RD Sharma Class 11 Solutions Chapter - Mathematical Reasoning | Mathematics (Maths) Class 11 - Commerce PDF Download

 Page 1


31. Mathematical Reasoning
Exercise 31.1
1 A. Question
Find out which of the following sentences are statements and which are not. Justify your answer.
Listen to me, Ravi!
Answer
Concept Used: A statement is an assertive (declarative) sentence if it is either true or false but not both.
The sentence “Listen to me, Ravi ! “ is an exclamatory sentence.
Hence, it is not a statement.
1 B. Question
Find out which of the following sentences are statements and which are not. Justify your answer.
Every set is a finite set.
Answer
Concept Used: A statement is an assertive (declarative) sentence if it is either true or false but not both.
So, This sentence is always false, because there are sets which are not finite.
Hence, it is a statement.
1 C. Question
Find out which of the following sentences are statements and which are not. Justify your answer.
Two non-empty sets have always a non-empty intersection.
Answer
Concept Used: A statement is an assertive (declarative) sentence if it is either true or false but not both.
So, This sentence is always false, because there are non-empty sets whose intersection is empty.
Hence, it is a statement.
1 D. Question
Find out which of the following sentences are statements and which are not. Justify your answer.
The cat pussy is black.
Answer
Concept Used: A statement is an assertive (declarative) sentence if it is either true or false but not both.
Here, Some cats are black, and some cat is not black, So, the given sentence may or may not be true.
Hence, it is not a statement.
1 E. Question
Find out which of the following sentences are statements and which are not. Justify your answer.
Are all circles round?
Answer
Concept Used: A statement is an assertive (declarative) sentence if it is either true or false but not both.
Here, The given sentence is an interrogative sentence.
Hence, it is not a statement.
Page 2


31. Mathematical Reasoning
Exercise 31.1
1 A. Question
Find out which of the following sentences are statements and which are not. Justify your answer.
Listen to me, Ravi!
Answer
Concept Used: A statement is an assertive (declarative) sentence if it is either true or false but not both.
The sentence “Listen to me, Ravi ! “ is an exclamatory sentence.
Hence, it is not a statement.
1 B. Question
Find out which of the following sentences are statements and which are not. Justify your answer.
Every set is a finite set.
Answer
Concept Used: A statement is an assertive (declarative) sentence if it is either true or false but not both.
So, This sentence is always false, because there are sets which are not finite.
Hence, it is a statement.
1 C. Question
Find out which of the following sentences are statements and which are not. Justify your answer.
Two non-empty sets have always a non-empty intersection.
Answer
Concept Used: A statement is an assertive (declarative) sentence if it is either true or false but not both.
So, This sentence is always false, because there are non-empty sets whose intersection is empty.
Hence, it is a statement.
1 D. Question
Find out which of the following sentences are statements and which are not. Justify your answer.
The cat pussy is black.
Answer
Concept Used: A statement is an assertive (declarative) sentence if it is either true or false but not both.
Here, Some cats are black, and some cat is not black, So, the given sentence may or may not be true.
Hence, it is not a statement.
1 E. Question
Find out which of the following sentences are statements and which are not. Justify your answer.
Are all circles round?
Answer
Concept Used: A statement is an assertive (declarative) sentence if it is either true or false but not both.
Here, The given sentence is an interrogative sentence.
Hence, it is not a statement.
1 F. Question
Find out which of the following sentences are statements and which are not. Justify your answer.
All triangles have three sides.
Answer
Concept Used: A statement is an assertive (declarative) sentence if it is either true or false but not both.
The given sentence is a true declarative sentence.
Hence, it is a true statement.
1 G. Question
Find out which of the following sentences are statements and which are not. Justify your answer.
Every rhombus is a square.
Answer
Concept Used: A statement is an assertive (declarative) sentence if it is either true or false but not both.
So, This sentence is always false, because Rhombuses are not a square.
