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Edited By Kuldeep Maurya | Updated on Jan 21, 2022 01:20 PM IST

The RD Sharma solutions are the most prescribed solution books by the CBSE schools to their students. Students would not face challenges in doing their mathematics sums when they possess this set of books. Especially when it comes to chapter 15 MCQs, students tend to confuse the options. The RD Sharma Class 12th MCQ will lend a helping hand in those circumstances.

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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-FBQ

Chapter 15 -Tangents and Normals Ex-VSA

Tangents and Normals Exercise Multiple Choice Questions Question 1

Equation of normal: slop of normal

Equation of normal at

Is the requried

Equation of normals to curve at

Tangents and Normals Exercise Multiple Choice Questions Question 2

Therefore, slope of tangent is 0. this means that the normal is a vertical line of the

Since the point has x-coordinate , the equation of the line is

is requried

Tangents and Normals Exercise Multiple Choice Questions Question 3

T he given curve is

i.e.

Now the equation of the tangent at the point

Or

So, the slope of the tangent is

Therefore the slope of the normal=

So, the equation of the normal at the point is

I.e. is required

Tangents and Normals Exercise Multiple Choice Questions Question 4

Slope of the tangent at that point of the curve

Given curve is

Point

Point is requried.

Tangents and Normals Exercise Multiple Choice Questions Question 6

Differentiate both the side w.r.t x we get

Let be the required point

It is given that point lies on the curve

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

Slope of the tangent=

The tangent is perpendicular to the line

Slope of the tangent =-1/slope of the line=

Now,

Tangents and Normals Exercise Multiple Choice Questions Question 7

Slope of the tangent at that point of the curve

Given curve is

Keep then

Point

Point

Tangents and Normals Exercise Multiple Choice Question Question 9

**Hint:**

** **** **Use slop of the tangent **Given:**** **The curve and at **Solution:**** **

Slope of the tangent to curve

At is

Slope of the tangent to the curve

At is

Angle between tangents to the curve is

Or

Tangents and Normals Exercise Multiple Choice Question Question 10

Is parallel to

Given the equation of the line now differentiating both sides with respect to x, we get

Now applying the sum rule of differentiation an differentiation of constant =0,so we get

So, this is the slope of the given curve. We know the slope of the normal to the curve is

(1)

Now the given equation of the line

Differentiating w.r.t x we get

So, the slope of the line is

Now, as the normal to the curve is parallel to this line, hence the slope of the line should be equal to the slope of the normal to the given curve,

On substituting this value of the given equation of the curve, we get

When x=2 the equation of the curve becomes,

When x=-2, the equation of the curve becomes,

So, the points at which normal is parallel to the given line are

And required equation of the normal to the curve at is

Hence the equation of normal to the curve

Which is parallel to the line is

Tangents and Normals Exercise Multiple Choice Questions Question 15

Slope of the tangent at (1,1) to is

Slope of the tangent at (1,1) to is

Curve cuts through orthogonally

Tangents and Normals Exercise Multiple Choice Question Question 16

Clearly ,

(1)

Differentiate w.r.t, we get

Putting the point and m=1 in above curve

Putting the point in (1)

Tangents and Normals Exercise Multiple Choice Questions Question 17

(1)

Differentiate w.r.t t

Tangents and Normals Exercise Multiple Choice Question Question 18

(1)

Differentiating w.r.t x, we get

And (2)

Since equation (1) and equation (2) intersect orthogonally

Tangents and Normals Exercise Multiple Choice Questions Question 19

Differentiating w.r.t

Differentiating w.r.t

Slope of normal of the given curve

Equation of normal at

Tangents and Normals Exercise 15 point 1 Question 20

(1)

Differentiating w.r.t x, we get

Again (2)

Differentiating w.r.t x we get

Since (1) and (2) intersect orthogonally

Tangents and Normals Exercise Multiple Choice Question , Question 21

Answer : (b) is the correct optionHint :

Given :

Solution :

(1)

Differentiating (1) w.r.t x we get

And (2)

Since

Taking

Putting the values of x in (1) we get .

Point is

Tangents and Normals Exercise Multiple Choice Question , Question 22

Answer : (c) is the correct option.Hint :

Given :

Solution :

Differentiating (1) w.r.t x we get

And

Tangents and Normals Exercise Multiple Choice Question , Question 23

Answer : (c) is the correct optionHint :

Given :

Solution :

(1)

Differentiating (1) w.r.t x, we get

And (2)

Differentiating (2) w.r.t y, we get

Tangents and Normals Exercise Multiple Choice Question , Question 24

Answer : (c) is the correct optionHint : Obtain the slope

Given :

Solution :

(1)

Differentiating (1) w.r.t x

Here the slope cannot be zero. Hence it make an acute angle with x-axis

Tangents and Normals Exercise Multiple Choice Question , Question 25

Answer : (a) is the correct optionHint : Slope of the mormal is 1 or -1

Given :

Solution :

(1)

Differentiating (1) w.r.t x, we get

Putting the values of x in (1) we get

Similarly in case (2) if

The point

Tangents and Normals Exercise Multiple Choice Question , Question 26

Answer : (b) is the correct optionHint :

Solution :

(1)

(2)

(3)

Putting in (1)

Putting in (2)

Putiing in (3)

The slope of tangent is

Tangents and Normals Exercise Multiple Choice Question , Question 27

Answer : (a) is the correct optionHint :

Use

Given : The line is the tangent of the curve

Solution :

(1)

From equation (1) we equate with so that we get the value of , the y intercept of the parabola

