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Linear Inequalities

Short Answer Type

21. Find all pairs of consecutive odd positive integers which are smaller than 10 such that their sum is more than 11.

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22. Find pairs of consecutive even positive integers which are larger than 5 and are such that their sum is less than 20.

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23. The marks obtained by a student of class XI in first terminal and second terminal examination are 62 and 48 respectively. Find the number of minimum marks he should get in the annual examination to have an average of at least 60 marks.

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24. The longest side of a triangle is 3 times the shortest side and the third side is 2 cm shorter than the longest side. If the perimeter of the triangle is at least 61cm, find the minimum length of the shortest side.

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25. A man wants to cut three lengths from a single piece of board of length 91 cm. The second length is to be 3 cm longer than the shortest and the third length is to be twice as long as the shortest. What are the possible lengths of the shortest board if the third piece is to be at least 5 cm longer than the second ?

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26. How many litres of water will have to be added to 1125 litres of the 45%, solution of acid so that the resulting mixture will contain more than 25% but less than 30% acid content?

Quantity of 45% acid solution = 1125 litres

Quantity of acid = $fraction numerator 1125 cross times 45 over denominator 100 end fraction litres space equals space 2025 over 4 space litres$

Suppose x litres of water is added

∴ Quantity of acid = $2025 over 4 space litres$

Total quantity of mixture = (1125 + x) litres

% of acid content = $space space fraction numerator 2025 over denominator 4 space left parenthesis 1125 plus straight x right parenthesis end fraction cross times 100$
According to the problem,

$space space space space space 25 less or equal than fraction numerator 2025 over denominator 4 space left parenthesis 1125 plus straight x right parenthesis end fraction cross times 100 less or equal than 30$

$rightwards double arrow$ $25 less or equal than fraction numerator 50625 over denominator 1125 plus straight x end fraction less or equal than 30$

Now, $25 less or equal than fraction numerator 50625 over denominator 1125 plus straight x end fraction space space rightwards double arrow space space 1 less or equal than fraction numerator 2025 over denominator 1125 plus straight x end fraction space rightwards double arrow space 1125 plus straight x space less or equal than space 2025 space rightwards double arrow space straight x space less or equal than space 900$ ...(i)

Also, $fraction numerator 50625 over denominator 1125 plus straight x end fraction less or equal than 30 space rightwards double arrow space fraction numerator 10125 over denominator 1125 plus straight x end fraction less or equal than 6 space space rightwards double arrow space 10125 space less or equal than space 6750 space plus space 6 straight x space rightwards double arrow space 3375 less or equal than 6 straight x$

$rightwards double arrow$ $6 straight x greater or equal than 3375$

$rightwards double arrow$ $2 straight x greater or equal than 1125 space rightwards double arrow space straight x greater or equal than 562.5$ ...(ii)

From (i) and (ii), we have $562.5 less or equal than straight x less or equal than 900$

Hence, a minimum of 562.5 litres and a maximum of 900 litres of water be added so as to move a solution with 25% to 30% of acid content.

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27. In drilling world’s deepest hole, it was found that the temperature T in degree celsius, x km below the surface of the earth, was given by

straight T space equals space 30 plus space 25 left parenthesis straight x minus 3 right parenthesis comma space space space 3 less than straight x less than 15

At what depth will the temperature be between

200 degree straight C

and

300 degree straight C

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28. IQ of a person is given by the formula:

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where MA is mental age and CA is chronological age. If

80 less or equal than IQ less or equal than 140.

For a group of 12 year children, find the range of their mental age.

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29. In an experiment, a solution of a hydrochloric acid is to be kept between 30° and 35° Celsius. What is the range of the temperature in degree Fahrenheit, if conversion formula is given by

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where C and F represent temperature in degree Celsius and degree Fahrenheit, respectively.

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30. To receive Grade ‘A’ in a course, one must obtain an average of 90 marks or more in five examinations (each of 100 marks). If Sunita’s marks in the first four examinations are 87, 92, 94 and 95, find minimum marks that Sunita must obtain in fifth examination to get grade ‘A’ in the course.

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