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Quadratic Equations

Long Answer Type

181. Seven years ago Varun’s age was five times the square of Swati’s age. Three years hence Swati’s age will be two fifth of Varun’s age. Find their present ages.

Seven years ago, let Swati’s age be x years.
Then, seven years ago Varun’s age was 5x² years.
∴ Swati’s present age = (x + 7) years
Varun’s present age = (5x² + 7) years
Three years hence,
Swati’s age = (x + 7 + 3) years
= (x + 10) years
and Varun’s age = (5x² + 7 + 3) years
= (5x² + 10) years
According to given condition,
∴ $straight x plus 10 space equals space 2 over 5 left parenthesis 5 straight x squared plus 10 right parenthesis$
$rightwards double arrow space space space space space space space space space space space space space space space space straight x plus 10 space equals space 2 straight x squared plus 4 rightwards double arrow space space space space space space space space 2 straight x squared minus straight x minus 6 space equals space 0 rightwards double arrow space space space 2 straight x squared minus 4 straight x plus 3 straight x minus 6 space equals space 0 rightwards double arrow space space space 2 straight x left parenthesis straight x minus 2 right parenthesis space plus space 3 left parenthesis straight x minus 2 right parenthesis space equals space 0 rightwards double arrow space space space space space space space space left parenthesis 2 straight x plus 3 right parenthesis thin space left parenthesis straight x minus 2 right parenthesis space equals space 0 rightwards double arrow space space space space space space space space space space space space space space space space space space space space space space space space space space space straight x minus 2 space equals space 0 space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space left square bracket because space space space space 2 straight x plus 3 not equal to 0 space as space straight x greater than 0 right square bracket rightwards double arrow space space space space space space space space space space space space space space space space space space space space space straight x space equals space 2 space space space space space space space space space space space space space space space space space$
Hence, Swati's present age = (2 + 7) years = 9 years
Varun's present age = $space left parenthesis 5 cross times 2 squared plus 7 right parenthesis space years space equals space 27 space years.$

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182. The sum of ages of a father and his son is 45 years. Five years ago, the product of their ages (in years) was 124. Determine their present ages.

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183. One year ago, the father was 8 times as old as his son. Now his age is the square of his son’s age. Find their present ages.

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184. Two years ago, a man's age was three times the square of his son's age. In three years time, his age will be four times his son's age. Find their present ages.

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185. The age of a Father is twice the square of the age of his son. Eight years hence the age of the father will be 4 years more than three times the age of his son. Find their present ages.

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186. The hypotenuse of a right angled triangle is 6 metres more than twice the shortest side. If the third side is 2 m less than the hypotenuse, find the sides of the triangle.

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187. The hypotenuse of a right triangle is 1 m less than twice the shortest side. If the third side is 1 m more than the shortest side, find the sides of the triangle.

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188. The hypotenuse of a right triangle is 25 cm. The difference between the lengths of the other two sides of the triangle is 5 cm. Find the lengths of these sides.

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189. The length of the hypotenuse of a right triangle exceeds the base by 2 cm and exceeds twice the length of altitude by 1 cm. Find each side.

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190. Manisha wishes to fit three rods together in the shape of a right triangle. The hypotenuse is to be 2 cm longer than the base and 4 cm longer than the altitude. What should be the length of the rods?

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