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Decimal representation of rational numbers - class-IX

Description: decimal representation of rational numbers
Number of Questions: 23
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Tags: maths rational and irrational numbers
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Number of ordered pairs $(x, y)$ of real numbers satisfying the equation $x^{2} + y^{2} - 24x - 26y + 313 = 0$ is equal to

decimal representation of rational numbers rational and irrational numbers maths

  1. Infinite

  2. Finite but more than one

  3. Exactly one

  4. Zero

Show answer
Correct Option: D

The number having no reciprocal is

decimal representation of rational numbers rational and irrational numbers maths

  1. $2$

  2. $1$

  3. $-1$

  4. $0$

Show answer
Correct Option: D
Explanation:

If a number is 2 then its reciprocal is $\dfrac{1}{2}$

The number '0' has no reciprocal.

The decimal part of $9.99$ is

decimal representation of rational numbers rational and irrational numbers maths

  1. $0.99$

  2. $9$

  3. $8$

  4. $0.98$

Show answer
Correct Option: A
Explanation:

Decimal part of $9.99$ is $0.99$.

So, option A is correct.

The integral part of $3.89$ is

decimal representation of rational numbers rational and irrational numbers maths

  1. $3$

  2. $0.89$

  3. $4$

  4. None of these

Show answer
Correct Option: A
Explanation:

Integral part of $3.89$ is $3$.

The integral part of $4.20$ is

decimal representation of rational numbers rational and irrational numbers maths

  1. $4$

  2. $2$

  3. $3$

  4. None of these

Show answer
Correct Option: A
Explanation:

The integral part of $4.20$ is $4$.

So, option A is correct.

The decimal part of $4.20$ is

decimal representation of rational numbers rational and irrational numbers maths

  1. $4$

  2. $0.20$

  3. $2$

  4. None of these

Show answer
Correct Option: B
Explanation:

The decimal part of $4.20$ is $0.20$.

The decimal part of $3.89$ is

decimal representation of rational numbers rational and irrational numbers maths

  1. $3$

  2. $0.79$

  3. $0.89$

  4. None of these

Show answer
Correct Option: C
Explanation:

A decimal may have both a whole-number part and a fractional part. 

The whole-number part of a decimal are those digits to the left of the decimal point. 

The fractional part(decimal part) of a decimal is represented by the digits to the right of the decimal point.

Decimal part of $3.89$ is $0.89$.

The integral part of $9.99$ is

decimal representation of rational numbers rational and irrational numbers maths

  1. $9$

  2. $8$

  3. $7$

  4. None of these

Show answer
Correct Option: A
Explanation:

Fractional part of $9.99$ is $9$.

So, option A is correct.

The integral part of $63.281$ is?

decimal representation of rational numbers rational and irrational numbers maths

  1. $63$

  2. $0.281$

  3. $26$

  4. $5$

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.

$\therefore$ Integral part of $63.281$ is $63$

Fractional Part of $2.056$ is?

decimal representation of rational numbers rational and irrational numbers maths

  1. $0.056$

  2. $0.05$

  3. $0.5$

  4. $0.66$

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 $2.056$ is $0.056$

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

If a and b are any two such real numbers that ab $ = 0 $ , then

decimal representation of rational numbers rational and irrational numbers maths

  1. $a = 0, b \leq 0$

  2. $b = 0, a \leq 0$

  3. a = 0 or b = 0 or both

  4. $a = b$ and $b = 0$

Show answer
Correct Option: C
Explanation:

if both number are real the either a or b or both should be zero.
then only ab will be 0.
if any real number is multiplied by 0 then result will be zero.
So, answer is
C a = 0 or b = 0 or both

If $f(x)-2f(1-x) = x^2+2$, then what is $f(x)$?

decimal representation of rational numbers rational and irrational numbers maths

  1. $f(x)=-x^2+\dfrac{4}{3}x-\dfrac{3}{8}$

  2. $f(x)=−x^2+\dfrac{4}{3}x−\dfrac{8}{3}$

  3. $f(x)=−x^2+\dfrac{8}{3}x−\dfrac{4}{3}$

  4. $f(x)=−x^2+\dfrac{3}{8}x−\dfrac{3}{4}$

Show answer
Correct Option: B
Explanation:
$f\left(x\right)-2f\left(1-x\right)={x}^{2}+2$      .......$(1)$

Setting $x=1-x$ then we get

$f\left(1-x\right)-2f\left(1-1+x\right)={\left(1-x\right)}^{2}+2$ 

$f\left(1-x\right)-2f\left(x\right)={x}^{2}-2x+3$ 

$2f\left(1-x\right)-4f\left(x\right)=2{x}^{2}-4x+6$    .......$(2)$

Adding $(1)$ and $(2)$ we get

$-3f\left(x\right)=3{x}^{2}-4x+8$ 

$\therefore f\left(x\right)=-{x}^{2}+\dfrac{4}{3}x-\dfrac{8}{3}$ 
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