Hence, it is a statement.
1 H. Question
Find out which of the following sentences are statements and which are not. Justify your answer.
x
2
 + 5|x| + 6 = 0 has no real roots.
Answer
Concept Used: A statement is an assertive (declarative) sentence if it is either true or false but not both.
Here, If x>0,
x
2
+5|x|+6=0
x
2
+5x+6=0
x = -3 or x = -2
But, Since x>0, the equation has no roots.
If x<0,
x
2
+5|x|+6=0
x
2
 - 5x+6=0
But, Since x<0, the equation has no real roots.
Hence,it is a statement.
1 I. Question
Find out which of the following sentences are statements and which are not. Justify your answer.
This sentences is a statement
Answer
Concept Used: A statement is an assertive (declarative) sentence if it is either true or false but not both.
Here, We cannot find the truth values of this sentence, because either value contradict the sense of the
sentence.
Hence, it is not a statement.
Page 3


31. Mathematical Reasoning
Exercise 31.1
1 A. Question
Find out which of the following sentences are statements and which are not. Justify your answer.
Listen to me, Ravi!
Answer
Concept Used: A statement is an assertive (declarative) sentence if it is either true or false but not both.
The sentence “Listen to me, Ravi ! “ is an exclamatory sentence.
Hence, it is not a statement.
1 B. Question
Find out which of the following sentences are statements and which are not. Justify your answer.
Every set is a finite set.
Answer
Concept Used: A statement is an assertive (declarative) sentence if it is either true or false but not both.
So, This sentence is always false, because there are sets which are not finite.
Hence, it is a statement.
1 C. Question
Find out which of the following sentences are statements and which are not. Justify your answer.
Two non-empty sets have always a non-empty intersection.
Answer
Concept Used: A statement is an assertive (declarative) sentence if it is either true or false but not both.
So, This sentence is always false, because there are non-empty sets whose intersection is empty.
Hence, it is a statement.
1 D. Question
Find out which of the following sentences are statements and which are not. Justify your answer.
The cat pussy is black.
Answer
Concept Used: A statement is an assertive (declarative) sentence if it is either true or false but not both.
Here, Some cats are black, and some cat is not black, So, the given sentence may or may not be true.
Hence, it is not a statement.
1 E. Question
Find out which of the following sentences are statements and which are not. Justify your answer.
Are all circles round?
Answer
Concept Used: A statement is an assertive (declarative) sentence if it is either true or false but not both.
Here, The given sentence is an interrogative sentence.
Hence, it is not a statement.
1 F. Question
Find out which of the following sentences are statements and which are not. Justify your answer.
All triangles have three sides.
Answer
Concept Used: A statement is an assertive (declarative) sentence if it is either true or false but not both.
The given sentence is a true declarative sentence.
Hence, it is a true statement.
1 G. Question
Find out which of the following sentences are statements and which are not. Justify your answer.
Every rhombus is a square.
Answer
Concept Used: A statement is an assertive (declarative) sentence if it is either true or false but not both.
So, This sentence is always false, because Rhombuses are not a square.
Hence, it is a statement.
1 H. Question
Find out which of the following sentences are statements and which are not. Justify your answer.
x
2
 + 5|x| + 6 = 0 has no real roots.
Answer
Concept Used: A statement is an assertive (declarative) sentence if it is either true or false but not both.
Here, If x>0,
x
2
+5|x|+6=0
x
2
+5x+6=0
x = -3 or x = -2
But, Since x>0, the equation has no roots.
If x<0,
x
2
+5|x|+6=0
x
2
 - 5x+6=0
But, Since x<0, the equation has no real roots.
Hence,it is a statement.
1 I. Question
Find out which of the following sentences are statements and which are not. Justify your answer.
This sentences is a statement
Answer
Concept Used: A statement is an assertive (declarative) sentence if it is either true or false but not both.
