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Class- X-CBSE-Mathematics Quardatic Equations

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CBSE NCERT Solutions for Class 10

Mathematics

Chapter 4 Back of Chapter Questions

1.C heck whether the following are quadratic equations: (i) (ݔ+ 1) = 2(ݔ െ3) (ii)ݔ െ2ݔ=(െ2)(3െ ݔ) (iii) (ݔ െ2)(ݔ+ 1)=(ݔ െ1)(

ݔ+ 3)

(iv) (ݔ െ3)(2ݔ+ 1)=ݔ(ݔ+ 5) (v) (2ݔ െ1)(ݔ െ3)=(ݔ+ 5)(ݔ െ1) + 3ݔ+ 1= (ݔ െ2) vii) (ݔ+ 2) = 2ݔ(ݔ (viii)ݔ െ4ݔ െ ݔ+ 1= (ݔ െ2) So lution: (i)W e know that any equation of the form ܽ

ܾ+T+ܿ

quadratic equation, where ܿ,ܾ,ܽ are real numbers and ܽ ven equation (ݔ+ 1) = 2(ݔ െ3) Us ing the formula + 2ܽ>+ܾ + 2ݔ+ 1= 2ݔ െ6 Her e, ܽ= 1,ܾ= 0 and ܿ Thus, the given equation is a quadratic equation as ܽ (i i)We know that any equation of the form ܽ

ܾ+T+ܿ

quadratic equation, where

ܿ,ܾ,ܽ are real numbers and ܽ

ven equation: െ2ݔ=(െ2)(3െ ݔ) െ2ݔ=െ6 +2 ݔ െ4ݔ+ 6= 0

Here, ܽ= 1,ܾ=െ4 and ܿ

Thus, the given equation is a quadratic equation as ܽ (iii)W e know that any equation of the form

ܾ+T+ܿ

quadratic equation, where

ܿ,ܾ,ܽ are real numbers and ܽ

Class- X-CBSE-Mathematics Quardatic Equations

Practice more on Quardatic Equations Page - 2 www.embibe.com Given equation: (ݔ െ2)(ݔ+ 1)=(ݔ െ1)(ݔ+ 3) െ ݔെ 2 =ݔ + 2ݔ െ3

But, here

So, the given equation is not a quadratic equation. iv) We know that any equation of the form ܽ

ܾ+T+ܿ

quadratic equation, where

ܿ,ܾ,ܽ are real numbers and ܽ

G iven equation: (ݔ െ3)(2ݔ+ 1)=ݔ(ݔ+ 5) െ5ݔ െ3 =ݔ + 5ݔ െ10ݔ െ3 =0 Here

ܽ, = 1,ܾ=െ10 and ܿ

Thus, the given equation is a quadratic equation as ܽ v) We know that any equation of the form ܽ

ܾ+T+ܿ

quadratic equation, where

ܿ,ܾ,ܽ are real numbers and ܽ

G iven equation: (2ݔ െ1)(ݔ െ3)=(ݔ+ 5)(ݔ െ1) െ7ݔ+ 3= ݔ + 4ݔ െ5 െ11ݔ+ 8= 0

Here, ܽ= 1,ܾ=െ11 and ܿ

Thus, the given equation is a quadratic equation as ܽ vi) We know that any equation of the form ܽ

ܾ+T+ܿ

quadratic equation, where

ܿ,ܾ,ܽ are real numbers and ܽ

G iven equation: + 3ݔ+ 1= (ݔ െ2) U sing the formula െ2ܽ>+ܾ + 3ݔ+ 1= ݔ െ4ݔ+ 4 B ut, here So, t he given equation is not a quadratic equation. vii) We know that any equation of the form ܽ

ܾ+T+ܿ

quadratic equation, where

ܿ,ܾ,ܽ are real numbers and ܽ

G iven equation: (ݔ+ 2) = 2ݔ(ݔ െ1)

Using the formula

+ 3ܽ

ܾ+ 3ܾܽ

Class- X-CBSE-Mathematics Quardatic Equations

Practice more on Quardatic Equations Page - 3 www.embibe.com + 8+ 6ݔ +12ݔ= 2ݔ െ2ݔ െ14ݔ െ6ݔ െ8 =0 T his equation is not of the form ܽ

ܾ+T+ܿ

So, t he given equation is not a quadratic equation. viii) We know that any equation of the form ܽ

