NCERT Solutions class-11 Maths Miscellaneous

Miscellaneous Exercise

1. Write the negation of the following statements:

(i) For every positive real number , the number is also positive.

(ii) All cats scratch.

(iii) For every real number , either or

(iv) There exists a number such that

Ans. (i) There exists a positive real number such that is not positive.

(ii) There exists a cat which does not scratch.

(iii) There exists a real number such that or

(iv) There does not exist a number such that

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2. State the converse and contrapositive of each of the following statements:

(i) A positive integer is prime only if it has no divisors other than 1 and itself.

(ii) I go to a beach whenever it is a sunny day.

(iii) If it is not outside, then you feel thirsty.

Ans. (i) Contrapositive: If a positive integer has divisors other than 1 and itself then it is not prime.

Converse: If a positive integer has no divisors other than 1 and itself then it is a prime.

(ii) Contrapositive: If I do not go to a beach then it is not a sunny day.

Converse: If I go to a beach then it is a sunny day.

(iii) Contrapositive: If you do not feel thirsty then it is not hot outside.

Converse: If you feel thirsty then it is hot outside.

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3. Rewrite each of the following statements in the form “ it and only if ”.

(i) It is necessary to have a password to log on to the server.

(ii) There is traffic Jam whenever it rains.

(iii) You can access the website only if you pay a subscription fee.

Ans. (i) If you log on to the server then you have a password.

(ii) If it rains, then there is traffic jam.

(iii) If you can access the website, then you pay a subscription fee.

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4. Rewrite each of the following statements in the form “ it and only if ”.

(i) If you watch television, then your mind is free and if your mind is free, then you watch television.

(ii) For you to get an A grade, it is necessary and sufficient that you do all the homework regularly.

(iii) If a quadrilateral is equiangular, then it is a rectangle and if a quadrilateral is a rectangle, then it is equiangular.

Ans. (i) You watch television if and only if your mind is free.

(ii) You get an A grade if and only if you do all the homework regularly.

(iii) A quadrilateral is equiangular if and only if it is a rectangle.

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5. Given below are two statements:

25 is a multiple of 5.

25 is a multiple of 8.

Write the compound statements connecting these two statements with “and” and “or”. In both cases check the validity of the compound statement.

Ans. The compound statement with “and” is: 25 is a multiple of 5 and 8.

Since, is true and is false therefore the compound statement with “and” is not true.

Therefore, the statement “ and ” is not valid.

Now the compound statement with “or” is: 25 is a multiple of 5 or 8.

Since, is true and is false therefore the compound statement with “or” is true.

Therefore, the statement “ and ” is valid.

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6. Check the validity of the statements given below by the method given against it:

(i) The sum of an irrational number and a rational number is irrational (by contradiction method)

(ii) If is a real number with then (by contradiction method)

Ans. (i) Let us assume that is not true.

Sum of an irrational and a rational number is not irrational.

There exists an irrational number and a rational number such that is not irrational.

(say) is a rational number.

is rational.

But is irrational, which is contradiction, therefore our supposition is wrong.

Therefore, is true.

(ii) Let and be the statements given by

is a real number with

If possible let is not true, then is true.

and is true.

is a real number with and which is contradiction, therefore our supposition is wrong.

Therefore, is true.

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7. Write the following statement in five different ways, conveying the same meaning:

If a triangle is equiangular, then it is an obtuse angled triangle.

Ans. (i) A triangle is equiangular implies that it is an obtuse angled triangle.

(ii) A triangle is equiangular only if it an obtuse angled triangle.

(iii) For a triangle to be equiangular it is necessary it is an obtuse angled triangle.

(iv) For a triangle to be obtuse angled triangle it is sufficient that the triangle is equiangular.

(v) Is a triangle is not equiangular then it is not an obtuse angled triangle.