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Class- X-CBSE-Mathematics Quardatic Equations
Practice more on Quardatic Equations Page - 1 www.embibe.com
CBSE NCERT Solutions for Class 10 Mathematics Chapter 4 Back of Chapter Questions
1. Check whether the following are quadratic equations:
(i) (๐ฅ๐ฅ + 1)2 = 2(๐ฅ๐ฅ โ 3)
(ii) ๐ฅ๐ฅ2 โ 2๐ฅ๐ฅ = (โ2)(3โ ๐ฅ๐ฅ)
(iii) (๐ฅ๐ฅ โ 2)(๐ฅ๐ฅ + 1) = (๐ฅ๐ฅ โ 1)(๐ฅ๐ฅ + 3)
(iv) (๐ฅ๐ฅ โ 3)(2๐ฅ๐ฅ + 1) = ๐ฅ๐ฅ(๐ฅ๐ฅ + 5)
(v) (2๐ฅ๐ฅ โ 1)(๐ฅ๐ฅ โ 3) = (๐ฅ๐ฅ + 5)(๐ฅ๐ฅ โ 1)
(vi) ๐ฅ๐ฅ2 + 3๐ฅ๐ฅ + 1 = (๐ฅ๐ฅ โ 2)2
(vii) (๐ฅ๐ฅ + 2)3 = 2๐ฅ๐ฅ(๐ฅ๐ฅ2 โ 1)
(viii) ๐ฅ๐ฅ3 โ 4๐ฅ๐ฅ2 โ ๐ฅ๐ฅ + 1 = (๐ฅ๐ฅ โ 2)3
Solution:
(i) We know that any equation of the form ๐๐๐ฅ๐ฅ2 + ๐๐๐ฅ๐ฅ + ๐๐ = 0 is called aquadratic equation, where ๐๐, ๐๐, ๐๐ are real numbers and ๐๐ โ 0.
Given equation: (๐ฅ๐ฅ + 1)2 = 2(๐ฅ๐ฅ โ 3)
Using the formula (๐๐ + ๐๐)2 = ๐๐2 + 2๐๐๐๐ + ๐๐2
โ ๐ฅ๐ฅ2 + 2๐ฅ๐ฅ + 1 = 2๐ฅ๐ฅ โ 6
โ ๐ฅ๐ฅ2 + 7 = 0
Here, ๐๐ = 1, ๐๐ = 0 and ๐๐ = 7.
Thus, the given equation is a quadratic equation as ๐๐ โ 0.
(ii) We know that any equation of the form ๐๐๐ฅ๐ฅ2 + ๐๐๐ฅ๐ฅ + ๐๐ = 0 is called aquadratic equation, where ๐๐, ๐๐, ๐๐ are real numbers and ๐๐ โ 0.
Given equation: ๐ฅ๐ฅ2 โ 2๐ฅ๐ฅ = (โ2)(3โ ๐ฅ๐ฅ)
โ ๐ฅ๐ฅ2 โ 2๐ฅ๐ฅ = โ6 + 2๐ฅ๐ฅ
โ ๐ฅ๐ฅ2 โ 4๐ฅ๐ฅ + 6 = 0
Here, ๐๐ = 1, ๐๐ = โ4 and ๐๐ = 6.
Thus, the given equation is a quadratic equation as ๐๐ โ 0.
(iii) We know that any equation of the form ๐๐๐ฅ๐ฅ2 + ๐๐๐ฅ๐ฅ + ๐๐ = 0 is called aquadratic equation, where ๐๐, ๐๐, ๐๐ are real numbers and ๐๐ โ 0.
Class- X-CBSE-Mathematics Quardatic Equations
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Given equation: (๐ฅ๐ฅ โ 2)(๐ฅ๐ฅ + 1) = (๐ฅ๐ฅ โ 1)(๐ฅ๐ฅ + 3)
โ ๐ฅ๐ฅ2 โ ๐ฅ๐ฅ โ 2 = ๐ฅ๐ฅ2 + 2๐ฅ๐ฅ โ 3
โ 3๐ฅ๐ฅ โ 1 = 0
But, here ๐๐ = 0.
So, the given equation is not a quadratic equation.
(iv) We know that any equation of the form ๐๐๐ฅ๐ฅ2 + ๐๐๐ฅ๐ฅ + ๐๐ = 0 is called a quadratic equation, where ๐๐, ๐๐, ๐๐ are real numbers and ๐๐ โ 0.
Given equation: (๐ฅ๐ฅ โ 3)(2๐ฅ๐ฅ + 1) = ๐ฅ๐ฅ(๐ฅ๐ฅ + 5)
โ 2๐ฅ๐ฅ2 โ 5๐ฅ๐ฅ โ 3 = ๐ฅ๐ฅ2 + 5๐ฅ๐ฅ
โ ๐ฅ๐ฅ2 โ 10๐ฅ๐ฅ โ 3 = 0
Here, ๐๐ = 1, ๐๐ = โ10 and ๐๐ = โ3.
Thus, the given equation is a quadratic equation as ๐๐ โ 0.
(v) We know that any equation of the form ๐๐๐ฅ๐ฅ2 + ๐๐๐ฅ๐ฅ + ๐๐ = 0 is called a quadratic equation, where ๐๐, ๐๐, ๐๐ are real numbers and ๐๐ โ 0.
Given equation: (2๐ฅ๐ฅ โ 1)(๐ฅ๐ฅ โ 3) = (๐ฅ๐ฅ + 5)(๐ฅ๐ฅ โ 1)
โ 2๐ฅ๐ฅ2 โ 7๐ฅ๐ฅ + 3 = ๐ฅ๐ฅ2 + 4๐ฅ๐ฅ โ 5
โ ๐ฅ๐ฅ2 โ 11๐ฅ๐ฅ + 8 = 0
Here, ๐๐ = 1, ๐๐ = โ11 and ๐๐ = 8.
Thus, the given equation is a quadratic equation as ๐๐ โ 0.
(vi) We know that any equation of the form ๐๐๐ฅ๐ฅ2 + ๐๐๐ฅ๐ฅ + ๐๐ = 0 is called a quadratic equation, where ๐๐, ๐๐, ๐๐ are real numbers and ๐๐ โ 0.
Given equation: ๐ฅ๐ฅ2 + 3๐ฅ๐ฅ + 1 = (๐ฅ๐ฅ โ 2)2
Using the formula (๐๐ โ ๐๐)2 = ๐๐2 โ 2๐๐๐๐ + ๐๐2
โ ๐ฅ๐ฅ2 + 3๐ฅ๐ฅ + 1 = ๐ฅ๐ฅ2 โ 4๐ฅ๐ฅ + 4
โ 7๐ฅ๐ฅ โ 3 = 0
But, here ๐๐ = 0.
So, the given equation is not a quadratic equation.
(vii) We know that any equation of the form ๐๐๐ฅ๐ฅ2 + ๐๐๐ฅ๐ฅ + ๐๐ = 0 is called a quadratic equation, where ๐๐, ๐๐, ๐๐ are real numbers and ๐๐ โ 0.
Given equation: (๐ฅ๐ฅ + 2)3 = 2๐ฅ๐ฅ(๐ฅ๐ฅ2 โ 1)
Using the formula (๐๐ + ๐๐)3 = ๐๐3 + ๐๐3 + 3๐๐2๐๐ + 3๐๐๐๐2
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โ ๐ฅ๐ฅ3 + 8 + 6๐ฅ๐ฅ2 + 12๐ฅ๐ฅ = 2๐ฅ๐ฅ3 โ 2๐ฅ๐ฅ
โ ๐ฅ๐ฅ3 โ 14๐ฅ๐ฅ โ 6๐ฅ๐ฅ2 โ 8 = 0
This equation is not of the form ๐๐๐ฅ๐ฅ2 + ๐๐๐ฅ๐ฅ + ๐๐ = 0
So, the given equation is not a quadratic equation.
