Tag: decimal representation of rational numbers

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Questions Related to decimal representation of rational numbers

Multiple choice decimal representation of rational numbers rational and irrational numbers maths

The fractional part of $28.13$ is?

  1. $0.13$
  2. $28.1$
  3. $2.81$
  4. $13.28$
Reveal answer Fill a bubble to check yourself
A Correct answer
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 $0.13$

Multiple choice decimal representation of rational numbers rational and irrational numbers maths

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

  1. $1.45$
  2. $10.45$
  3. $11.45$
  4. $1.045$
Reveal answer Fill a bubble to check yourself
A Correct answer
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 $= 1.45$

Multiple choice decimal representation of rational numbers rational and irrational numbers maths

Integral part of $034.098$ is :

  1. $34$
  2. $09$
  3. $098$
  4. $0.098$
Reveal answer Fill a bubble to check yourself
A Correct answer
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.

Multiple choice decimal representation of rational numbers rational and irrational numbers maths

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

  1. $35.26$
  2. $28.25$
  3. $26.25$
  4. $16.25$
Reveal answer Fill a bubble to check yourself
B Correct answer
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 $= 28.25$

Multiple choice decimal representation of rational numbers rational and irrational numbers maths

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

  1. $23.05$
  2. $23.5$
  3. $32.05$
  4. $23.4$
Reveal answer Fill a bubble to check yourself
B Correct answer
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 $= 23.50 = 23.5$

Multiple choice decimal representation of rational numbers rational and irrational numbers maths

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

  1. $18.026$
  2. $260.81$
  3. $81.26$
  4. $81.62$
Reveal answer Fill a bubble to check yourself
C Correct answer
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 $= 81.26$

Multiple choice decimal representation of rational numbers rational and irrational numbers maths

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

  1. $\lambda <1+2\sqrt { 2 } $
  2. $\lambda >2\sqrt { 2 } -1$
  3. $\lambda <-1-2\sqrt { 2 } $
  4. $\lambda >1+2\sqrt { 2 } $
Reveal answer Fill a bubble to check yourself
C Correct answer
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 } $
Multiple choice decimal representation of rational numbers rational and irrational numbers maths

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

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

  1. $1$
  2. $2$
  3. $3$
  4. $0$
Reveal answer Fill a bubble to check yourself
C Correct answer
Explanation

Solving [2x] - 3{2x} = 1. Let y = 2x. Then [y] - 3{y} = 1. Since {y} = y - [y], we have [y] - 3(y - [y]) = 1, which simplifies to 4[y] - 3y = 1, or y = (4[y] - 1)/3. Testing integer values for [y] gives valid solutions for y in the range [n, n+1).