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The CBSE schools recommend the student use the RD Sharma solution books to prepare for their tests and exams. The students might face numerous doubts while finding the answers for the FBQs. The concept of Tangents and Normals make it even harder to find accurate answers. As the FBQ section is where students tend to lose marks, extra practice is required to avoid it. This is where the RD Sharma Class 12th FBQ comes to help.

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Chapter 15 - Tangents and Normals Ex 15.1

Chapter 15 - Tangents and Normals Ex 15.2

Chapter 15 - Tangents and Normals Ex 15.3

Chapter 15 -Tangents and Normals Ex-MCQ

Chapter 15 -Tangents and Normals Ex-VSA

Tangents and Normals exercise Fill in the blanks question 1

Equation of normal,

Use equation of normal formula,

Here given the curve ,

We have to find the equation of normal to the curve at

Here given,

Differentiating both side with respect to , we get

Slope of normal at

Now, equation of normal to the curve at is

Hence, this is the required equation of normal.

Tangents and Normals exercise Fill in the blanks question 2

Since the given curve touches axis. i.e.

… (i)

We have to find the value of for which the given curve touches axis.

We know that,

Since the given curve,

touches the axis

Differentiating equation (i) with respect to , we get

Putting the value of in equation (i), we get

Since

Hence, the required value of are

Tangents and Normals exercise Fill in the blanks question 3

Gradient is zero. i.e.

Given curve,

We have to find the points on the given curve at which the gradient is zero.

Here … (i)

Given that gradient is zero

i.e.

Differentiating equation (i), we get

Since

Putting the value in equation (i), we get

Hence the points are and

Tangents and Normals Exercise Fill in the blanks Question 4

- Slope of normal
- Slope of line is . i.e.

Given curve, and the line

We have to find the co-ordinates of a point on the given curve at which the normal is parallel to the line

We have,

…(i)

Differentiating both side with respect to , we get

The Slope of normal

The line,

Comparing this equation with the formula , we can get the slope .

Here the slope of line is

As the slope of parallel lines are equal,

Therefore,

Putting the value of in equation (i)

Therefore the coordinate of the point is

Tangents and Normals Exercise Fill in the blanks Question 5

First we find slope of curve then equate with given slope

Given curve, and tangent has slope

We have to find the coordinates of the point on the given curve where tangent has slope .

We have,

… (i)

Differentiating equation (i) on both side with respect to , we get

And tangent has slope

Squaring on both sides,

Substituting the value of in equation (i), we get

Hence the required coordinate is

Tangents and Normals Exercise Fill in the blanks Question 6

Slope of tangent =

First put in equation , then find and .

Given curve,

and

We have to find the slope of tangent to the given curve at

Given,

… (i)

… (ii)

Put in equation (i), we get

Differentiating equation (i) and (ii) with respect to , we get

Thus we get the slope of tangent

Tangents and Normals Exercise Fill in the blanks Question 7

First we need to find their point of intersection. So solve these given equations.

Given curves,

and

We have to find the angle of intersection of the given curves at

Here … (i)

… (ii)

From equation (i) and (ii), we get

Which gives

Therefore points of intersection of the curves are and

On differentiating equation (i) and (ii), we get

Similarly,

To find the angle of intersection at

So, slope of tangent to curve at

So tangent is parallel to axis.

= slope of tangent to curve at

which is not defined.

So tangent is parallel to axis.

Now, one tangent is parallel to axis and other is parallel to axis.

Hence angle between tangents is right angle. i.e.

Tangents and Normals Exercise Fill in the blanks Question 8

Use is the point where tangent to the curve crosses axis, then proceed next to find slope of tangent.

Given curve,

We have to find the slope of tangent where the curves crosses axis.

Given equation of curve is,

… (i)

Let be the point where tangent to curve crosses axis.

So the point is

Differentiating equation (i), we get

Hence slope of tangent at is .

Tangents and Normals Exercise Fill in the blanks Question 9

Point

- Using equation of tangent is
- Tangent line meets the axis. i.e.

Given equation of curve,

We have to find the point where curve at cut axis.

The given equation of curve,

… (i)

Differentiating equation (i), we get

Slope of tangent to the curve =2

So equation of tangent is

Above tangent line cuts the axis, where

Hence the required point is

Tangents and Normals Excercise Fill in the blanks Question 10

Slope of normal

Slope of normal

Given equation of curve,

We have to find the slope of normal to the given curve at the point

Given curve,

… (i)

On differentiating with respect to , we get

Hence the slope of the normal is

Tangents and Normals Excercise Fill in the blanks Question 11

First find the equation of normal then comparing with

The given equation of curve,

We have to find the value of

We have,

… (i)

Differentiating equation (i) with respect to , we get

Then the equation of normal at point is

… (ii)

As the normal is of the form … (iii)

Comparing equation (ii) and (iii), we get

and

Hence

Tangents and Normals Excercise Fill in the blanks Question 12

Should be negative. i.e.

First find the slope of normal to the given curve then compare with slope of

Given the equation of curve,

We have to find the set of if the line is normal to the given curve.

Given,

Differentiating with respect to , we get

Thus slope of normal

Which is always positive and it is given is normal

Slope =

So and are of opposite sign.

Hence

=> should be negative. i.e

Tangents and Normals Excercise Fill in the blanks Question 13

First find the slope of given curve at point , then compare with

Given curve,

We have to find , if the normal to the given curve at makes an angle with positive x-axis.

