Tag: maths

Questions Related to maths

State whether True or False :


All the following numbers are irrationals.
(i) $\dfrac { 2 }{ \sqrt { 7 }  } $ (ii) $\dfrac { 3 }{ 2\sqrt { 5 }  }$ (iii) $4+\sqrt { 2 } $ (iv) $5\sqrt { 2 } $

maths rational numbers proof of irrationality of numbers proofs of irrationality introduction to irrational numbers

  1. True

  2. False

Show answer
Correct Option: A
Explanation:

In all of the above questions $\sqrt7,\sqrt5,\sqrt2$ are a $irrational$ numbers


And $\text{the addition, subtraction, division and product between  rational and irrational gives  an irrational number}$

So that all of the above are irrational numbers.

hence option A is correct.

State whether the given statement is True or False :

$2-3\sqrt { 5 }$ is an irrational number.
maths rational numbers proof of irrationality of numbers proofs of irrationality introduction to irrational numbers

  1. True

  2. False

Show answer
Correct Option: A
Explanation:

Here 2 is a rational number

and $3\sqrt5$ is an irrational number  and $\text{difference of rational and irrational is always an irrational number}$


So that $2 - 3\sqrt5 $  is an $irrational$ number

hence option A is correct.

State whether the given statement is True or False :

$\sqrt { 3 } +\sqrt { 4 } $ is an irrational number.
maths rational numbers proof of irrationality of numbers proofs of irrationality introduction to irrational numbers

  1. True

  2. False

Show answer
Correct Option: A
Explanation:

$\sqrt4= 2 \text{ is a  rational number and }$$\sqrt3$ is a $irrational$ number


And $\text{the addition of rational and irrational is always an irrational number}$

So that $(\sqrt3+\sqrt4)$ is an irrational number.

hence option A is correct.

State whether the given statement is True or False :

The number $6+\sqrt { 2 } $ is irrational.
maths rational numbers proof of irrationality of numbers proofs of irrationality introduction to irrational numbers

  1. True

  2. False

Show answer
Correct Option: A
Explanation:

$Here \  6 \text{ is a  rational number and }$$\sqrt2$ is a $irrational$ number


And $\text{the addition of rational and irrational is always an irrational number}$

So that $(6+\sqrt2)$ is an irrational number.

hence option A is correct

State whether the given statement is True or False :
If $p,  q $ are prime positive integers, then $\sqrt { p } +\sqrt { q } $ is an irrational number.
maths rational numbers proof of irrationality of numbers proofs of irrationality introduction to irrational numbers

  1. True

  2. False

Show answer
Correct Option: A
Explanation:
$\sqrt{p}+\sqrt{q}$ is rational ........ assumption
$\sqrt{p}+\sqrt{q}=\dfrac{a}{b}$
squaring, we get
$p+q+2\sqrt{pq}=\left(\dfrac{a}{b}\right)^{2}$

$\sqrt{pq}=\dfrac{1}{2}\left[\left(\dfrac{a}{b}\right)^{2}-p-q\right] - (i)$

Now, $p$ & $q$ are prime positive numbers so, $\sqrt{p}$ and $\sqrt{q}$ is irrational also $\sqrt{pq}$ 

so in (i)
Irrational $=$ rational $\Rightarrow$ which is a contradiction

$\Rightarrow\ \sqrt p+\sqrt q$ is a irrational number if $p,q$ are prime positive numbers

Prove following equation as irrational 

maths rational numbers proof of irrationality of numbers proofs of irrationality introduction to irrational numbers

  1. $2+\sqrt {3}$

  2. $2-\sqrt {3}$

  3. $3\sqrt {2}+\sqrt {3}$

  4. $\dfrac {1}{\sqrt {2}}$

  5. $\dfrac {1}{\sqrt {3}-\sqrt {2}}$

Show answer
Correct Option: A

$1$ billion $=$ _______ crores.

maths 5-digit numbers expanded form introduction to numbers and number systems numbers in general form

  1. $1$

  2. $10$

  3. $100$

  4. $1000$

Show answer
Correct Option: C
Explanation:

$ 10 $ crores $ = 10,00,00,000 $

$ 1 $ billion $ = 1,000,000,000 $

So, $ 1 $ billion  $ = 100 \times 10,00,00,000 = 100 $ crore

Compare $1,00,002 \square 10,00,002$

maths 5-digit numbers expanded form introduction to numbers and number systems numbers in general form

  1. $>$

  2. $<$

  3. $=$

  4. none of these

Show answer
Correct Option: B
Explanation:

Since,  $1,00,002$ is six digit number and $10,00,002$ is seven digit number.
$\therefore 1,00,002 < 10,00,002$
Option $B$ is correct.

The greater number in the subtraction is called

maths 5-digit numbers expanded form introduction to numbers and number systems numbers in general form

  1. minuend

  2. difference

  3. subtrahend

  4. none of these

Show answer
Correct Option: A
Explanation:

In a subtraction minuend is a greater number and subtrahend is smaller number.
$\therefore$ greater number in the subtraction is called minuend.
Option A is correct.

Place value and face value are always equal for

maths 5-digit numbers expanded form introduction to numbers and number systems numbers in general form

  1. $0$

  2. $10$

  3. any digit

  4. $100$

Show answer
Correct Option: A
Explanation:

Place value and face value are always equal for $ 0 $