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.
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.
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
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}}}\\)
If a > 0 is a real number and m and n are rational numbers, the following laws of exponents hold good.
Question 1
Prime Factorise & Rationalise Denominator:
\\frac{14}{{\sqrt {108}} - {\sqrt {96}} + {\sqrt {192}} - {\sqrt {54}}}\\)
Question 2
Question 3
Express as Fractions
Express 1.363636... in the form \\frac{p}{q}), where p and q are integers and q ≠ 0.
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.
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)
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}})
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}}})
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}\\)?
Question 9
Rationalise & find value of cubic expression
If x = \\frac{1}{8-\sqrt{60}}\\), what is the value of (x^{3} - 5x^{2} + 8x - 4) ?
Question 10
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