NCERT Solutions for Class 11 Maths Chapter 14 Mathematical Reasoning

Ans : (i) This sentence is incorrect because the maximum number of days in a month is 31. Hence, it is a statement. 
(ii) This sentence is subjective in the sense that for some people, mathematics can be easy and for some others, it can be difficult. Hence, it is not a statement. 
(iii) The sum of 5 and 7 is 12, which is greater than 10. Therefore, this sentence is always correct. Hence, it is a statement. 
(iv) This sentence is sometimes correct and sometimes incorrect. For example, the square of 2 is an even number. However, the square of 3 is an odd number. Hence, it is not a statement. 
(v) This sentence is sometimes correct and sometimes incorrect. For example, squares and rhombus have sides of equal lengths. However, trapezium and rectangles have sides of unequal lengths. Hence, it is not a statement. 
(vi) It is an order. Therefore, it is not a statement. 
(vii) The product of (–1) and 8 is (–8). Therefore, the given sentence is incorrect. Hence, it is a statement. 
(viii) This sentence is correct and hence, it is a statement. 
(ix) The day that is being referred to is not evident from the sentence. Hence, it is not a statement. 
(x) All real numbers can be written as a × 1 + 0 × i. Therefore, the given sentence is always correct. Hence, it is a statement. 

Ans : The three examples of sentences, which are not statements, are as follows.
(i) He is a doctor. 
It is not evident from the sentence as to whom ‘he’ is referred to. Therefore, it is not a statement. 
(ii) Geometry is difficult. 
This is not a statement because for some people, geometry can be easy and for some others, it can be difficult. 
(iii) Where is she going? 
This is a question, which also contains ‘she’, and it is not evident as to who ‘she’ is. Hence, it is not a statement. 

Ans : (i) Chennai is not the capital of Tamil Nadu. 
(ii) \(\sqrt{2}\) is a complex number. 
(iii) All triangles are equilateral triangles. 
(iv) The number 2 is not greater than 7. 
(v) Every natural number is not an integer. 

Ans : (i) The negation of the first statement is “the number x is a rational number”. 
This is same as the second statement. This is because if a number is not an irrational number, then it is a rational number.
Therefore, the given statements are negations of each other.
(ii) The negation of the first statement is “the number x is not a rational number”. This means that the number x is an irrational number, which is the same as the second statement.
Therefore, the given statements are negations of each other. 

Ans : (i) The component statements are as follows. 
p: Number 3 is prime. 
q: Number 3 is odd. 
Both the statements are true. 
(ii) The component statements are as follows. 
p: All integers are positive. 
q: All integers are negative. 
Both the statements are false. 
(iii) The component statements are as follows. 
p: 100 is divisible by 3. 
q: 100 is divisible by 11. 
r: 100 is divisible by 5. 
Here, the statements, p and q, are false and statement r is true.