Tag: rational and irrational numbers

Questions Related to rational and irrational numbers

The fractional part of $28.13$ is?

decimal representation of rational numbers rational and irrational numbers maths

  1. $0.13$

  2. $28.1$

  3. $2.81$

  4. $13.28$

Show answer
Correct Option: A
Explanation:

The integer part, or integral part of a decimal number is the part to the left of the decimal separator. The part from the decimal separator to the right is the fractional part.

$\therefore$ Fractional part of $28.13$ is $0.13$

The integral part of $78.027$ is?

decimal representation of rational numbers rational and irrational numbers maths

  1. $24$

  2. $0.27$

  3. $78$

  4. $38$

Show answer
Correct Option: C
Explanation:

The integer part, or integral part of a decimal number is the part to the left of the decimal separator. The part from the decimal separator to the right is the fractional part.

$\therefore$ Integral part of $78.027$ is $78$

The number obtained by interchanging integral part and fractional part of $45.01$ is?

decimal representation of rational numbers rational and irrational numbers maths

  1. $1.45$

  2. $10.45$

  3. $11.45$

  4. $1.045$

Show answer
Correct Option: A
Explanation:

Integral part of a decimal number is the part to the left of the decimal separator. The part from the decimal separator to the right is the fractional part.


$\Rightarrow$ Integral part $=45$ Fractional part $=0.01$

$\therefore$ New number obtained by interchanging $= 1.45$

Integral part of $034.098$ is :

decimal representation of rational numbers rational and irrational numbers maths

  1. $34$

  2. $09$

  3. $098$

  4. $0.098$

Show answer
Correct Option: A
Explanation:

Let $x=34.098$

$\left[ x \right] =x-\left{ x \right} \ \left[ 34.098 \right] =34.098-.098\ \Rightarrow \left[ 34.098 \right] =34$
where $\left[ . \right] $ is integral part and ${.}$ is fractional part function
So, option A is correct.

Number obtained by incrementing an integral part of $27.25$ by $1$ is?

decimal representation of rational numbers rational and irrational numbers maths

  1. $35.26$

  2. $28.25$

  3. $26.25$

  4. $16.25$

Show answer
Correct Option: B
Explanation:

Integral part of a decimal number is the part to the left of the decimal separator. 


$\Rightarrow$ Integral part $=27$
By incrementing integral part by $1$ we get $28$

$\therefore$ New number obtained $= 28.25$

The Number obtained by interchanging the integral and fractional part of $50.23$ is?

decimal representation of rational numbers rational and irrational numbers maths

  1. $23.05$

  2. $23.5$

  3. $32.05$

  4. $23.4$

Show answer
Correct Option: B
Explanation:

Integral part of a decimal number is the part to the left of the decimal separator. The part from the decimal separator to the right is the fractional part.


$\Rightarrow$ Integral part $=50$ Fractional part $=0.23$

$\therefore$ New number obtained by interchanging $= 23.50 = 23.5$

The number obtained on interchanging the integral and fractional part of $26.081$ is

decimal representation of rational numbers rational and irrational numbers maths

  1. $18.026$

  2. $260.81$

  3. $81.26$

  4. $81.62$

Show answer
Correct Option: C
Explanation:

Integral part of a decimal number is the part to the left of the decimal separator. The part from the decimal separator to the right is the fractional part.


$\Rightarrow$ Integral part $= 26$ Fractional part $=0.081$

New number obtained $= 81.26$

The point $\left( \sin { \theta  } ,\cos { \theta  }  \right) ,\theta $ being any real number, lie inside the circle ${ x }^{ 2 }+{ y }^{ 2 }-2x-2y+\lambda =0$, if

decimal representation of rational numbers rational and irrational numbers maths

  1. $\lambda <1+2\sqrt { 2 } $

  2. $\lambda >2\sqrt { 2 } -1$

  3. $\lambda <-1-2\sqrt { 2 } $

  4. $\lambda >1+2\sqrt { 2 } $

Show answer
Correct Option: C
Explanation:
${ x }^{ 2 }+{ y }^{ 2 }-2x-2y+\lambda =0$
Radius of circle$=\sqrt { 1+1-\lambda  } $
$=\sqrt { 2-\lambda  } $
Maximum distance of $\left( \sin { \theta  } ,\cos { \theta  }  \right) $
From center of above circle is when $\theta =\cfrac { 5\pi  }{ 4 } $
Thus distance$=\sqrt { { \left( 1+\cfrac { 1 }{ \sqrt { 2 }  }  \right)  }^{ 2 }+{ \left( 1+\cfrac { 1 }{ \sqrt { 2 }  }  \right)  }^{ 2 } } $\
$=\left( \sqrt { 2 } +1 \right) $
$\therefore \sqrt { 2-\lambda  } >\sqrt { 2+1 } $ [For all points to lie in center]
$\therefore 2-\lambda >2+1+2\sqrt { 2 } $
$\lambda <2-3-2\sqrt { 2 } $
$\lambda <-1-2\sqrt { 2 } $

Number of solutions of the equation $[2x]-3{2x}=1$ is?

(where $[\cdot]$ and ${\cdot }$ denote greatest integer an fractional part function respectively).

decimal representation of rational numbers rational and irrational numbers maths

  1. $1$

  2. $2$

  3. $3$

  4. $0$

Show answer
Correct Option: C

Integral part of $05.89$ is?

decimal representation of rational numbers rational and irrational numbers maths

  1. $5$

  2. $89$

  3. $2$

  4. $3$

Show answer
Correct Option: A
Explanation:

The integer part, or integral part of a decimal number is the part to the left of the decimal separator. 

So. Integral Part of given number 05.89 = 5.89 is 5