Mathematics · Quantitative Aptitude

Real Number System

277 Questions

The real number system encompasses rational and irrational numbers, forming the basis of arithmetic and number theory. It is a key area in the quantitative aptitude and mathematics sections of school level and competitive exams. Practice these questions to master the properties and identification of different types of numbers.

Identifying irrational numbersOrdering rational numbersProperties of square rootsPerfect number propertiesRational and irrational products

Real Number System Questions

Multiple choice maths square and square root perfect square or square number squares and triangles powers and roots

Find the square of rational number: $\dfrac{7 \times 7\times 4}{28 \times 14}$

  1. $\dfrac{3}{4}$
  2. $\dfrac{1}{5}$
  3. $\dfrac{5}{4}$
  4. $\dfrac{1}{4}$
Reveal answer Fill a bubble to check yourself
D Correct answer
Explanation

Square of rational numbers = $(\cfrac{7 \times 7\times 4}{28 \times 14})^2$
= $\cfrac{49 \times 49\times 16}{784 \times 196}$
= $\cfrac{7 \times 7\times 4}{28 \times 14}$
= $\cfrac{1}{4}$

Multiple choice maths square and square root perfect square or square number squares and triangles powers and roots

Find the square of rational number: $\dfrac{17 \times 18}{2 \times 9}$

  1. $144$
  2. $81$
  3. $289$
  4. $324$
Reveal answer Fill a bubble to check yourself
C Correct answer
Explanation

Square of rational number $ = (\dfrac{17 \times 18}{2 \times 9})^2$
                                            

                                            $ = \dfrac{289 \times 324}{4 \times 81}$
                                             $= 289$

Multiple choice irrational numbers irrational numbers and their operations rational and irrational numbers real numbers (rational and irrational numbers) maths

State whether the following statement is True or False:
A rational and Irrational number between $2.357$ and $3.121$ is $3, 3.101101110 ...$

  1. True

  2. False
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

Any terminating decimal between 2.357 and 3.121 will be a rational number like, ${3}$

Any non-terminating and non-recurring decimal between  2.357 and 3.121 will be an irrational number like $3.101101110...$

Multiple choice irrational numbers irrational numbers and their operations rational and irrational numbers real numbers (rational and irrational numbers) maths

State true or false:
The three rational numbers between $\displaystyle \sqrt{3}$ and $\displaystyle \sqrt{5}$ are 1.6, 1.8, 2.2

  1. True

  2. False
Reveal answer Fill a bubble to check yourself
B Correct answer
Explanation

We know that, $ \sqrt {3} = 1.732 $ and $ \sqrt {5} = 2.236 $





Hence three rational numbers between $ 1.732 $ and $ 2.236 $  can be $

1.8( = \frac {18}{10}) ; 2 $ and $ 2.2 (= \frac {22}{10}) $


Multiple choice irrational numbers irrational numbers and their operations rational and irrational numbers real numbers (rational and irrational numbers) maths

State whether the given statement is true/false.
An irrational number between two numbers $\dfrac{1}{7}$ and $\dfrac{2}{7}$ is $0.1501500 15000...$ .

  1. True

  2. False
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation
Let us first find the decimal forms of the given numbers as follows: 
 
$\dfrac { 1 }{ 7 } =0.\overline { 142857 } ,\dfrac { 2 }{ 7 } =0.\overline { 285714 }$

We find a number which is non-terminating non-recurring lying between them.
So, we can find infinite many such numbers. For example, $0.150150015000...$ and $0.20200200020000....$

Hence, an irrational number between two numbers $\dfrac {1}{7}$ and $\dfrac {2}{7}$ is $0.150150015000...$
Multiple choice business maths functions and graphs some functions and their graphs -i graphs of the form y=ax^2+bx+c introduction to sets

Write a rational function $f$ that has vertical asymptote at $x=4$, a horizontal asymptote at $y=5$ and a zero at $x=-7$.

