Answer: $\begin{array}{l} \frac{2}{{\sqrt 7 }} = \frac{2}{{\sqrt 7 }} \times \frac{{\sqrt 7 }}{{\sqrt 7 }}\\ =  > \frac{2}{7}\sqrt 7 \end{array}$ is a rational number. p and q are some integers...

Answer: Consider, 5√2 - rational number. ∴ 5√2 = p/q, where p and q are some integers and HCF(p, q) = 1 5√2q = p (5√2q)2 = p2 2(25q2) = p2 p2 is divisible by 2 p is divisible by 2 p = 2m, where m is...

Answer: Consider, x = 5 - 2√3 be a rational number. x = 5 - 2√3 x2 = (5 - 2√3)2 x2 = 52 + (2√3) 2 – 2(5) (2√3) x2 = 25 + 12 – 20√3 x2 – 37 = – 20√3 => $\frac{37-{{x}^{2}}}{20}$ => √3 x...

Answer: Consider, x = 4 - 5√2 be a rational number. x = 4 - 5√2 x2 = (4 - 5√2)2 x2 = 42 + (5√2)2 – 2(4) (5√2) x2 = 16 + 50 – 40√2 x2 – 66 x2 = – 40√2 => $\frac{66-{{x}^{2}}}{40}$ => √2 x is a...

Answer: Consider, x = 2 √3 – 1 be a rational number. x = 2 √3 – 1 x2 = (2 √3 – 1)2 x2 = (2 √3 )2 + (1)2 – 2(2 √3)(1) x2 = 12 + 1 - 4 √3 x2 – 13 = - 4 √3 $\frac{13-{{x}^{2}}}{4}=>\sqrt{3}$ x -...

Answers: (i) The sum of a rational and an irrational is irrational The given statement is True. (ii) The product of a rational and an irrational is irrational The given statement is True....

Answers: (i) The sum of two irrationals is an irrational The given statement is False. Reason: 2 + √3 and 2 - √3 are two irrational numbers. But their sum is 4, which is a rational number. (ii) The...

Answers: (i) The sum of two rationals is always rational The given statement is True. (ii) The product of two rationals is always rational The given statement is True.

Answers: (i) Consider, (2 + √3), (2 - √3) be two irrationals. ∴ (2 + √3) + (2 - √3) = 4 = rational number (ii) Consider, 2 √3, 3 √3 be two irrationals. ∴ 2 √3 × 3 √3 = 18 = rational...

Answer: Consider,  $\frac{1}{\sqrt{3}}$ be rational. $\frac{1}{\sqrt{3}}=\frac{a}{b}$ , where a, b are positive integers having no common factor other than 1 ∴√3 = $\frac{b}{a}$ Since a, b are...

Answer: Consider, √3 + √5 be rational. ∴√3 + √5 = a, where a is rational. ∴ √3 = a - √5 Square on both sides, 3 = (a - √5) 2 = a 2 + 5 - 2√5a $\sqrt{5}=\frac{{{a}^{2}}+2}{2a}$ This is impossible...

Answers: (i) Consider, 3/√5 be rational. 1/3 × 3/√5 = 1/√5 is rational This contradicts the fact that 1/√5 is irrational. ∴1 × √5/√5 × √5 = 1/5 √5 If 1/√5 is irrational, then 1/5√5 is rational...

Answers: (i) Consider, 5 + 3√2 be rational. 5 and 5 + 3√2 are rational. ∴ (5 + 3√2 – 5) = 3√2 = rational ∴ 1/3 × 3√2 = √2 = rational This contradicts the fact that √2 is irrational. The...

Answers: (i) Consider, 3 + √2 be rational. 3 and 3 + √2 are rational. ∴ 3 + √2 – 3 = √2 = rational This contradicts the fact that √2 is irrational. The contradiction arises by assuming 3 + √2 is...

Answers: (i) Consider, √6 = √2 × √3 be rational. √2, √3 are both rational. This contradicts the fact that √2, √3 is irrational. The contradiction arises by assuming √6 is rational. √6 is irrational....

Answers: (i) √21 = √3 × √7 is an irrational number because √3 and √7 are irrational and prime numbers. (ii) ${}^{3}\sqrt{3}$ is an irrational number because 3 is a prime number. So, √3 is an...

Answers: (i) 1.535335333… is an irrational number because it is a non-terminating and non-repeating decimal. (ii) 3.121221222… is an irrational number because it is a non-terminating and...

Answers: (i) 5.636363… is a rational number because it is a non-terminating and non-repeating decimal. (ii) 2.040040004… is an irrational number because it is a non-terminating and non-repeating...

Answers: (i) π is an irrational number because it is a non-repeating and non-terminating decimal. (ii) $3.\overline{142857}$ is a rational number because it is a repeating decimal.  

Answers: (i) $\frac{22}{7}$ is a rational number because it is of the form of $\frac{p}{q}$ where q ≠ 0. (ii) 3.1416 is a rational number because it is a terminating decimal.

Real numbers: The numbers which are positive or negative, whole numbers or decimal numbers and rational numbers or irrational number are called real numbers. Example: 2, 1/3 ,√2 , -3 etc.

(i) Rational numbers: The numbers of the form ????/???? where ????, ???? are integers and ???? ≠ 0 are called rational numbers. Example: 2/3 (ii) Irrational numbers: The numbers which when expressed...

Answer: Total number of pens = 1001 Total number pencils = 910 ∴ Maximum number of students who get the same number of pens and pencils = HCF (1001, 910) Using prime factorization, 1001 = 11 × 91...