Here, We cannot find the truth values of this sentence, because either value contradict the sense of the
sentence.
Hence, it is not a statement.
1 J. Question
Find out which of the following sentences are statements and which are not. Justify your answer.
Is the earth round?
Answer
Concept Used: A statement is an assertive (declarative) sentence if it is either true or false but not both.
The given sentence is an interrogative sentence.
Hence, it is not a statement.
1 K. Question
Find out which of the following sentences are statements and which are not. Justify your answer.
Go!
Answer
Concept Used: A statement is an assertive (declarative) sentence if it is either true or false but not both.
The given sentence “Go!” is an exclamatory sentence.
Hence, it is not a statement.
1 L. Question
Find out which of the following sentences are statements and which are not. Justify your answer.
The real number x is less than 2.
Answer
Concept Used: A statement is an assertive (declarative) sentence if it is either true or false but not both.
It is not a statement because its truth and false value cannot be determined without knowing the value of x.
Hence, It is not a statement.
1 M. Question
Find out which of the following sentences are statements and which are not. Justify your answer.
There are 35 days in a month.
Answer
Concept Used: A statement is an assertive (declarative) sentence if it is either true or false but not both.
The sentence “There are 35 days in a month” is a false declarative statement, so it is a false statement.
Hence, It is a statement.
1 N. Question
Find out which of the following sentences are statements and which are not. Justify your answer.
Mathematics is difficult.
Answer
Concept Used: A statement is an assertive (declarative) sentence if it is either true or false but not both.
The given sentence is true for some people, and it is not true for some people, So the given sentence may or
may not be true.
Hence, It is not a statement.
1 O. Question
Find out which of the following sentences are statements and which are not. Justify your answer.
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31. Mathematical Reasoning
Exercise 31.1
1 A. Question
Find out which of the following sentences are statements and which are not. Justify your answer.
Listen to me, Ravi!
Answer
Concept Used: A statement is an assertive (declarative) sentence if it is either true or false but not both.
The sentence “Listen to me, Ravi ! “ is an exclamatory sentence.
Hence, it is not a statement.
1 B. Question
Find out which of the following sentences are statements and which are not. Justify your answer.
Every set is a finite set.
Answer
Concept Used: A statement is an assertive (declarative) sentence if it is either true or false but not both.
So, This sentence is always false, because there are sets which are not finite.
Hence, it is a statement.
1 C. Question
Find out which of the following sentences are statements and which are not. Justify your answer.
Two non-empty sets have always a non-empty intersection.
Answer
Concept Used: A statement is an assertive (declarative) sentence if it is either true or false but not both.
So, This sentence is always false, because there are non-empty sets whose intersection is empty.
Hence, it is a statement.
1 D. Question
Find out which of the following sentences are statements and which are not. Justify your answer.
The cat pussy is black.
Answer
Concept Used: A statement is an assertive (declarative) sentence if it is either true or false but not both.
Here, Some cats are black, and some cat is not black, So, the given sentence may or may not be true.
Hence, it is not a statement.
1 E. Question
Find out which of the following sentences are statements and which are not. Justify your answer.
Are all circles round?
Answer
Concept Used: A statement is an assertive (declarative) sentence if it is either true or false but not both.
Here, The given sentence is an interrogative sentence.
Hence, it is not a statement.
1 F. Question
Find out which of the following sentences are statements and which are not. Justify your answer.
All triangles have three sides.
Answer
Concept Used: A statement is an assertive (declarative) sentence if it is either true or false but not both.
The given sentence is a true declarative sentence.
Hence, it is a true statement.
1 G. Question
Find out which of the following sentences are statements and which are not. Justify your answer.
Every rhombus is a square.
Answer
Concept Used: A statement is an assertive (declarative) sentence if it is either true or false but not both.
So, This sentence is always false, because Rhombuses are not a square.
Hence, it is a statement.
