NCERT Solutions for Class 10 Maths Chapter 4 Quadratic Equations
NCERT solutions for class 10 maths chapter 4 Quadratic Equations:- In the previous class you have studied polynomials. A polynomial having degree two is called a quadratic equation. NCERT solutions for class 10 maths chapter 4 Quadratic Equations can be a good assistant for you while preparing this chapter. When we equate the quadratic polynomial to zero, then we get a quadratic equation. That is an equation of the form ax^{2 }+bx+c=0 with 'a' not equal to zero is known as quadratic equations. There are many real-life problems where we can form quadratic equations to solve it. NCERT solutions for class 10 maths chapter 4 Quadratic Equations introduces the concepts of quadratic equations and different methods to solve it. NCERT solutions for class 10 maths chapter 4 Quadratic Equations is providing an in-depth and step by step solution to each question. For example, Ramu has a rectangular plot. He forgot the length and breadth of the plot. But he remembers that area of the plot is 400 m^{2 } and length is 9m more than the breadth. What should be the length and breadth of the plot? To solve this question let's consider the length to be x then the breadth will be x-9 and the area is (x-9)x=400. This is how a word problem is converted to a quadratic equation. Here you will get NCERT Solutions from class 6 to 12 for science and maths explained in detailed manner. Here you will get NCERT solutions for class 10 also.
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Types of questions asked from class 10 maths chapter 4 Quadratic Equations
- Representation of situation in a quadratic equation.
- Checking of an equation is quadratic or not.
- Solving a quadratic equation using factorization/ roots of quadratic solutions.
- The solution of the quadratic equation using completing the square method.
- Solving a quadratic equation using the Sridharacharya formula.
- Product of roots in a quadratic equation.
- Sum of roots in a quadratic equation.
NCERT solutions for class 10 maths chapter 4 Quadratic Equations Excercise: 4.1
Q1 (i) Check whether the following are quadratic equations :
Answer:
We have L.H.S.
Therefore, can be written as:
i.e.,
Or
This equation is of type: .
Hence, the given equation is a quadratic equation.
Q1 (ii) Check whether the following are quadratic equations :
Answer:
Given equation can be written as:
i.e.,
This equation is of type: .
Hence, the given equation is a quadratic equation.
Q1 (iii) Check whether the following are quadratic equations :
Answer:
L.H.S. can be written as:
and R.H.S can be written as:
i.e.,
The equation is of the type: .
Hence, the given equation is not a quadratic equation since a=0.
Q1 (iv) Check whether the following are quadratic equations :
Answer:
L.H.S. can be written as:
and R.H.S can be written as:
i.e.,
This equation is of type: .
Hence, the given equation is a quadratic equation.
Q1 (v) Check whether the following are quadratic equations :
Answer:
L.H.S. can be written as:
and R.H.S can be written as:
i.e.,
This equation is of type: .
Hence, the given equation is a quadratic equation.
Q1 (vi) Check whether the following are quadratic equations :
Answer:
L.H.S.
and R.H.S can be written as:
i.e.,
This equation is NOT of type: .
Here a=0, hence, the given equation is not a quadratic equation.
Q1 (vii) Check whether the following are quadratic equations :
Answer:
L.H.S. can be written as:
and R.H.S can be written as:
i.e.,
This equation is NOT of type: .
Hence, the given equation is not a quadratic equation.
Q1 (viii) Check whether the following are quadratic equations :
Answer:
L.H.S. ,
and R.H.S can be written as:
i.e.,
This equation is of type: .
Hence, the given equation is a quadratic equation.
Q2 (i) Represent the following situations in the form of quadratic equations : The area of a rectangular plot is . The length of the plot (in meters) is one more than twice its breadth. We need to find the length and breadth of the plot.
Answer:
Given the area of a rectangular plot is .
Let the breadth of the plot be .
Then, the length of the plot will be: .
Therefore the area will be:
which is equal to the given plot area .
Hence, the length and breadth of the plot will satisfy the equation
Q2 (ii) Represent the following situations in the form of quadratic equations : The product of two consecutive positive integers is 306. We need to find the integers.
Answer:
Given the product of two consecutive integers is
Let two consecutive integers be and .
Then, their product will be:
Or .
Hence, the two consecutive integers will satisfy this quadratic equation .
Q2 (iii) Represent the following situations in the form of quadratic equations: Rohan’s mother is 26 years older than him. The product of their ages (in years) 3 years from now will be 360. We would like to find Rohan’s present age.
Answer:
Let the age of Rohan be years.
Then his mother age will be: years.
After three years,
Rohan's age will be years and his mother age will be years.
Then according to question,
The product of their ages 3 years from now will be:
Or
Hence, the age of Rohan satisfies the quadratic equation .
