# RBSE Class 9 Maths Important Questions Chapter 1 Number Systems

Rajasthan Board RBSE Class 9 Maths Important Questions Chapter 1 Number Systems Important Questions and Answers.

Rajasthan Board NCERT New Syllabus RBSE Solutions for Class 9 Guide Pdf free download in Hindi Medium and English Medium are part of RBSE Solutions. Here we have given RBSE Class 9th Books Solutions.

## RBSE Class 9 Maths Chapter 1 Important Questions Number Systems

I. Multiple Choice Questions :
Choose the correct answer from the given options.

Question 1.
Which of the following is an irrational number?
(a) $$\sqrt{\frac{4}{9}}$$
(b) $$\frac{\sqrt{12}}{\sqrt{3}}$$
(c) √7
(d) √81
(c) √7

Question 2.
A rational number lying between √2 and √3 is :
(a) $$\frac{\sqrt{2}+\sqrt{3}}{2}$$
(b) √6
(c) 1.6
(d) 1.9
(c) 1.6

Question 3.
An irrational number lying between 2 and 3 is :
(a) √5
(b) √13
(c) 2.41
(d) √2
(a) √5

Question 4.
√10 × √l5 is equal to :
(a) 6√5
(b) 5√6
(c) √25
(d) 10√5
(b) 5√6

Question 5.
The product $$\sqrt[3]{2} \cdot \sqrt[4]{2} \cdot \sqrt[12]{32}$$ is equal to :
(a) √2
(b) 2
(c) $$\sqrt[12]{2}$$
(d) $$12 \sqrt{32}$$
(b) 2

Question 6.
Decimal representation of a rational number cannot be :
(a) terminating
(b) non-terminating
(c) non-terminating
(d) non-terminating non-repeating
(d) non-terminating non-repeating

Question 7.
$$\sqrt[4]{\sqrt[3]{2^{2}}}$$ is equal to :
(a) 2-1/6
(b) 2-6
(c) 21/6
(d) 26
(c) 21/6

Question 8.
If √2 = 1.4142, then $$\sqrt{\frac{\sqrt{2}-1}{\sqrt{2}+1}}$$ equal to :
(a) 2.4142
(b) 5.8284
(c) 0.4142
(d) 0.1718
(c) 0.4142

Question 9.
$$\frac{1}{\sqrt{9}-\sqrt{8}}$$
(a) $$\frac{1}{2}$$(3 - 2√2)
(b) $$\frac{1}{3+2 \sqrt{2}}$$
(c) 3 - 2√2
(d) 3 + 2√2
(d) 3 + 2√2

Question 10.
2√3 + √3 is equal to :
(a) 2√6
(b) 6
(c) 3√3
(d) 4√6
(c) 3√3

Question 11.
The value of 1.999 in the form of $$\frac{p}{q}$$, where p and q are integers and q ≠ 0, is:
(a) $$\frac{19}{20}$$
(b) $$\frac{1999}{1000}$$
(c) 2
(d) $$\frac{1}{9}$$
(c) 2

Question 12.
π is:
(a) a rational number
(b) an integer
(c) an irrational number
(d) a whole number
(c) an irrational number

Question 13.
If $$x^{\frac{1}{12}}$$ = $$49^{\frac{1}{24}}$$, then x is equal to :
(a) 49
(b) 2
(c) 12
(d) 7
(d) 7

Question 14.
Which of the following is not equal to $$\left[\left(\frac{5}{6}\right)^{1 / 5}\right]^{-1 / 6}$$
(a)

(b) $$\frac{1}{\left[\left(\frac{5}{6}\right)^{1 / 5}\right]^{1 / 6}}$$
(c) $$\left(\frac{6}{5}\right)^{1 / 30}$$
(d) $$\left(\frac{5}{6}\right)^{-1 / 30}$$
(a)

Question 15.
Which of the following is equal to x?
(a) $$x^{\frac{12}{7}}-x^{\frac{5}{7}}$$
(b) $$\sqrt[12]{\left(x^{4}\right)^{1 / 3}}$$
(c) $$\left(\sqrt{x^{3}}\right)^{2 / 3}$$
(d) $$x^{\frac{12}{7}} \times x^{\frac{7}{12}}$$
(c) $$\left(\sqrt{x^{3}}\right)^{2 / 3}$$

