NCERT MATHS CLASS 10TH CHAPTER 4 EXERCISE 4.3 QUESTION 7 AND 8
प्रश्न - (7) दो संख्याओं के वर्गो का अन्तर 180 हे। छोटी संख्या का वर्ग बड़ी संख्या का आठ गुना हे दोनों संख्याये ज्ञात कीजिये।
हल : माना की बड़ी संख्या का X हे। तब प्रश्नानुसार ,
( छोटी संख्या )^ 2 = 8X
=> छोटी संख्याओं = \/8X
एवं , X^2 - 8X =180
=> X^2 -8X -180 = 0
=> X^2 -18X + 10X -180 =0
=> X (X -18 ) + 10 (X - 18 )= 0
=> (X - 18 )(X + 10 ) = 0
=> X - 18 = 0 => X = 18 =बड़ी संख्या
तब छोटी संख्या = \/8X = \/8 * 18 = \/144 = + 12 और -12 होगा।
या X +10 = 0 => X = -10 (असंभव )
अभीष्ट संख्या 18 , 12 अथवा 18 , -12 होगी।
प्रश्न - (8 ) एक रेलगाड़ी एक समान चाल से 360 KM की दूरी तय करती हे। यदि यह चाल 5 KM / H अधिक होती हे। , तो वह उसी यात्रा में 1 घंटा कम समय लेती। रेलगाड़ी की चाल ज्ञात कीजिये।
हल : माना की रेलगाड़ी की चाल X KM / H हे ,
तब 360 KM की दूरी तय करने में लगा समय = 360 / X h
प्रश्नानुसार , 360 / (x + 5 ) = 360 / x - 1
=> 1 = 360 / x - 360 / ( x + 5 )
=> x (x + 5 ) = 360 (x + 5 ) - 360 x
=> x^2 + 5x - 1800 = 0
=> x^2 + 45 x - 40x -1800 = 0
=> x (x +45 ) - 40 (x + 45 )=0
=> ( x + 45 ) (x -40 ) = 0
=> x + 45 = 0 => x = -45 (असंभव )
या x - 40 = 0 => x = 40 = रेलगाड़ी की चाल।
ENGLISH TRANSLATION :
Question - (7) The difference of the squares of two numbers is 180. The square of the smaller number is eight times the larger number. Find both the numbers.
Solution : Let X be the largest number. Then according to the question,
(smaller number)^ 2 = 8X
=> small numbers = \/8X
and X^2 - 8X =180
=> X^2 -8X -180 = 0
=> X^2 -18X + 10X -180 =0
=> X (X -18 ) + 10 (X - 18 )=0
=> (X - 18 )(X + 10 ) = 0
=> X - 18 = 0 => X = 18 = Large number
Then the smaller number = \/8X = \/8 * 18 = \/144 = + 12 and -12.
or X +10 = 0 => X = -10 (impossible)
The required number would be 18, 12 or 18, -12.
Question - ( 8 ) A train travels a distance of 360 KM with a uniform speed. If this speed was 5 KM / H more. , then she would have taken 1 hour less in the same journey. Find the speed of the train.
Solution : Let the speed of the train be X KM / H O,
Then time taken to cover a distance of 360 KM = 360 / X h
According to the question,
360 / (x + 5 ) = 360 / x - 1
=> 1 = 360 / x - 360 / ( x + 5 )
=> x (x + 5 ) = 360 (x + 5 ) - 360 x
=> x^2 + 5x - 1800 = 0
=> x^2 + 45 x - 40x -1800 = 0
=> x (x +45 ) - 40 (x + 45 )=0
=> ( x + 45 ) (x -40 ) = 0
=> x + 45 = 0 => x = -45 (impossible)
or x - 40 = 0 => x = 40 = speed of the train.