1 H. Question
Find out which of the following sentences are statements and which are not. Justify your answer.
x
2
 + 5|x| + 6 = 0 has no real roots.
Answer
Concept Used: A statement is an assertive (declarative) sentence if it is either true or false but not both.
Here, If x>0,
x
2
+5|x|+6=0
x
2
+5x+6=0
x = -3 or x = -2
But, Since x>0, the equation has no roots.
If x<0,
x
2
+5|x|+6=0
x
2
 - 5x+6=0
But, Since x<0, the equation has no real roots.
Hence,it is a statement.
1 I. Question
Find out which of the following sentences are statements and which are not. Justify your answer.
This sentences is a statement
Answer
Concept Used: A statement is an assertive (declarative) sentence if it is either true or false but not both.
Here, We cannot find the truth values of this sentence, because either value contradict the sense of the
sentence.
Hence, it is not a statement.
1 J. Question
Find out which of the following sentences are statements and which are not. Justify your answer.
Is the earth round?
Answer
Concept Used: A statement is an assertive (declarative) sentence if it is either true or false but not both.
The given sentence is an interrogative sentence.
Hence, it is not a statement.
1 K. Question
Find out which of the following sentences are statements and which are not. Justify your answer.
Go!
Answer
Concept Used: A statement is an assertive (declarative) sentence if it is either true or false but not both.
The given sentence “Go!” is an exclamatory sentence.
Hence, it is not a statement.
1 L. Question
Find out which of the following sentences are statements and which are not. Justify your answer.
The real number x is less than 2.
Answer
Concept Used: A statement is an assertive (declarative) sentence if it is either true or false but not both.
It is not a statement because its truth and false value cannot be determined without knowing the value of x.
Hence, It is not a statement.
1 M. Question
Find out which of the following sentences are statements and which are not. Justify your answer.
There are 35 days in a month.
Answer
Concept Used: A statement is an assertive (declarative) sentence if it is either true or false but not both.
The sentence “There are 35 days in a month” is a false declarative statement, so it is a false statement.
Hence, It is a statement.
1 N. Question
Find out which of the following sentences are statements and which are not. Justify your answer.
Mathematics is difficult.
Answer
Concept Used: A statement is an assertive (declarative) sentence if it is either true or false but not both.
The given sentence is true for some people, and it is not true for some people, So the given sentence may or
may not be true.
Hence, It is not a statement.
1 O. Question
Find out which of the following sentences are statements and which are not. Justify your answer.
All real numbers are complex numbers.
Answer
Concept Used: A statement is an assertive (declarative) sentence if it is either true or false but not both.
The given sentence is always true.
Hence, It is a statement.
1 P. Question
Find out which of the following sentences are statements and which are not. Justify your answer.
The product of (-1) and 8 is 8.
Answer
Concept Used: A statement is an assertive (declarative) sentence if it is either true or false but not both.
The given sentence is always false because,
(-1) × (8) = -8
Hence, It is a statement.
2. Question
Give three examples of sentences which are not statements. Give reasons for the answers.
Answer
(i) “Who is the Chancellor of your University”?
Concept Used: A statement is an assertive (declarative) sentence if it is either true or false but not both.
The given sentence is an interrogative sentence.
Hence, It is not a statement.
(ii) There are 31 days in a month
Concept Used: A statement is an assertive (declarative) sentence if it is either true or false but not both.
The given sentence is true for some particular months, and it is not true for others, so it may be true of false.
Hence, It is a not statement.
(iii) Hurray! We won the match.
Concept Used: A statement is an assertive (declarative) sentence if it is either true or false but not both.
The given sentence is in an exclamatory sentence.
Hence, It is a not statement.
Exercise 31.2
1 A. Question
Write the negation of the following statement:
Banglore is the capital of Karnataka.
Answer
Negation of statement p is "not p." The negation of p is symbolized by "~p." The truth value of ~p is the
opposite of the truth value of p.
