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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.

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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.

Let present age of son = x years
Two years ago, Age of son = (x – 2) years
Age of man = 3(x – 2)²∴ Man’s present age = 3(x – 2)² + 2
= 3(x² – 4x + 4) + 2
= 3x² – 12x + 14
Age of man = 3(x – 2)²∴ Man’s present age = 3(x – 2)² + 2
= 3(x² – 4x + 4) + 2
= 3x² – 12x + 14
After 3 year
The age of son = (x + 3) years
Age of man $equals space left parenthesis 3 straight x squared minus 12 straight x plus 14 right parenthesis plus 3 equals space left parenthesis 3 straight x squared minus 12 straight x plus 17 right parenthesis space years$
According to given condition,
$3 straight x squared minus 12 straight x plus 17 space equals space 4 left parenthesis straight x plus 3 right parenthesis$
$rightwards double arrow space space space space space space space space space space 3 straight x squared minus 12 straight x plus 17 space equals space 4 straight x plus 12 rightwards double arrow space space space space space space space space space space space 3 straight x squared minus 16 straight x plus 5 space space space equals space space space 0 rightwards double arrow space space space space space space space space space space left parenthesis straight x space minus 5 right parenthesis space left parenthesis 3 straight x minus 1 right parenthesis space equals space 0 rightwards double arrow space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space straight x space equals space 5 space space or space straight x space equals space 1 third$
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∴ The present age of man
$equals space 3 straight x squared minus 12 straight x plus 14 equals space 3 cross times 5 squared minus 12 cross times 5 space plus space 14 equals space 29 space years$

And, the present age of son = x = 5 years. Ans.

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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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