Here we have,

On differentiating with respect to , we get

Slope of tangent at

Therefore, slope of normal

But [Given]

Hence

Tangents and Normals Exercise Fill in the blanks Question 14

Equation of tangent is

Tangent is parallel to X-axis so slope becomes 0

Given curve

We have to find the equation of the tangent to the given curve that is parallel to X-axis

Given tangent is parallel to X-axis so slope is 0

… (i)

Here … (ii)

On differentiate both side with respect to we get

Substituting the value of in equation (i), we get

From (ii) we get

Hence equation of tangent is [ equation of tangent ]

Tangents and Normals Exercise Fill in the blanks Question 15

Point:

- First we will find the slope of line then evaluate with the slope of curve
- If tangent is perpendicular to the line then

Given curve where the tangent is perpendicular to the line

We have to find the co-ordinate of the point on the given curve.

We have,

… (i)

Comparing equation (i) with equation

Again, if the line is perpendicular to the tangent then

Given curve,

… (ii)

On differentiating both side with respect to , we get

Here slope of tangent is

Substituting the value of in equation (ii), we get

Hence the point is

Tangents and Normals Exercise Fill in the blanks Question 16

Points are and

If the slope of tangent is equal to ordinate of point that means

The slope of tangent to curve at a point is equal to ordinate of point.

We have to find the point

We have,

… (i)

On differentiating with respect to , we get

Now, we know that the given slope of tangent to the given curve at a point is equal to ordinate of point

… (ii)

Putting value of in equation (i), we get

and [From equation (i)]

Thus the two points are and

Tangents and Normals Excercise Fill in the blanks Question 17

Equation of normal,

Use equation of normal,

Given curve,

We have to find the slope of normal to the given curve at point

Given equation is

On differentiating both side with respect to , we get

[Taking common from each term]

Now, equation of normal is

Substituting these value, we get

i.e.

Hence, is required equation of normal.

Tangents and Normals Excercise Fill in the blanks Question 18

First find the slope of curve , then compare with slope with x-axis.

i.e. slope

Given curve, , the tangent at which makes an angle of with x-axis.

We have to find the point on the given curve.

We have

… (i)

Differentiate both side with respect to , we get

… (ii)

Also tangent makes an angle of with x-axis

… (iii)

From equation (ii) and (iii), we get

Put value of in equation

Hence required point is

Tangents and Normals Excercise Fill in the blanks Question 19

When both curves touch then slope of both curves should be same.

Given curves,

and touch each other.

We have to find the point where the given curves touch each.

The curves, and

For first curve say

For second curve

When both curves touch the slope both curves should be same

Solving the quadratic equation, we get

Now consider,

For first curve,

For second curve,

…(here there should be 10 inplace of 13)

Thus at both curves touch

Now consider,

For first curve,

For second curve,

Thus at both curves do not meet.

But their tangent are parallel

Hence the only point where both curves touch is

Tangents and Normals Excercise Fill in the blanks Question 20

**Answer:**

is the required equation of normal**Hint:**

Equation of normal,**Given:**

Given curve,**To find:**

We have to find the equation of normal to the given curve at the origin.**Solution:**

We have,

On differentiating both side with respect to , we get

At origin i.e.

Here the equation of normal,

Hence the required equation of normal is .

Chapter 15, Tangents and Normals, has three exercises, ex 15.1, ex15.2, and ex 15.3. RD Sharma Class 12 Solutions Tangents and Normals FBQ include the equation of the normal to the curve, coordinates of a point, slope of a tangent, angle of intersection curves, and slope of normal to the curve. There are around 20 FBQ questions asked in the mathematics textbook. Students can utilize the RD Sharma Class 12 Chapter 15 FBQ book to find and verify their answers.

Experts provide accurate answers for these FBQs in the RD Sharma solution books with a broad knowledge of their respective domains. The NCERT pattern is strictly followed for the CBSE board students to adapt it quickly. Apart from the solutions for the textbook, the RD Sharma Class 12th FBQ also contains additional practice questions for the students to work out. This makes them increase their speed in finding answers during the examinations.

You will not find FBQs challenging to solve once you commence your practice with the Class 12 RD Sharma Chapter 15 FBQ resource material. The questions are solved in simple methods using shortcuts to find the answers quickly. This saves time during examinations. Moreover, once the students achieve full marks in FBQs, they can easily cross their benchmark scores.

The added benefit of using the RD Sharma Class 12 Solutions FBQ is that it is available for free of cost on the Career 360 website. Hence, you need not worry about its affordability. You can also download this resource material onto your device to refer to whenever necessary. As no kind of monetary charge is required, every student can use it easily.

Class 12 RD Sharma Chapter 15 FBQ Solution are widely used, and hence questions for the public exams are likely to be taken from this book. Preparing with the RD Sharma Class 12th FBQ book will prepare the students for their public exams. Many students of the previous set have been benefitted from using the RD Sharma Class 12 Solutions Chapter 15 FBQ book. Download your own set of RD Sharma books and start preparing for your exams from today.

- 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 prescribed book for the students to clarify their doubts regarding the mathematics chapter 15 FBQ?

The students can utilize the RD Sharma Class 12th FBQ book to clarify their doubts regarding the portions in the mathematics chapter 15.

2. Is it possible to find RD Sharma Class 12 Solutions Chapter 15 FBQ for free?

It is a boon that the students can now download the RD Sharma solution from the Career 360 website for free of cost. Hence, the students need not pay money to utilize this best resource material.

3. How can I get the RD Sharma solution book from the Career 360 website?

Visit the Career 360 website and type for the name of the solutions that you require. For example, if you search for RD Sharma Class 12th FBQ, you can download the solutions for the FBQs.

4. Are the solutions given in the RD Sharma books verified?

Experts in the teaching field prepare all the solutions provided in the RD Sharma books. Therefore, the answers are accurate and verified for the welfare of the students.

5. Do the RD Sharma books contain solutions only for the questions given in the exercises?

The RD Sharma books provide solutions for the exercise questions, MCQs, FBQs, and VSA portions.

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