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NCERT Class 10 Mathematics Exercise 1.2 Solutions: Real Numbers

Step-by-step working for every question of Exercise 1.2 in NCERT's Class 10 Mathematics book, chapter “Real Numbers”, written by Shishya. Each question is in our own short words; open the NCERT chapter for the exact wording and figures.

1. Ex 1.2 Q1

   Prove that √5 is irrational.

   1. Assume the contrary that √5 is a rational number.
      why?

      Proof by contradiction: assume the opposite of what is to be proved and show that it leads to something impossible; then the assumption must be false.

   2. Then √5 can be written as p/q, where p and q are coprime integers and q ≠ 0.
   3. Squaring both sides, we get 5 = p²/q², which implies p² = 5q².
      why?

      Doing the same thing to both sides of an equation (adding, subtracting, multiplying or dividing by the same non-zero number) keeps the two sides equal, so the equation stays true while the unknown is left on its own.

   4. This equation shows that 5 divides p², and since 5 is prime, 5 must divide p.
   5. Let p = 5m for some integer m.
   6. Substituting p = 5m into p² = 5q² gives (5m)² = 5q², so 25m² = 5q².
      why?

      Putting the known values in place of the letters turns the formula or equation into plain arithmetic — the letters stood for exactly these values.

   7. Dividing by 5, we get 5m² = q², which means 5 divides q² and hence 5 divides q.
   8. Thus, 5 is a common factor of both p and q, which contradicts the assumption that p and q are coprime.
      why?

      Proof by contradiction: assume the opposite of what is to be proved and show that it leads to something impossible; then the assumption must be false.

   9. Therefore, the initial assumption is false, and √5 is irrational.

   Answer: √5 is irrational.

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2. Ex 1.2 Q2

   Prove that 3 + 2√5 is irrational.

   1. Assume 3 + 2√5 is rational, so 3 + 2√5 = a/b where a and b are coprime integers and b ≠ 0.
      why?

      Proof by contradiction: assume the opposite of what is to be proved and show that it leads to something impossible; then the assumption must be false.

   2. Rearrange the equation to isolate the radical: 2√5 = a/b − 3.
   3. Combine the terms on the right side: 2√5 = (a − 3b)/b.
   4. Divide both sides by 2 to get √5 = (a − 3b)/(2b).
      why?

      Doing the same thing to both sides of an equation (adding, subtracting, multiplying or dividing by the same non-zero number) keeps the two sides equal, so the equation stays true while the unknown is left on its own.

   5. Since a, b, and 3 are integers, the expression (a − 3b)/(2b) is a ratio of integers, making it rational.
   6. This implies that √5 is rational, which contradicts the known fact that √5 is irrational.
      why?

      Proof by contradiction: assume the opposite of what is to be proved and show that it leads to something impossible; then the assumption must be false.

   7. Therefore, the initial assumption that 3 + 2√5 is rational is false.

   Answer: 3 + 2√5 is irrational.

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3. Ex 1.2 Q3

   Prove that these are irrational: (i) 1/√2 (ii) 7√5 (iii) 6 + √2.

   1. (i) Assume 1/√2 is rational, so 1/√2 = a/b where a and b are coprime integers and b ≠ 0.
      why?

      Proof by contradiction: assume the opposite of what is to be proved and show that it leads to something impossible; then the assumption must be false.

   2. (i) Since 1/√2 ≠ 0, a ≠ 0, so √2 = b/a, a ratio of integers — √2 would be rational.
   3. (i) This contradicts the known fact that √2 is irrational, so 1/√2 is irrational.
      why?

      Proof by contradiction: assume the opposite of what is to be proved and show that it leads to something impossible; then the assumption must be false.

   4. (ii) Assume 7√5 is rational, so 7√5 = a/b where a and b are coprime integers and b ≠ 0.
      why?

      Proof by contradiction: assume the opposite of what is to be proved and show that it leads to something impossible; then the assumption must be false.

   5. (ii) This implies √5 = a/(7b), which means √5 is rational.
   6. (ii) This contradicts the known fact that √5 is irrational, so 7√5 is irrational.
      why?

      Proof by contradiction: assume the opposite of what is to be proved and show that it leads to something impossible; then the assumption must be false.

   7. (iii) Assume 6 + √2 is rational, so 6 + √2 = a/b where a and b are coprime integers and b ≠ 0.
      why?

      Proof by contradiction: assume the opposite of what is to be proved and show that it leads to something impossible; then the assumption must be false.

   8. (iii) This implies √2 = a/b − 6 = (a − 6b)/b, which means √2 is rational.
   9. (iii) This contradicts the known fact that √2 is irrational, so 6 + √2 is irrational.
      why?

      Proof by contradiction: assume the opposite of what is to be proved and show that it leads to something impossible; then the assumption must be false.

   Answer: (i) 1/√2 is irrational; (ii) 7√5 is irrational; (iii) 6 + √2 is irrational

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