NCERT Exemplar solutions for Mathematics [English] Class 9 chapter 1 - Number Systems [Latest edition]

Every rational number is ______.

Between two rational numbers ______.

Decimal representation of a rational number cannot be ______.

The product of any two irrational numbers is ______.

The decimal expansion of the number `sqrt(2)` is ______.

Which of the following is irrational?

Which of the following is irrational?

A rational number between `sqrt(2)` and `sqrt(3)` is ______.

The value of 1.999... in the form `p/q`, where p and q are integers and q ≠ 0, is ______.

`2sqrt(3) + sqrt(3)` is equal to ______.

`sqrt(10) xx sqrt(15)` is equal to ______.

The number obtained on rationalising the denominator of `1/(sqrt(7) - 2)` is ______.

`1/(sqrt(9) - sqrt(8))` is equal to ______.

After rationalising the denominator of `7/(3sqrt(3) - 2sqrt(2))`, we get the denominator as ______.

The value of `(sqrt(32) + sqrt(48))/(sqrt(8) + sqrt(12))` is equal to ______.

If `sqrt(2) = 1.4142`, then `sqrt((sqrt(2) - 1)/(sqrt(2) + 1))` is equal to ______.

`root(4)root(3)(2^2)` equals to ______.

The product `root(3)(2) * root(4)(2) * root(12)(32)` equals to ______.

Value of `root(4)((81)^-2)` is ______.

Value of (256)^0.16 × (256)^0.09 is ______.

Which of the following is equal to x?

Let x and y be rational and irrational numbers, respectively. Is x + y necessarily an irrational number? Give an example in support of your answer.

Let x be rational and y be irrational. Is xy necessarily irrational? Justify your answer by an example.

`sqrt(2)/3` is a rational number.

There are infinitely many integers between any two integers.

Number of rational numbers between 15 and 18 is finite.

There are numbers which cannot be written in the form `p/q, q ≠ 0, p, q` both are integers.

The square of an irrational number is always rational.

`sqrt(12)/sqrt(3)` is not a rational number as `sqrt(12)` and `sqrt(3)` are not integers.

`sqrt(15)/sqrt(3)` is written in the form `p/q, q ≠ 0` and so it is a rational number.

Classify the following number as rational or irrational with justification:

`sqrt(196)`

Classify the following number as rational or irrational with justification:

`3sqrt(18)`

Classify the following number as rational or irrational with justification:

`sqrt(9/27)`

Classify the following number as rational or irrational with justification:

`sqrt(28)/sqrt(343)`

Classify the following number as rational or irrational with justification:

`- sqrt(0.4)`

Classify the following number as rational or irrational with justification:

`sqrt(12)/sqrt(75)`

Classify the following number as rational or irrational with justification:

0.5918

Classify the following number as rational or irrational with justification:

`(1 + sqrt(5)) - (4 + sqrt(5))`

Classify the following number as rational or irrational with justification:

10.124124...

Classify the following number as rational or irrational with justification:

1.010010001...

Find whether the variable x represents a rational or an irrational number:

x² = 5

Find whether the variable y represents a rational or an irrational number:

y² = 9

Find whether the variable z represents a rational or an irrational number:

z² = 0.04

Find whether the variable u represents a rational or an irrational number:

`u^2 = 17/4`

Find three rational numbers between –1 and –2

Find three rational numbers between 0.1 and 0.11

Find three rational numbers between `5/7` and `6/7`

Find three rational numbers between `1/4` and `1/5`

Insert a rational number and an irrational number between the following:

2 and 3

Insert a rational number and an irrational number between the following:

0 and 0.1

Insert a rational number and an irrational number between the following:

`1/3` and `1/2`

Insert a rational number and an irrational number between the following:

`(-2)/5` and `1/2`

Insert a rational number and an irrational number between the following:

0.15 and 0.16

Insert a rational number and an irrational number between the following:

`sqrt(2)` and `sqrt(3)`

Insert a rational number and an irrational number between the following:

2.357 and 3.121

Insert a rational number and an irrational number between the following:

0.0001 and 0.001

Insert a rational number and an irrational number between the following:

3.623623 and 0.484848

Insert a rational number and an irrational number between the following:

6.375289 and 6.375738

Represent the following number on the number line:

7

Represent the following number on the number line:

7.2

Represent the following number on the number line:

`(-3)/2`

Represent the following number on the number line:

`(-12)/5`

Locate `sqrt(5), sqrt(10)` and `sqrt(17)` on the number line.

Represent geometrically the following number on the number line:

`sqrt(4.5)`

Represent geometrically the following number on the number line:

`sqrt(5.6)`

Represent geometrically the following number on the number line:

`sqrt(8.1)`

Represent geometrically the following number on the number line:

`sqrt(2.3)`

Express the following in the form `p/q`, where p and q are integers and q ≠ 0:

0.2

Express the following in the form `p/q`, where p and q are integers and q ≠ 0:

0.888...

Express the following in the form `p/q`, where p and q are integers and q ≠ 0:

`5.bar2`

Express the following in the form `p/q`, where p and q are integers and q ≠ 0:

`0.bar001`

Express the following in the form `p/q`, where p and q are integers and q ≠ 0:

0.2555...

Express the following in the form `p/q`, where p and q are integers and q ≠ 0:

`0.1bar34`

Express the following in the form `p/q`, where p and q are integers and q ≠ 0:

0.00323232...

