Exercise 2.9 - Chapter 2 Real Numbers 9th Maths Guide Samacheer Kalvi Solutions - SaraNextGen [2024-2025]

Updated By SaraNextGen

On April 24, 2024, 11:35 AM

Ex $2.9$
Multiple Choice Questions:
Question $1 .$
If $\mathrm{n}$ is a natural number then $\sqrt{n}$ is
(1) always a natural number
(2) always an irrational number
(3) always a rational number
(4) may be rational or irrational
Solution:
(4) may be rational or irrational
 

Question $2 .$
Which of the following is not true?
(1) Every rational number is a real number.
(2) Every integer is a rational number.
(3) Every real number is an irrational number.
(4) Every natural number is a whole number.
Solution:
(3) Every real number is an irrational number
Hint:
Real numbers contain rationals and irrationals.
 

Question $3 .$
Which one of the following, regarding sum of two irrational numbers, is true?
(1) always an irrational number
(2) may be a rational or irrational number.
(3) always a rational number
(4) always an integer.
Solution:
(2) may be a rational or irrational number

Question $4 .$
Which one of the following has a terminating decimal expansion?
(1) $\frac{5}{64}$
(2) $\frac{8}{9}$
(3) $\frac{14}{15}$
(4) $\frac{1}{12}$
Solution:
(1) $\frac{5}{64}$

Question $5 .$
Which one of the following is an irrational number?
(1) $\sqrt{25}$
(2) $\sqrt{\frac{9}{4}}$
(3) $\frac{7}{11}$
(4) $\pi$
Solution:
(4) $\pi$
Hint:
$\pi$ represents a irrational number
 

Question $6 .$
An irrational number between 2 and $2.5$ is
(1) $\sqrt{11}$
(2) $\sqrt{5}$
(3) $\sqrt{2.5}$
(4) $\sqrt{8}$
Solution:
(2) $\sqrt{5}$
Hint:
$2^{2}=4$ and $2.5^{2}=6.25$

Question $7 .$
The smallest rational number by which - should be multiplied so that its decimal expansion terminates after one place of decimal is
(1) $\frac{1}{10}$
(2) $\frac{3}{10}$
(3) 3
(4) 30
Solution:

(2) $\frac{3}{10}$
Hint:
$\frac{3}{10}$ is the small number.
 

Question $8 .$
If $\frac{1}{7}=0 . \overline{142857}$ then the value of $\frac{5}{7}$ is
(1) $0 . \overline{142857}$
(2) $0 . \overline{714285}$
(3) $1 . \overline{571428}$
(4) $0.714285$
Solution:
(2) $0 . \overline{714285}$
Hint:
$5 \times \frac{1}{7}=5 \times 0 . \overline{142857}=0 . \overline{714285}$
 

Question $9 .$
Find the odd one out of the following.
(1) $\sqrt{32} \times \sqrt{2}$
(2) $\frac{\sqrt{27}}{\sqrt{3}}$
(3) $\sqrt{72} \times \sqrt{8}$
(4) $\frac{\sqrt{54}}{\sqrt{18}}$
Solution:
(4) $\frac{\sqrt{54}}{\sqrt{18}}$
Hint:
$\sqrt{72} \times \sqrt{8}=\sqrt{9 \times 8} \times \sqrt{8}=3 \times 8=24$

Question $10 .$
$0 . \overline{34}+0.3 \overline{4}=$
(1) $0.6 \overline{87}$
(2) $0 . \overline{68}$
(3) $0.6 \overline{8}$
(4) $0.68 \overline{7}$
Solution:
(1) $0.6 \overline{87}$
Hint:
$0.343434 \ldots+0.344444 \ldots=0.6 \overline{87}$
 

Question 11.
Which of the following statement is false?
(1) The square root of 25 is 5 or $-5$
(2) $\sqrt{25}=5$
(3) $-\sqrt{25}=-5$
(4) $\sqrt{25}=\pm 5$
Solution:
(4) $\sqrt{25}=\pm 5$
 

Question $12 .$
Which one of the following is not a rational number?

(1) $\sqrt{\frac{8}{18}}$
(2) $\frac{7}{3}$
(3) $\sqrt{0.01}$
(4) $\sqrt{13}$
Solution:
(4) $\sqrt{13}$
Hint:
(1) $\sqrt{\frac{8}{18}}=\sqrt{\frac{4}{9}}=\frac{2}{3}$ is a arational number
(2) $\frac{7}{3}$ is a rational number
(3) $\sqrt{0.01}=\sqrt{\frac{1}{100}}=\frac{2}{3}$ is a rational number
(4) $\sqrt{13}$ is a rational number
 

Question 13.
$\sqrt{27}+\sqrt{12}=$
(1) $\sqrt{39}$
(2) $5 \sqrt{6}$
(3) $5 \sqrt{3}$
(4) $3 \sqrt{5}$
Solution:
(3) $5 \sqrt{3}$
Hint:
$\sqrt{27}+\sqrt{12}=\sqrt{9 \times 3}+\sqrt{4 \times 3}=3 \sqrt{3}+2 \sqrt{3}=5 \sqrt{3}$

Question $14 .$
if $\sqrt{80}=\mathrm{k} \sqrt{5}$, then $\mathrm{k}=$
(1) 2
(2) 4

(3) 8
(4) 16
Solution:
(2) 4
Hint: $\sqrt{80}=\sqrt{16 \times 5}=4 \sqrt{5}=k \sqrt{5} \Rightarrow \mathrm{k}=4$
 

