Real Numbers Test

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Real Numbers Test - 24

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Weekly Quiz Competition

Question 1
1 / -0

Find HCF of 25 and 55:

A
$$5$$

B
$$3$$

C
$$2$$

D
$$4$$

Solution

Prime factors of $$25 = 5 \times 5$$
Prime factors of $$55 = 5 \times 11$$
Since the common factor is only $$5$$.
Therefore,
$$HCF(25,55)=5$$
Hence, the correct option $$A$$
Question 2
1 / -0

What is the HCF of $$13$$ and $$22$$?

A
$$13$$

B
$$22$$

C
$$1$$

D
$$286$$

Solution

Let's first write the prime factorization of given numbers
$$13=13$$
$$22=2\times 11$$
So, we see that there is no common factor in the prime factorization of $$13$$ and $$22$$
$$\implies 1$$ is the only factor that is common to $$13$$ and $$22$$
$$\implies $$ H.C.F is $$1$$
Question 3
1 / -0

$$HCF$$ of $$(10224, 1608)$$ is _________.

A
$$12$$

B
$$24$$

C
$$48$$

D
$$96$$

Solution

First step is to write $$10224$$ and $$1608$$ as the product of its prime factors.
$$10224 = 2^{4}\times3^{2}\times71$$
$$1608 \;\;= 2^{3}\times3\times67$$

$$\therefore HCF(10224,1608) = 2^{3}\times 3$$
$$=24$$
Question 4
1 / -0

The fact that $$3+2\sqrt{5}$$ is irrational is because

A
Sum of two irrational numbers is rational

B
Sum of two irrational numbers is irrational

C
Sum of a rational and an irrational number is irrational

D
Sum of a rational and an irrational number is rational

Solution

$$3$$ is a rational number and $$2\sqrt5$$ is an irrational number

Sum of a rational and irrational number is irrational, so $$3+2\sqrt5$$ is irrational.
Question 5
1 / -0

Using fundamental theorem of Arithmetic find L.C.M. and H.C.F of $$816$$ and $$170$$.

A
L.C.M. $$=4080$$ and H.C.F $$=34$$

B
L.C.M. $$=8160$$ and H.C.F $$=68$$

C
L.C.M. $$=2048$$ and H.C.F $$=62$$

D
L.C.M. $$=2040$$ and H.C.F $$=72$$
Solution
According to the fundamental theorem of arithmetic every composite number can be factorised as a product of primes and this factorization is unique apart from the order in which the prime factor occurs.
Fundamental theorem of arithmetic is also called unique factorization theorem.
Composite number = product of prime numbers.
Any Integer greater than 1, either be a prime number or can be written as a product of prime factors.
The prime factors of $$816=2\times2\times2\times2\times3\times17=2^4\times3\times17$$

The prime factors of $$170=2\times5\times17$$

LCM of $$816$$ and $$170=2^4\times3\times5\times17=4080$$

HCF of $$816$$ and $$170=2\times17=34$$
Option A.
Question 6
1 / -0

$$\sqrt { 7 }$$ is a

A
an integer

B
an irrational number

C
a rational number

D
none of these

Solution

$$\sqrt 7 $$ is a irrational number
Question 7
1 / -0

Which of the following is an irrational number?

A
$$0.14$$

B
$$0.14 \overline{16}$$

C
$$1.1 {416}$$

D
$$0.4014001400014....$$

Solution

The decimal expansion of a rational number is either terminating or non-terminating repeating.

(A) $$0.14$$ is terminating, so it is a rational number

(B) $$0.14\bar{16}=0.141616....$$

is also rational ( non-terminating repeating ), where digits $$16$$ are repeating.

(C) $$0.1416$$ is terminating, so it is a rational number

(D) $$0.4014001400014.....$$

is an irrational number because it is neither terminating nor repeating.
Question 8
1 / -0

Which of the following is always true

A
$$irrational + irrational =irrational $$

B
$$\dfrac{rational }{rational }=rational $$

C
$$\dfrac{integer }{integer}=integer$$

D
None of these

Solution

Counter-example for A: $$(\sqrt{2}) + (4-\sqrt{2}) = 4$$

Counter-example for C: $$\dfrac{1}{2}=0.5$$

Proof for B:

Let $$q_1, q_2$$ be two rational numbers such that $$q_2\neq0$$.