Q2 (iv) Represent the following situations in the form of quadratic equations : A train travels a distance of 480 km at a uniform speed. If the speed had been 8 km/h less, then it would have taken 3 hours more to cover the same distance. We need to find the speed of the train.
Answer:
Let the speed of the train be km/h.
The distance to be covered by the train is .
The time taken will be
If the speed had been less, the time taken would be: .
Now, according to question
Dividing by 3 on both the side
Hence, the speed of the train satisfies the quadratic equation
NCERT solutions for class 10 maths chapter 4 Quadratic Equations Excercise: 4.2
Q1 (i) Find the roots of the following quadratic equations by factorization:
Answer:
Given the quadratic equation:
Factorization gives,
Hence, the roots of the given quadratic equation are .
Q1 (ii) Find the roots of the following quadratic equations by factorization:
Answer:
Given the quadratic equation:
Factorisation gives,
Hence, the roots of the given quadratic equation are
Q1 (iii) Find the roots of the following quadratic equations by factorization:
Answer:
Given the quadratic equation:
Factorization gives,
Hence, the roots of the given quadratic equation are
Q1 (iv) Find the roots of the following quadratic equations by factorization:
Answer:
Given the quadratic equation:
Solving the quadratic equations, we get
Factorization gives,
Hence, the roots of the given quadratic equation are
Q1 (v) Find the roots of the following quadratic equations by factorization:
Answer:
Given the quadratic equation:
Factorization gives,
Hence, the roots of the given quadratic equation are
.
Q2 Solve the problems given in Example 1. (i) (ii)
Answer:
From Example 1 we get:
Equations:
(i)
Solving by factorization method:
Given the quadratic equation:
Factorization gives,
Hence, the roots of the given quadratic equation are .
Therefore, John and Jivanti have 36 and 9 marbles respectively in the beginning.
(ii)
Solving by factorization method:
Given the quadratic equation:
Factorization gives,
Hence, the roots of the given quadratic equation are .
Therefore, the number of toys on that day was
Q3 Find two numbers whose sum is 27 and the product is 182.
Answer:
Let two numbers be x and y .
Then, their sum will be equal to 27 and the product equals 182.
...............................(1)
.................................(2)
From equation (2) we have:
Then putting the value of y in equation (1), we get
Solving this equation:
Hence, the two required numbers are .
Q4 Find two consecutive positive integers, the sum of whose squares is 365.
Answer:
Let the two consecutive integers be
Then the sum of the squares is 365.
.
Hence, the two consecutive integers are .
Answer:
Let the length of the base of the triangle be .
Then, the altitude length will be: .
Given if hypotenuse is .
Applying the Pythagoras theorem; we get
So,
Or
But, the length of the base cannot be negative.
Hence the base length will be .
Therefore, we have
Altitude length and Base length
Answer:
Let the number of articles produced in a day
The cost of production of each article will be
Given the total production on that day was .
Hence we have the equation;
But, x cannot be negative as it is the number of articles.
Therefore, and the cost of each article
Hence, the number of articles is 6 and the cost of each article is Rs.15.
NCERT solutions for class 10 maths chapter 4 Quadratic Equations Excercise: 4.3
Q1 (i) Find the roots of the following quadratic equations, if they exist, by the method of completing the square
Answer:
Given equation:
On dividing both sides of the equation by 2, we obtain
Q1 (ii) Find the roots of the following quadratic equations, if they exist, by the method of completing the square
Answer:
Given equation:
On dividing both sides of the equation by 2, we obtain
Adding and subtracting in the equation, we get
Q1 (iii) Find the roots of the following quadratic equations, if they exist, by the method of completing the square
Answer:
Given equation:
On dividing both sides of the equation by 4, we obtain
Adding and subtracting in the equation, we get
Hence there are the same roots and equal:
Q2 (iv) Find the roots of the following quadratic equations, if they exist, by the method of completing the square
Answer:
Given equation:
On dividing both sides of the equation by 2, we obtain
Adding and subtracting in the equation, we get
Here the real roots do not exist (in the higher studies we will study how to find the root of such equations).
Q2 Find the roots of the quadratic equations given in Q.1 above by applying the quadratic formula.
Answer:
(i)
The general form of a quadratic equation is : , where a, b, and c are arbitrary constants.
Hence on comparing the given equation with the general form, we get
And the quadratic formula for finding the roots is:
Substituting the values in the quadratic formula, we obtain
Therefore, the real roots are:
(ii)
The general form of a quadratic equation is : , where a, b, and c are arbitrary constants.
Hence on comparing the given equation with the general form, we get
And the quadratic formula for finding the roots is:
Substituting the values in the quadratic formula, we obtain
Therefore, the real roots are:
(iii)
The general form of a quadratic equation is : , where a, b, and c are arbitrary constants.
