Extra Questions For Class 9 Maths Chapter 1

Chapter 1 of CBSE NCERT Class 9 Math covers number systems. Concepts covered in chapter 1 include rational numbers, irrational numbers, rationalizing irrational numbers by multiplying with their conjugates, decimal expansion of real numbers, operations on real numbers and laws of exponents or rules of indices. The extra questions given below include questions akin to HOTS (Higher Order Thinking Skills) questions and exemplar questions of NCERT.

Here is a quick recap of the key concepts that are covered in this chapter in the CBSE NCERT Class 9 Math text book.

What are rational numbers?

A number that can be written in the form \\frac{p}{q}\\) where p and q are integers and p ≠ 0 is a rational number.

Possibility 1: If the decimal expansion of the number is terminating it is a rational number. Note: Integers are terminating decimals and are therefore, rational numbers.

Possibility 2: If the decimal expansion of the number is non-terminating but is recurring, it is rational. Example \\frac{1}{3}\\) = 0.333.. is a non-terminating recurring decimal and is a rational number.

What are irrational numbers?

A number that CANNOT be written in the form \\frac{p}{q}\\) where p and q are integers and p ≠ 0 is an irrational number.

If the decimal expansion of the number is non-terminating AND non-recurring it is an irrational number. Example: \\sqrt{2}\\), π

How to Rationalize Irrational Numbers?

For an irrational number of the form a + √b, a - √b is its conjugate. And for an irrational number of the from a - √b, a + √b is its conjugate.

To rationalize the denominator of \\frac {1}{a + {\sqrt {b}}}\\), multiply and divide it by the conjugate of the denominator.
i.e., multiply and divide \\frac {1}{a + {\sqrt{b}}}\\) with \\frac {a - {\sqrt{b}}}{a - {\sqrt{b}}}\\)

Important Laws of Exponents (Rules of Indices)

If a > 0 is a real number and m and n are rational numbers, the following laws of exponents hold good.

1. a^m × aⁿ = a ^{m + n} Example.: 10³ × 10² = 10^{3 + 2} = 10⁵
2. (a^m)ⁿ = a ^mn Example: (10³)² = 10^{(3 \\times\\) 2)} = 10⁶
3. \\frac{a^m}{a^n}\\) = a^{(m - n)} Example: \\frac{10^3}{10^2}\\) = 10^{(3 - 2)} = 10
4. a^mb^m = (ab)^m Example: 2² × 5² = (2 × 5)² = 10²

Extra Questions for Class 9 Maths - Number Systems

1. Question 1

   Prime Factorise & Rationalise Denominator:
   \\frac{14}{{\sqrt {108}} - {\sqrt {96}} + {\sqrt {192}} - {\sqrt {54}}}\\)

   Explanation Video Solution

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2. Question 2

   Rational numbers - Fractions:
   Find 5 rational numbers between \\frac{3}{4}) and \\frac{4}{5}).
   

   Explanation Video Solution

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3. Question 3

   Express as Fractions
   Express 1.363636... in the form \\frac{p}{q}), where p and q are integers and q ≠ 0.

   Explanation Video Solution

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4. Question 4

   Express in the form \\frac{p}{q})
   Express 0.4323232… in the form \\frac{p}{q}), where p and q are integers and q ≠ 0.

   Explanation Video Solution

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5. Question 5

   Simplify the following
   (a) \({8 + \sqrt{5})}) \({8 - \sqrt{5})})
   (b) \({10 + \sqrt{3})}) \({6 + \sqrt{2})})
   (c) \{(\sqrt {3} + \sqrt {11})}^2) + \{(\sqrt {3} - \sqrt {11})}^2)

   Explanation Video Solution

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6. Question 6

   Rationalize the denominator:
   (a) \\frac{2}{\sqrt{3} - 1})
   (b) \\frac{7}{\sqrt{12} - \sqrt{5}})
   (c) \\frac{1}{8 + 3\sqrt{5}})
   (d) \\frac{1}{4 + \sqrt{2} + \sqrt{5}})

   Explanation Video Solution 1 Video Solution 2

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7. Question 7

   Simplify and find the value of
   (a) \{(729)}^{\frac{1}{6}})
   (b) \{(64)}^{\frac{2}{3}})
   (c) \{(243)}^{\frac{6}{5}})
   (d) \{(21)}^{\frac{3}{2}} \times {(21)}^{\frac{5}{2}})
   (e) \\frac{{(81)}^{\frac{1}{3}}}{{(81)}^{\frac{1}{12}}})

   Explanation Video Solution

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8. Question 8

   Operation on real numbers & Algebraic identities
   If x = \\frac{3 - {\sqrt{13}}}{2}\\), what is the value of \x^2 + \frac{1}{x^2}\\)?

   Explanation Video Solution

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9. Question 9

   Rationalise & find value of cubic expression
   If x = \\frac{1}{8-\sqrt{60}}\\), what is the value of (x³ - 5x² + 8x - 4) ?

   Explanation Video Solution

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10. Question 10

    Rationalise the denominator
    \\frac{1}{9 + {\sqrt{5} + \sqrt{6}}}\\)

    Explanation Video Solution

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