

ANSWER: Equation of tangent,Equation of normal,

HINTS:

Differentiating the given equation

GIVEN:

.....(1)

SOLUTION:

Since lies on the curve(i)

.......(ii)

Equation of tangent at is

Equation of Normal at is,

Tangents and Normals Exercise 15.2 Question 3 Sub Question 13

Equation of normal ,

HINTS:

Differentiating the given curve with respect to x and find its slope.

Since lies on the curve(i)

Slope of tangent,

Equation of tangent is

Again, slope of normal,

Equation of Normal is ,

Tangents and Normals Exercise 15.2 Question 3 Sub Question 14

Equation of normal ,

HINTS:

Differentiating the given curve with respect to x and find its slope first.

Slope of tangent,

Equation of tangent is

Equation of Normal is ,

Tangents and Normals Exercise 15.2 Question 3 Sub Question 15

Equation of normal ,

HINTS:

Differentiating the given curve with respect to x and find its slope .

Slope of tangent,

Equation of tangent is

Equation of Normal is ,

Tangents and Normals Exercise 15.2 Question 3 Sub Question 16

Equation of normal ,

HINTS:

Differentiating the given curve with respect to x .

**SOLUTION:**

Slope of tangent =

Slope of normal =

Equation of tangent is

Equation of Normal is ,

Tangents and Normals Exercise 15.2 Question 3 Sub Question 17

Equation of normal ,

HINTS:

Differentiating the given curve with respect to x

Slope of tangent ,

Equation of tangent is,

Equation of Normal is,

Tangents and Normals Exercise 15.2 Question 3 Sub Question 18

Equation of normal ,

HINTS:

Differentiating with respect to x and find its slope.

**SOLUTION:**

Equation of tangent is

Equation of Normal is ,

Tangents and Normals Exercise 15.2 Question 4

HINTS:

To find equation, first find slope and one point on the tangent.

To find two slope, we find

Thus

Solving above

Thus

Slope,

Now apply point slope form to get equation of tangent,

Tangents and Normals Exercise 15.2 Question 5 Sub Question 1

Equation of normal ,

HINTS:

Differentiate the given equation with respect to and to get the slope of the tangent.

Given as

On differentiating ,

The normal is perpendicular to tangent, therefore ,

The equation of tangent is given by,

The equation of Normal is given by ,

Tangents and Normals Exercise 15.2 Question 5 Sub Question 2

Equation of normal ,

HINTS:

Differentiate the given equation with respect to t and to get the slopes.

Upon differentiation,

The normal is perpendicular to tangent, therefore ,

The equation of tangent is given by,

The equation of Normal is given by ,

Tangents and Normals Exercise 15.2 Question 6

Differentiating with respect to x to get its slope.

Upon differentiation

Finding the y coordinate by substitute x in the given curve

or

The normal is perpendicular to tangent, therefore ,

Which is Undefined

The equation of Normal is given by ,

when y=2

Slope of tangent =

Equation of normal is

In both cases the equation of normal is x=2

Tangents and Normals Exercise 15.2 Question 5 Sub Question 3

Answer:: Equation of tangent,Equation of normal ,

Hint:

Differentiating with respect to x to get its slope.

Upon differentiation

M(tangent) at t=1 is 1

The normal is perpendicular to tangent, therefore ,

The equation of tangent is given by,

The equation of Normal is given by ,

Tangents and Normals Exercise 15.2 Question 5 Sub Question 4

Equation of normal ,

Differentiating with respect to x to get its slope.

Upon differentiation

The normal is perpendicular to tangent, therefore ,

The equation of tangent is given by,

The equation of Normal is given by ,

Tangents and Normals Exercise 15.2 Question 5 Sub Question 6

Equation of normal ,

Differentiating with respect to to get its slope.

