Real Numbers Test

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Real Numbers Test - 23

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Weekly Quiz Competition

Question 1
1 / -0

Which option will have a terminating decimal expansion?

A
$$\dfrac {77}{210}$$

B
$$\dfrac {23}{30}$$

C
$$\dfrac {125}{441}$$

D
$$\dfrac {23}{8}$$

Solution

The decimal expansion of a number is terminating if the denominator of the number can be expressed in the form of $$2^m\times 5^n$$ where $$m$$ and $$n$$ are non-negative integers
A) $$\dfrac{77}{210},$$ prime factorisation of the denominator is given as
$$210=2\times 3\times 5\times 7$$
$$\implies$$ denominator is not of the form $$2^m \times 5^n $$
B) $$\dfrac{23}{30},$$ prime factorisation of the denominator is given as
$$30=2\times 3\times 5$$
$$\implies$$ denominator is not of the form $$2^m \times 5^n $$
C) $$\dfrac{125}{441},$$ prime factorisation of the denominator is given as
$$441=21^2$$
$$\implies$$ denominator is not of the form $$2^m \times 5^n $$
D) $$\dfrac{23}{8},$$ prime factorisation of the denominator is given as
$$8=2^3$$
$$\implies$$ denominator is of the form $$2^m \times 5^n $$ where $$m=3$$ and $$n=0$$
Hence the decimal expansion of $$\dfrac{23}{8}$$ is terminating
Question 2
1 / -0

We need blocks to build a building. In the same way _______ are the basic blocks to form all natural numbers.

A
prime numbers

B
real numbers

C
unique numbers

D
negative numbers

Solution

Let us consider some examples.
$$10 = 2 \times 5$$, we need $$2$$ and $$5$$ which are prime numbers to get a new number $$10$$.
$$12= 2 \times 2 \times 3$$, we need $$2$$ and $$3$$ which are prime numbers to get a new number $$12$$.
Therefore$$,$$ $$A$$ is the correct answer.
Question 3
1 / -0

$$2\times 2\times 2\times 3\times 3\times 13 = 2^{3} \times 3^{2} \times 13$$ is equal to

A
$$1004$$

B
$$828$$

C
$$724$$

D
$$936$$

Solution

$$936 = 2\times 2\times 2\times 3\times 3\times 13 = 2^{3} \times 3^{2} \times 13$$
Therefore$$, D$$ is the correct answer.
Question 4
1 / -0

The non-terminating non-recurring decimal cannot be represented as:

A
irrational numbers

B
rational numbers

C
real numbers

D
none of these

Solution

Irrational numbers are non-terminating and non-recurring decimal, and they can not be represented as a quotient of two integers, i.e. as a rational number.
Therefore, $$B$$ is the correct answer.
Question 5
1 / -0

Euclids division lemma can be used to find the $$...........$$ of any two positive integers and to show the common properties of numbers.

A
None of the common factors

B
Lowest common factor

C
Highest common factor

D
Common factor

Solution

As seen by dividing the two positive integers we get a quotient and a remainder and using this lemma, we get the highest common factor at the end of the procedure.
Therefore$$,$$ $$C$$ is the correct answer.
Question 6
1 / -0

$$m$$ is not a perfect square, then $$\sqrt {m}$$ is

A
An irrational number

B
A composite number

C
A rational number

D
None of these

Solution

$$\sqrt {m}$$ will be an irrational number if $$m$$ is not a perfect square number.

Therefore, option $$A$$ is the correct.
Question 7
1 / -0

What is the square of $$(2 + \sqrt {2})$$?

A
A rational number

B
An irrational number

C
A natural number

D
A whole number

Solution

We know that , $$(a+b)^2=a^2+2ab+b^2$$
$$\therefore (2+√2)^2= (4+2\times 2\times√2+2) $$
$$= 6+4√2$$
which is an irrational number
Question 8
1 / -0

$$\dfrac {p}{q}$$ form of $$0.0875$$ is _______

A
$$\dfrac {7}{2^{4}\times 5}$$

B
$$\dfrac {7}{2\times 5^{4}}$$

C
$$\dfrac {7}{2^{4}\times 5^{4}}$$

D
$$\dfrac {5^{3}\times 7}{2^{3}\times 5^{4}}$$

Solution

Since, $$0.0875=\displaystyle \frac {875}{10000}=\frac {175}{2000}=\frac{35}{400} =\frac {7}{80}=\frac 7{16\times 5}=\frac 7{2^4\times 5}$$
Option $$A$$ is correct.
Question 9
1 / -0

Let $$x$$ be an irrational number then what can be said about $${x}^{2}$$

A
It is rational

B
It can be irrational.

C
It can be rational.

D
Both $$B$$ and $$C$$

Solution

$$x$$ is any irrational number
Let $$x=\sqrt [ 4 ]{ 3 } $$
$$\Rightarrow { x }^{ 2 }=\sqrt { 3 } $$
which is irrational so option $$B$$ is correct.
Now let $$x=\sqrt 3$$
$$\Rightarrow {x}^{2}=3$$
which is rational so option $$C$$ is correct.
So correct answer is $$D$$
Question 10
1 / -0

To get the terminating decimal expansion of a rational number $$\dfrac{p}{q}$$. if $$q = 2^m 5^n$$ then $$m$$ and $$n$$ must belong to .................

A
$$Z$$

B
$$N \cup$${$$0$$}

C
$$N$$

D
$$R$$

Solution

To get the terminating decimal expansion of a rational number $$\dfrac{p}{q}$$, $$q$$ have to be an integer or natural number.
Now, $$q = 2^m 5^n$$ will be an integer or natural if $$m$$ and $$n$$ both are naturals. So, to get $$q = 2^m 5^n$$ as integer $$m$$ and $$n$$ must belong to $$N$$.
Hence, option C is correct.

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Real Numbers Test - 23

Result

Real Numbers Test - 23

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Question 1

$$\dfrac {77}{210}$$

$$\dfrac {23}{30}$$

$$\dfrac {125}{441}$$

$$\dfrac {23}{8}$$

Solution

Question 2

prime numbers

real numbers

unique numbers

negative numbers

Solution

Question 3

$$1004$$

$$828$$

$$724$$

$$936$$

Solution

Question 4

irrational numbers

rational numbers

real numbers

none of these

Solution

Question 5

None of the common factors

Lowest common factor

Highest common factor

Common factor

Solution

Question 6

An irrational number

A composite number

A rational number

None of these

Solution

Question 7

A rational number

An irrational number

A natural number

A whole number

Solution

Question 8

$$\dfrac {7}{2^{4}\times 5}$$

$$\dfrac {7}{2\times 5^{4}}$$

$$\dfrac {7}{2^{4}\times 5^{4}}$$

$$\dfrac {5^{3}\times 7}{2^{3}\times 5^{4}}$$

Solution

Question 9

It is rational

It can be irrational.

It can be rational.

Both $$B$$ and $$C$$

Solution

Question 10

$$Z$$

$$N \cup$${$$0$$}

$$N$$

$$R$$

Solution

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