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Class 11 RD Sharma Solutions – Chapter 31 Derivatives – Exercise 31.1

  • Last Updated : 21 Dec, 2020

Question 1. Find out which of the following sentences are statements and which are not. Justify your answer.

(i) Listen to me, Ravi!

(ii) Every set is a finite set.

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(iii) Two non-empty sets have always a non-empty intersection.

(iv) The cat pussy is black.

(v) Are all circles round?

(vi) All triangles have three sides.

(vii) Every rhombus is a square.

(viii) x2 + 5|x| + 6 = 0 has no real roots.

(ix) This sentences is a statement.

(x) Is the earth round?

(xi) Go!

(xii) The real number x is less than 2.

(xiii) There are 35 days in a month.

(xiv) Mathematics is difficult. 

(xv) All real numbers are complex numbers. 

(xvi) The product of (-1) and 8 is 8.

Solution:

Note: Any declarative sentence which can have either an affirmative(true/ yes) or negative (false/no), but not both answers can be considered as a statement or a proposition. 

(i) Exclamatory sentences are not statements. Therefore, the line, “Listen to me, Ravi!” is not a statement.

(ii) There exists infinite sets also, and therefore, this sentence is false. Hence, it is a statement.

(iii) This sentence will always hold a false value, because there are non-empty sets whose intersection is empty. Therefore, it is a statement.



(iv) Some of the cats are not black, hence, this sentence may have a yes or no answer depending on the cat.  Hence, it is not a statement.

(v) Interrogative sentences are not statements. Therefore, the question, “Are all circles round?” is not a statement.

(vi) Every triangle has three sides, therefore, this is a declarative statement with a true value. 

(vii) A rhombus with not all equal sides are not squares. Therefore, this sentence is false and hence a statement,

(viii) Solving the equation, we have, 

For x > 0

x2 + 5 |x| + 6 = 0

⇒ x2 + 5x + 6 = 0

We obtain, 

⇒ x = – 3 or x = – 2



But, since x > 0, the equation has no roots.

For x < 0

x2 + 5 |x| + 6 = 0

⇒ x2 – 5x + 6 = 0

There are no possible real roots, which makes the sentence to be always true. Hence, it is a statement.

(ix)The sentence “This sentence is a statement: cannot be assigned a truth value of either true or false. Because either assignment contradicts the sense of the sentence.

(x) Interrogative sentences are not statements. Therefore, the question, “Is the earth round?”  is not a statement.

(xi) Exclamatory sentences are not statements. Therefore, the line, “Go!” is not a statement.

(xii) The sentence may hold either true or false value depending on the value of x. Therefore, it is not a statement, because its value can’t be known without knowing the value of x.

(xiii) There are either 30 or 31 days in a month, therefore, this statement is false. 

(xiv) Some people may find mathematics difficult, and some may like it. Therefore, the true or false value is dependent on the person and hence, it is not a statement.  

(xv) This sentence is always true. Hence, it is a statement.

(xvi) This sentence is always false, because the product of -1 and 8 is -8. Hence, it is a statement.

Question 2. Give three examples of sentences which are not statements. Give reasons for the answers.

Solution:

A statement or a proposition is an assertive (or a declarative) sentence which is either true or false but not both.

Example 1: “Aha! What a bird ! ” – Exclamatory sentences are not statements. Therefore, this line is not a statement.

Example 2: “Are you enrolled on GFG?” – Interrogative sentences are not statements. Therefore, this question, is not a statement.

Example 3: “Girls are more sweet than boys” – This depends on the each person’s outlook, hence it may vary. Therefore, it is not a statement. 

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