Ncert Solutions Maths class 11th

This is a  Mathematical Reasoning Solutions Type Questions as classified in NCERT Exemplar

(i) The 'or' in this statement is conclusive because it is not possible for the sun to rise and the moon to set together.

(ii) The 'or' in this statement is inclusive since a person can have both a ration card and a passport to apply for a driving license.

(iii) The 'or' in this statement is exclusive since all integers cannot be both positive and negative.

This is a  Mathematical Reasoning Solutions Type Questions as classified in NCERT Exemplar

Negation of statement (i) is:

There exists real numbers x and y for which x + y = y + x.

Now, this statement is not same as statement (ii).

Therefore, the given statements are not negation of each other.

This is a  Mathematical Reasoning Solutions Type Questions as classified in NCERT Exemplar

(i) Here, the quantifier is 'there exists'.

The negation of this statement is

There does not exist a number which is equal to its square.

(ii) Here, the quantifier is 'for every'.

The negation of the statement is

There exist a real number x, such that x is not less than x+1

(iii) Here, the quantifier is 'there exists'.

The negation of this statement is

In India, there exists a state which does not have a capital.

This is a  Mathematical Reasoning Solutions Type Questions as classified in NCERT Exemplar

(i) The component statements are

(a) Number 3 is prime

(b) Number 3 is odd

Here, both the statements are true.

(ii) The component statements are

(a) All integers are positive

(b) All integers are negative

Here, both the statements are false.

(iii) The component statements are

(a) 100 is divisible by 3

(b) 100 is divisible by 11

(c) 100 is divisible by 5

Here, the statements (a) and (b) are false and c) is true.

This is a  Mathematical Reasoning Solutions Type Questions as classified in NCERT Exemplar

(i) The negation of the first statement is 'the number x is a rational number'.

This is same as the second statement because if a number is not an irrational number then the number is a rational number.

Hence, 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 same as the second statement.

Hence, the given statements are negations of each other.

This is a  Mathematical Reasoning Solutions Type Questions as classified in NCERT Exemplar

(i) Chennai is not the capital of Tamil Nadu.

(ii) √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.

This is a  Mathematical Reasoning Solutions Type Questions as classified in NCERT Exemplar

The three examples of sentences which are not statements are:

(1) He is an engineer.

Here, in this sentence, it is not evident as to whom 'he' is referred to. Hence, it is not a statement.

(2) Painting is easy for some people, painting can be easy and for some others, it can be difficult. Hence, it is not a statement.

(3) Where is my hat?

This sentence is a question which is not considered as a statement in mathematical language. Hence, it is not a statement.

15. ${\left[3x^2-2ax+3a^2\right]}^3$

= ${\left[3x^2+a\left(-2x+3a\right)\right]}^3$

We know that by binomial theorem,

${(a+b)}^3$ = $a^3+b^3+3ab\left(a+b\right)$

= $a^3+b^3+3a^{2}b+3a{b}^2$

Then,

${\left[3x^2+a\left(-2x+3a\right)\right]}^3$

= (3x²)³ + ${\left[a\left(-2x+3a\right)\right]}^3$ + $\left[3(3x^2){}^2 a\left(-2x+3a\right)\right]$ + $\left[ 3(3x^2){\left\{a\left(-2x+3a\right)\right\}}^2\right]$

= 27x⁶ + $\left[a^3{\left(-2x+3a\right)}^3\right]$ + $\left[3(9x^4)\left(-2ax+3a^2\right)\right]$ + $\left[3(3x^2)\left\{a^2{\left(3a-2x\right)}^2\right\}\right]$

= 27x⁶ + $\left[a^3\left\{-8x^3+27a^3+3\left(4x^2\right)\left(3a\right)+3\left(-2x\right)\left(9a^2\right)\right\}\right]$ + $\left[-54a{x}^5+81a^2{x}^4\right]$ + [ $\left(9a^2{x}^2\right)$ $\left(9a^2+4x^2-12ax\right)$ ]

= 27x⁶ + [ $-8a^3{x}^3+27a^6+36a^4{x}^2-54a^{5}x$ ] + [ $-54a{x}^5+81a^2{x}^4$ ] + [ $(81a^4{x}^2+36a^2{x}^4-108a^3{x}^3$ ]

= 27x⁶ $- 8a^3{x}^3+27a^6+36a^4{x}^2-54a^{5}x$ $- 54a{x}^5+81a^2{x}^4$ + $81a^4{x}^2+36a^2{x}^4-108a^3{x}^3$

= 27x⁶– 54ax⁵ $+ 117a^2{x}^4$ $- 116a^3{x}^3$ + $117a^4{x}^2$ $- 54a^{5}x$ $+ 27a^6$