The negation of the statement is “Banglore is not the capital of Karnataka.”
1 B. Question
Write the negation of the following statement:
Page 5


31. Mathematical Reasoning
Exercise 31.1
1 A. Question
Find out which of the following sentences are statements and which are not. Justify your answer.
Listen to me, Ravi!
Answer
Concept Used: A statement is an assertive (declarative) sentence if it is either true or false but not both.
The sentence “Listen to me, Ravi ! “ is an exclamatory sentence.
Hence, it is not a statement.
1 B. Question
Find out which of the following sentences are statements and which are not. Justify your answer.
Every set is a finite set.
Answer
Concept Used: A statement is an assertive (declarative) sentence if it is either true or false but not both.
So, This sentence is always false, because there are sets which are not finite.
Hence, it is a statement.
1 C. Question
Find out which of the following sentences are statements and which are not. Justify your answer.
Two non-empty sets have always a non-empty intersection.
Answer
Concept Used: A statement is an assertive (declarative) sentence if it is either true or false but not both.
So, This sentence is always false, because there are non-empty sets whose intersection is empty.
Hence, it is a statement.
1 D. Question
Find out which of the following sentences are statements and which are not. Justify your answer.
The cat pussy is black.
Answer
Concept Used: A statement is an assertive (declarative) sentence if it is either true or false but not both.
Here, Some cats are black, and some cat is not black, So, the given sentence may or may not be true.
Hence, it is not a statement.
1 E. Question
Find out which of the following sentences are statements and which are not. Justify your answer.
Are all circles round?
Answer
Concept Used: A statement is an assertive (declarative) sentence if it is either true or false but not both.
Here, The given sentence is an interrogative sentence.
Hence, it is not a statement.
1 F. Question
Find out which of the following sentences are statements and which are not. Justify your answer.
All triangles have three sides.
Answer
Concept Used: A statement is an assertive (declarative) sentence if it is either true or false but not both.
The given sentence is a true declarative sentence.
Hence, it is a true statement.
1 G. Question
Find out which of the following sentences are statements and which are not. Justify your answer.
Every rhombus is a square.
Answer
Concept Used: A statement is an assertive (declarative) sentence if it is either true or false but not both.
So, This sentence is always false, because Rhombuses are not a square.
Hence, it is a statement.
1 H. Question
Find out which of the following sentences are statements and which are not. Justify your answer.
x
2
 + 5|x| + 6 = 0 has no real roots.
Answer
Concept Used: A statement is an assertive (declarative) sentence if it is either true or false but not both.
Here, If x>0,
x
2
+5|x|+6=0
x
2
+5x+6=0
x = -3 or x = -2
But, Since x>0, the equation has no roots.
If x<0,
x
2
+5|x|+6=0
x
2
 - 5x+6=0
But, Since x<0, the equation has no real roots.
Hence,it is a statement.
1 I. Question
Find out which of the following sentences are statements and which are not. Justify your answer.
This sentences is a statement
Answer
Concept Used: A statement is an assertive (declarative) sentence if it is either true or false but not both.
Here, We cannot find the truth values of this sentence, because either value contradict the sense of the
sentence.
Hence, it is not a statement.
1 J. Question
Find out which of the following sentences are statements and which are not. Justify your answer.
Is the earth round?
Answer
Concept Used: A statement is an assertive (declarative) sentence if it is either true or false but not both.
The given sentence is an interrogative sentence.
Hence, it is not a statement.
1 K. Question
Find out which of the following sentences are statements and which are not. Justify your answer.
Go!
Answer
Concept Used: A statement is an assertive (declarative) sentence if it is either true or false but not both.
The given sentence “Go!” is an exclamatory sentence.
Hence, it is not a statement.
1 L. Question
Find out which of the following sentences are statements and which are not. Justify your answer.
The real number x is less than 2.
Answer
Concept Used: A statement is an assertive (declarative) sentence if it is either true or false but not both.
