RBSE Solutions for Class 11 Maths Chapter 14 Mathematical Reasoning Miscellaneous Exercise

Rajasthan Board RBSE Solutions for Class 11 Maths Chapter 14 Mathematical Reasoning Miscellaneous Exercise Textbook Exercise Questions and Answers.

RBSE Class 11 Maths Solutions Chapter 14 Mathematical Reasoning Miscellaneous Exercise

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) Given statement: For every positive real number x, the number x -1 is also positive.
Negation : For every positive real number x, the number x -1 is not a positive number.
(ii) Given Statement: All cats scratch.
Negation : All cats does not scratch
or
There exist at least one cat who do not scratch.
(iii) Given statement: For every real number x, either x > 1 or x <1
Negation : For every real number x neither x > 1 nor x < 1 or
There exists at least one real number x, for which neither x < 1, nor;c<l
(iv) Given Statement : There exist a number x such that 0 < x < 1
Negation : There exist a number x, for which 0 < x < 1 is not true.
or
There exist no number x for which 0 < x < 1

Question 2.
State the converse and contrapositive 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) Given statement: A positive integer is prime Only if it has no divisors other than 1 and itself.
Component statement p : A positive integer is prime number. ,
Component statement q : It has no divisors other than 1 and itself.
Converse : A positive integer is not prime number if it has not divisors other than 1 and itself.
Contrapositive: A positive integer is not prime number if it has divisors other than 1 and itself.

(ii) Given Statement : I go to a beach whenever it is a Sunny day.
We can rewrite this statement as,:
If there is sunny day then I go to beach.
Component statement p: If there is sunny day Component statement q: I go to a beach.
Converse : If I go to a beach then there is sunny day Contrapositive : If I do not go to a beach then there is no sunny day.

(iii) Given Statement : If it is hot outside, then you feel thirsty
Component statement p: It is hot outside.
Component statement q: You feel thirsty.
Converse: If you feel thirsty then it is not outside. Contrapositive : If you did 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) Given statement: It is necessary to have a password to log on to the server.
Component Statement p: To log on the server. 
Component statement q: It is necessary to have a password. Statement ‘If pthen q’
If log on to the server then to have a password.

(ii) Given statement: There is traffic jam whenever it rains Component statement p : There is rain.
Component statement q : There is traffic jam.
Statement ‘If p then q’
If there is rain then there is traffic jam.

(iii) Given statement: You can access the website only if you pay a subscription fee.
Component statement p: You have pay a subscription fee.
Component statement q : You can access the website. Statement: ‘If p then q'
If you have pay a subscription fee then you can access the website.

Question 4.
Rewrite 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) Given statement: If you watch television, then your mind is free, then you watch television.
Component Statement p : If you watch television, then your mind is free.
Component statement q : If your mind is free, then you watch television.
Statement ‘ p if only if q’
You watch television ‘if and only if’ your mind is free.

(ii) Given statement : For you to get an A grade, it is necessary and sufficient that you do all the homework regurlarly.
Component statement p : You have got an A grade. Component statement q : You do all the homework regularly.
Statement ‘p if and only if q
You will get A grade ‘if and only if’ you do all the homework regularly.

(iii) Given statement: If a quadrilateral is equiangular, then it is a rectangle and if a quadrilateral is a rectangle, then it is equiangular.
Component statement p : If a quadrilateral is equiangular then it is a rectangle.
Component statement q: A quadrilateral is rectangle, then it is an equiangular.
Statement ‘p if and only If q’
A quadrilateral is equiangular ‘if and only if’ it is 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:
Component statements are: 
p: 25 is a multiple of 5
q : 25 is a multiple of 8

Compound statement formed by connective ‘And’ is as follows:
“25 is a multiple of number 5 and 8 ”
Test of validity of compound statement:
Since, component statement q is false and connective is ‘And’.
Thus, above statement is false.
Compound statement for by connective ‘Or’ is as follows 
“25, is a multiple of numbers 5 or 8”

Test of validity of statement:
Since, here connective is ‘Or’
and one statement p is true.
Thus, statement is true.

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) Given statement p : The sum of an irrational number and a rational number is irrational.
Let a be an irrational number and
a = \(\frac{\mathrm{km}}{\mathrm{kn}}\)
and b is a rational number, whereas
b = \(\frac{c}{t}\)
where c and t are integers and some of their factors are not common.
Then, sum of rational number b and irrational number a
(a + b) = \(\frac{\mathrm{km}}{\mathrm{kn}}+\frac{c}{t}\)
(a + b) = \(\frac{k m t+k n c}{k n t}\)
Clearly, if sum (a + b) will be written in the fraction form then factor k will be common in numerator and denominator. Then (a + b) will not be rational, i.e., (a + bj will be irrational.
Thus, given statement is true.
NOTE Sum of rational number 2 and irrational number √2 is 2 + √2 which is. a irrational number.

(ii) Given statement q : If n is a real number such that n > 3 then n² >9
Let n is not greater then number 3. i.e., 3
i.e n ≯ 3
Then, either n = 3 or n < 3 If n = 3 then n² = 9 and If n <3 i.e. n =3 - k, then 
n² = (3 - k)²
⇒ n² = 32 - 6k + k²
⇒ n² = 9 - 6 k + k²
⇒ n² = 9 - k (6 - k)
Clearly, n cannot be more than 9 until n is less than number 3.
Thus, when n > 3 then n² > 9. 

Question 7.
Write the following statement in five different ways, conveying the same meaning. p:If a triangle is equiangular, then it is an obtuse angled triangle.
Answer:
Given Statement: If a triangle is equiangular, then it is an obtuse angled triangle.
Five different ways of given statement conveying the same meaning:
(i) A triangle is equiangular’ It implies that it is obtuse angled triangle.
(ii) A triangle is equiangular, only when it is obtuse angled triangle.
(iii) For equiangular triangle it is necessary that triangle will be obtuse angled triangle.
(iv) To be equiangular triangle it is sufficient that it should be obtuse angled triangle.
(v) If a triangle is not equiangular then it is not obtuse angled triangle.