Hence on comparing the given equation with the general form, we get
And the quadratic formula for finding the roots is:
Substituting the values in the quadratic formula, we obtain
Therefore, the real roots are:
(iv)
The general form of a quadratic equation is : , where a, b, and c are arbitrary constants.
Hence on comparing the given equation with the general form, we get
And the quadratic formula for finding the roots is:
Substituting the values in the quadratic formula, we obtain
Here the term inside the root is negative
Therefore there are no real roots for the given equation.
Q3 (i) Find the roots of the following equations:
Answer:
Given equation:
So, simplifying it,
Comparing with the general form of the quadratic equation: , we get
Now, applying the quadratic formula to find the roots:
Therefore, the roots are
Q3 (ii) Find the roots of the following equations:
Answer:
Given equation:
So, simplifying it,
or
Can be written as:
Hence the roots of the given equation are:
Answer:
Let the present age of Rehman be years.
Then, 3 years ago, his age was years.
and 5 years later, his age will be years.
Then according to the question we have,
Simplifying it to get the quadratic equation:
Hence the roots are:
However, age cannot be negative
Therefore, Rehman is 7 years old in the present.
Answer:
Let the marks obtained in Mathematics be 'm' then, the marks obtain in English will be '30-m'.
Then according to the question:
Simplifying to get the quadratic equation:
Solving by the factorizing method:
We have two situations when,
The marks obtained in Mathematics is 12, then marks in English will be 30-12 = 18.
Or,
The marks obtained in Mathematics is 13, then marks in English will be 30-13 = 17.
Answer:
Let the shorter side of the rectangle be x m.
Then, the larger side of the rectangle wil be .
Diagonal of the rectangle:
It is given that the diagonal of the rectangle is 60m more than the shorter side.
Therefore,
Solving by the factorizing method:
Hence, the roots are:
But the side cannot be negative.
Hence the length of the shorter side will be: 90 m
and the length of the larger side will be
Answer:
Given the difference of squares of two numbers is 180.
Let the larger number be 'x' and the smaller number be 'y'.
Then, according to the question:
and
On solving these two equations:
Solving by the factorizing method:
As the negative value of x is not satisfied in the equation:
Hence, the larger number will be 18 and a smaller number can be found by,
putting x = 18, we obtain
.
Therefore, the numbers are or .
Answer:
Let the speed of the train be
Then, time taken to cover will be:
According to the question,
Making it a quadratic equation.
Now, solving by the factorizing method:
However, the speed cannot be negative hence,
The speed of the train is .
Answer:
Let the time taken by the smaller pipe to fill the tank be
Then, the time taken by the larger pipe will be: .
The fraction of the tank filled by a smaller pipe in 1 hour:
The fraction of the tank filled by the larger pipe in 1 hour.
Given that two water taps together can fill a tank in hours.
Therefore,
Making it a quadratic equation:
Hence the roots are
As time is taken cannot be negative:
Therefore, time is taken individually by the smaller pipe and the larger pipe will be and hours respectively.
Answer:
Let the average speed of the passenger train be .
Given the average speed of the express train
also given that the time taken by the express train to cover 132 km is 1 hour less than the passenger train to cover the same distance.
Therefore,
Can be written as quadratic form:
Roots are:
As the speed cannot be negative.
Therefore, the speed of the passenger train will be and
The speed of the express train will be .
Answer:
Let the sides of the squares be . (NOTE: length are in meters)
And the perimeters will be: respectively.
Areas respectively.
It is given that,
.................................(1)
.................................(2)
Solving both equations:
or putting in equation (1), we obtain
Solving by the factorizing method:
Here the roots are:
As the sides of a square cannot be negative.
Therefore, the sides of the squares are and .
NCERT solutions for class 10 maths chapter 4 Quadratic Equations Excercise: 4.4
Answer:
For a quadratic equation, the value of discriminant determines the nature of roots and is equal to:
If D>0 then roots are distinct and real.
If D<0 then no real roots.
If D= 0 then there exists two equal real roots.
Given the quadratic equation, .
Comparing with general to get the values of a,b,c.
Finding the discriminant:
Here D is negative hence there are no real roots possible for the given equation.
Q1 (ii) Find the nature of the roots of the following quadratic equations. If the real roots exist, find them:
Answer:
Here the value of discriminant =0, which implies that roots exist and the roots are equal.
The roots are given by the formula
So the roots are
Answer:
The value of the discriminant
The discriminant > 0. Therefore the given quadratic equation has two distinct real root
roots are
So the roots are
Answer:
For two equal roots for the quadratic equation:
The value of the discriminant .
Given equation:
Comparing and getting the values of a,b, and, c.
The value of
Or,
Q2 (ii) Find the values of k for each of the following quadratic equations so that they have two equal roots
Answer:
For two equal roots for the quadratic equation:
The value of the discriminant .
Given equation:
Can be written as:
Comparing and getting the values of a,b, and, c.
