Real Numbers Test

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Real Numbers Test - 29

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Weekly Quiz Competition

Question 1
1 / -0

The HCF of the numbers $$0.48, 0.72$$ and $$0.108$$ is

A
$$1$$

B
$$12$$

C
$$0.12$$

D
$$0.012$$

Solution

$$480=2^5\times 3\times 5$$
$$720=2^4\times 3^2\times 5$$
$$108=2^2\times 3^2$$
$$HCF=2^2\times 3=12$$

HCF of $$0.48, 0.72$$ and $$0.108 = 0.012$$
So, option $$D$$ is correct.
Question 2
1 / -0

If the denominator of a fraction has only factors of $$2$$ and factors of $$5$$, the decimal expression .............

A
has equal numerator and denominator

B
becomes a whole number

C
does not terminate

D
terminates

Solution

When the prime factorization of the denominator of a fraction has only factors of $$2$$ and factors of $$5$$, we can always express the decimal as terminating decimal. Reason is we can always multiply the denominator with required factors and make it to the power of $$10$$ which will always result in a terminating decimal.
For example, $$\dfrac {1}{25} = \dfrac {4}{25\times 4}=\dfrac {4}{100}=0.04$$
Therefore, $$D$$ is the correct answer.
Question 3
1 / -0

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

A
rational

B
whole

C
integer

D
irrational

Solution

$$\sqrt {5} = \dfrac {a}{b}$$
$$b\sqrt {5} = a$$ $$(a$$ and $$b$$ are co-prime i.e. they have no common factors$$) ...(1)$$
$$5b^{2} = a^{2}$$ (squaring both sides)
Therefore $$5$$ divides $$a^{2}$$
As per Fundamental Theorem of Arithmetic, $$5$$ divides $$a.$$
Let's take it as $$a = 5c,$$
$$5b^{2} = 25 c^{2}$$
$$b^{2} = 5c^{2}$$
As per Fundamental Theorem of Arithmetic, $$5$$ divides $$a.$$
So $$a$$ and $$b$$ have $$5$$ as a common factor but $$a$$ and $$b$$ have only $$1$$ common factor $$1$$ from equation $$(1),$$ so it is not rational.
So, we conclude that $$\sqrt {5}$$ is irrational.
Therefore, $$D$$ is the correct answer.
Question 4
1 / -0

............. states that for any two positive integers $$a$$ and $$b$$ we can find two whole numbers $$q$$ and $$r$$ such that $$a = b \times q + r$$ where $$0 \leq r < b .$$

A
Euclid's addition lemma

B
Euclid's subtraction lemma

C
Euclid's multiplication lemma

D
Euclid's division lemma

Solution

Let the two positive integers be $$a$$, a dividend and $$b$$, a divisor.
We can find two whole numbers, $$q$$ and $$r$$ such that $$q$$ is the quotient and $$r$$ is the remainder.
So, $$a = b \times q + r$$ where $$0 \leq r < b.$$
This is Euclid's Division Lemma.

Hence, option D is correct.
Question 5
1 / -0

The H.C.F. of the numbers $$16.5, 0.90$$ and $$15$$ is

A
$$16.5$$

B
$$0.90$$

C
$$15$$

D
$$0.3$$

Solution

The given numbers are $$16.5, 0.90$$ and $$15$$
Multiplying each number by $$100$$ we get $$1650,90$$ and $$1500$$

Let us consider the numbers $$1650,90$$ and $$1500$$

By prime factorization, we get,
$$1650 = 2\times 3\times 5^{2}\times 11$$
$$90 = 2\times 3^{2} \times 5$$
$$1500 = 2^{2} \times 3\times 5^{3}$$

H.C.F. of $$1650, 90$$ and $$1500$$ is $$2\times 3\times 5 = 30$$

Therefore, H.C.F. of $$16.5, 0.90$$ and $$15$$ will be $$\dfrac{30}{100}$$$$=0.30$$

So, option $$D$$ is correct.
Question 6
1 / -0

$$\dfrac {1}{2} = 0.5$$
It is a terminating decimal because the denominator has a factor as ...........

A
$$0$$

B
$$1$$

C
$$2$$

D
$$6$$

Solution

When the prime factorization of the denominator of a fraction has only factors of $$2$$ and factors of $$5$$, we say the decimal as terminating decimal.
Here, the denominator has $$2$$ as a factor and hence it is terminating.
Therefore, $$C$$ is the correct answer.
Question 7
1 / -0

Find the HCF of $$26$$ and $$455$$

A
$$11$$

B
$$12$$

C
$$13$$

D
none of the above

Solution

$$26=13\times 2$$
$$455=5\times 13\times 7$$
HCF is $$13.$$
Question 8
1 / -0

If the denominator of a fraction has factors other than $$2$$ and $$5$$, the decimal expression ..............

A
repeats

B
is that of a whole number

C
has equal numerator and denominator

D
terminates

Solution

If there are prime factors in the denominator other than $$2$$ or $$5$$, then a block of digit repeats.
For example,$$\dfrac {1}{24} = \dfrac {1}{3\times 2\times 2\times 2}$$ (there is a factor of $$3$$, therefore decimal will repeat.)
$$=0.416666...$$
Therefore, $$A$$ is the correct answer.
Question 9
1 / -0

For finding the greatest common divisor of two given integers. A method based on the division algorithm is used called ............

A
Euclid's division algorithm

B
Euclid's addition algorithm

C
Euclid's subtraction algorithm

D
Euclid's multiplication algorithm

Solution

Euclid's division algorithm is a way to find the HCF of two numbers by using Euclid's division lemma. It states that if there are any two integers $$a$$ and $$b$$, there exists $$q$$ and $$r$$ such that it satisfies the given condition $$a = bq + r$$ where $$0 ≤ r < b$$..
Therefore, $$A$$ is the correct answer.
Question 10
1 / -0

The HCF of two consecutive odd numbers is

A
$$28$$

B
$$30$$

C
$$1$$

D
$$5$$

Solution

$$\textbf{Check for the HCF of consecutive odd numbers by writing down the first few odd numbers.}$$

$$\textbf{Step 1: Writing out the first few odd numbers}$$
$$\text{The first few odd numbers are 1, 3, 5, 7, 9, 11}$$
$$\textbf{Step 2: Finding out the HCF of consecutive odd numbers}$$
$$\text{It is evident that there is no common factors among any 2 consecutive odd numbers except 1}$$
$$\text{For example the HCF of 3 and 5 is 1, and that of 5 and 7 is also 1}$$
$$\textbf{Hence option C is correct}$$

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Real Numbers Test - 29

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Real Numbers Test - 29

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SHARING IS CARING

Question 1

$$1$$

$$12$$

$$0.12$$

$$0.012$$

Solution

Question 2

has equal numerator and denominator

becomes a whole number

does not terminate

terminates

Solution

Question 3

rational

whole

integer

irrational

Solution

Question 4

Euclid's addition lemma

Euclid's subtraction lemma

Euclid's multiplication lemma

Euclid's division lemma

Solution

Question 5

$$16.5$$

$$0.90$$

$$15$$

$$0.3$$

Solution

Question 6

$$0$$

$$1$$

$$2$$

$$6$$

Solution

Question 7

$$11$$

$$12$$

$$13$$

none of the above

Solution

Question 8

repeats

is that of a whole number

has equal numerator and denominator

terminates

Solution

Question 9

Euclid's division algorithm

Euclid's addition algorithm

Euclid's subtraction algorithm

Euclid's multiplication algorithm

Solution

Question 10

$$28$$

$$30$$

$$1$$

$$5$$

Solution

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