NCERT Solutions for Class 9 Math | Exercise 1.5

CBSE Chapter 1 | Number Systems | Rationalizing Irrational Numbers

Chapter 1 of CBSE NCERT Class 9 Math covers number system. Concepts covered in exercise 1.5 include rational numbers, rationalizing irrational numbers by multiplying with the conjugate of the denominator to simplify the expressions.

Solutions to Question 1 and Question 3 are given in this page. For the other questions, click on the solution links given adjacent to the questions.

NCERT Solutions for Class 9 Math | Exercise 1.5

  1. Question 1

    Classify the following numbers as rational or irrational:
    i) 2 - √5
    ii) (3 + √23) - √23
    iii) \\frac{2√7}{7√7})
    iv) \\frac{1}{√2})
    v) 2π

    Answer

    i) 2 - √5 is irrational as it contains the irrational number √5 in it. √5 cannot be expressed in the form \\frac{p}{q}) where p and q are integers and q ≠ 0.
    ii) (3 + √23) - √23 = 3 + √23 - √23 = 3 is a natural number.
    All natural numbers are rational numbers. 3 can be expressed in the form \\frac{p}{q}).
    iii) \\frac{2√7}{7√7}) = \\frac{2}{7}) which is rational as it can be expressed in the form \\frac{p}{q}).
    Property used for (iv) and (v): The product or quotient of a rational number and an irrational number is an irrational number.
    iv) \\frac{1}{√2}) contains the irrational number √2. So, \\frac{1}{√2}) is irrational. It cannot be expressed in the form \\frac{p}{q}) where p and q are integers and q ≠ 0.
    v) 2π is irrational because π is irrational.


  2. Question 2

    Simplify each of the following expressions:
    (i) (3 + √3) (2 + √2)
    (ii) (3 + √3) (3 - √3)
    (iii) (√5 + √2)2
    (iv) (√5 - √2)(√5 + √2)


  3. Question 3

    Recall, π is defined as the ratio of the circumference (c) of a circle to its diameter (d). That is, π = \\frac{c}{d}). This seems to contradict the fact that π is irrational. How can you resolve this contradiction?

    Answer

    There is no contradiction. The measure of c will be irrational if d is rational or the measure of d will be irrational if c is rational or both c and d will be irrational. Essentially, at least one of c or d will be irrational. Because at least one of c or d is not an integer, though it is of the form \\frac{c}{d}), π is not a rational number.


  4. Question 5

    Rationalise the denominators of the following:
    (i) \\frac{1}{√7})
    (ii) \\frac{1}{√7 - √6})
    (iii) \\frac{1}{√5 + √2})
    (iv) \\frac{1}{√7 - 2})



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