Express the following in the form `p/q`, where p and q are integers and q ≠ 0:

0.404040...

Show that 0.142857142857... = `1/7`

Simplify the following: 

`sqrt(45) - 3sqrt(20) + 4sqrt(5)`

Simplify the following: 

`sqrt(24)/8 + sqrt(54)/9`

Simplify the following: 

`4sqrt12 xx 7sqrt6`

Simplify the following:

`4sqrt(28) ÷ 3sqrt(7) ÷ root(3)(7)`

Simplify the following:

`3sqrt(3) + 2sqrt(27) + 7/sqrt(3)`

Simplify the following: 

`(sqrt(3) - sqrt(2))^2`

Simplify the following: 

`root(4)(81) - 8root(3)(216) + 15root(5)(32) + sqrt(225)`

Simplify the following:

`3/sqrt(8) + 1/sqrt(2)`

Simplify the following:

`(2sqrt(3))/3 - sqrt(3)/6`

Rationalise the denominator of the following:

`2/(3sqrt(3)`

Rationalise the denominator of the following:

`sqrt(40)/sqrt(3)`

Rationalise the denominator of the following:

`(3 + sqrt(2))/(4sqrt(2))`

Rationalise the denominator of the following:

`16/(sqrt(41) - 5)`

Rationalise the denominator of the following:

`(2 + sqrt(3))/(2 - sqrt(3))`

Rationalise the denominator of the following:

`sqrt(6)/(sqrt(2) + sqrt(3))`

Rationalise the denominator of the following:

`(sqrt(3) + sqrt(2))/(sqrt(3) - sqrt(2))`

Rationalise the denominator of the following:

`(3sqrt(5) + sqrt(3))/(sqrt(5) - sqrt(3))`

Rationalise the denominator of the following:

`(4sqrt(3) + 5sqrt(2))/(sqrt(48) + sqrt(18))`

Find the value of a and b in the following:

`(5 + 2sqrt(3))/(7 + 4sqrt(3)) = a - 6sqrt(3)`

Find the value of a and b in the following:

`(3 - sqrt(5))/(3 + 2sqrt(5)) = asqrt(5) - 19/11`

Find the value of a and b in the following:

`(sqrt(2) + sqrt(3))/(3sqrt(2) - 2sqrt(3)) = 2 - bsqrt(6)`

Find the value of a and b in the following:

`(7 + sqrt(5))/(7 - sqrt(5)) - (7 - sqrt(5))/(7 + sqrt(5)) = a + 7/11 sqrt(5)b`

If  `a = 2 + sqrt(3)`, then find the value of `a - 1/a`.

Rationalise the denominator in the following and hence evaluate by taking `sqrt(2) = 1.414, sqrt(3) = 1.732` and `sqrt(5) = 2.236`, upto three places of decimal.

`4/sqrt(3)`

Rationalise the denominator in the following and hence evaluate by taking `sqrt(2) = 1.414, sqrt(3) = 1.732` and `sqrt(5) = 2.236`, upto three places of decimal.

`6/sqrt(6)`

Rationalise the denominator in the following and hence evaluate by taking `sqrt(2) = 1.414, sqrt(3) = 1.732` and `sqrt(5) = 2.236`, upto three places of decimal.

`(sqrt(10) - sqrt(5))/2`

Rationalise the denominator in the following and hence evaluate by taking `sqrt(2) = 1.414, sqrt(3) = 1.732` and `sqrt(5) = 2.236`, upto three places of decimal.

`sqrt(2)/(2 + sqrt(2)`

Rationalise the denominator in the following and hence evaluate by taking `sqrt(2) = 1.414, sqrt(3) = 1.732` and `sqrt(5) = 2.236`, upto three places of decimal.

`1/(sqrt(3) + sqrt(2))`

Simplify:

`(1^3 + 2^3 + 3^3)^(1/2)`

Simplify:

`(3/5)^4 (8/5)^-12 (32/5)^6`

Simplify:

`(1/27)^((-2)/3)`

Simplify:

`[((625)^(-1/2))^((-1)/4)]^2`

Simplify:

`(9^(1/3) xx 27^(-1/2))/(3^(1/6) xx 3^(- 2/3))`

Simplify:

`64^(-1/3)[64^(1/3) - 64^(2/3)]`

Simplify:

`(8^(1/3) xx 16^(1/3))/(32^(-1/3))`

Express `0.6 + 0.bar7 + 0.4bar7` in the form `p/q`, where p and q are integers and q ≠ 0.

Simplify:

`(7sqrt(3))/(sqrt(10) + sqrt(3)) - (2sqrt(5))/(sqrt(6) + sqrt(5)) - (3sqrt(2))/(sqrt(15) + 3sqrt(2))`

If `sqrt(2) = 1.414, sqrt(3) = 1.732`, then find the value of `4/(3sqrt(3) - 2sqrt(2)) + 3/(3sqrt(3) + 2sqrt(2))`.

If `a = (3 + sqrt(5))/2`, then find the value of `a^2 + 1/a^2`.

If `x = (sqrt(3) + sqrt(2))/(sqrt(3) - sqrt(2))` and `y = (sqrt(3) - sqrt(2))/(sqrt(3) + sqrt(2))`, then find the value of x² + y².

Simplify:

`(256)^(-(4^((-3)/2))`

Find the value of `4/((216)^(-2/3)) + 1/((256)^(- 3/4)) + 2/((243)^(- 1/5))`