As they are rational, they can be written as $$a/b, c/d$$ respectively for some integers $$a, b, c, d$$. $$(b,c,d\neq0)$$

$$\dfrac{q_1}{q_2}=\dfrac{a/b}{c/d}=\dfrac{ad}{bc}$$

Since, $$a,b,c,d$$ were integers, even $$ad$$ and $$bc(\neq0)$$ are integers and therefore, the above expression is rational.
Question 9
1 / -0

The product of a non-zero rational and an irrational number is

A
always irrational

B
always rational

C
rational or irrational

D
one

Solution

Let $$x$$ be a rational number and $$y$$ be an irrational number.
Let $$xy = a $$.

Let us assume that $$a$$ is rational.

Since, $$a$$ is rational it can be expressed as $$\dfrac{p}{q} $$, where $$p$$ and $$q$$ are integers.

Let $$ x =\dfrac{m}{n}$$ , where $$m$$ and $$n$$ are integers.
Now, $$xy = a $$

$$\dfrac{my}{n} = \dfrac{p}{q}$$.
On cross multipliying we get,

$$\Rightarrow y =\dfrac{pn}{qm}$$.

Now, $$pn$$ and $$qm$$ are integers.
Hence, $$\dfrac{pn}{qm}$$ is a rational number.

However, $$y$$ is irrational.
Hence, our assumption is incorrect.
Hence, the product of a non-zero rational and an irrational number is always an Irrational number.

So, option $$A$$.
Question 10
1 / -0

According to Euclid's division algorithm, using Euclid's division lemma for any two positive integers $$a$$ and $$b$$ with $$a > b$$ enables us to find the

A
HCF

B
LCM

C
Decimal expansion

D
Quotient

Solution

According to Euclid's division lemma,

For each pair of positive integers $$a$$ and $$b$$, we can find unique integers $$q$$ and $$r$$ satisfying the relation
$$a = bq + r , 0 ≤ r < b$$
If $$r = 0$$ then q will be HCF of $$a$$ and $$b$$

The basis of the Euclidean division algorithm is Euclid’s division lemma.

To calculate the Highest Common Factor (HCF) of two positive integers $$a$$ and $$b$$ we use Euclid’s division algorithm.

HCF is the largest number which exactly divides two or more positive integers.

By exactly we mean that on dividing both the integers $$a$$ and $$b$$ by HCF, the remainder is zero.

So correct answer is option A

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Real Numbers Test - 24

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Real Numbers Test - 24

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Question 1

$$5$$

$$3$$

$$2$$

$$4$$

Solution

Question 2

$$13$$

$$22$$

$$1$$

$$286$$

Solution

Question 3

$$12$$

$$24$$

$$48$$

$$96$$

Solution

Question 4

Sum of two irrational numbers is rational

Sum of two irrational numbers is irrational

Sum of a rational and an irrational number is irrational

Sum of a rational and an irrational number is rational

Solution

Question 5

L.C.M. $$=4080$$ and H.C.F $$=34$$

L.C.M. $$=8160$$ and H.C.F $$=68$$

L.C.M. $$=2048$$ and H.C.F $$=62$$

L.C.M. $$=2040$$ and H.C.F $$=72$$

Solution

Question 6

an integer

an irrational number

a rational number

none of these

Solution

Question 7

$$0.14$$

$$0.14 \overline{16}$$

$$1.1 {416}$$

$$0.4014001400014....$$

Solution

Question 8

$$irrational + irrational =irrational $$

$$\dfrac{rational }{rational }=rational $$

$$\dfrac{integer }{integer}=integer$$

None of these

Solution

Question 9

always irrational

always rational

rational or irrational

one

Solution

Question 10

HCF

LCM

Decimal expansion

Quotient

Solution

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