(viii) We know that any equation of the form ๐๐๐ฅ๐ฅ2 + ๐๐๐ฅ๐ฅ + ๐๐ = 0 is called a quadratic equation, where ๐๐, ๐๐, ๐๐ are real numbers and ๐๐ โ 0.
Given equation: ๐ฅ๐ฅ3 โ 4๐ฅ๐ฅ2 โ ๐ฅ๐ฅ + 1 = (๐ฅ๐ฅ โ 2)3
Using the formula (๐๐ โ ๐๐)3 = ๐๐3 โ 3๐๐2๐๐ + 3๐๐๐๐2 โ ๐๐3
โ ๐ฅ๐ฅ3 โ 4๐ฅ๐ฅ2 โ ๐ฅ๐ฅ + 1 = ๐ฅ๐ฅ3 โ 8 โ 6๐ฅ๐ฅ2 + 12๐ฅ๐ฅ
โ 2๐ฅ๐ฅ2 โ 13๐ฅ๐ฅ + 9 = 0
Here, ๐๐ = 2, ๐๐ = โ13 and ๐๐ = 9.
Thus, the given equation is a quadratic equation as ๐๐ โ 0.
2. Represent the following situations in the form of quadratic equations:
(i) The area of a rectangular plot is 528 m2. 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) Rohanโs 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 Rohanโs present age.
(iv) A train travels a distance of 480 km at a uniform speed. If the speed had been 8 km hโ less, then it would have taken 3 hours more to cover the same distance. We need to find the speed of the train.
Solution:
(i) Let the breadth of the plot be ๐ฅ๐ฅ m.
Hence, the length of the plot is (2๐ฅ๐ฅ + 1) m. (Since, given that length is one more than twice its breadth)
Therefore, area of a rectangle = length ร breadth
Given: area of rectangle = 528 m2
โด 528 = ๐ฅ๐ฅ(2๐ฅ๐ฅ + 1)
โ 2๐ฅ๐ฅ2 + ๐ฅ๐ฅ โ 528 = 0 โฆโฆโฆ(i), which is of the form ๐๐๐ฅ๐ฅ2 + ๐๐๐ฅ๐ฅ + ๐๐ = 0
Class- X-CBSE-Mathematics Quardatic Equations
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Here ๐๐ = 2(โ 0), ๐๐ = 1 and ๐๐ = โ528
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 ๐ฅ๐ฅ and ๐ฅ๐ฅ + 1.
Given that their product is 306.
โด ๐ฅ๐ฅ(๐ฅ๐ฅ + 1) = 306
โ ๐ฅ๐ฅ2 + ๐ฅ๐ฅ โ 306 = 0 โฆโฆโฆ.. (i), which is of the form ๐๐๐ฅ๐ฅ2 + ๐๐๐ฅ๐ฅ + ๐๐ = 0
Here ๐๐ = 1(โ 0),๐๐ = 1 and ๐๐ = โ306
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 ๐ฅ๐ฅ,
His mother's age = ๐ฅ๐ฅ + 26 (given that Rohanโs mother is 26 years older than him)
3 years from now:
Rohan's age will be = ๐ฅ๐ฅ + 3
Mother's age will be = ๐ฅ๐ฅ + 26 + 3 = ๐ฅ๐ฅ + 29
Also given that the product of their ages after 3 years is 360.
โด (๐ฅ๐ฅ + 3)(๐ฅ๐ฅ + 29) = 360
On simplification, we get
๐ฅ๐ฅ2 + 32๐ฅ๐ฅ โ 273 = 0โฆโฆโฆ (i), which is of the form ๐๐๐ฅ๐ฅ2 + ๐๐๐ฅ๐ฅ + ๐๐ = 0
Here ๐๐ = 1(โ 0),๐๐ = 32 and ๐๐ = โ273
Thus, quadratic equation (i) represents the situation given in the question and positive root of this equation will represent the Rohanโs present age.
(iv) In first case,
Let the speed of train be ๐ฅ๐ฅ km/h.
Total time taken to travel 480 km = 480๐ฅ๐ฅ
hrs
In second case,
Given: speed became 8 km/h less
So, the speed of train = (๐ฅ๐ฅ โ 8)km/h
Class- X-CBSE-Mathematics Quardatic Equations
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Also given that the train will take 3 more hours to cover the same distance.
Therefore, time take to travel 480 km = ๏ฟฝ480๐ฅ๐ฅ
+ 3๏ฟฝhrs
Speed ร Time = Distance
(๐ฅ๐ฅ โ 8) ๏ฟฝ480๐ฅ๐ฅ + 3๏ฟฝ = 480
โ 480 + 3๐ฅ๐ฅ โ3840๐ฅ๐ฅ โ 24 = 480
โ 3๐ฅ๐ฅ โ3840๐ฅ๐ฅ = 24
โ 3๐ฅ๐ฅ2 โ 24๐ฅ๐ฅ โ 3840 = 0
โ ๐ฅ๐ฅ2 โ 8๐ฅ๐ฅ โ 1280 = 0โฆโฆ.. (i), which is of the form ๐๐๐ฅ๐ฅ2 + ๐๐๐ฅ๐ฅ + ๐๐ = 0
Here ๐๐ = 1(โ 0),๐๐ = โ8 and ๐๐ = โ1280
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) ๐ฅ๐ฅ2 โ 3๐ฅ๐ฅ โ 10 = 0
(ii) 2๐ฅ๐ฅ2 + ๐ฅ๐ฅ โ 6 = 0
(iii) โ2 ๐ฅ๐ฅ2 + 7๐ฅ๐ฅ + 5โ2 = 0
(iv) 2๐ฅ๐ฅ2 โ ๐ฅ๐ฅ + 18
= 0
(v) 100๐ฅ๐ฅ2 โ 20๐ฅ๐ฅ + 1 = 0
Solution:
(i) To find the roots of given quadratic equation, lets first factorise the given quadratic expression ๐ฅ๐ฅ2 โ 3๐ฅ๐ฅ โ 10. The given quadratic expression can be written as follows:
๐ฅ๐ฅ2 โ 3๐ฅ๐ฅ โ 10
= ๐ฅ๐ฅ2 โ 5๐ฅ๐ฅ + 2๐ฅ๐ฅ โ 10 (we factorise by method of splitting the middle term)
= ๐ฅ๐ฅ(๐ฅ๐ฅ โ 5) + 2(๐ฅ๐ฅ โ 5)
= (๐ฅ๐ฅ โ 5)(๐ฅ๐ฅ + 2)
Class- X-CBSE-Mathematics Quardatic Equations
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Now, the roots of this quadratic equation are the values of ๐ฅ๐ฅ for which (๐ฅ๐ฅ โ 5)(๐ฅ๐ฅ + 2) = 0
โด ๐ฅ๐ฅ โ 5 = 0 or ๐ฅ๐ฅ + 2 = 0
๐๐. ๐๐. ,๐ฅ๐ฅ = 5 or ๐ฅ๐ฅ = โ2
Hence, the roots of this quadratic equation are 5 andโ2.
(ii) To find the roots of given quadratic equation, lets first factorise the given quadratic expression 2๐ฅ๐ฅ2 + ๐ฅ๐ฅ โ 6 . The given quadratic expression can be written as follows:
2๐ฅ๐ฅ2 + ๐ฅ๐ฅ โ 6
= 2๐ฅ๐ฅ2 + 4๐ฅ๐ฅ โ 3๐ฅ๐ฅ โ 6 (we factorise by method of splitting the middle term)
= 2๐ฅ๐ฅ(๐ฅ๐ฅ + 2)โ 3(๐ฅ๐ฅ + 2)
= (๐ฅ๐ฅ + 2)(2๐ฅ๐ฅ โ 3)
Now, the roots of this quadratic equation are the values of ๐ฅ๐ฅ for which (๐ฅ๐ฅ + 2)(2๐ฅ๐ฅ โ 3) = 0
โด ๐ฅ๐ฅ + 2 = 0 or 2๐ฅ๐ฅ โ 3 = 0
๐๐. ๐๐. ,๐ฅ๐ฅ = โ2 or ๐ฅ๐ฅ = 32
Hence, the roots of this quadratic equation are โ2 and 32.