  1. $f(x)=\dfrac{5(x-7)}{(x-4)}$
  2. $f(x)=\dfrac{5(x+7)}{(x-4)}$
  3. $f(x)=\dfrac{5(x-7)}{(x+4)}$
  4. $f(x)=\dfrac{(x+7)}{(x+4)}$
Reveal answer Fill a bubble to check yourself
B Correct answer
Explanation

Since the rational function $f$ has the vertical asymptote at $x=4$, then the denominator of $f$ contains the term $(x-4)$.

Thus function $f(x)$ is of the form $f=\dfrac{g(x)}{x-4}$.
Since the horizontal asymptote exists $y=5$, the numerator $g(x)$ of $f(x)$ has to be of the same degree as the denominator with a leading coefficient equal to $5$.
Also $g(x)$ must contain the term $(x+7)$ since $f$ has zero at $x=-7$.
Hence, $f(x)=\dfrac{5(x+7)}{(x-4)}$

Multiple choice maths square and square root scientific notation use of exponents power of 10

Which one of the following statements is correct?

  1. There can be a real number which is both rational and irrational.

  2. The sum of two irrational numbers is always irrational.
  3. For any real numbers x and y, $x < y \Rightarrow x^{2} < y^{2}$

  4. Every integer is a rational number.
Reveal answer Fill a bubble to check yourself
D Correct answer
Explanation

Every integer can be expressed as a rational number by dividing itself by 1.
Now, it becomes a rational number with the numerator the number itself and denominator 1.

Multiple choice maths equations and simple functions further equations solving linear equations with variable on both sides solution of linear equations in one variable

The denominator of a rational number is greater from its numerators by 10. If the numerators is increased by 19 & the denominator is decreased by 1. The number obtained is $\dfrac { 3 }{ 2 } $. Find the rational number.

  1. $\dfrac{1}{11}$
  2. $\dfrac{7}{17}$
  3. $\dfrac{11}{21}$

  4. None of these
Reveal answer Fill a bubble to check yourself
C Correct answer
Explanation

Let the numerator be $x$.

$\Rightarrow$  Then, denominator will be $=x+10$
According to the question,
$\Rightarrow$ $\dfrac{x+19}{x+10-1}=\dfrac{3}{2}$

$\Rightarrow$  $\dfrac{x+19}{x+9}=\dfrac{3}{2}$

$\Rightarrow$  $2(x+19)=3(x+9)$

$\Rightarrow$  $2x+38=3x+27$

$\Rightarrow$  $x=38-27$

$\Rightarrow$  $x=11$

$\Rightarrow$  The required rational number $=\dfrac{x}{x+10}=\dfrac{11}{11+10}=\dfrac{11}{21}$

Multiple choice maths operations on rational numbers rational numbers on number line rational numbers and their decimal expansions rational numbers between two rational numbers

A rational number between $\dfrac {-2}{3}$ and $\dfrac {1}{2}$ is

  1. $-\dfrac {3}{6}$
  2. $\dfrac {-1}{12}$
  3. $\dfrac {-5}{6}$
  4. $\dfrac {5}{6}$
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

To find a rational number between -2/3 and 1/2, we need a value satisfying -2/3 < x < 1/2. Option A: -3/6 = -1/2, which satisfies -2/3 < -1/2 < 1/2. Option B: -1/12 ≈ -0.083 also satisfies the condition, making both A and B technically correct.

Multiple choice maths operations on rational numbers rational numbers on number line rational numbers and their decimal expansions rational numbers between two rational numbers

The three rational number between $5$ and $6$ are $ [\displaystyle\frac{21}{4},\frac{22}{4},\frac{23}{4}]$.

  1. True

  2. False
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

$5=\frac { 20 }{ 4 } \quad and\quad 6=\frac { 24 }{ 4 } ,\ \frac { 21 }{ 4 } ,\frac { 22 }{ 4 } and\frac { 23 }{ 4 } \quad are\quad three\quad rational\quad numbers\quad lying\quad between\quad 5\quad and\quad 6.\quad \quad \ $