Upon differentiation

The normal is perpendicular to tangent, therefore ,

m(normal0 at is

The equation of tangent is given by,

The equation of Normal is given by ,

Tangents and Normals Exercise 15.2 Question 7

ANSWER: The equation of normalHINTS:

Differentiating with respect to x to get its slope of tangent

GIVEN:

SOLUTION:

Upon differentiation

m(tangent) at

The normal is perpendicular to tangent, therefore ,

m(tangent) at at

The equation of Normal is given by ,

Tangents and Normals Exercise 15.2 Question 8

Differentiating the curve to get its slope of tangent

Upon differentiation

The equation of tangent is given by

Comparing the slopes of a tangent with the given equation

(2,3) lies on the curve, these points must satisfy

Tangents and Normals Exercise 15.2 Question 9

Differentiate with respect to x to get its slope

Which is parallel to

Upon differentiation

The equation of tangent is given by

Comparing the slopes of a tangent with the given equation

Substitutes the value of x in the curve to find y

So,the equation of tangent is parallel to the given line is

Tangents and Normals Exercise 15.2 Question 10

ANSWER: Equation of normal:HINTS:

Differentiate the given curve, to get slope of tangent

GIVEN:

Which is parallel to

SOLUTION:

Upon differentiation

The normal is perpendicular to tangent, therefore ,

m(normal) at

The equation of Normal is given by ,

On comparing the slope of normal with the given equation

m(normal)

Thus ,the corresponding value of y is 18 or -6

Therefore ,the equation of normal are

Tangents and Normals Exercise 15.2 Question 11

ANSWER: Equation of tangent:HINTS:

Slope of tangent = slope of perpendicular line

GIVEN:

Which is perpendicular to

SOLUTION:

Let be a point on the curve which are used to find the tangents

Slope of the given line =

Since tangent is perpendicular to the given line ,

Slope of the tangent=

Let be a point where the tangent is drawn to this curve

Since, the point lies on the curve

Hence,

Slope of the tangent =slope of the perpendicular line

Tangents and Normals Exercise 15.2 Question 12

First find the coordinates

Which is parallel to

Upon differentiation

Tangents and Normals Exercise 15.2 Question 13 Sub Question 1

Answer:Equation of tangent

Hint: Differentiate both with respect to x

Given:

parallel to

Solution:

The equation of the given curve is

upon differentiation

This is in the form

If a tangent is parallel to the line then the slope of the tangent is equal to the slope of the line. Therefore we have:

Thus the equation of the tangent passing through (2,7) is given by

Hence the equation of the tangent line to the given curve (which is parallel to the line ) is

(Ans)

Tangents and Normals Exercise 15.2 Question 13 Sub Question 2

Equation of tangent

perpendicular to

the equation of the line is

This is of the form

If a tangent is perpendicular to the line , then the slope of the tangent is

Now,

Thus, the equation of the tangent passing through [5/6,217/36] is given by

(Ans)

Tangents and Normals Exercise 15.2 Question 14

Slope of the curve is

Which has no real roots

Hence: there is no tangent to the given curve having slope 2.

Tangents and Normals Exercise 15.2 Question 16

Answer: Equation of the tangent isHint:

Given: Which is paralle to

Solution: The equation of the given curve is

The slope of the tangent to the given curve at any point (x,y) is given by

The Equation of the given line is

Now the tangent to the given curve is parallel to the line , if slope of tangent is equal to the slope of line

the equation of tangent at point is given by

Tangents and Normals Exercise 15.2 Question 17

Answer : Equation of the tangent isHint: Differentiate on both sides

Given: which is parallel to

Solution: Given equation of the curve is

....()

On differentiating on both sides

or

the slope of tangent

Given the equation of the line

or

or or

on putting in the Eq we get

so,the tanent is passing through point and it has slope 4

Hence the required equation of the tangent is

(Ans)

Tangents and Normals Exercise 15.2 Question 18

Answer: To Prove (R.H.S=L.H.S)Hint: Differentiate

Given:

Solution:

....(ii)

The equation of the tangent is

So the given line touched the given curve at the given point

Tangents and Normals Exercise 15.2 Question 19

Now,

Equation of the tangent is

or

Tangents and Normals Exercise 15.2 Question 20

Answer: and y =7 at points (2,7),(3,6)Hint: Differentiate both side with respect to x

Given :

Differentiating both sides w.r.t x, we get

For the points on the curve, where tangents are parallel to x-axis

The equation of tangent at (2,7) parallel to x-axis is

The equation of tangent at (2,6) parallel to x-axis is

or

Tangents and Normals Exercise 15.2 Question 21

Answer: The equation of tangentHint: Differentiate with respect to .