ܾ+T+ܿ

quadratic equation, where

ܿ,ܾ,ܽ are real numbers and ܽ

G iven equation: െ4ݔ െ ݔ+ 1= (ݔ െ2)

Using the formula

െ3ܽ

ܾ+ 3ܾܽ

െ4ݔ െ ݔ+ 1= ݔ െ8െ6ݔ +12ݔ െ13ݔ+ 9= 0

Here, ܽ= 2,ܾ=െ13 and ܿ

Thus, the given equation is a quadratic equation as ܽ

2. Represent the following situations in the form of quadratic equations:

(i) The area of a rectangular plot is 528 m . The length of the plot (in metres) is one more than twice its breadth. We need to find the length and breadth of the plot. (ii) The product of two consecutive positive integers is 306. We need to find the integers. (iii) 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. (iv) A train travels a distance of 480 km at a uniform speed. If the speed had distance. We need to find the speed of the train.

Solution:

(i) Let the breadth of the plot be ݔ m.

Hence, the length of the plot is

(2ݔ+ 1) m. (Since, given that length is one more than twice its breadth) Therefore, area of a rectangle = length × breadth

Given: area of rectangle

=528 m 2ݔ +ݔ െ528= 0 .........(i), which is of the form ܽ

ܾ+T+ܿ

Class- X-CBSE-Mathematics Quardatic Equations

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Here ܽ= 2(്0),ܾ= 1 and ܿ

Thus, quadratic equation (i) represents the situation given in the question and roots of this equation will represent the breadth of the plot. (ii) We know that the difference between two consecutive positive integers is 1. So, let the consecutive positive integers be ݔ and ݔ+ 1.

Given that their product is 306.

(ݔ+ 1)=306 +ݔ െ306= 0 ........... (i), which is of the form ܽ

ܾ+T+ܿ

H ere ܽ= 1(്0),ܾ= 1 and ܿ Thus, quadratic equation (i) represents the situation given in the question and roots of this equation will represent the smaller positive integer. (iii) Let Rohan's age be ݔ, His mother's age =ݔ+26 (given that Rohan's mother is 26 years older than him)

3 years from now:

Rohan's age will be =ݔ+ 3

Mother's age will be =ݔ+26+ 3= ݔ+29

Also given that the product of their ages after 3 years is 360.

On simplification, we get

+32ݔ െ273= 0......... (i), which is of the form ܽ

ܾ+T+ܿ

H ere ܽ= 1(്0),ܾ=32 and ܿ Thus, quadratic equation (i) represents the situation given in the question and positive root of this equation will represent the Rohan's present age (iv) In first case,

Let the speed of train

be ݔ km/h.

Total time taken to travel 480 km=

hrs

In second

case

Given: speed became 8 km/h less

So, the speed of train =(ݔ െ8)km/h

Class- X-CBSE-Mathematics Quardatic Equations

Practice more on Quardatic Equations Page - 5 www.embibe.com Also given that the train will take 3 more hours to cover the same distance.

Therefore, time take to travel 480 km=ቀ

+ 3ቁhrs

Speed × Time = Distance

ݔ െ8)൬480

+ 3൰=480

F24=480

=24 െ24ݔ െ3840= 0 െ8ݔ െ1280= 0........ (i), which is of the form ܽ

ܾ+T+ܿ

H ere ܽ= 1(്0),ܾ=െ8 and ܿ Thus, quadratic equation (i) represents the situation given in the question and positive root of this equation will represent the speed of train.

EXERCISE 4.2

1. Find the roots of the following quadratic equations by factorisation:

(i) ݔ െ3ݔ െ10= 0 (ii) 2ݔ +ݔ െ6 =0 (iii) ξ2 ݔ + 7ݔ+ 5ξ2= 0 (iv) 2ݔ = 0 (v) 100ݔ െ20ݔ+ 1= 0

Solution:

(i) To find the roots of given quadratic equation, lets first factorise the given quadratic expression ݔ െ3ݔ െ10. The given quadratic expression can be written as follows: െ3ݔ െ10 െ5ݔ+ 2ݔ െ10 (we factorise by method of splitting the middle term) =ݔ(ݔ െ5)+ 2(ݔ െ5) (ݔ െ5)(ݔ+ 2)

