Tag: existence of irrational numbers

Questions Related to existence of irrational numbers

$\pi$ is a(n) ________ while $\dfrac{22}{7}$ is rational.

maths rational and irrational numbers existence of irrational numbers irrational numbers properties of irrational numbers

  1. Integer

  2. Whole Number

  3. Rational Number

  4. Irrational Number

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Correct Option: D
Explanation:

The value $\dfrac{22}7$ is a rational number, as it can be expressed in the form $\dfrac pq$. 

We consider it as an approximate value of $\pi$ because $\pi$ is close to $\dfrac{22}7$. 
But actually its value is $3.14159....$, which is neither terminating nor repeating. 
Thus, $\pi $ is irrational, but $\dfrac{22}7$ is rational.

$\sqrt{5}$ is an irrational number.

maths rational and irrational numbers existence of irrational numbers irrational numbers properties of irrational numbers

  1. True

  2. False

Show answer
Correct Option: A
Explanation:

An irrational number is any real number that cannot be expressed as a ratio a/b, where a and b are integers and b is non-zero.
$\sqrt5$ is irrational as it can never be expressed in the form a/b

Check whether following statement is true or false.
$7\sqrt{5}$ is a rational number.

maths rational and irrational numbers existence of irrational numbers irrational numbers properties of irrational numbers

  1. True

  2. False

Show answer
Correct Option: B
Explanation:

An irrational number is any real number that cannot be expressed as a ratio a/b, where a and b are integers and b is non-zero.
$7\sqrt5$ is irrational as it can never be expressed in the form a/b

$\dfrac{1}{\sqrt{2}}$ is an irrational number.

maths rational and irrational numbers existence of irrational numbers irrational numbers properties of irrational numbers

  1. True

  2. False

Show answer
Correct Option: A
Explanation:

An irrational number is any real number that cannot be expressed as a ratio a/b, where a and b are integers and b is non-zero.
$1/(\sqrt2)$ is irrational as it can never be expressed in the form a/b

$3+2\sqrt{5}$ a rational number.

maths rational and irrational numbers existence of irrational numbers irrational numbers properties of irrational numbers

  1. True

  2. False

Show answer
Correct Option: B
Explanation:

Let's assume that $3+2\sqrt5$ is rational..... 

then 

$3+2\sqrt5 = p/q $

$\sqrt5 =( p-3q)/(2q) $ 

now take $p-3q$ to be P and $2q$ to be Q........where P and Q are integers 

which means, $\sqrt5= P/Q$...... 

But this contradicts the fact that $\sqrt5$ is rational 

So our assumption is wrong and $3+2\sqrt5$ is irrational.

$\sqrt { 2 } ,\sqrt { 3 }$ are

maths rational and irrational numbers existence of irrational numbers irrational numbers properties of irrational numbers

  1. Whole numbers

  2. Rational numbers

  3. Irrational numbers

  4. Integers

Show answer
Correct Option: C
Explanation:

A rational number is any number that can be expressed as a fraction $\dfrac pq$ of two integers with $q$ not equal to zero.
As in the case of $\sqrt2$ and $\sqrt3$, it cannot be expressed as a fraction $\dfrac pq$.

Hence, option $A$ is the correct answer.

If $p$ is prime, then $\sqrt{p}$ is irrational and if $a, b$ are two odd prime numbers, then $a^2 -b^2$ is composite. As per the above passage mark the correct answer to the following question.
$\sqrt{7}$ is:

maths rational and irrational numbers existence of irrational numbers irrational numbers properties of irrational numbers

  1. a rational number

  2. an irrational number

  3. not a real number

  4. terminating decimal

Show answer
Correct Option: B
Explanation:

The basic definition for a rational number is that it can be represented in the form of $p/q$, where p and q are integers and q is a non-zero integer. Here, $\sqrt7$ is not a perfect square and thus cannot be expressed in the form of $p/q$, thus it is an irrational number.

Consider the given statements:
I. All surds are irrational numbers.
II. All irrationals numbers are surds.
Which of the following is true.

maths rational and irrational numbers existence of irrational numbers irrational numbers properties of irrational numbers

  1. Only I

  2. Only II

  3. Both I and II

  4. Neither I nor II

Show answer
Correct Option: A
Explanation:

A surd, by its very definition is an irrational number.

However, not every irrational number can be expressed as a surd.
Hence, only statement 1 is true.

Which of the following numbers is different from others?

maths rational and irrational numbers existence of irrational numbers irrational numbers properties of irrational numbers

  1. $\sqrt{6}$

  2. $\sqrt{11}$

  3. $\sqrt{15}$

  4. $\sqrt{16}$

Show answer
Correct Option: D
Explanation:

$16^{\frac{1}{2}}=4$, which is a rational number.
The other options are irrational  numbers.
So, $16^{\frac{1}{2}}$ is different from others.

If $a\neq 1$ and $ln{ a }^{ 2 }+{ \left( ln{ a }^{ 2 } \right)  }^{ 2 }+{ \left( ln{ a }^{ 2 } \right)  }^{ 3 }+........=3\left( lna+{ \left( ln{ a } \right)  }^{ 2 }+{ \left( ln{ a } \right)  }^{ 3 }+{ \left( ln{ a } \right)  }^{ 4 }+...... \right)$ then $a$ is

maths rational and irrational numbers existence of irrational numbers irrational numbers properties of irrational numbers

  1. $an\ irrational\ number$

  2. $a\ transcendental\ number$

  3. $an\ algeberaic\ number$

  4. $a\ surd$

Show answer
Correct Option: A
Explanation:

Given, for $a\ne 1$,

$ln{ a }^{ 2 }+{ \left( ln{ a }^{ 2 } \right)  }^{ 2 }+{ \left( ln{ a }^{ 2 } \right)  }^{ 3 }+........=3\left( lna+{ \left( ln{ a } \right)  }^{ 2 }+{ \left( ln{ a } \right)  }^{ 3 }+{ \left( ln{ a } \right)  }^{ 4 }+...... \right)$
or, $\dfrac{\ln a^2}{1-\ln a^2}=3\times \dfrac{\ln a}{1-\ln a}$
or, $\dfrac{2\ln a}{1-2\ln a}=3\times \dfrac{\ln a}{1-\ln a}$
or, $2(1-\ln a)=3(1-2\ln a)$ [Since $a\ne 1\Rightarrow \ln a \ne 0$ ]
or, $4\ln a =1$
or, $a=\sqrt[4]{e}$.
So clearly $a$ is an irrational number.