The value of
But is NOT possible because it will not satisfy the given equation.
Hence the only value of is 6 to get two equal roots.
Answer:
Let the breadth of mango grove be .
Then the length of mango grove will be .
And the area will be:
Which will be equal to according to question.
Comparing to get the values of .
Finding the discriminant value:
Here,
Therefore, the equation will have real roots.
And hence finding the dimensions:
As negative value is not possible, hence the value of breadth of mango grove will be 20m.
And the length of mango grove will be:
Answer:
Let the age of one friend be
and the age of another friend will be:
4 years ago, their ages were, and .
According to the question, the product of their ages in years was 48.
or
Now, comparing to get the values of .
Discriminant value
As .
Therefore, there are no real roots possible for this given equation and hence,
This situation is NOT possible.
Answer:
Let us assume the length and breadth of the park be respectively.
Then, the perimeter will be
The area of the park is:
Given :
Comparing to get the values of a, b and c.
The value of the discriminant
As .
Therefore, this equation will have two equal roots.
And hence the roots will be:
Therefore, the length of the park,
and breadth of the park .
NCERT solutions for class 10 maths chapter wise
Chapter No. | Chapter Name |
Chapter 1 | |
Chapter 2 | |
Chapter 3 | NCERT solutions for class 10 maths chapter 3 Pair of Linear Equations in Two Variables |
Chapter 4 | NCERT solutions for class 10 maths chapter 4 Quadratic Equations |
Chapter 5 | NCERT solutions for class 10 chapter 5 Arithmetic Progressions |
Chapter 6 | |
Chapter 7 | NCERT solutions for class 10 maths chapter 7 Coordinate Geometry |
Chapter 8 | NCERT solutions for class 10 maths chapter 8 Introduction to Trigonometry |
Chapter 9 | NCERT solutions for class 10 maths chapter 9 Some Applications of Trigonometry |
Chapter 10 | |
Chapter 11 | |
Chapter 12 | NCERT solutions for class 10 chapter maths chapter 12 Areas Related to Circles |
Chapter 13 | NCERT solutions class 10 maths chapter 13 Surface Areas and Volumes |
Chapter 14 | |
Chapter 15 |
NCERT solutions of class 10 subject wise
How to use NCERT solutions for class 10 maths chapter 4 Quadratic Equations?
First of all list down all the questions in which you need assistance and go through the NCERT solution of that particular question.
When you complete the first step then your next target should be previous papers. You can pick past year papers and practice them thoroughly.
Once you complete NCERTs and previous year papers, try to solve the questions of that particular chapter from different state board books.
Keep working hard & happy learning!
Frequently Asked Question (FAQs) - NCERT Solutions for Class 10 Maths Chapter 4 Quadratic Equations
Question: What is the weightage of the chapter quadratic equations for CBSE board exam
Answer:
CBSE doesn't provides the marks distributions chapter-wise but it provides the total weightage of a unit( upto 4-5 chapters). As per CBSE the total weightage of algebra ( 4 chapters) is 20 marks in the final board exam.
Question: Where can I find the complete solutions of NCERT class 10 maths
Answer:
Here you will get the detailed NCERT solutions for class 10 maths by clicking on the link.
Question: What are the important topics of NCERT class 10 maths chapter 4 quadratic equations ?
Answer:
Representation of statement in a quadratic equation, solving a quadratic equation using different methods, solving a quadratic equation using the "Sridharacharya" formula, sum and product of roots in a quadratic equation are important topics in this chapter.
Question: How many chapters are there in the class 10 maths ?
Answer:
There are 15 chapters in the class 10 maths NCERT. Chapter 1- Real Numbers, Chapter 2- Polynomials, Chapter 3- Pair of Linear Equations in Two Variables, Chapter 4- Quadratic Equations, Chapter 5- Arithmetic Progressions, Chapter 6- Triangles, Chapter 7- Coordinate Geometry, Chapter 8- Introduction to Trigonometry, Chapter 9- Some Applications of Trigonometry, Chapter 10- Circles, Chapter 11- Constructions, Chapter 12- Areas Related to Circles, Chapter 13- Surface Areas and Volumes, Chapter 14- Statistics, Chapter 15- Probability are the chapters in the NCERT class 10 maths.
Question: Which is the official website of NCERT ?
Answer:
http://ncert.nic.in/ is the official website of the NCERT where you can get NCERT textbooks from class 1 to 12 and syllabus from class 1 to 12 for all the subjects.
Question: What are the benefits of NCERT solutions ?
Answer:
Benefits of NCERT Solutions -
- It will help you to get the conceptual clarity about the subject.
- You will get to know, how best to write in board exam.
- These solutions are provided in very simple language, so it will very easy for you to understand the concepts.
- These NCERT solutions for class 10 will help you to score good marks in board exams.
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