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Let first positive integer be x So, other positive integer = x + 1 According to question, x 2 + (x + 1) 2 = 365 x 2 + x 2 + 1 + 2x = 365 ⇒ 2x 2 + 2x – 364 = 0 ⇒ x 2 + x – 182 = 0 ⇒ x 2 + 14x – 13x – 182 = 0 ⇒ x(x + 14) – 13(x + 14) = 0 ⇒ (x – 13) (x + 14) = 0 ⇒ x – 13 = 0 or x + 14 = 0 ⇒ x = 13 or x = –14 Here x = – 14 does not exist. Therefore, required positive integers are : x = 13 and x + 1 = 13 + 1 = 14.

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Quadratic Equations

Long Answer Type

21. Find two consecutive positive integers, sum of whose square is 365.

Let first positive integer be x
So, other positive integer = x + 1
According to question,
x² + (x + 1)² = 365 x² + x² + 1 + 2x = 365
⇒ 2x² + 2x – 364 = 0
⇒ x² + x – 182 = 0
⇒ x²+ 14x – 13x – 182 = 0
⇒ x(x + 14) – 13(x + 14) = 0
⇒ (x – 13) (x + 14) = 0
⇒ x – 13 = 0
or x + 14 = 0
⇒ x = 13
or x = –14
Here x = – 14 does not exist.
Therefore, required positive integers are : x = 13 and x + 1 = 13 + 1 = 14.

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22. The altitude of a right triangle is 7 cm less than its base. If the hypotenuse is 13 cm. Find the other two sides.

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Short Answer Type

23. A cottage industry produces a certain number of pottery articles in a day. It was observed on a particular day that the cost of production of each article (in rupees) was 3 more than twice the number of articles produced on that day. If the total cost of production on that day was Rs. 90. Find the number of articles produced and the cost of each article.

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24. Find the roots of the following quadratic equations, if they exist by the method of completing the square:
2x² –7x + 3 = 0

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25. Find the roots of the following quadratic equations, if they exist by the method of completing the square:

2 straight x squared plus straight x minus 4 space equals space 0

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26. Find the roots of the following quadratic equations, if they exist by the method of completing the square:

4 straight x squared plus 4 square root of 3 straight x plus 3 equals 0

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27. Find the roots of the following quadratic equations, if they exist by the method of completing the square:
2x²+x+ 4 = 0.

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Long Answer Type

28. Find the roots of the quadratic equations given in Q. 1, above by applying the quadratic formula.

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29. Find the roots of the factors of equations:

straight x minus 1 over straight x equals 3 comma space space straight x not equal to 0.

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30. The sum of the reciprocals of Rehman’s age (in years) 3 years ago and 5 years from now is

space 1 third.

Find his present age.

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