(iii) To find the roots of given quadratic equation, lets first factorise the given quadratic expression โ2๐ฅ๐ฅ2 + 7๐ฅ๐ฅ + 5โ2 . The given quadratic expression can be written as follows:
โ2๐ฅ๐ฅ2 + 7๐ฅ๐ฅ + 5โ2
= โ2๐ฅ๐ฅ2 + 5๐ฅ๐ฅ + 2๐ฅ๐ฅ + 5โ2 (we factorise by method of splitting the middle term)
= ๐ฅ๐ฅ๏ฟฝโ2๐ฅ๐ฅ + 5๏ฟฝ + โ2๏ฟฝโ2๐ฅ๐ฅ + 5๏ฟฝ
= ๏ฟฝโ2๐ฅ๐ฅ + 5๏ฟฝ๏ฟฝ๐ฅ๐ฅ + โ2๏ฟฝ
Now, the roots of this quadratic equation are the values of ๐ฅ๐ฅ for which
(โ2๐ฅ๐ฅ + 5)๏ฟฝ๐ฅ๐ฅ + โ2๏ฟฝ = 0
โด โ2๐ฅ๐ฅ + 5 = 0 or ๐ฅ๐ฅ + โ2 = 0
๐๐. ๐๐. ,๐ฅ๐ฅ = โ 5โ2
or ๐ฅ๐ฅ = โโ2
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Hence, the roots of this quadratic equation are โ 5โ2
and โโ2.
(iv) To find the roots of given quadratic equation, lets first factorise the given quadratic expression 2๐ฅ๐ฅ2 โ ๐ฅ๐ฅ + 1
8 . The given quadratic expression can be
written as follows:
2๐ฅ๐ฅ2 โ ๐ฅ๐ฅ +18
=18
(16๐ฅ๐ฅ2 โ 8๐ฅ๐ฅ + 1)
= 18
(16๐ฅ๐ฅ2 โ 4๐ฅ๐ฅ โ 4๐ฅ๐ฅ + 1) (we factorise by method of splitting the middle term)
=18๏ฟฝ4๐ฅ๐ฅ(4๐ฅ๐ฅ โ 1)โ 1(4๐ฅ๐ฅ โ 1)๏ฟฝ
=18
(4๐ฅ๐ฅ โ 1)2
Now, the roots of this quadratic equation are the values of ๐ฅ๐ฅ for which (4๐ฅ๐ฅ โ 1)2 = 0
Thus, (4๐ฅ๐ฅ โ 1) = 0 or (4๐ฅ๐ฅ โ 1) = 0
๐๐. ๐๐. ,๐ฅ๐ฅ = 14 or ๐ฅ๐ฅ = 1
4
Hence, the roots of this quadratic equation are 14 and 1
4.
(v) To find the roots of given quadratic equation, lets first factorise the given quadratic expression 100๐ฅ๐ฅ2 โ 20๐ฅ๐ฅ + 1. The given quadratic expression can be written as follows:
100๐ฅ๐ฅ2 โ 20๐ฅ๐ฅ + 1
= 100๐ฅ๐ฅ2 โ 10๐ฅ๐ฅ โ 10๐ฅ๐ฅ + 1 (we factorise by method of splitting the middle term)
= 10๐ฅ๐ฅ(10๐ฅ๐ฅ โ 1)โ 1(10๐ฅ๐ฅ โ 1)
= (10๐ฅ๐ฅ โ 1)2
Now, the roots of this quadratic equation are the values of ๐ฅ๐ฅ for which (10๐ฅ๐ฅ โ 1)2 = 0
Thus, (10๐ฅ๐ฅ โ 1) = 0 or (10๐ฅ๐ฅ โ 1) = 0
i. e., ๐ฅ๐ฅ = 110
or ๐ฅ๐ฅ = 110
Hence, the roots of this quadratic equation are 110
and 110
.
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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 โน 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 ๐ฅ๐ฅ.
So, the number of Jivanti's marbles = 45โ ๐ฅ๐ฅ
If both lost 5 marbles each,
Then number of marbles left with John = ๐ฅ๐ฅ โ 5
Then number of marbles left with Jivanti = 45โ ๐ฅ๐ฅ โ 5 = 40 โ ๐ฅ๐ฅ
Given that the product of their marbles is 124.
โด (๐ฅ๐ฅ โ 5)(40โ ๐ฅ๐ฅ) = 124
โ ๐ฅ๐ฅ2 โ 45๐ฅ๐ฅ + 324 = 0
โ ๐ฅ๐ฅ2 โ 36๐ฅ๐ฅ โ 9๐ฅ๐ฅ + 324 = 0
โ ๐ฅ๐ฅ(๐ฅ๐ฅ โ 36)โ 9(๐ฅ๐ฅ โ 36) = 0
โ (๐ฅ๐ฅ โ 36)(๐ฅ๐ฅ โ 9) = 0
Either ๐ฅ๐ฅ = 36 = 0 or ๐ฅ๐ฅ โ 9 = 0
๐๐. ๐๐. ,๐ฅ๐ฅ = 36 or ๐ฅ๐ฅ = 9
If the number of John's marbles = 36
Then, the number of Jivanti's marbles = 45โ 36 = 9
If the number of John's marbles = 9
Then, the number of Jivanti's marbles = 45โ 9 = 36.
(ii) Let the number of toys produced on that day be ๐ฅ๐ฅ.
โด The cost of production of each toy that day = โน (55โ ๐ฅ๐ฅ)
So, the total cost of production that day = ๐ฅ๐ฅ(55โ ๐ฅ๐ฅ)
As per the question, the total cost of production of the toys = โน 750
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โด (55โ ๐ฅ๐ฅ)๐ฅ๐ฅ = 750
โ ๐ฅ๐ฅ2 โ 55๐ฅ๐ฅ + 750 = 0
โ ๐ฅ๐ฅ2 โ 25๐ฅ๐ฅ โ 30๐ฅ๐ฅ + 750 = 0
โ ๐ฅ๐ฅ(๐ฅ๐ฅ โ 25)โ 30(๐ฅ๐ฅ โ 25) = 0
โ (๐ฅ๐ฅ โ 25)(๐ฅ๐ฅ โ 30) = 0
Either ๐ฅ๐ฅ โ 25 = 0 or ๐ฅ๐ฅ โ 30 = 0
๐๐. ๐๐. ,๐ฅ๐ฅ = 25 or ๐ฅ๐ฅ = 30
Thus, the number of toys produced that day will be either 25 or 30.
3. Find two numbers whose sum is 27 and product is 182.
Solution:
Let the first number be ๐ฅ๐ฅ.
Then the second number is 27 โ ๐ฅ๐ฅ. (Given sum of two numbers = 27)
Thus, their product = ๐ฅ๐ฅ(27โ ๐ฅ๐ฅ)
According to the question, the product of these numbers is 182.
Therefore, ๐ฅ๐ฅ(27โ ๐ฅ๐ฅ) = 182
โ ๐ฅ๐ฅ2 โ 27๐ฅ๐ฅ + 182 = 0
โ ๐ฅ๐ฅ2 โ 13๐ฅ๐ฅ โ 14๐ฅ๐ฅ + 182 = 0
โ ๐ฅ๐ฅ(๐ฅ๐ฅ โ 13)โ 14(๐ฅ๐ฅ โ 13) = 0
โ (๐ฅ๐ฅ โ 13)(๐ฅ๐ฅ โ 14) = 0
Either ๐ฅ๐ฅ = 13 = 0 or ๐ฅ๐ฅ โ 14 = 0
i. e. , ๐ฅ๐ฅ = 13 or ๐ฅ๐ฅ = 14
If first number = 13, then
Second number = 27โ 13 = 14
If first number = 14, then
Second number = 27 โ 14 = 13
Hence, the numbers are 13 and 14.
4. Find two consecutive positive integers, sum of whose squares is 365.
Solution:
We know that the difference between two consecutive positive integers is 1.