Given: , which passes through

Solution:

Diff w.r.t

The eq. of tangent at is

Tangent passes through the point _{}

_{}

_{}

_{}

the points are (2,2) and (2,-2)

The eq. of tangent at (2,2) is :

The eq.of tangent at (2,-2) is :

Tangents and Normals Exercise 15.2 Question 3 Sub Question 19

**Answer:** Equation of tangent,

Equation of normal, **Hint:** Differentiating the given curve with respect to x.**Given**: **Solution**:

Slope of tangent is

and of the normal is

The equation of tangent is

Equation of normal is,

The 15th chapter, Tangents, and Normals are not where students find the sums very easy to solve. Predominantly, the second exercise, ex 15.2, consists of sums that are tricky to solve. It includes concepts like finding the equation of the tangent, finding the normal of the tangents to specific points, equation of the normal to the curve, and equation of lines to the slope. There are around 45 questions in this exercise, including its subparts. Again, the RD Sharma Class 12 Chapter 15 Exercise 15.2 will lend a helping hand.

This second exercise might be a bit harder initially, but students can understand the concept and work it out with a bit of practice. RD Sharma Class 12 Solutions Tangents and Normals Ex 15.2 follow the NCERT pattern. This paves the way for the CBSE board students to use it. Apart from solutions for the textbook, the RD Sharma Class 12th Exercise 15.2 also contains numerous practice questions for the students to get exam-ready by practising.

If you are weak in understanding the concepts in the chapter Tangents and Normals, jump to the Class 12 RD Sharma Chapter 15 Exercise 15.2 Solution material to understand it better. The problems are solved in all possible methods making it easier for the students to choose the ones they feel are easy to work out. This gives them the confidence to face examinations, and eventually, they can find themselves crossing their benchmark scores.

And the main advantage is that you need not spend hundreds or thousands of rupees utilizing this best resource material. The RD Sharma Class 12 Solutions Tangents and Normals can be downloaded for free of cost from top educational websites like Career 360.

The RD Sharma books are authorized and are used by the teachers to prepare question papers too. The RD Sharma Class 12th Exercise 15.2 book's practice questions have been asked in the previous year's examinations and are expected to be asked in the current year. Hence, practising with the RD Sharma Class 12 Solutions Chapter 15 ex 15.2 will make you score full marks on any sums asked from this portion. Download your copy of RD Sharma solution books now.

- Chapter 1 - Relations
- Chapter 2 - Functions
- Chapter 3 - Inverse Trigonometric Functions
- Chapter 4 - Algebra of Matrices
- Chapter 5 - Determinants
- Chapter 6 - Adjoint and Inverse of a Matrix
- Chapter 7 - Solution of Simultaneous Linear Equations
- Chapter 8 - Continuity
- Chapter 9 - Differentiability
- Chapter 10 - Differentiation
- Chapter 11 - Higher Order Derivatives
- Chapter 12 - Derivative as a Rate Measurer
- Chapter 13 - Differentials, Errors and Approximations
- Chapter 14 - Mean Value Theorems
- Chapter 15 - Tangents and Normals
- Chapter 16 - Increasing and Decreasing Functions
- Chapter 17 - Maxima and Minima
- Chapter 18 - Indefinite Integrals
- Chapter 19 - Definite Integrals
- Chapter 20 - Areas of Bounded Regions
- Chapter 21 - Differential Equations
- Chapter 22 - Algebra of Vectors
- Chapter 23 - Scalar Or Dot Product
- Chapter 24 - Vector or Cross Product
- Chapter 25 - Scalar Triple Product
- Chapter 26 - Direction Cosines and Direction Ratios
- Chapter 27 - Straight Line in Space
- Chapter 28 - The Plane
- Chapter 29 - Linear programming
- Chapter 30- Probability
- Chapter 31 - Mean and Variance of a Random Variable

1. Which is the best guide for the students to refer to the doubts in the mathematics chapter 15 portions?

The students can utilize the RD Sharma Class 12th Exercise 15.2 books to clarify their doubts regarding the portions given in the mathematics chapter 15 portion.

2. Can everyone access the RD Sharma solution books?

Yes, anyone can access the RD Sharma solution books and download them to their device. There are no restrictions to this.

3. Is it enough if I work out the sums for chapter 15 that are provided in the RD Sharma book?

The RD Sharma solution book consists of solutions to every question given in the textbook and various other practice questions. Practising the sums in RD Sharma Class 12th Exercise 15.2 is enough for a student to achieve good marks in this chapter.

4. Are there any possibilities of finding the RD Sharma books for free?

The RD Sharma books are available for free of cost at the Career360 website. So, all you need to do is, visit the website, search for the book, and download it.

Mar 22, 2023

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