Q. 24.1( 453 Votes )
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) 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 been8 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.
(i)The area of a rectangular plot is 528 m2.
Let the breadth of the plot be x 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.
Thus, the length of the plot is (2x + 1) m.
Area of a rectangle = Length × Breadth
∴ 528 = x (2x + 1)
⇒ 2x2 + x – 528 = 0 (required quadratic form)
(ii)The product of two consecutive positive integers is 306. We need to find the integers.
Let the consecutive integers be x and x + 1.
It is given that their product is 306.
∴ x (x+1) = 306
⇒ x2 + x – 306 = 0 (required quadratic form)
Let Rohan’s present age be x.
Given, Rohan's Mother is 26 years older than him
Hence, his mother’s age = x+26
3 years hence,
Rohan’s age = x + 3
Mother’s age = x + 26 + 3
= x + 29
It is given that the product of their ages after 3 years is 360.
∴ (x+3) (x+29) = 360x2 + 3x + 29x + 87 = 360
⇒ x2 +32x – 273 = 0 (required quadratic form)
Let the speed of train be x km/h.
As speed = distance / time
⇒ Time taken for travel 480 km =
In the second condition,
speed of train = (x – 8) km/h
Given that the train will take 3 hours more to cover the same distance.
Therefore, Time taken for traveling 480 km =
Speed × Time = Distance
⇒ 480 + 3x - - 24 = 480
⇒ 3x - = 24⇒ 3x2 - 3840 = 24x
⇒ 3x2 – 24x - 3840 = 0
⇒ x2 – 8x - 1280 = 0 (required quadratic form)
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PREVIOUSCheck whether the following are quadratic equations : (i) (x + 1)2 = 2 (x - 3) (ii) x2 – 2x = (-2)(3 – x ) (iii) (x – 2 )(x + 1) = (x – 1)(x + 3) (iv) (x – 3 )(2x + 1) = x(x + 5) (v) (2x – 1)(x – 3 ) = (x + 5)(x – 1 ) (vi) x2 + 3x + 1 = (x – 2 )2 (vii) (x + 2)3 = 2x (x2 - 1) (viii) x3 – 4x2 – x + 1 = (x - 2)3NEXTFind the roots of the following quadratic equations by factorisation:(i) (ii) (iii) (iv) (v)
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