Tag: introduction to numbers and number systems

Questions Related to introduction to numbers and number systems

The locus of point of trisections of the focal chords of the parabola, ${y^2} = 4x$ :

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

  1. ${y^2} = x - 1$

  2. $9{y^2} = 4\left( {3x - 4} \right)$

  3. ${y^2} = 2\left( {1 - x} \right)$

  4. None of these

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Correct Option: B

Prove that $\dfrac{a^{-1}}{(a^{-1}+b^{-1})}$ is equal to $\dfrac{b}{(a+b)}$

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

  1. True

  2. False

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Correct Option: A
Explanation:
$\cfrac{{ a }^{ -1 }}{{ a }^{ -1 }+{ b }^{ -1 }}\Leftrightarrow \cfrac{{ a }^{ -1 }}{\cfrac{1}{a}+\cfrac{1}{b}}$
$\Rightarrow$ $\cfrac{{ a }^{ -1 }}{\cfrac{b+a}{a.b}}$
$\Rightarrow$ $\cfrac{a.b}{a(b+a)}$
$\Rightarrow$ $\cfrac{b}{b+a}$
$\Rightarrow$ $\cfrac{b}{a+b}$
$\therefore$ $\cfrac{{ a }^{ -1 }}{{ a }^{ -1 }+{ b }^{ -1 }}=\cfrac{b}{a+b}$

The number of digits in $5^{30}$ is ,$(\log _{10}2=0.3010)$

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

  1. $30$

  2. $22$

  3. $21$

  4. $none\ of\ these$

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Correct Option: A

Numeral for ninety million ninety thousand ninety is

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

  1. $9090095$

  2. $90090090$

  3. $909090$

  4. None of these

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Correct Option: B
Explanation:
We know that,

$1$ million $= 1000000$, therefore, $90$ million $= 90000000$

$1$ thousand $= 1000$, therefore, $90$ thousand $= 90000$

Thus, ninety million ninety thousand ninety is

$=90000000+90000+90=90090090$

Hence, numeral for ninety million ninety thousand ninety is $90090090$.

How many hundreds are there in 5127900 ?

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

  1. 9

  2. 900

  3. 90

  4. 9000

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Correct Option: A

The positive two-digit integers $x$ and $y$ have the same digits, but in reverse order. Which of the following must be a factor of $x + y$? 

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

  1. $6$

  2. $9$

  3. $10$

  4. $11$

  5. $14$

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Correct Option: D
Explanation:

$\Rightarrow$  Let two positive integers are $x=17$ and $y=71$.

$\Rightarrow$  $x+y=17+71=88$
$\Rightarrow$  $88=11\times 2 \times 2\times 2$
$\therefore$   From the factors given in the options, $11$ will be the factor of $x+y$.

The  expanded form of $67$ is

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

  1. $66+1$

  2. $65+2$

  3. $60 \times 1 + 7$

  4. None of the above

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Correct Option: C
Explanation:

$\Rightarrow$  The given number is $67$.

$\Rightarrow$  The expand form of $67$ = $60\times 1+7$

Expanded form of $78.059$ is:

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

  1. $\displaystyle 78+\frac{5}{10}+\frac{9}{100}$

  2. $\displaystyle 70+8+0+\frac{5}{100}+\frac{9}{1000}$

  3. $\displaystyle 70+8+\frac{5}{10}+\frac{9}{100}$

  4. none

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Correct Option: B
Explanation:

The expanded form of $78.059$ is

$78+0.059= 70+8+0+\dfrac{5}{100}+\dfrac{9}{1000}$

The general form of $129$ is

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

  1. $100 \times 1 + 10 \times 2 + 9 \times 1$

  2. $120 \times 1 + 9 \times 1$

  3. $100 \times 1 + 30 \times 1 - 1$

  4. All of the above

Show answer
Correct Option: A
Explanation:

General form of a number is denoted by expansion of itself written as sum of multiplication of the digit by its place value.

Therefore, $129$ can be written as $1 \times 100 + 2 \times 10 + 9 \times 1$

The general form of $6.234$ is

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

  1. $6 + \dfrac {200}{10} + \dfrac {3}{100} + \dfrac {4}{1000}$

  2. $6 + \dfrac {2}{10} + \dfrac {30}{100} + \dfrac {4}{1000}$

  3. $60 + \dfrac {2}{10} + \dfrac {3}{100} + \dfrac {4}{1000}$

  4. $6 + \dfrac {2}{10} + \dfrac {3}{100} + \dfrac {4}{1000}$

Show answer
Correct Option: D
Explanation:
Representing the given number in base $10$.
$6.234 = 6\times { 10 }^{ 0 }+2\times { 10 }^{ -1 }+3\times { 10 }^{ -2 }+4\times { 10 }^{ -3 }$
$\Rightarrow 6.234 = 6+ \dfrac { 2 }{ 10 } +\dfrac { 3 }{ 100 } +\dfrac { 4 }{ 1000 } $