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 meters) 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 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.
Solution:
(i) We know that the area of a rectangle can be expressed as the product of its length and breadth.
Since we don’t know the length and breadth of the given rectangle, we assume the breadth of the plot to be a variable (x meters). Then, we use the given relationship between length and breadth: length = 1 + 2 times breadth.
Therefore, Length = 2x + 1
Area of rectangle = Length × Breadth
Breadth = x
Length = 2x + 1
Area of Rectangular Plot = Length × Breadth = 528 m2
(2x + 1) × (x) = 528
2x2 + x = 528
2x2 + x - 528 = 0
Thus, the quadratic equation is 2x2 + x - 528 = 0 , where x is the breadth of the rectangular plot.
(ii) Let the first integer be x.
Since the integers are consecutive, the next integer is (x + 1).
It is given that
First integer × Next integer = 306
Therefore,
x (x + 1) = 306
x (x +1) = 306
x2 + x = 306
x2 + x - 306 = 0
Thus, the quadratic equation is x2 + x - 306 = 0 where x is the first integer.
(iii) Let's assume that Rohan’s present age is x years. Then, from the first condition, his mother’s age is (x + 26) years.
Three years from now, Rohan’s age will be (x + 3) and Rohan’s mother age will be (x + 3) + 26 = (x + 29). The product of their ages is 360.
Therefore,
(x + 3) × (x + 29) = 360
x2 + 29x + 3x + 87 = 360
x2 + 32x + 87 - 360 = 0
x2 + 32x - 273 = 0
Thus, the quadratic equation is x2 + 32x - 273 = 0 where x is the present age of Rohan.
(iv) Distance is equal to speed multiplied by time. Let the speed be s km/h and time be t hours.
D = st
480 = st …(i)
t = 480/s …(ii)
As per the given conditions, for the same distance covered at a speed reduced by 8 km/h, the time taken would have increased by 3 hours.
Therefore,
(s - 8)(t + 3) = 480
st + 3s - 8t - 24 = 480
480 + 3s - 8 (480/s) - 24 = 480 [From equations (i) and (ii)]
3s - 3840/s - 24 = 0
3s (s) - 3840 - 24 (s) = 0
3s2 - 24s - 3840 = 0
(3s2 - 24s - 3840)/3 = 0
s2 - 8s - 1280 = 0
Thus, the quadratic equation is s2 - 8s - 1280 = 0, where s is the speed of the train.
☛ Check: NCERT Solutions for Class 10 Maths Chapter 4
Video Solution:
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 meters) 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 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.
NCERT Solutions Class 10 Maths Chapter 4 Exercise 4.1 Question 2
Summary:
The quadratic equations for (i) The area of a rectangular plot is 528 m². 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. (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 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 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 are i) 2x2 + x − 528 = 0 , where x is the breadth of the rectangular plot , ii) x2 + x − 306 = 0 where x is a positive integer, iii) x2 + 32x − 273 = 0 where x is the present age of Rohan and, iv) s2 − 8s − 1280 = 0 where s is the speed of the train.
☛ Related Questions:
- Represent the following situation in the form of quadratic equations:(i) The area of a rectangular plot is 528 m². 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.
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