Tag: maths

Questions Related to maths

State whether the following statement are true or false? Justify your answers.

Every real number is an irrational number.

maths rational and irrational numbers existence of irrational numbers irrational numbers properties of irrational numbers

  1. True

  2. False

Show answer
Correct Option: B
Explanation:

The statement is false since real numbers consists of both rational and irrational numbers. $5,65,8/9...$ are all real numbers which are rational.

Classify the following numbers as rational or irrational : $2-\sqrt{5}$

maths rational and irrational numbers existence of irrational numbers irrational numbers properties of irrational numbers

  1. Irrational number

  2. Rational number

  3. Less Data

  4. None of the above

Show answer
Correct Option: A
Explanation:

$2$ is rational

$\sqrt 5 =2.035.........$ which is non terminating and non repeating hence irrational number.
We know that rational- irrational= irrational number.
Hence $2-\sqrt 5= irrational \,  number$
Hence, option A is the correct answer.

Decimal representation of an irrational number is always

maths rational and irrational numbers existence of irrational numbers irrational numbers properties of irrational numbers

  1. Terminating

  2. Terminating, Repeating

  3. Non-Terminating, Repeating

  4. Non-Terminating, Non-Repeating

Show answer
Correct Option: D
Explanation:

Decimal representation of an irrational number is always non terminating non repeating.

 For example,$\sqrt{2}$ $=1.41421356237309504880168872420969807856967187537694807317667973799...$

Are the square roots of all positive integers irrational?

maths rational and irrational numbers existence of irrational numbers irrational numbers properties of irrational numbers

  1. True

  2. False

Show answer
Correct Option: B
Explanation:

No. Not all square roots of integers are irrational. Examples are $\sqrt{4}=2$, $\sqrt{1}=1$, $\sqrt{9}=3$, etc.

Which among the following is true?

maths rational and irrational numbers existence of irrational numbers irrational numbers properties of irrational numbers

  1. There is no rational number between two irrational numbers.

  2. If ${x}^{2}=0.4$,then x is a rational number.

  3. The only real numbers are rational numbers.

  4. The reciprocal of an irrational number is irrational.

Show answer
Correct Option: D

The decimal expansion of the number $\sqrt{2}$ is 

maths rational and irrational numbers existence of irrational numbers irrational numbers properties of irrational numbers

  1. A finite decimal

  2. 1.4121

  3. Non-Terminating, Recurring

  4. Non-Terminating, Non-Recurring

Show answer
Correct Option: D
Explanation:

$\sqrt{2} = 1.4142136...$
The decimal expansion of the number $\sqrt{2}$ is Non-terminating non recurring

State True or False.

${(\sqrt{2}-2)}^{2}$ is an irrational number.

maths rational and irrational numbers existence of irrational numbers irrational numbers properties of irrational numbers

  1. True

  2. False

Show answer
Correct Option: A
Explanation:

$\ { (\sqrt { 2 } -2) }^{ 2 }=2+4-4\sqrt { 2 } =6-4\sqrt { 2 } \ \sqrt { 2 } =1.41421356237........\ \ \sqrt { 2 } is\quad an\quad irrational\quad number,\quad since\quad its\quad decimal\quad representaion\quad is\quad non\quad terminating\quad non\quad repeating.\ Multiplication\quad and\quad subtration\quad of\quad rational\quad with\quad irrational\quad is\quad irrational.\ Hence,\quad { (\sqrt { 2 } -2) }^{ 2 }\quad is\quad an\quad irrational\quad number.\ \quad $

State True or False.

(√2+√3)2(2√+3√)2 is an irrational number.

maths rational and irrational numbers existence of irrational numbers irrational numbers properties of irrational numbers

  1. True

  2. False

Show answer
Correct Option: A
Explanation:

$\ { (\sqrt { 2 } +\sqrt { 3 } ) }^{ 2 }=2+3+2\sqrt { 6 } =5+2\sqrt { 6 } \ \sqrt { 6 } =2.44948974278........\ \ \sqrt { 6 } is\quad an\quad irrational\quad number,\quad since\quad its\quad decimal\quad representaion\quad is\quad non\quad terminating\quad non\quad repeating.\ Multiplication\quad ans\quad addition\quad of\quad rational\quad with\quad irrational\quad is\quad irrational.\ Hence,\quad { (\sqrt { 2 } +\sqrt { 3 } ) }^{ 2 }\quad is\quad an\quad irrational\quad number.\ \quad $

State True or False.

$2+\sqrt{3}$ is an irrational number.

maths rational and irrational numbers existence of irrational numbers irrational numbers properties of irrational numbers

  1. True

  2. False

Show answer
Correct Option: A
Explanation:

$\ \sqrt { 3 } =1.73205080757......\ \sqrt { 3 } is\quad an\quad irrational\quad number,\quad since\quad it's\quad decimal\quad representaion\quad is\quad non\quad terminating\quad non\quad repeating.\ And\quad addition\quad of\quad a\quad rational\quad and\quad irrational\quad number\quad is\quad irrational.\ Hence,\quad 2+\sqrt { 3 } \quad is\quad an\quad irrational\quad number.\ \quad $

State True or False.

$\sqrt{3}+\sqrt{2}$ is an irrational number.

maths rational and irrational numbers existence of irrational numbers irrational numbers properties of irrational numbers

  1. True

  2. False

Show answer
Correct Option: A
Explanation:

$\ \sqrt { 3 } =1.73205080757......\ Also\quad \sqrt { 2 } =1.41421356237........\ \sqrt { 3 } and\quad \sqrt { 2 } are\quad irrational\quad numbers,\quad since\quad their\quad decimal\quad representaion\quad is\quad non\quad terminating\quad non\quad repeating.\ And\quad addition\quad of\quad two\quad irrational\quad numbers\quad is\quad irrational.\ Hence,\quad \sqrt { 2 } +\sqrt { 3 } \quad is\quad an\quad irrational\quad number.\ \quad $