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So, let the consecutive positive integers be ๐ฅ๐ฅ and ๐ฅ๐ฅ + 1.
As per the question, ๐ฅ๐ฅ2 + (๐ฅ๐ฅ + 1)2 = 365
โ ๐ฅ๐ฅ2 + ๐ฅ๐ฅ2 + 1 + 2๐ฅ๐ฅ = 365
โ 2๐ฅ๐ฅ2 + 2๐ฅ๐ฅ โ 364 = 0
โ ๐ฅ๐ฅ2 + ๐ฅ๐ฅ โ 182 = 0
โ ๐ฅ๐ฅ2 + 14๐ฅ๐ฅ โ 13๐ฅ๐ฅ โ 182 = 0
โ ๐ฅ๐ฅ(๐ฅ๐ฅ + 14)โ 13(๐ฅ๐ฅ + 14) = 0
โ (๐ฅ๐ฅ + 14)(๐ฅ๐ฅ โ 13) = 0
Either ๐ฅ๐ฅ + 14 = 0 or ๐ฅ๐ฅ โ 13 = 0, i. e. , ๐ฅ๐ฅ = โ14 or ๐ฅ๐ฅ = 13
Since given that integers are positive, ๐ฅ๐ฅ can only be 13.
โด ๐ฅ๐ฅ + 1 = 13 + 1 = 14
Hence, the two consecutive positive integers are 13 and 14.
5. The altitude of a right triangle is 7 cm less than its base. If the hypotenuse is 13 cm, find the other two sides.
Solution: As per the question, hypotenuse is 13 cm
Let the base of the right triangle be ๐ฅ๐ฅ cm.
Its altitude = (๐ฅ๐ฅ โ 7)cm
From Pythagoras theorem,
Base2 + Altitude2 = Hypotenuse2
โด ๐ฅ๐ฅ2 + (๐ฅ๐ฅ โ 7)2 = 132
โ ๐ฅ๐ฅ2 + ๐ฅ๐ฅ2 + 49 โ 14๐ฅ๐ฅ = 169
โ 2๐ฅ๐ฅ2 โ 14๐ฅ๐ฅ โ 120 = 0
โ ๐ฅ๐ฅ2 โ 7๐ฅ๐ฅ โ 60 = 0
โ ๐ฅ๐ฅ2 โ 12๐ฅ๐ฅ + 5๐ฅ๐ฅ โ 60 = 0
โ ๐ฅ๐ฅ(๐ฅ๐ฅ โ 12) + 5(๐ฅ๐ฅ โ 12) = 0
โ (๐ฅ๐ฅ โ 12)(๐ฅ๐ฅ + 5) = 0
Either ๐ฅ๐ฅ โ 12 = 0 or ๐ฅ๐ฅ + 5 = 0, i. e. , ๐ฅ๐ฅ = 12 or ๐ฅ๐ฅ = โ5
Since sides of a triangle are positive, ๐ฅ๐ฅ can only take 12.
Hence, the base of the right triangle is 12 cm and the altitude of this triangle is (12โ 7)cm = 5 cm.
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6. A cottage industry produces a certain number of pottery articles in a day. It was observed on a particular day that the cost of production of each article (in rupees) was 3 more than twice the number of articles produced on that day. If the total cost of production on that day was โน. 90, find the number of articles produced and the cost of each article.
Solution:
Let the number of articles produced on that day be ๐ฅ๐ฅ.
So, the cost of production of each article = โน(2๐ฅ๐ฅ + 3)
According to the question, the total cost of production on that day was โน 90.
We know that
Total cost of production = Cost of each article ร Number of articles produced
โด ๐ฅ๐ฅ(2๐ฅ๐ฅ + 3) = 90
โ 2๐ฅ๐ฅ2 + 3๐ฅ๐ฅ โ 90 = 0
โ 2๐ฅ๐ฅ2 + 15๐ฅ๐ฅ โ 12๐ฅ๐ฅ โ 90 = 0
โ ๐ฅ๐ฅ(2๐ฅ๐ฅ + 15) โ 6(2๐ฅ๐ฅ + 15) = 0
โ (2๐ฅ๐ฅ + 15)(๐ฅ๐ฅ โ 6) = 0
Either 2๐ฅ๐ฅ + 15 = 0 or ๐ฅ๐ฅ โ 6 = 0, ๐๐. ๐๐. ,๐ฅ๐ฅ = โ152
or ๐ฅ๐ฅ = 6
Itโs clear that number of articles produced can only be a positive integer, so, ๐ฅ๐ฅ can only be 6.
Therefore, number of articles produced on that day = 6
Cost of each article = (2 ร 6) + 3 = โน 15
EXERCISE 4.3
1. Find the roots of the following quadratic equations, if they exist, by the method of completing the square:
(i) 2๐ฅ๐ฅ2 โ 7๐ฅ๐ฅ + 3 = 0
(ii) (ii) 2๐ฅ๐ฅ2 + ๐ฅ๐ฅ โ 4 = 0
(iii) 4๐ฅ๐ฅ2 + 4โ3๐ฅ๐ฅ + 3 = 0
(iv) 2๐ฅ๐ฅ2 + ๐ฅ๐ฅ + 4 = 0
Solution:
(i) Given quadratic equation: 2๐ฅ๐ฅ2 โ 7๐ฅ๐ฅ + 3 = 0
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โ 2๐ฅ๐ฅ2 โ 7๐ฅ๐ฅ = โ3
On dividing both sides of the equation by 2, we obtain
โ ๐ฅ๐ฅ2 โ72 ๐ฅ๐ฅ = โ
32
โ ๐ฅ๐ฅ2 โ 2 ร ๐ฅ๐ฅ ร74 = โ
32
On adding ๏ฟฝ74๏ฟฝ2 to both sides of equation by, we obtain
โ (๐ฅ๐ฅ)2 โ 2 ร ๐ฅ๐ฅ ร74 + ๏ฟฝ
74๏ฟฝ
2
= ๏ฟฝ74๏ฟฝ
2
โ32
โ ๏ฟฝ๐ฅ๐ฅ โ74๏ฟฝ
2
=4916โ
32
โ ๏ฟฝ๐ฅ๐ฅ โ74๏ฟฝ
2
=2516
โ ๏ฟฝ๐ฅ๐ฅ โ 74๏ฟฝ = ยฑ 5
4 (Cancelling square both the sides)
โ ๐ฅ๐ฅ =74 ยฑ
54
โ ๐ฅ๐ฅ = 74
+ 54 or ๐ฅ๐ฅ = 7
4โ 5
4
โ ๐ฅ๐ฅ = 124
or ๐ฅ๐ฅ = 24
โ ๐ฅ๐ฅ = 3 or 12
Hence, the roots of this quadratic equation are 3 and 12 .
(ii) 2๐ฅ๐ฅ2 + ๐ฅ๐ฅ โ 4 = 0
โ 2๐ฅ๐ฅ2 + ๐ฅ๐ฅ = 4
On dividing both sides of the equation by 2, we obtain
โ ๐ฅ๐ฅ2 +12 ๐ฅ๐ฅ = 2
On adding ๏ฟฝ14๏ฟฝ2 to both sides of the equation, we obtain
โ (๐ฅ๐ฅ)2 + 2 ร ๐ฅ๐ฅ ร14 + ๏ฟฝ
14๏ฟฝ
2
= 2 + ๏ฟฝ14๏ฟฝ
2
โ ๏ฟฝ๐ฅ๐ฅ +14๏ฟฝ
2
=3316
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โ ๐ฅ๐ฅ + 14
= ยฑ โ334
(Cancelling square both the sides)
โ ๐ฅ๐ฅ = ยฑโ33
4 โ14
โ ๐ฅ๐ฅ =ยฑโ33โ 1
4
โ ๐ฅ๐ฅ = โ33โ14
or โโ33โ14
Hence, the roots of this quadratic equation are โ1+โ334
and โ1โโ334
.