It is not a statement because its truth and false value cannot be determined without knowing the value of x.
Hence, It is not a statement.
1 M. Question
Find out which of the following sentences are statements and which are not. Justify your answer.
There are 35 days in a month.
Answer
Concept Used: A statement is an assertive (declarative) sentence if it is either true or false but not both.
The sentence “There are 35 days in a month” is a false declarative statement, so it is a false statement.
Hence, It is a statement.
1 N. Question
Find out which of the following sentences are statements and which are not. Justify your answer.
Mathematics is difficult.
Answer
Concept Used: A statement is an assertive (declarative) sentence if it is either true or false but not both.
The given sentence is true for some people, and it is not true for some people, So the given sentence may or
may not be true.
Hence, It is not a statement.
1 O. Question
Find out which of the following sentences are statements and which are not. Justify your answer.
All real numbers are complex numbers.
Answer
Concept Used: A statement is an assertive (declarative) sentence if it is either true or false but not both.
The given sentence is always true.
Hence, It is a statement.
1 P. Question
Find out which of the following sentences are statements and which are not. Justify your answer.
The product of (-1) and 8 is 8.
Answer
Concept Used: A statement is an assertive (declarative) sentence if it is either true or false but not both.
The given sentence is always false because,
(-1) × (8) = -8
Hence, It is a statement.
2. Question
Give three examples of sentences which are not statements. Give reasons for the answers.
Answer
(i) “Who is the Chancellor of your University”?
Concept Used: A statement is an assertive (declarative) sentence if it is either true or false but not both.
The given sentence is an interrogative sentence.
Hence, It is not a statement.
(ii) There are 31 days in a month
Concept Used: A statement is an assertive (declarative) sentence if it is either true or false but not both.
The given sentence is true for some particular months, and it is not true for others, so it may be true of false.
Hence, It is a not statement.
(iii) Hurray! We won the match.
Concept Used: A statement is an assertive (declarative) sentence if it is either true or false but not both.
The given sentence is in an exclamatory sentence.
Hence, It is a not statement.
Exercise 31.2
1 A. Question
Write the negation of the following statement:
Banglore is the capital of Karnataka.
Answer
Negation of statement p is "not p." The negation of p is symbolized by "~p." The truth value of ~p is the
opposite of the truth value of p.
The negation of the statement is “Banglore is not the capital of Karnataka.”
1 B. Question
Write the negation of the following statement:
It rained on July 4, 2005.
Answer
Negation of statement p is "not p." The negation of p is symbolized by "~p." The truth value of ~p is the
opposite of the truth value of p.
The negation of the statement is “It didn’t rain on July 4, 2005”.
1 C. Question
Write the negation of the following statement:
Ravish is honest.
Answer
Negation of statement p is "not p." The negation of p is symbolized by "~p." The truth value of ~p is the
opposite of the truth value of p.
So,
The negation of the statement is “Ravish is not honest.”
1 D. Question
Write the negation of the following statement:
The earth is round.
Answer
Negation of statement p is "not p." The negation of p is symbolized by "~p." The truth value of ~p is the
opposite of the truth value of p.
So, The negation of the statement is “The earth is not round.”
1 E. Question
Write the negation of the following statement:
The sun is cold.
Answer
Negation of statement p is "not p." The negation of p is symbolized by "~p." The truth value of ~p is the
opposite of the truth value of p.
So, The negation of the statement is “The sun is not cold.”
2 A. Question
All birds sing.
Answer
Negation of statement p is "not p." The negation of p is symbolized by "~p." The truth value of ~p is the
opposite of the truth value of p.
So, The negation of the statement is “Not all birds sing.”
2 B. Question
Some even integers are prime.
Answer
Negation of statement p is "not p." The negation of p is symbolized by "~p." The truth value of ~p is the
opposite of the truth value of p.
So, The negation of the statement is “No Even integer is prime.”
2 C. Question