##### Tallentex 2025 - ALLEN's Talent Encouragement Exam

ApplyRegister for Tallentex '25 - One of The Biggest Talent Encouragement Exam

Edited By Kuldeep Maurya | Updated on Jan 21, 2022 01:19 PM IST

Even though a teacher guides the students by teaching the concepts, there arises doubt when they try to solve an answer or a sum. This led to the need for high-quality solution books like the RD Sharma books. Especially while working out sums in maths, it is recommended to have a good set of solution books. RD Sharma solutions The widely used solution book for the students to understand the concepts is the RD Sharma Class 12th exercise 15.3 material.** **

**JEE Main 2025: Sample Papers | Mock Tests | PYQs | Study Plan 100 Days**

**JEE Main 2025: Maths Formulas | Study Materials**

**JEE Main 2025: Syllabus | Preparation Guide | High Scoring Topics **

Chapter 15 - Tangents and Normals Ex 15.1

Chapter 15 - Tangents and Normals Ex 15.2

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 15.3 , question 1 sub question 1

Hint – The angle of intersection of curves is

m

First curve is y

Differentiating above with respect to x,

As we know,

Second curve is

Differentiating above with respect to x,

Substituting (1) in (2),we get

Substituting y=0 or y=1 in (1)

x=y

When, y = 0, x = 0

y = 1, x = 1

Substituting the values of (y = 0, x = 0),(y = 1 , x = 1) for m

When, y = 0

When, y = 1

Value of m

When x = 0

When x = 1

Values of m

As we know, Angle of intersection of two curves is given by

When m

As we know

When and

Tangents and Normals exercise 15.3 , question 1 sub question 2

Hint – The angle of intersection of curves is

m

Given-

First curve is

Differentiating above with respect to x,

As we know,

Second curve is

Differentiating above with respect to x,

Substituting (1) in (2),we get

By factorization method

=y=-5 and y=4

Substituting y=-5 and y=4 in (1)

When, y = -5

This is not possible

When y=4

Substituting the values for m

When, x=2

When x=-2

Value of m

When y=4 & x=2

When y=4 & x=-2

Values of m

As we know, Angle of intersection of two curves is given by

When m

Then,

When and

Tangents and Normals exercise 15.3 , question 1 sub question 3

Hint – The angle of intersection of curves is

m

Given-

First curve is

Differentiating above with respect to x,

As we know,

Second curve is

Differentiating above with respect to x,

Substituting (1) in (2),we get

Substituting x=0 or in (2)

This is not possible

When x=0

When x=8

Substituting the values for m

When, x=0

When x=8,y=16

Value of m

When x=0 & y=0

When y=16

Values of m

As we know, Angle of intersection of two curves is given by

When m

As we know

When and

Tangents and Normals exercise 15.3 , question 1 sub question 4

Answer:Hint – The angle of intersection of curves is

m

Given – &

First curve is

Subtracting (2) from (1) we get ,

Substituting y=2x-4 in (3), we get

so When x =3

when x=1

The point of intersection of two curves are (3, 2) & (1,-2).

Differentiating curves (1) & (2) with respect to x,

Second curve is

At (3, 2) in eq (4), we get

At (3, 2) in eq (5), we get

At (1,-2) in eq (4), we get

Value of

Value of

Angle of intersection of two curves is given by

When m

When m

Tangents and Normals exercise 15.3 , question 1 sub question 5

Hint - The angle of intersection of curves is

m

Given –

Considering second curve

Substituting this in eq (1)

Since,

Since curves are

Differentiating above with respect to x,

Second curve is

Substituting (3) in (4) above values for m

At in eq (6), we get

When and

Angle of intersection of two curves is given by

Tangents and Normals exercise 15.3 , question 1 sub question 6

Hint - The angle of intersection of curves is

m

Given –

Considering second curve

Substituting this in eq (1)

Now putting the value of y in

When y = 1

Point of intersection are (2,1) and (-2,-1)

Since curves are

Differentiating above with respect to x

When (2,1)

When (-2,-1)

Angle of intersection of two curves is given by

When m

Tangents and Normals exercise 15.3 question 1 sub question 7

Hint - The angle of intersection of curves is

m

Given –

Considering second curve

Substituting this in eq (1)

Now putting the value of y=0 & 12 in

When y = 0

When y = 12

Point of intersection are (0, 0) and (18, 12)

Since curves are

Differentiating above with respect to x

Now consider

When (0,0)

When (18, 12)

Value of m

Angle of intersection of two curves is given by

When m

Tangents and Normals exercise 15.3 question 1 sub question 8

Hint - The angle of intersection of curves is

m

Given –

Considering second curve

Substituting this in eq (1)

Now putting the value of x = 0 & 1 in

When x = 0

When x = 1

Point of intersection are (0, 0) and (1, 1)

Since curves are

Differentiating above with respect to x

As we know,

Now consider

When (0, 0)

When (1, 1)

Angle of intersection of two curves is given by

When m

Tangents and Normals exercise 15.3 question 1 sub question 9

Hint - The angle of intersection of curves is

m

Given –

Substituting eq (2) in (1) we get

,

Put in eq (2), we get

When

When

Thus two curves intersect at and

Since curves are and

Differentiating above with respect to x

As we know,

Consider first curve

Now consider second curve

When

When

Angle of intersection of two curves is given by

Tangents and Normals exercise 15.3 question 2 sub question 1

Hint - Two curves intersects orthogonally if , where m

Given –

Substituting y= x

Since , we have to find f(x)=0, so that x is a factor of f(x).