(iii) 4๐ฅ๐ฅ2 + 4โ3๐ฅ๐ฅ + 3 = 0
โ (2๐ฅ๐ฅ)2 + 2 ร 2๐ฅ๐ฅ ร โ3 + ๏ฟฝโ3๏ฟฝ2
= 0
โ ๏ฟฝ2๐ฅ๐ฅ + โ3๏ฟฝ2
= 0
โ ๏ฟฝ2๐ฅ๐ฅ + โ3๏ฟฝ
= 0 and ๏ฟฝ2๐ฅ๐ฅ + โ3๏ฟฝ
= 0
โ ๐ฅ๐ฅ = โโ32
and ๐ฅ๐ฅ = โโ32
Hence, the roots of this quadratic equation are โโ32
and โ โ32
.
(iv) 2๐ฅ๐ฅ2 + ๐ฅ๐ฅ + 4 = 0
โ 2๐ฅ๐ฅ2 + ๐ฅ๐ฅ = โ4
On dividing both sides of the equation by 2, we obtain
โ ๐ฅ๐ฅ2 +12 ๐ฅ๐ฅ = โ2
โ ๐ฅ๐ฅ2 + 2 ร ๐ฅ๐ฅ ร14 = โ2
On adding ๏ฟฝ14๏ฟฝ2to both sides of the equation, we obtain
โ (๐ฅ๐ฅ)2 + 2 ร ๐ฅ๐ฅ ร14 + ๏ฟฝ
14๏ฟฝ
2
= ๏ฟฝ14๏ฟฝ
2
โ 2
โ ๏ฟฝ๐ฅ๐ฅ +14๏ฟฝ
2
=1
16โ 2
โ ๏ฟฝ๐ฅ๐ฅ +14๏ฟฝ
2
= โ3116
Since, the square of a number cannot be negative.
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Thus, there is no real root for the given equation.
2. Find the roots of the quadratic equations given by applying the quadratic formula.
(i) 2๐ฅ๐ฅ2 โ 7๐ฅ๐ฅ + 3 = 0
(ii) 2๐ฅ๐ฅ2 + ๐ฅ๐ฅ โ 4 = 0
(iii) 4๐ฅ๐ฅ2 + 4โ3๐ฅ๐ฅ + 3 = 0
(iv) 2๐ฅ๐ฅ2 + ๐ฅ๐ฅ + 4 = 0
Solution:
(i) 2๐ฅ๐ฅ2 โ 7๐ฅ๐ฅ + 3 = 0
On comparing this equation with ๐๐๐ฅ๐ฅ2 + ๐๐๐ฅ๐ฅ + ๐๐ = 0, we obtain
๐๐ = 2,๐๐ = โ7, ๐๐ = 3
By using quadratic formula, we obtain
๐ฅ๐ฅ =โ๐๐ ยฑ โ๐๐2 โ 4๐๐๐๐
2๐๐
โ ๐ฅ๐ฅ =7 ยฑ โ49โ 24
2๐๐
โ ๐ฅ๐ฅ =7 ยฑ โ25
4
โ ๐ฅ๐ฅ =7 ยฑ 5
4
โ ๐ฅ๐ฅ = 7+54
or 7โ54
โ ๐ฅ๐ฅ = 124
or 24
โด ๐ฅ๐ฅ = 3 or 12
Hence, the roots of this quadratic equation are 3 and 12 .
(ii) 2๐ฅ๐ฅ2 + ๐ฅ๐ฅ โ 4 = 0
On comparing this equation with ๐๐๐ฅ๐ฅ2 + ๐๐๐ฅ๐ฅ + ๐๐ = 0, we obtain
๐๐ = 2,๐๐ = 1, ๐๐ = โ4
By using quadratic formula, we obtain
๐ฅ๐ฅ =โ๐๐ ยฑ โ๐๐2 โ 4๐๐๐๐
2๐๐
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โ ๐ฅ๐ฅ =โ1 ยฑ โ1 + 32
4
โ ๐ฅ๐ฅ =โ1 ยฑ โ33
4
โด ๐ฅ๐ฅ = โ1+โ334
or โ1โโ334
Hence, the roots of this quadratic equation are โ1+โ334
and โ1โโ334
.
(iii) 4๐ฅ๐ฅ2 + 4โ3๐ฅ๐ฅ + 3 = 0
On comparing this equation ๐๐๐ฅ๐ฅ2 + ๐๐๐ฅ๐ฅ + ๐๐ = 0, we obtain ๐๐ = 4,๐๐ =4โ3, ๐๐ = 3
By using quadratic formula, we obtain
๐ฅ๐ฅ =โ๐๐ ยฑ โ๐๐2 โ 4๐๐๐๐
2๐๐
โ ๐ฅ๐ฅ =โ4โ3 ยฑ โ48โ 48
8
โ ๐ฅ๐ฅ =โ4โ3 ยฑ 0
8
โด ๐ฅ๐ฅ = โโ32
or โ โ32
Hence, the roots of this quadratic equation are โโ32
and โ โ32
.
(iv) 2๐ฅ๐ฅ2 + ๐ฅ๐ฅ + 4 = 0
On comparing this equation with ๐๐๐ฅ๐ฅ2 + ๐๐๐ฅ๐ฅ + ๐๐ = 0, we obtain
๐๐ = 2,๐๐ = 1, ๐๐ = 4
By using quadratic formula, we obtain
๐ฅ๐ฅ =โ๐๐ ยฑ โ๐๐2 โ 4๐๐๐๐
2๐๐
โ ๐ฅ๐ฅ =โ1 ยฑ โ1 โ 32
4
โ ๐ฅ๐ฅ =โ1 ยฑ โโ31
4
Hence, roots do not exist for this quadratic equation as ๐ท๐ท = ๐๐2 โ 4๐๐๐๐ = โ31 < 0.
3. Find the roots of the following equations:
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(i) ๐ฅ๐ฅ โ 1๐ฅ๐ฅ
= 3, ๐ฅ๐ฅ โ 0
(ii) 1๐ฅ๐ฅ+4
โ 1๐ฅ๐ฅโ7
= 1130
, ๐ฅ๐ฅ โ โ4, 7
Solution:
(i) ๐ฅ๐ฅ โ 1๐ฅ๐ฅ
= 3 โ ๐ฅ๐ฅ2 โ 3๐ฅ๐ฅ โ 1 = 0
On comparing this equation with ๐๐๐ฅ๐ฅ2 + ๐๐๐ฅ๐ฅ + ๐๐ = 0, we obtain
๐๐ = 1,๐๐ = โ3, ๐๐ = โ1
By using quadratic formula, we obtain
๐ฅ๐ฅ =โ๐๐ ยฑ โ๐๐2 โ 4๐๐๐๐
2๐๐
โ ๐ฅ๐ฅ =3 ยฑ โ9 + 4
2
โ ๐ฅ๐ฅ =3 ยฑ โ13
2
Therefore, ๐ฅ๐ฅ = 3+โ132
or 3โโ132
(ii) 1๐ฅ๐ฅ+4
โ 1๐ฅ๐ฅโ7
= 1130
โ๐ฅ๐ฅ โ 7โ ๐ฅ๐ฅ โ 4
(๐ฅ๐ฅ + 4)(๐ฅ๐ฅ โ 7) =1130
โโ11
(๐ฅ๐ฅ + 4)(๐ฅ๐ฅ โ 7) =1130
โ (๐ฅ๐ฅ + 4)(๐ฅ๐ฅ โ 7) = โ30
โ ๐ฅ๐ฅ2 โ 3๐ฅ๐ฅ โ 28 = โ30
โ ๐ฅ๐ฅ2 โ 3๐ฅ๐ฅ + 2 = 0
โ ๐ฅ๐ฅ2 โ 2๐ฅ๐ฅ โ ๐ฅ๐ฅ + 2 = 0
โ ๐ฅ๐ฅ(๐ฅ๐ฅ โ 2)โ 1(๐ฅ๐ฅ โ 2) = 0
โ (๐ฅ๐ฅ โ 2)(๐ฅ๐ฅ โ 1) = 0
โ ๐ฅ๐ฅ = 1 or 2
4. The sum of the reciprocals of Rehmanโs ages, (in years) 3 years ago and 5 years from now is 1
3. Find his present age.
Solution:
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Let Rehmanโs present age = ๐ฅ๐ฅ years.