Question 15 .
$4 \sqrt{7} \times 2 \sqrt{3}=$
(1) $6 \sqrt{10}$
(2) $8 \sqrt{21}$
(3) $8 \sqrt{10}$
(4) $6 \sqrt{21}$
(2) $8 \sqrt{21}$
(2) $8 \sqrt{2}$
$4 \sqrt{7} \times 2 \sqrt{3}=8 \times \sqrt{7 \times 3}=8 \sqrt{21}$
 

Question $16 .$
When written with a rational denominator, the expression $\frac{2 \sqrt{3}}{3 \sqrt{2}}$ can be simplified as
(1) $\frac{\sqrt{2}}{3}$
(2) $\frac{\sqrt{3}}{2}$
(3) $\frac{\sqrt{6}}{3}$
(4) $\frac{2}{3}$
Solution:
(3) $\frac{\sqrt{6}}{3}$
Hint:
$\frac{2 \sqrt{3}}{3 \sqrt{2}}=\frac{2 \sqrt{3}}{3 \sqrt{2}} \times \frac{\sqrt{2}}{\sqrt{2}}=\frac{2 \sqrt{6}}{3 \times 2}=\frac{2 \sqrt{6}}{63}$

Question $17 .$
When $(2 \sqrt{5}-\sqrt{2})^{2}$ is simplified, we get
(1) $4 \sqrt{5}+2 \sqrt{2}$
(2) $22-4 \sqrt{10}$
(3) $8-4 \sqrt{10}$
(4) $2 \sqrt{10}-2$
Solution:
(2) $22-4 \sqrt{10}$
Hint:
$\begin{aligned}
&(2 \sqrt{5}-\sqrt{2})^{2}=(2 \sqrt{5})^{2}-2 \times 2 \sqrt{5} \times \sqrt{2}+\sqrt{2^{2}} \\
&=4 \times 5-4 \sqrt{10}+2=22-4 \sqrt{10}
\end{aligned}$
 

Question $18 .$
$(0.000729)^{\frac{-3}{4}} \times(0.09)^{\frac{-3}{4}}=$
(1) $\frac{10^{3}}{3^{3}}$
(2) $\frac{10^{5}}{3^{5}}$
(3) $\frac{10^{2}}{3^{2}}$
(4) $\frac{10^{6}}{3^{6}}$

Solution:
(4) $\frac{10^{6}}{3^{6}}$
Hint :
$\begin{aligned}
&(0.000729)^{\frac{-3}{4}} \times(0.09)^{\frac{-3}{4}} \\
&=\left(7.29 \times 10^{-4}\right)^{\frac{-3}{4}} \times\left(9 \times 10^{-2}\right)^{\frac{-3}{4}}=(7.29)^{-\frac{3}{4}} \times 10-4 \times \frac{-3}{A} \times 9^{\frac{-3}{4}} \times 10-2 \times \frac{3}{A} \\
&=(7.29)^{\frac{-3}{4}} \times 10^{+3} \times 9^{\frac{-3}{4}} \times 10^{\frac{3}{2}}=\left(729 \times 10^{-2}\right)^{\frac{-3}{4}} \times 10^{3+\frac{3}{2}} \times 9^{\frac{-3}{4}} \\
&=\left(9^{3} \times 10^{2}\right)^{\frac{-3}{4}} \times 10^{\frac{9}{2}} \times 9^{\frac{-3}{4}}=\left(9^{3}\right)^{\frac{-3}{4}} \times 10^{-2} \times \frac{-3}{4} \times 10^{\frac{9}{2}} \times 9^{\frac{-3}{4}} \\
&=9^{\frac{-9}{4}} \times 10^{\frac{3}{2}+\frac{9}{2}} \times 9^{\frac{-3}{4}}=9^{\frac{-9}{4} \frac{3}{4}} \times 10^{\frac{12}{z}} \\
&=9^{\frac{-12}{4}} \times 10^{6}=9^{-3} \times 10^{6}=\frac{10^{6}}{9^{3}}=\frac{10^{6}}{\left(3^{2}\right)^{3}}=\frac{10^{6}}{3^{6}}
\end{aligned}$

Question $19 .$
If $\sqrt{9^{x}}=\sqrt[3]{9^{2}}$, than $\mathrm{x}=$
(1) $\frac{2}{3}$
(2) $\frac{4}{3}$
(3) $\frac{1}{3}$
(4) $\frac{5}{3}$
Solution:
(2) $\frac{4}{3}$

Question 20 .
The length and breadth of a rectangular plot are $5 \times 105$ and $4 \times 104$ metres respectively. Its area is .
(1) $9 \times 10^{1} \mathrm{~m}^{2}$
(2) $9 \times 10^{9} \mathrm{~m}^{2}$
(3) $2 \times 10^{10} \mathrm{~m}^{2}$
(4) $20 \times 10^{20} \mathrm{~m}^{2}$
Solution:
(3) $2 \times 10^{10} \mathrm{~m}^{2}$
Hint:
$\begin{aligned}
&1=5 \times 10^{5} \text { metres; b=4 } \times 10^{4} \text { metres } \\
&\therefore \text { Area }=1 \times b=5 \times 10^{5} \times 4 \times 10^{4} \\
&=20 \times 10^{5+4}=20 \times 10^{9}=2.0 \times 10^{1} \times 10^{9}=2 \times 10^{10} \mathrm{~m}^{2}
\end{aligned}$