When x = 1

Hence, x = 1 is a factor of f(x)

Substituting x = 1 in y= x

The point of intersection of two curves is (1, 1)

First curve is

Differentiating above with respect to x,

As we know,

Second curve is

Differentiating above with respect to x,

Now, put (1,1) in m

When m

Two curves intersects orthogonally if

Hence, two curves & intersect orthogonally.

Tangents and Normals exercise 15.3 question 2 sub question 2

Hence, two curves intersect orthogonally.

Hint - Two curves intersect orthogonally if , where m

Given –

Adding (1) & (2), we get

Substituting x=y in eq(1), we get

Since x=y (y=1)

The point of intersection of two curves is (1, 1)

First curve is

Differentiating above with respect to x,

As we know,

Second curve is

Differentiating above with respect to x,

When and

Two curves intersects orthogonally if

Two curves intersect orthogonally.

Tangents and Normals exercise 15.3 question 2 sub question 3

Hence, two curves intersect orthogonally.

Hint - Two curves intersects orthogonally if , where m

Given –

Solving (1) & (2),we get

From (2) curve,

Substituting in (1)

Substituting in , we get

The point of intersection of two curves is

First curve is

Differentiating above with respect to x,

As we know,

Second curve is

Differentiating above with respect to x,

At in eq (3), we get

At in eq (4), we get

When and

Two curves intersects orthogonally if

Two curves intersect orthogonally.

Tangents and Normals exercise 15.3 question 3 sub question 1

Hence, two

curves intersect orthogonally.

Hint - Two curves intersects orthogonally if , where m

Given –

The point of intersection of two curve (2,1).

First curve is

Differentiating above with respect to x,

As we know,

Second curve is

Differentiating above with respect to x,

Substituting (2, 1) for m

When m

Two curves intersects orthogonally if

Hence, two curves intersect orthogonally.

Tangents and Normals exercise 15.3 question 3 sub question 2

Hence, two

curves intersect orthogonally.

Hint - Two curves intersects orthogonally if , where m

Given –

The point of intersection of two curve (1,1).

First curve is

Differentiating above with respect to x,

As we know,

Second curve is

Differentiating above with respect to x,

Substituting (1, 1) for m

When m

Two curves intersects orthogonally if

Hence, two curves intersect orthogonally.

Tangents and Normals exercise 15.3 question 3 sub question 3

Hence, two

curves intersect orthogonally.

Hint - Two curves intersects orthogonally if , where m

Given –

The point of intersection of two curve .

First curve is

Differentiating above with respect to x,

As we know,

Second curve is

Differentiating above with respect to x,

Substituting for m

When and

Two curves intersects orthogonally if

Hence, two curves intersect orthogonally.

Tangents and Normals exercise 15.3 question 4

Hint - Two curves intersect orthogonally if , where m

Given –

Prove -

Two curves cut at right angle ,if

Consider First curve is

Differentiating above with respect to x,

As we know,

Second curve is

Differentiating above with respect to x,

Two curves intersects orthogonally if

Since m

Now solving (1) & (2), we get

&

Substituting in ,we get

Substituting in we get

and put x=2

, cube both sides

Tangents and Normals exercise 15.3 question 5

Hint - Two curves intersect orthogonally if , where m1 and m2 are the slopes of two curves.

Given –

Prove -

Two curves cut at right angle ,if

Consider First curve is

Differentiating above with respect to x,

As we know,

Second curve is

Differentiating above with respect to x,

Two curves intersects orthogonally if

Since m1& m2 cuts orthogonally

Now solving (1) & (2), we get

&

Substituting in ,we get

Substituting in and x=1 we get

, cube both sides

Hence, prove two curves cut at right angles if

Tangents and Normals exercise 15.3 question 6

Hint - Two curves intersect orthogonally if , where m1 and m2 are the slopes of two curves.

Given –

Prove -

Consider First curve is

Substituting in eq (2), we get

Now substituting value of in eq

When y = 2

When y = -2

Thus, two curves intersect at (2, 2) and (-2,-2)

Consider first curve xy = 4

Differentiating above with respect to y,

As we know,

Second curve is

Differentiating above with respect to y,

At (2,2) in eq (3) , we get

At (2, 2) in eq (4) , we get

Clearly, at

At (-2,-2) in eq (3) , we get

At (-2,-2) in eq (4) , we get

Clearly, at

So, given two curves touch each other at (2,2).