His age three years ago = (๐ฅ๐ฅ โ 3) years.
His age five years from now = (๐ฅ๐ฅ + 5) years.
As per the question, the sum of the reciprocals of Rehman's ages 3 years ago and 5 years from now is 1
3.
โด1
๐ฅ๐ฅ โ 3 +1
๐ฅ๐ฅ + 5 =13
๐ฅ๐ฅ + 5 + ๐ฅ๐ฅ โ 3(๐ฅ๐ฅ โ 3)(๐ฅ๐ฅ + 5) =
13
2๐ฅ๐ฅ + 2(๐ฅ๐ฅ โ 3)(๐ฅ๐ฅ + 5) =
13
โ 3(2๐ฅ๐ฅ + 2) = (๐ฅ๐ฅ โ 3)(๐ฅ๐ฅ + 5)
โ 6๐ฅ๐ฅ + 6 = ๐ฅ๐ฅ2 + 2๐ฅ๐ฅ โ 15
โ ๐ฅ๐ฅ2 โ 4๐ฅ๐ฅ โ 21 = 0
โ ๐ฅ๐ฅ2 โ 7๐ฅ๐ฅ + 3๐ฅ๐ฅ โ 21 = 0
โ ๐ฅ๐ฅ(๐ฅ๐ฅ โ 7) + 3(๐ฅ๐ฅ โ 7) = 0
โ (๐ฅ๐ฅ โ 7)( ๐ฅ๐ฅ + 3) = 0
โ ๐ฅ๐ฅ = 7 or ๐ฅ๐ฅ = โ3
Itโs clear that age is always positive.
Thus, Rehman's present age is 7 years.
5. In a class test, the sum of Shefaliโs marks in Mathematics and English is 30. Had she got 2 marks more in Mathematics and 3 marks less in English, the product of their marks would have been 210. Find her marks in the two subjects.
Solution:
Let Shefaliโs marks in Mathematics = ๐ฅ๐ฅ.
Then, her marks in English = 30โ ๐ฅ๐ฅ. (Given in question)
As per the question, we get
(๐ฅ๐ฅ + 2)(30โ ๐ฅ๐ฅ โ 3) = 210
(๐ฅ๐ฅ + 2)(27โ ๐ฅ๐ฅ) = 210
โ โ๐ฅ๐ฅ2 + 25๐ฅ๐ฅ + 54 = 210
โ ๐ฅ๐ฅ2 โ 25๐ฅ๐ฅ + 156 = 0
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โ ๐ฅ๐ฅ2 โ 12๐ฅ๐ฅ โ 13๐ฅ๐ฅ + 156 = 0
โ ๐ฅ๐ฅ(๐ฅ๐ฅ + 2)โ 13(๐ฅ๐ฅ โ 12) = 0
โ (๐ฅ๐ฅ โ 12)(๐ฅ๐ฅ โ 13) = 0
โ ๐ฅ๐ฅ = 12 or ๐ฅ๐ฅ = 13
If the marks in Mathematics is 12, then marks in English will be 30โ 12 = 18
If the marks in Mathematics is 13,then marks in English will be 30โ 13 = 17
6. The diagonal of a rectangular field is 60 metres more than the shorter side. If the longer side is 30 meters more than the shorter side, find the sides of the field.
Solution:
Let the shorter side of the rectangle be ๐ฅ๐ฅ m.
Then, longer side of the rectangle = (๐ฅ๐ฅ + 30)m
Diagonal of the rectangle = ๏ฟฝ๐ฅ๐ฅ2 + (๐ฅ๐ฅ + 30)2 (By Pythagoras theorem)
But in question, it is given that the diagonal of the rectangular field is 60 ๐๐ more than the shorter side.
โด ๏ฟฝ๐ฅ๐ฅ2 + (๐ฅ๐ฅ + 30)2 = ๐ฅ๐ฅ + 60
โ ๐ฅ๐ฅ2 + (๐ฅ๐ฅ + 30)2 = (๐ฅ๐ฅ + 60)2 (By squaring on both the sides)
โ ๐ฅ๐ฅ2 + ๐ฅ๐ฅ2 + 900 + 60๐ฅ๐ฅ = ๐ฅ๐ฅ2 + 3600 + 120๐ฅ๐ฅ
โ ๐ฅ๐ฅ2 โ 60๐ฅ๐ฅ โ 2700 = 0
โ ๐ฅ๐ฅ2 โ 90๐ฅ๐ฅ + 30๐ฅ๐ฅ โ 2700 = 0
โ ๐ฅ๐ฅ(๐ฅ๐ฅ โ 90) + 30(๐ฅ๐ฅ โ 90)
โ (๐ฅ๐ฅ โ 90)(๐ฅ๐ฅ + 30) = 0
โ ๐ฅ๐ฅ = 90 or ๐ฅ๐ฅ = โ30
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But, side of a rectangle cannot be negative. So, the length of the shorter side is 90 m. Thus, the length of the longer side will be (90 + 30) m = 120 m
7. The difference of squares of two numbers is 180. The square of the smaller number is 8 times the larger number. Find the two numbers.
Solution:
Let the larger and smaller number be ๐ฅ๐ฅ and ๐ฆ๐ฆ respectively.
It is given in the question that,
๐ฅ๐ฅ2 โ ๐ฆ๐ฆ2 = 180 and ๐ฆ๐ฆ2 = 8๐ฅ๐ฅ
โ ๐ฅ๐ฅ2 โ 8๐ฅ๐ฅ = 180
โ ๐ฅ๐ฅ2 โ 8๐ฅ๐ฅ โ 180 = 0
โ ๐ฅ๐ฅ2 โ 18๐ฅ๐ฅ + 10๐ฅ๐ฅ โ 180 = 0
โ ๐ฅ๐ฅ(๐ฅ๐ฅ โ 18) + 10(๐ฅ๐ฅ โ 18) = 0
โ (๐ฅ๐ฅ โ 18)(๐ฅ๐ฅ + 10) = 0
โ ๐ฅ๐ฅ = 18,โ10
If larger number, ๐ฅ๐ฅ = โ10
then smaller number, ๐ฆ๐ฆ = ยฑโ8๐ฅ๐ฅ
= ยฑ๏ฟฝ8(โ10)
= ยฑโโ80
Since we cannot have negative number in roots
๐ฅ๐ฅ = โ10 is not possible
Therefore, the larger number will be 18 only.
๐ฅ๐ฅ = 18
โด ๐ฆ๐ฆ2 = 8๐ฅ๐ฅ = 8 ร 18 = 144
โ ๐ฆ๐ฆ = ยฑโ144 = ยฑ12
โด Smaller number = ยฑ12
Therefore, the numbers are 18 and 12 or 18 and โ12.
8. A train travels 360 km at a uniform speed. If the speed had been 5 km hโ more, it would have taken 1 hour less for the same journey. Find the speed of the train.
Solution:
Let the speed to the train be ๐ฅ๐ฅ km/h.
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Total time taken to cover 360 km = 360๐ฅ๐ฅ
hr
It is given in question,
(๐ฅ๐ฅ + 5) ๏ฟฝ360๐ฅ๐ฅโ 1๏ฟฝ = 360 (Distance = Speed ร Time)
โ 360โ ๐ฅ๐ฅ +1800๐ฅ๐ฅ โ 5 = 360
โ ๐ฅ๐ฅ2 + 5๐ฅ๐ฅ โ 1800 = 0
โ ๐ฅ๐ฅ2 + 45๐ฅ๐ฅ โ 40๐ฅ๐ฅ โ 1800 = 0
โ ๐ฅ๐ฅ(๐ฅ๐ฅ + 45)โ 40(๐ฅ๐ฅ + 45) = 0
โ (๐ฅ๐ฅ + 45)(๐ฅ๐ฅ โ 40) = 0
โ ๐ฅ๐ฅ = 40 or ๐ฅ๐ฅ = โ45
But, speed cannot be negative.