Class- X-CBSE-Mathematics Quardatic Equations

Practice more on Quardatic Equations Page - 6 www.embibe.com Now, the roots of this quadratic equation are the values of ݔ for which

ݔ െ5)(ݔ+ 2)= 0

݅.݁.,ݔ= 5 or ݔ=െ2

Hence, the roots of this quadratic equation are 5 andെ2. (ii) To find the roots of given quadratic equation, lets first factorise the given quadratic expression 2ݔ +ݔ െ6 . The given quadratic expression can be written as follows: 2ݔ +ݔ െ6 = 2ݔ + 4ݔ െ3ݔ െ6 (we factorise by method of splitting the middle term) = 2ݔ(ݔ+ 2)െ3(ݔ+ 2) = (ݔ+ 2)(2ݔ െ3) Now, the roots of this quadratic equation are the values of

ݔ for which

ݔ+ 2)(2ݔ െ3)= 0

݅.݁.,ݔ=െ2 or ݔ=

Hence, the roots of this quadratic equation are

െ2 and (iii) To find the roots of given quadratic equation, lets first factorise the given quadratic expression ξ2 + 7ݔ+ 5ξ2 . The given quadratic expression can be written as follows:

ξ2ݔ

+ 7ݔ+ 5ξ2 =ξ2ݔ + 5ݔ+ 2ݔ+ 5ξ2 (we factorise by method of splitting the middle term) =ݔ൫ξ2

ݔ+ 5൯+ξ2൫ξ2ݔ+ 5൯

൫ξ2

ݔ+ 5൯൫ݔ+ξ2൯

Now, the roots of this quadratic equation are the values of

ݔ for which

ξ2

ݔ+ 5)൫ݔ+ξ2൯= 0

or ݔ=െξ2

Class- X-CBSE-Mathematics Quardatic Equations

Practice more on Quardatic Equations Page - 7 www.embibe.com Hence, the roots of this quadratic equation are െ and െξ2. (iv) To find the roots of given quadratic equation, lets first factorise the given quadratic expression 2ݔ . The given quadratic expression can be written as follows: 2ݔ െ ݔ+1 8 1 8 (16ݔ െ8ݔ+ 1) (16ݔ െ4ݔ െ4ݔ+ 1) (we factorise by method of splitting the middle term) 1 8 ൫4ݔ(4ݔ െ1)െ1(4ݔ െ1)൯ 1 8 (4ݔ െ1) Now, the roots of this quadratic equation are the values of

ݔ for which

4ݔ െ1)

= 0

Thus, (4ݔ െ1)= 0 or (4ݔ െ1)= 0

or ݔ=

Hence, the roots of this quadratic equation are

and (v) To find the roots of given quadratic equation, lets first factorise the given quadratic expression 100
െ20ݔ+ 1. The given quadratic expression can be written as follows: 100
െ20ݔ+ 1 =100ݔ െ10ݔ െ10ݔ+ 1 (we factorise by method of splitting the middle term) =10ݔ(10ݔ െ1)െ1(10ݔ െ1) (10ݔ െ1) Now, the roots of this quadratic equation are the values of

ݔ for which

10

ݔ െ1)

= 0

Thus, (10ݔ െ1)= 0 or (10ݔ െ1)= 0

i.e., ݔ= or ݔ=

Hence, the roots of this quadratic equation are

and

Class- X-CBSE-Mathematics Quardatic Equations

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2. Solve the problems given below.

Represent the following situations mathematically: (i) John and Jivanti together have 45 marbles. Both of them lost 5 marbles each, and the product of the number of marbles they now have is 124. We would like to find out how many marbles they had to start with. (ii) A cottage industry produces a certain number of toys in a day. The cost of production of each toy (in rupees) was found to be 55 minus the number of toys produced in a day. On a particular day, the total cost of production was

750. We would like to find out the number of toys produced on that day.

Solution:

(i) Let the number of John's marbles be ݔ. So, the number of Jivanti's marbles =45െ ݔ

If both lost 5 marbles each,

Then n

umber of marbles left with John =ݔ െ5

Then n

umber of marbles left with Jivanti =45െ ݔെ 5 =40െ ݔ

Given that the product of their marbles is 124.

െ45ݔ+324= 0 െ36ݔ െ9ݔ+324= 0 (ݔ െ36)െ9(ݔ െ36)= 0quotesdbs_dbs17.pdfusesText_23

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