Question 2 of 11intermediate🔧 ApplyShort Answer2 marks

Represent the following situations in the form of quadratic equations:

(i)

The area of a rectangular plot is 528 m2528\text{ m}^2. 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.

Answer

Let breadth = x metres. Then length = (2x + 1) metres. Area = Length × Breadth = x(2x + 1) = 528 Therefore, 2x² + x – 528 = 0 is the required quadratic equation.

(ii)

The product of two consecutive positive integers is 306. We need to find the integers.

Answer

Let the first positive integer = x. Then the next consecutive integer = (x + 1). Product = x(x + 1) = 306 Therefore, x² + x – 306 = 0 is the required quadratic equation.

(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.

Answer

Let Rohan's present age = x years. Then his mother's age = (x + 26) years. 3 years from now: Rohan's age = (x + 3), Mother's age = (x + 29) Product = (x + 3)(x + 29) = 360 Simplifying: x² + 32x + 87 = 360 Therefore, x² + 32x – 273 = 0 is the required quadratic equation.

(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.

Answer

Let the uniform speed of train = x km/h. Time taken = 480/x hours. If speed is 8 km/h less, new speed = (x – 8) km/h, new time = 480/(x – 8) hours. According to question: 480/(x – 8) – 480/x = 3 Simplifying: 3840/(x² – 8x) = 3 Therefore, x² – 8x – 1280 = 0 is the required quadratic equation.

Explanation

This question tests the ability to translate word problems into mathematical equations. Each situation requires: (1) identifying the unknown quantity and assigning a variable, (2) expressing related quantities in terms of this variable, (3) forming an equation using the given condition, and (4) simplifying to standard quadratic form ax² + bx + c = 0. The context shows similar examples where equations are simplified to standard form before concluding they are quadratic equations.

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