Thus, the speed of the train is 40 km/h
9. Two water taps together can fill a tank in 9 38 hours. The tap of larger diameter takes
10 hours less than the smaller one to fill the tank separately. Find the time in which each tap can separately fill the tank.
Solution:
Let the time taken by the smaller diameter tap to fill the tank be ๐ฅ๐ฅ hr.
Time taken by the larger diameter tap = (๐ฅ๐ฅ โ 10)hr
Volume of tank filled by smaller tap in 1hour = 1๐ฅ๐ฅ
Volume of tank filled by larger tap in 1 hour = 1๐ฅ๐ฅโ10
As per the question, the tank can be filled in 9 38
= 758
hours by both the taps together.
Hence,
1๐ฅ๐ฅ +
1๐ฅ๐ฅ โ 10 =
875
๐ฅ๐ฅ โ 10 + ๐ฅ๐ฅ๐ฅ๐ฅ(๐ฅ๐ฅ โ 10) =
875
โ2๐ฅ๐ฅ โ 10๐ฅ๐ฅ(๐ฅ๐ฅ โ 10) =
875
โ 75(2๐ฅ๐ฅ โ 10) = 8๐ฅ๐ฅ2 โ 80๐ฅ๐ฅ
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โ 150๐ฅ๐ฅ โ 750 = 8๐ฅ๐ฅ2 โ 80๐ฅ๐ฅ
โ 8๐ฅ๐ฅ2 โ 230๐ฅ๐ฅ + 750 = 0
โ 8๐ฅ๐ฅ2 โ 200๐ฅ๐ฅ โ 30๐ฅ๐ฅ + 750 = 0
โ 8๐ฅ๐ฅ(๐ฅ๐ฅ โ 25) โ 30(๐ฅ๐ฅ โ 25) = 0
โ (๐ฅ๐ฅ โ 25)(8๐ฅ๐ฅ โ 30) = 0
๐๐. ๐๐. ,๐ฅ๐ฅ = 25 or ๐ฅ๐ฅ = 308
= 154
Taking ๐ฅ๐ฅ = 154
Time taken by smaller tap
= ๐ฅ๐ฅ = 154
hrs
Time taken by larger tap = ๐ฅ๐ฅ โ 10
= 154โ 10 = 15โ40
4= โ 25
4
Since time is negative,
๐ฅ๐ฅ = 154
is not the solution
Thus, the time taken separately by the smaller diameter tap and the larger diameter tap will be 25 hours and (25โ 10) = 15 hours respectively.
10. An express train takes 1 hour less than a passenger train to travel 132 km between Mysore and Bangalore (without taking into consideration the time they stop at intermediate stations). If the average speed of the express train is 11 km hโ more than that of the passenger train, find the average speed of the two trains.
Solution:
Let the average speed of passenger train be ๐ฅ๐ฅ km/h.
Average speed of express train = (๐ฅ๐ฅ + 11) km/h (Given in question)
According to the question, the time taken by the express train to cover 132 km is
1 hour less than a passenger train.
Therefore, time taken by passenger train โ time taken by express train = 1 hour
โด 132๐ฅ๐ฅโ 132
๐ฅ๐ฅ+11= 1 ๏ฟฝTotal time = Distance
Average Speed๏ฟฝ
โ 132 ๏ฟฝ๐ฅ๐ฅ + 11 โ ๐ฅ๐ฅ๐ฅ๐ฅ(๐ฅ๐ฅ + 11) ๏ฟฝ = 1
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โ132 ร 11๐ฅ๐ฅ(๐ฅ๐ฅ + 11) = 1
โ 132 ร 11 = ๐ฅ๐ฅ(๐ฅ๐ฅ + 11)
โ ๐ฅ๐ฅ2 + 11๐ฅ๐ฅ โ 1452 = 0
โ ๐ฅ๐ฅ2 + 44๐ฅ๐ฅ โ 33๐ฅ๐ฅ โ 1452 = 0
โ ๐ฅ๐ฅ(๐ฅ๐ฅ + 44)โ 339๐ฅ๐ฅ + 44) = 0
โ (๐ฅ๐ฅ + 44)(๐ฅ๐ฅ โ 33) = 0
โ ๐ฅ๐ฅ = โ44, 33
Average speed of passenger train cannot be negative. Hence, the speed of the passenger train is 33km/h and thus, the speed of the express train will be 33 +11 = 44 km/h.
11. Sum of the areas of two squares is 468 m2. If the difference of their perimeters is 24 m, find the sides of the two squares.
Solution:
Let the sides of the two squares be ๐ฅ๐ฅ m and ๐ฆ๐ฆ m.
So, their perimeter will be 4๐ฅ๐ฅ and 4๐ฆ๐ฆ respectively and their areas will be ๐ฅ๐ฅ2and ๐ฆ๐ฆ2 respectively.
According to the question, 4๐ฅ๐ฅ โ 4๐ฆ๐ฆ = 24
โ ๐ฅ๐ฅ โ ๐ฆ๐ฆ = 6
โ ๐ฅ๐ฅ = ๐ฆ๐ฆ + 6
Also, ๐ฅ๐ฅ2 + ๐ฆ๐ฆ2 = 468
โ (6 + ๐ฆ๐ฆ)2 + ๐ฆ๐ฆ2 = 468
โ 36 + ๐ฆ๐ฆ2 + 12๐ฆ๐ฆ + ๐ฆ๐ฆ2 = 468
โ 2๐ฆ๐ฆ2 + 12๐ฆ๐ฆ โ 432 = 0
โ ๐ฆ๐ฆ2 + 6๐ฆ๐ฆ โ 216 = 0
โ ๐ฆ๐ฆ2 + 18๐ฆ๐ฆ โ 12๐ฆ๐ฆ โ 216 = 0
โ ๐ฆ๐ฆ(๐ฆ๐ฆ + 18)โ 12(๐ฆ๐ฆ + 18) = 0
โ (๐ฆ๐ฆ + 18)(๐ฆ๐ฆ โ 12) = 0
โ ๐ฆ๐ฆ = โ18 or 12.
But, side of a square cannot be negative.
Hence, ๐ฆ๐ฆ = 12 & ๐ฅ๐ฅ = 12 + 6 = 18
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Therefore, the sides of the squares are 12 m and 18 m
EXERCISE 4.4
1. Find the nature of the roots of the following quadratic equations. If the real roots exist, find them:
(i) 2๐ฅ๐ฅ2 โ 3๐ฅ๐ฅ + 5 = 0
(ii) 3๐ฅ๐ฅ2 โ 4โ3๐ฅ๐ฅ + 4 = 0
(iii) 2๐ฅ๐ฅ2 โ 6๐ฅ๐ฅ + 3 = 0
Solution:
We know that for a quadratic equation ๐๐๐ฅ๐ฅ2 + ๐๐๐ฅ๐ฅ + ๐๐ = 0, discriminant is
๐๐2 โ 4๐๐๐๐.
(A) If ๐๐2 โ 4๐๐๐๐ > 0 implies two distinct real roots
(B) If ๐๐2 โ 4๐๐๐๐ = 0 implies two equal real roots
(C) If ๐๐2 โ 4๐๐๐๐ < 0 implies imaginary roots
(i) 2๐ฅ๐ฅ2 โ 3๐ฅ๐ฅ + 5 = 0
Comparing this equation with ๐๐๐ฅ๐ฅ2 + ๐๐๐ฅ๐ฅ + ๐๐ = 0, we obtain ๐๐ = 2, ๐๐ =โ3, ๐๐ = 5
Discriminant = ๐๐2 โ 4๐๐๐๐ = (โ3)2 โ 4(2)(5) = 9โ 40 = โ31
As ๐๐2 โ 4๐๐๐๐ < 0,
Hence, no real root is possible for the given equation.
