# Class 11 RD Sharma Solutions – Chapter 31 Derivatives – Exercise 31.2

**Question 1. Write the negation of the following statement:**

**(i) Bangalore is the capital of Karnataka.**

**(ii) It rained on July 4, 2005.**

**(iii) Ravish is honest.**

**(iv) The earth is round.**

**(v) The sun is cold.**

**Solution:**

(i)Bangalore is not the capital of Karnataka or It is false that “Bangalore is the capital of Karnataka.”

(ii)It did not rain on July 4, 2005 or It is false that it rained on July 4, 2005.

(iii)Ravish is dishonest or It is false that “Ravish is honest”.

(iv)The earth is not round or It is false that “The earth is round.”

(v)The sun is not cold or It is false that “The sun is cold.”

**Question 2. Write the negation of the following statement:**

**(i) All birds sing.**

**(ii) Some even integers are prime.**

**(iii) There is a complex number which is not a real number.**

**(iv) I will not go to school.**

**(v) Both the diagonals of a rectangle have the same length.**

**(vi) All policemen are thieves.**

**Solution:**

(i)All birds do not sing or It is false that “All birds sing.”

(ii)Not all even integers are prime or It is false that “even integers are prime.”

(iii)All complex number are real numbers or It is false that “complex numbers are not a real number.”

(iv)I will go to school.

(v)There is at least one rectangle whose both diagonals of unequal length.

(vi)No policemen are thief.

**Question 3. Are the following pairs of statements are a negation of each other:**

**(i) The number x is not a rational number.**

**The number x is not an irrational number.**

**(ii) The number x is not a rational number.**

**The number x is an irrational number.**

**Solution:**

(i)The number x is not a irrational number, means that the number x is a rational number.Therefore, the second statement is negation of the first statement.

(ii)The number x is not a rational number means that the number x is an irrational number.Therefore, the second statement is similar to the first statement, and therefore, they are not negation of each other.

**Question 4. Write the negation of the following statements:**

**(i) p: For every positive real number x, the number (x – 1) is also positive.**

**(ii) q: For every real number x, either x > 1 or x < 1.**

**(iii) r: There exists a number x such that 0 < x < 1.**

**Solution:**

(i)We have,p: For every positive real number x, the number (x – 1) is also positive.

The negation of the statement is,

~p: There exists at least one positive real number x, such that the number (x – 1) is not positive.

(ii)The negation of the statement:~q: There exists at least one real number, s.t neither x>1 nor x<1.

(iii)The negation of the statement:~r: For all real numbers x, such that either x ≤ 0 or x ≥ 1.

**Question 5. Check whether the following pair of statements is a negation of each other. Give reasons for your answer.**

**(i) a + b = b + a is true for every real number a and b.**

**(ii) There exist real numbers a and b for which a + b = b + a.**

**Solution:**

The negation of the

(i)statement:There exist real numbers are ‘a’ and ‘b’ for which a + b ≠ b + a.

So,

(ii)is not negation of(i). Hence, these statements are not a negation of each other.