Class 11 - Mathematics
Mathematical Reasoning - Exercise 14-Misc
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Question 1:
Write the negation of the following statements:
(i) p: For every positive real number x, the number x – 1 is also positive.
(ii) q: All cats scratch.
(iii) r: For every real number x, either x > 1 or x < 1.
(iv) s: There exists a number x such that 0 < x < 1.
Answer:
(i) The negation of statement p is as follows.
There exists a positive real number x, such that x – 1 is not positive.
(ii)The negation of statement q is as follows.
There exists a cat that does not scratch.
(iii) The negation of statement r is as follows.
There exists a real number x, such that neither x > 1 nor x < 1.
(iv) The negation of statement s is as follows. There does not exist a number x, such that
0 < x < 1.
Question 2:
State the converse and contra positive of each of the following statements:
(i) p: A positive integer is prime only if it has no divisors other than 1 and itself.
(ii) q: I go to a beach whenever it is a sunny day.
(iii) r: If it is hot outside, then you feel thirsty.
Answer:
(i) Statement p can be written as follows:
If a positive integer is prime, then it has no divisors other than 1 and itself.
The converse of the statement is as follows:
If a positive integer has no divisors other than 1 and itself, then it is prime.
The contra positive of the statement is as follows:
If positive integer has divisors other than 1 and itself, then it is not prime.
(ii)The given statement can be written as follows:
If it is a sunny day, then I go to a beach.
The converse of the statement is as follows:
If I go to a beach, then it is a sunny day.
The contra positive of the statement is as follows:
If I do not go to a beach, then it is not a sunny day.
(iii) The converse of statement r is as follows:
If you feel thirsty, then it is hot outside.
The contra positive of statement r is as follows.
If you do not feel thirsty, then it is not hot outside.
Question 3:
Write each of the statements in the form “if p, then q”.
(i) p: It is necessary to have a password to log on to the server.
(ii) q: There is traffic jam whenever it rains.
(iii) r: You can access the website only if you pay a subscription fee.
Answer:
(i) Statement p can be written as follows:
If you log on to the server, then you have a password.
(ii) Statement q can be written as follows:
If it rains, then there is a traffic jam.
(iii) Statement r can be written as follows:
If you can access the website, then you pay a subscription fee.
Question 4:
Re write each of the following statements in the form “p if and only if q”.
(i) p: If you watch television, then your mind is free and if your mind is free, then you watch television.
(ii) q: For you to get an A grade, it is necessary and sufficient that you do all the homework regularly.
(iii) r: If a quadrilateral is equiangular, then it is a rectangle and if a quadrilateral is a rectangle, then it is equiangular.
Answer:
(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.
Question 5:
Given below are two statements p: 25 is a multiple of 5. q: 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.
Answer:
The compound statement with ‘And’ is “25 is a multiple of 5 and 8”.
This is a false statement, since 25 is not a multiple of 8.
The compound statement with ‘Or’ is “25 is a multiple of 5 or 8”. This is a true statement, since
25 is not a multiple of 8 but it is a multiple of 5.
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Question 6:
Check the validity of the statements given below by the method given against it.
(i) p: The sum of an irrational number and a rational number is irrational (by contradiction method).
(ii) q: If n is a real number with n > 3, then n2> 9 (by contradiction method).
Answer:
(i) The given statement is as follows.
p: the sum of an irrational number and a rational number is irrational.
Let us assume that the given statement, p, is false. That is, we assume that the sum of an
irrational number and a rational number is rational.
Therefore √a + b/c = d/e, where √a is irrational and b, c, d, e are integers.
d/e – b/c is a rational number and √a is an irrational number.
This is a contradiction. Therefore, our assumption is wrong.
Therefore, the sum of an irrational number and a rational number is rational.
Thus, the given statement is true.
(ii) The given statement, q, is as follows.
If n is a real number with n > 3, then n2> 9.
Let us assume that n is a real number with n > 3, but n2> 9 is not true.
That is, n2< 9
Then, n > 3 and n is a real number.
Squaring both the sides, we obtain n2> 32
=> n2< 9, which is a contradiction, since we have assumed that n2< 9.
Thus, the given statement is true. That is, if n is a real number with n > 3,
then n2> 9.
Question 7:
Write the following statement in five different ways, conveying the same meaning.
p: If triangle is equiangular, then it is an obtuse angled triangle.
Answer:
The given statement can be written in five different ways as follows.
(i) A triangle is equiangular implies that it is an obtuse-angled triangle.
(ii) A triangle is equiangular only if it is an obtuse-angled triangle.
(iii) For a triangle to be equiangular, it is necessary that the triangle is an obtuse-angled
triangle.
(iv) For a triangle to be an obtuse-angled triangle, it is sufficient that the triangle is
equiangular.
(v) If a triangle is not an obtuse-angled triangle, then the triangle is not equiangular.
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