(ii) 3๐ฅ๐ฅ2 โ 4โ3๐ฅ๐ฅ + 4 = 0
Comparing this equation with ๐๐๐ฅ๐ฅ2 + ๐๐๐ฅ๐ฅ + ๐๐ = 0, we obtain
๐๐ = 3,๐๐ = โ4โ3, ๐๐ = 4
Discriminant = ๐๐2 โ 4๐๐๐๐ = ๏ฟฝโ4โ3๏ฟฝ2โ 4(3)(4) = 48โ 48 = 0
As ๐๐2 โ 4๐๐๐๐ = 0,
So, real roots exist for the given equation and they are equal to each other and the roots will be โ ๐๐
2๐๐ and โ ๐๐
2๐๐.
โ๐๐
2๐๐ =โ๏ฟฝโ4โ3๏ฟฝ
2 ร 3 =4โ3
6 =2โ3
3 =2โ3
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Hence, the roots are 2โ3
and 2โ3
.
(iii) 2๐ฅ๐ฅ2 โ 6๐ฅ๐ฅ + 3 = 0
Comparing this equation with ๐๐๐ฅ๐ฅ2 + ๐๐๐ฅ๐ฅ + ๐๐ = 0, we obtain ๐๐ = 2, ๐๐ =โ6, ๐๐ = 3
Discriminant = ๐๐2 โ 4๐๐๐๐ = (โ6)2 โ 4(2)(3) = 36โ 24 = 12
As ๐๐2 โ 4๐๐๐๐ > 0,
So, two distinct real roots exist for this equation as follows:
๐ฅ๐ฅ =โ๐๐ ยฑ โ๐๐2 โ 4๐๐๐๐
2๐๐
=โ(โ6) ยฑ ๏ฟฝ(โ6)2 โ 4(2)(3)
2(2)
=6 ยฑ โ12
4 =6 ยฑ 2โ3
4
=3 ยฑ โ3
2
Hence, the roots are 3+โ32
or 3โโ32
.
2. Find the values of ๐๐ for each of the following quadratic equations, so that they have two equal roots.
(i) 2๐ฅ๐ฅ2 + ๐๐๐ฅ๐ฅ + 3 = 0
(ii) ๐๐๐ฅ๐ฅ(๐ฅ๐ฅ โ 2) + 6 = 0
Solution:
We know that if an equation ๐๐๐ฅ๐ฅ2 + ๐๐๐ฅ๐ฅ + ๐๐ = 0 has two equal roots, its discriminant (๐๐2 โ 4๐๐๐๐) will be 0.
(i) 2๐ฅ๐ฅ2 + ๐๐๐ฅ๐ฅ + 3 = 0
Comparing this equation with ๐๐๐ฅ๐ฅ2 + ๐๐๐ฅ๐ฅ + ๐๐ = 0, we get ๐๐ = 2, ๐๐ = ๐๐, ๐๐ =3
Discriminant = ๐๐2 โ 4๐๐๐๐ = (๐๐)2 โ 4(2)(3) = ๐๐2 โ 24
For equal roots,
Discriminant = 0
๐๐2 โ 24 = 0
โ ๐๐2 = 24
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โ ๐๐ = ยฑโ24 = ยฑ2โ6
(ii) ๐๐๐ฅ๐ฅ(๐ฅ๐ฅ โ 2) + 6 = 0 or ๐๐๐ฅ๐ฅ2 โ 2๐๐๐ฅ๐ฅ + 6 = 0
Comparing the equation with ๐๐๐ฅ๐ฅ2 + ๐๐๐ฅ๐ฅ + ๐๐ = 0, we get ๐๐ = ๐๐, ๐๐ =โ2๐๐, ๐๐ = 6
Discriminant = ๐๐2 โ 4๐๐๐๐ = (โ2๐๐)2 โ 4(๐๐)(6) = 4๐๐2 โ 24๐๐
For equal roots,
๐ท๐ท = ๐๐2 โ 4๐๐๐๐ = 0
4๐๐2 โ 24๐๐ = 0
4๐๐(๐๐ โ 6) = 0
Either 4๐๐ = 0 or ๐๐ โ 6 = 0
โ ๐๐ = 0 or ๐๐ = 6
But, if ๐๐ = 0, then the equation will not have the terms โ๐ฅ๐ฅ2โ and โ๐ฅ๐ฅโ.
Hence, if this quadratic equation has two equal roots, then ๐๐ should be 6 only.
3. Is it possible to design a rectangular mango grove whose length is twice its breadth, and the area is 800 m2? If so, find its length and breadth.
Solution:
Let the breadth of mango grove be ๐ฅ๐ฅ.
Length of mango grove will be 2๐ฅ๐ฅ.
Area of mango grove = (2๐ฅ๐ฅ)(๐ฅ๐ฅ) = 2๐ฅ๐ฅ2
Hence, 2๐ฅ๐ฅ2 = 800
โ ๐ฅ๐ฅ2 =800
2
โ ๐ฅ๐ฅ2 = 400
Cancelling square on both the sides, we get ๐ฅ๐ฅ = ยฑ20
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But length cannot be negative.
So, breadth of mango grove = 20 m
Length of mango grove = 2 ร 20 = 40 m
4. Is the following situation possible? If so, determine their present ages.
The sum of the ages of two friends is 20 years. Four years ago, the product of their ages in years was 48.
Solution:
Let the age of one friend be ๐ฅ๐ฅ years.
Age of the other friend will be = (20โ ๐ฅ๐ฅ) years. (Given in question)
Four years ago, age of 1st friend = (๐ฅ๐ฅ โ 4) years
and, age of 2nd friend = ((20โ ๐ฅ๐ฅ)โ 4) = (16 โ ๐ฅ๐ฅ) years
As per the question,
(๐ฅ๐ฅ โ 4)(16โ ๐ฅ๐ฅ) = 48
16๐ฅ๐ฅ โ 64 โ ๐ฅ๐ฅ2 + 4๐ฅ๐ฅ = 48
โ๐ฅ๐ฅ2 + 20๐ฅ๐ฅ โ 112 = 0
๐ฅ๐ฅ2 โ 20๐ฅ๐ฅ + 112 = 0
Comparing this equation with ๐๐๐ฅ๐ฅ2 + ๐๐๐ฅ๐ฅ + ๐๐ = 0, we obtain
๐๐ = 1,๐๐ = โ20, ๐๐ = 112
Discriminant = ๐๐2 โ 4๐๐๐๐ = (โ20)2 โ 4(1)(112)
= 400โ 448 = โ48
As ๐๐2 โ 4๐๐๐๐ < 0,
Thus, no real roots are possible for this equation and hence, this situation is not possible.
5. Is it possible to design a rectangular park of perimeter 80 m and area 400 m2? If so, find its length and breadth.
Solution:
Let the length and breadth of the rectangular park be ๐๐ and b.
Perimeter = 2(๐๐ + ๐๐) = 80
๐๐ + ๐๐ = 40
or, ๐๐ = 40โ ๐๐
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Area = ๐๐ ร ๐๐ = ๐๐(40โ ๐๐) = 40๐๐ โ ๐๐2
40๐๐ โ ๐๐2 = 400
๐๐2 โ 40๐๐ + 400 = 0
Comparing this equation with ๐๐๐๐2 + ๐๐๐๐ + ๐๐ = 0, we obtain
๐๐ = 1,๐๐ = โ40, ๐๐ = 400
Discriminant = ๐๐2 โ 4๐๐๐๐ = (โ40)2 โ 4(1)(400)
= 1600โ 1600 = 0
As ๐๐2 โ 4๐๐๐๐ = 0,
Thus, this equation has equal real roots, and hence, this situation is possible.
Root of this equation,
๐๐ = โ๐๐
2๐๐
๐๐ = โ(โ40)2(1) =
402 = 20
So, length of park, ๐๐ = 20 m
and breadth of park, ๐๐ = 40 โ ๐๐ = 40โ 20 = 20 m
โฆ โฆ โฆ