II. Fill in the blanks

Question 1.
Rational number between $$\frac{1}{9}$$ and $$\frac{2}{9}$$ is _______________ .
$$\frac{1}{6}$$

Question 2.
$$\frac{p}{q}$$ form of 0.6 + 0.$$\overline{7}$$ + 0.4$$\overline{7}$$ is _______________ .
$$\frac{167}{90}$$

Question 3.
The rationalizing factor of the denominator in $$\frac{1}{\sqrt{2}+\sqrt{3}}$$ is _______________ .
√2 - √3

Question 4.
If $$\frac{\sqrt{3}-1}{\sqrt{3}+1}$$ = a + b√3, then a = _____________ and b = _______________ .
2, - 1

Question 5.
If a = 2 and 6 = 3, then the value of (aa + bb)-1 = _______________ .
$$\frac{1}{31}$$

III. True/False:

State whether the following statements are True or False.
Question 1.
The decimal expansion of a rational number is non-terminating non-recurring.
False

Question 2.
1 is the smallest prime number.
False

Question 3.
$$\frac{\sqrt{18}}{\sqrt{2}}$$ is a rational number.
True

Question 4.
2.0200200200002..... is an irrational number.
True

Question 5.
Between two rational numbers there is exactly one rational number.
False

Question 6.
The product of any two irrational numbers is always an irrational number.
False

Question 7.
There are infinitely many integers between any two integers.
False

IV. Match the Columns:

Match the column I with the column II.

 Column I Column II (1) 6.$$\overline{54}$$ is ………. (i) 14 (2) π is ………. (ii) 6 (3) The length of period of $$\frac{1}{7}$$ is ……… (iii) a rational  number (4) If x = (2 - √3), then $$\left(x^{2}+\frac{1}{x^{2}}\right)$$........... . (iv) an irrational number

 Column I Column II (1) 6.$$\overline{54}$$ is ………. (iii) a rational number (2) π is ………. (iv) an rational number (3) The length of period of $$\frac{1}{7}$$ is ……… (ii) 6 (4) If x = (2 - √3), then $$\left(x^{2}+\frac{1}{x^{2}}\right)$$........... . (i) 14

V. Very Short Answer Type Questions:

Question 1.
Find the value of (256)0.16 × (256)0.09.
(256)0.16 × (256)0.09 = (256)0.16 + 0 09  (∵ am × an = am+n)
= (256)0.25
$$(256)^{\frac{25}{100}}=(256)^{\frac{1}{4}}=\left(4^{4}\right)^{\frac{1}{4}}=4$$

Question 2.
If a = 2, b = 3 then find the values of (aa + bb)-1.
(aa + bb)-1 = (22 + 33)-1 = (4 + 27)-1 = (31)-1$$\frac{1}{31}$$

Question 3.
Simplify: (7)-3 (9)-3
(7)-3 (9)-3 = (7 × 9)-3
= (63)-3$$\frac{1}{(63)^{3}}$$

Question 4.
Taking  √2 = 1.414 and π = 3.141, evaluate $$\frac{1}{\sqrt{2}}$$ + π upto three places of decimal.

= 0.707 + 3.141 = 3.848

Question 5.
$$\frac{\sqrt{147}}{\sqrt{75}}$$ is not a rational number as √l47 and √75 are not rational. State wheather it is true or false. Justify your answer.
$$\frac{\sqrt{147}}{\sqrt{75}}=\frac{\sqrt{3 \times 7 \times 7}}{\sqrt{3 \times 5 \times 5}}=\frac{7 \sqrt{3}}{5 \sqrt{3}}=\frac{7}{5}$$
Which is clearly a rational number.
Hence, the given statement is false.

Question 1.
Express 0.00323232..... in the form of  $$\frac{p}{q}$$.
Let x = 0.00323232...
On multiplying equation (1) by 100, we get:
100x = 0.323232.....
Again, multiplying equation (2) by 100, we get:
10000a: = 32.3232....
On subtracting equation (2) from equation (3), we get:
10000x - 100x = 32.3232..... - 0.3232.....
⇒ 9900x = 32
⇒ x = $$\frac{32}{9900}=\frac{8}{2475}$$
Hence, 0.003232..... = $$\frac{8}{2475}$$