Simillarly, it can be seen that two curves touch each other at (-2, -2)

Tangents and Normals exercise 15.3 question 8 sub question 1

Hint –

Two curves intersects orthogonally if , where and are the slopes of two curves.

Given –

Consider First curve is

Differentiating above with respect to x,

As we know,

Second curve is

Differentiating above with respect to x,

Two curves intersects orthogonally if & =-1

Since & cuts orthogonally

Tangents and Normals exercise 15.3 question 8 subquestion 2

Two curves intersects orthogonally if , where m

Given –

Consider First curve

Differentiating above with respect to x,

As we know,

Consider second curve

Differentiating above with respect to x,

Two curves intersects orthogonally if

Now, subtract eq (1) & (2), we get

Put value of in eq (5), we get

Tangents and Normals exercise 15.3 question 9

Hint –

Two curves intersects orthogonally if , where m

Given –

Consider First curve is

Differentiating above with respect to x,

As we know,

Second curve is

Differentiating above with respect to x,

Now, subtract eq (2) from (1), we get

Two curves intersect orthogonally if

From (3) & (4)

Now putting the value of from eq (5) in eq (6)

Then,

So, the curves intersect at right angles.

Tangents and Normals exercise 15.3 question 10

Hint –

The equation of tangent at

Given –

And the curve

Or

Let line and curve touches each other at point

And

Differentiating (ii), we get

As we know,

Also, slope of line (i) is =

According to question,

and

Putting these values in (iv), we get

The 15th chapter, Tangents, and Normals in class 12 mathematics, have three exercises. The last exercise, 5.3, consists of concepts like the angle of intersection, curves intersecting orthogonally, proving that the curves touch each other, condition of the set of curves, and curves intersecting at right angles. There are around 23 questions along with their subparts given in this exercise. With the help of RD Sharma Class 12 Chapter 15 exercise 15.3 solution book, the students can easily solve the sums.

Many experts have contributed their knowledge to prepare this RD Sharma Class 12th exercise 15.3 set of solution books for the welfare of the students. The students will be clear about the Tangent and Normals concept once they practice using the RD Sharma Class 12th exercise 15.3 material. They can use this resource material to do their homework, complete assignments, and prepare for the tests and exams. Practising the Tangents and Normals daily can help them gain more knowledge and clarity in working out sums in this chapter.

If you find yourself scoring low marks in the chapter Tangents and Normals, use the solutions given for the Class 12 RD Sharma Chapter 15 Exercise 15.3 Solution and understand the concepts. Once you know the concepts, work out the practice questions given in the solution book to increase your efficiency and speed.

With so many that this book offers, you might presume that it would cost a lot. But that is not the fact. The RD Sharma Class 12 Solutions Tangents and Normals is available for free on top educational websites like Career 360. Directly visit the Career 360 website, type the name of the book and download it.

The class 12 public exams questions have also been taken from the practice question section given in the RD Sharma solution books. Hence, preparing for the tests and exams using the RD Sharma Class 12 Solutions Chapter 15 ex 15.3 provides more confidence to face the board exams. Therefore, it is better to practice by working out sums every day from this book. Download your RD Sharma Class 12th exercise 15.1 solution books and start preparing from today to reach your goal.

- 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. What is the prescribed solution book for the students to clarify their doubts regarding the RD Sharma Class 12 Solutions Tangents and Normals Ex 15.3?

The students can use the RD Sharma Class 12th exercise 15.3 solution book to clear their doubts regarding the sums in this chapter.

2. Where can I download the RD Sharma Solution books from?

The RD Sharma solution books can be downloaded from the Career 360 website. All the answers are present in an order-wise manner as given in the textbook.

3. Are the methods used in the RD Sharma books challenging to solve?

The RD Sharma solution books contain the sums solved in various methods. Therefore, the students can decide the way that they wish to adapt.

4. How much should one pay to own the RD Sharma Solution book?

Anyone can download the RD Sharma solution books for free of cost from the Career 360 website. Not even a single rupee is charged for the people who use this resource material.

5. How many questions are answered in the RD Sharma solution book for chapter 15, exercise 15.3?

All the 23 questions given in the textbook in exercise 15.3 are answered in the RD Sharma Class 12th exercise 15.3 book.

Sep 09, 2024

Get answers from students and experts

Register for Tallentex '25 - One of The Biggest Talent Encouragement Exam

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

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

Accepted by more than 11,000 universities in over 150 countries worldwide

Register now for PTE & Unlock 10% OFF : Use promo code: 'C360SPL10'. Limited Period Offer!

As per latest 2024 syllabus. Study 40% syllabus and score upto 100% marks in JEE

News and Notifications

Back to top