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Published on: 20/11/2019
Number Systems
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Questions + Answers key
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1.
If \(x=\frac { \sqrt { 3 } -1 }{ \sqrt { 3 } +1 } \) and \(y=\frac { 3+2\sqrt { 2 } }{ 3-2\sqrt { 2 } } \) , then find the the value of x+y
2.
Classify the following numbers as rational or irrational: \(\frac { 2\sqrt { 7 } }{ 7\sqrt { 7 } } \)
3.
Express the following in the form p/q, where p and q are integers and \(q\neq 0\)
\(0.\overline { 47 } \)
4.
Write the following in decimal form and say what kind of decimal expansion each has:
\(\frac { 2 }{ 11 } \)
5.
State whether the following statements are true or false.Give reasons for your answers.
(i) Every natural number is a whole number.
(ii) Every integer is a whole number.
(iii) Every rational number is a whole number.
6.
Find four rational numbers between \(\frac { 3 }{ 7 } \)and \(\frac { 5 }{ 7 } \)
7.
Rationalize \(\frac { 5 }{ \sqrt { 3 } -\sqrt { 5 } } \left( -\frac { 5 }{ 2 } \right) \)
8.
Simplify: \(\frac { { 23 }^{ -10 } }{ { 23 }^{ 7 } } \)
9.
Simplify: (52)-7
10.
Give three rational numbers between -3 and -2.
11.
Find two rational numbers between 3/4 and 5/9.
12.
The value of \(\sqrt { \left( { 3 }^{ -2 } \right) } \)
1/9
9
-3
1/3
13.
If a = 2 and b = 3, then the value of ba is
4
9
2
3
14.
If \(x=\frac { \sqrt { 7 } }{ 5 } \) and \(\frac { 5 }{ x } =p\sqrt { 7 } \) , then the value of p is:
\(\frac { 5 }{ \sqrt { 7 } } \)
\(\frac { 25 }{ 7 } \)
\(\frac { 7 }{ 25 } \)
\(\frac { \sqrt { 7 } }{ 5 } \)
15.
The value of \(2.\overline { 9 } \) in the form \(\frac { p }{ q } \) where p and q are integers and \(q\neq 0\), is
\(\frac { 2999 }{ 1000 } \)
\(\frac { 19 }{ 10 } \)
3
\(\frac { 26 }{ 9 } \)
16.
Which of the following is an irrational number?
\(3.\overline { 3 } \)
3.763
\(3.\overline { 763 } \)
3.101 1001 10001...
17.
Which of the following is a rational number?
\(\sqrt { 5 } \)
\(\pi \)
0.101 001 0001 00001...
0.853 853 853....
18.
A number is an irrational if and only if its decimal representation is:
non-terminating
non-terminating and repeating
non terminating and non repeating
terminating
19.
A terminating decimal is:
natural number
a whole number
a rational number
an integer
20.
Two rational numbers between \(\frac { 2 }{ 3 } \) and \(\frac { 5 }{ 3 } \)are:
1/6 and 2/6
1/2 and 2/7
5/6 and 7/6
2/3 and 4/3
21.
Every rational number is:
a natural number
an integer
a real number
a whole number
22.
Find the value of [(16)1/2]1/2
23.
Simplify: √72+√800-18.
24.
Calculate the value of \(2.\bar{9}\) in the form of \(\frac{p}{q}\), where p and q are integers and q≠0.
25.
Express the rational number \(0.\bar{9}\) in the form \(\frac{p}{q}\), where p and q are integers and q≠0.
26.
Is \(\frac{\sqrt{98}}{\sqrt{2}}\) a rational number or not?
27.
If x = 3-2√2, find the value of √x+\(\frac{1}{\sqrt{x}}\)
1.
\(19-\sqrt { 3 } +12\sqrt { 2 } \)
2.
\(\frac { 2\sqrt { 7 } }{ 7\sqrt { 7 } } \)=\(\frac { 2 }{ 7 } \) which is a rational number.
3.
Let x=\(0.\overline { 47 } \) =0.4777....
Multiplying both sides by 10, we get
10x = 9.9999...
10x = 4.7777...
10x = 4.3+0.47777...
10x = 4.3+x
10x - x = 4.3
9x = 4.3
x = 4.3/9 = 43/90
Thus, \(0.\overline { 47 } \) = 43/90
Here p =4 3
q = 90(\(\neq 0\))
4.
\(\frac { 2 }{ 11 } \)= 0.1818...= \(0.\overline { 18 } \)
5.
(i) True, since the collection of whole numbers contains all the natural numbers.
(ii) False, for example,-2 is an integer but not a whole number.
(iii) False, for example,.\(\frac { 1 }{ 2 } \)is a rational number but not a whole number.
6.
\(\frac { 3 }{ 7 } =\frac { 30 }{ 70 } \)
\(\\ \frac { 5 }{ 7 } =\frac { 50 }{ 70 } \)
30<31<32<33<34<35
\(\frac { 30 }{ 70 } <\frac { 31 }{ 70 } <\frac { 32 }{ 70 } <\frac { 33 }{ 70 } <\frac { 34 }{ 70 } <\frac { 50 }{ 70 } \)
So four rational numbers between \(\frac { 3 }{ 7 } \) and \(\frac { 5 }{ 7 } \)
\(\frac { 31 }{ 70 } ,\frac { 32 }{ 70 } ,\frac { 33 }{ 70 } \) and \(\frac { 34 }{ 70 } \)
\(\frac { 31 }{ 70 } ,\frac { 16 }{ 35 } ,\frac { 33 }{ 70 } \)and \(\frac { 17 }{ 35 } \)
7.
\(\left( \sqrt { 3 } +\sqrt { 5 } \right) \)
8.
23-17
9.
5-14
10.
-11/4, -5/2, -9/4
11.
101/144, 47/72
12.
(d)
1/3
13.
(b)
9
14.
(b)
\(\frac { 25 }{ 7 } \)
15.
(c)
3
16.
(d)
3.101 1001 10001...
17.
(d)
0.853 853 853....
18.
(c)
non terminating and non repeating
19.
(c)
a rational number
20.
(c)
5/6 and 7/6
21.
(c)
a real number
22.
( )
[(16)1/2]1/2 = \([(4^{2})^{\frac{1}{2}}]^\frac{1}{2}=[4]^{\frac{1}{2}}\)
= 2
23.
( )
√72 + √800-√18
=\(\sqrt{36\times2}+\sqrt{400\times2}-\sqrt{9\times2}\)
= 6√2 + 20√2 - 3√2 = 23√2
24.
( )
Let x = \(2.\bar{9}\) = 2.9999...
⇒ 10x = 29.999...
⇒ 10x-x = (29.999...)-(2.999...)
9x = 27
x = \(\frac{27}{9}\) = 3
25.
( )
Let, x = 0.999...
⇒ 10x = 9.999...
⇒ 10x - x = (9.999...)-(0.999...)
⇒ 9x = 9
⇒ x = 1
26.
( )
\(\frac{\sqrt{98}}{\sqrt{2}}\)=\(\frac{\sqrt{98}}{\sqrt{2}}\) = √49 = 7
So, it is a rational number.
27.
\(x=3-2\sqrt { 2 } \Rightarrow \frac { 1 }{ x } =3+2\sqrt { 2 } \)
\({ \left( \sqrt { x } +\frac { 1 }{ \sqrt { x } } \right) }^{ 2 }=8\)
\(\Rightarrow \sqrt { x } +\frac { 1 }{ \sqrt { x } } =\pm 2\sqrt { 2 } \)
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