Where m is slope

From (1) , the value of a is 1

The slope of the curve is 1

Hence the slope of the tangent is 1

Tangents and Normals Exercise Multiple Choice Question , Question 28

Answer : (b) is the correct optionHint :

Equation of normal

Given :

Solution :

(1)

Differentiating equation (1) w.r.t y

Equation of the normal at the point is

Tangents and Normals Exercise Multiple Choice Question , Question 29

Answer : (a) is the correct optionHint :

Slope

Given :

Solution :

(1)

Differentiating (1) w.r.t x

Slope of normal

Since the normal passes through

Putting the value of in (1) , was

Point of contact

Equation of normal

Tangents and Normals Exercise Multiple Choice Question , Question 30

Answer : (a) is the correct optionHint : Equation of the normal,

Given :

Solution :

Let be the point on the curve at which normal passes through origin

....(1)

Differentiate (1) w.r.t x, we get

Slope of the tangent at P is

Slope of normal at P is

Equation of normal at P is

Since it passes through the origin

.....(2)

On solving eqn 1 and 2

We get

As we know, abscissa is the x-coordinate

So, option (a) satisfies above equation i.e., 1

Tangents and Normals Exercise Multiple Choice Question, Question 31

Answer : (b) is the correct optionHint : Multiply the slope of both curve

Given :

Solution :

Given (1)

Differentiating (1) we get

And (2)

Differentiating (2), we get

Equation (1) and (2) at right angle

Tangents and Normals Exercise Multiple Choice Question, Question 32

Answer : (d) is the correct optionHint :

Given :

Solution :

(1)

Differentiating (1) w.r.t t, we get

(2)

Differentiating (2) w.r.t t, we get

Tangents and Normals Exercise Multiple Choice Question, Question 33

Answer : (b) is the correct optionHint : Equation of tangent

Given :

Solution :

(1)

Differentiating (1) w.r.t x, we get

Equation of the tangent at the point (0, 1) and having the slope 2 is

Since it (2) meets in x-axis

Putting in (2)

The tangent meets at the point

Tangents and Normals Exercise Multiple Choice Question, Question 34

Answer : (a) is the correct optionHint :

Equation of tangent

Given :

Solution :

(1)

Since it crosses x -axis

The point of contact

Differentiating (1) w.r.t x

Equation of tangent at

Tangents and Normals Exercise Multiple Choice Question, Question 35

Answer : (d) is the correct optionHint :

Given :

Solution :

(1)

Differentiating (1) w.r.t x, we get

Since it's parallel to x-axis

When

When

The points are and

Tangents and Normals Exercise Multiple Choice Question, Question 36

Answer : (a) is the correct optionHint :

Given :

Solution :

(1)

Differentiating (1) w.r.t x, we get

Has vertical tangent

Chapter 15, Tangents and Normals, is where many students lose marks due to a lack of clarity. This chapter consists of three exercises, ex 15.1 to ex 15.3. RD Sharma Class 12 Solutions Tangents and Normals MCQ contains questions from concepts like an equation to the normal curve, the point of the curve when the tangent is perpendicular, the angle between curves, the slope of the tangent, and the angle of intersection. There are around 36 MCQ questions present in this chapter. The RD Sharma Class 12 Chapter 15 MCQ will lend a helping hand.

Experts in the teaching field provide the given solutions in the RD Sharma Class 12th MCQ book. It follows the NCERT pattern making it easier for the CBSE school students to utilize. The sums are solved in shortcut methods that make the students understand how the answer has arrived. Hence, they can learn the concept easily.

If you are confused between the options given in the MCQ, try answering it and check with the Class 12 RD Sharma Chapter 15 MCQ solutions book. You can use it while doing homework, preparing assignments, and even while preparing for exams. With a bit of regular practice in MCQ questions, you will become familiar with it. Hence, you would not lose marks in MCQs during the exams, which eventually makes you cross the benchmark score.

The best part is that the RD Sharma Class 12 Solutions Tangents and Normals can be downloaded from the Career360 website for free of cost. Many students have benefitted by using these solution books to prepare for their examinations. The solutions are given in shortcut methods to find the answers for the MCQs, and it saves a lot of time during exams.

There are chances of questions being taken from the RD Sharma Class 12th MCQ book’s practice questions. Hence, RD Sharma books help a lot when a student prepares for their exam with it. So, download the RD Sharma Class 12 Solutions Chapter 15 MCQ from the Career360 website and start preparing for your exams. You can very well score good marks when you have the best guide to prepare for the challenging mathematics exams.

- 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 solution book can I use to clarify the doubts in the mathematics chapter 15 MCQ?

The best solution book that clears all your doubts in the mathematics chapter 15 multiple choice questions is the RD Sharma Class 12th MCQ book

2. Where can I find the RD Sharma solution books for free?

You can find the RD Sharma solutions book on the Career360 website for free of cost. It is accessible as well as can be downloaded for later use.

3. Why is the RD Sharma Class 12th MCQ book the most prescribed one for the students?

Most of the schools recommend the RD Sharma Class 12 Chapter 15 FBQ solution books due to the high quality of their answers. As experts provide these answers, the students can understand and can trust the accuracy of it.

4. Is it enough if I can solve the MCQs given in the RD Sharma books?

The RD Sharma has every question and answer as given in the Class 12 textbook. Furthermore, it also contains extra sums for practice. Hence, the students can become familiar with the concept if they use this book.

5. Can everyone access the RD Sharma solution books given on the Career 360 website?

The students utilize this resource material to clarify their doubts and the teachers use it to prepare question papers in most cases. Hence, the RD Sharma books can be accessible by anyone.

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