Question 2.
If 125x = $$\frac{25}{\mathbf{5}^{x}}$$, then find the value of x.
Given,  25x = $$\frac{25}{\mathbf{5}^{x}}$$
⇒ 53x = 52 - x
On equating power from both sides, we get:
3x = 2 - x
⇒ 4x = 2
∴ x =  $$\frac{2}{4}=\frac{1}{2}$$

Question 3.
If a = 7 - 4√3, then find the value of √a + $$\frac{1}{\sqrt{a}}$$

Question 4.
How many irrational numbers lie between √2 and √3? Find any three irrational numbers lying between √2 and √3.
Infinitely many irrational numbers lie between √2 and √3.
One irrational number between √2 and √3 is $$\sqrt{\sqrt{2} \sqrt{3}}$$ = $$\sqrt{\sqrt{6}}$$ = $$6^{\frac{1}{4}}$$
Another irrational number between √2 and √3 is $$\sqrt{\sqrt{2} 6^{\frac{1}{4}}}$$
$$2^{\frac{1}{4}} \cdot 6^{\frac{1}{8}}=2^{\frac{1}{4}} \cdot 2^{\frac{1}{8}} \cdot 3^{\frac{1}{8}}=2^{\frac{1}{4}+\frac{1}{8}} \cdot 3^{\frac{1}{8}}=2^{\frac{3}{8}} \cdot 3^{\frac{1}{8}}$$
Third irrational number between √2 and √3 is
$$\sqrt{\sqrt{2} 2^{\frac{3}{8}} \cdot 3^{\frac{1}{8}}}$$ = $$2^{\frac{1}{4}} \cdot 2^{\frac{3}{16}} \cdot 3^{\frac{1}{16}}$$$$2^{\frac{7}{16}} \cdot 3^{\frac{1}{16}}$$

Question 5.
Show that

Question 6.
If 2x = 3y = 6z, then prove that $$\frac{1}{x}+\frac{1}{y}-\frac{1}{z}$$= 0 or z = $$\frac{x y}{x+y}$$
Let 2x = 3y = 6z = k
⇒ 2x = k,3y = k and 6Z = k
⇒ 2 = $$k^{1 / x}$$, 3 = $$k^{1 / y}$$ and 6 = $$k^{1 / z}$$
Now, 6 = $$(k)^{1 / z}$$
⇒ 2 × 3 = $$(k)^{1 / z}$$
$$(k)^{1 / x} \times(k)^{1 / y}=(k)^{1 / z}$$
$$(k)^{1 / x+1 / y}=(k)^{1 / z}$$
On equating the power from both sides, we get:

Question 7.

Question 1.
Write the following in decimal form and say, what kind of decimal expansion each has?
(i) $$\frac{36}{100}$$
$$\frac{36}{100}$$ = 0.36, so it has terminating decimal expansion

(ii) 5$$\frac{1}{8}$$
5$$\frac{1}{8}$$ = $$\frac{41}{8}$$ = 5.125, so it has terminating decimal expansion.

(iii) $$\frac{4}{13}$$
$$\frac{4}{13}$$ = 0 . $$\overline{307692}$$, so it has non-terminating repeating decimal expansion.

(iv) $$\frac{2}{15}$$
$$\frac{2}{15}$$ = 0.1$$\overline{3}$$, so it has non-terminating repeating decimal expansion.

(v) $$\frac{329}{500}$$
$$\frac{329}{500}$$ = 0.658, so it has terminating decimal expansion.

Question 2.
Express $$1.\overline{32}$$ + $$0.\overline{35}$$ in the form $$\frac{p}{q}$$ where p and q are integers and q ≠ 0.
Let x = $$1.\overline{32}$$
Then  x  =  1.3232...
⇒ 10x  =  13.232... ........... (1)
⇒ 100x = 132.323... ........... (2)
Subtracting equations (1) from (2), we get:
90x = 119
⇒ x = $$\frac{119}{90}$$ ............... (3)
Let x = $$0.\overline{35}$$
Then x = 0.35 35 35 ... ............... (4)
⇒ 100x = 35.35 35 35 ... ............... (5)
Subtracting equations (4) from (5), we get
99x  = 35

⇒ x = $$\frac{35}{99}$$
Here, p = 553, q = 330(≠ 0)

Question 3.
Simplify the following into a fraction with rational denominator $$\frac{1}{\sqrt{5}+\sqrt{6}-\sqrt{11}}$$