Number Systems Test

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Number Systems Test - 21

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Weekly Quiz Competition

Question 1
1 / -0

The reciprocal of a positive rational number is positive.

A
True

B
False

C
Cannot be determined

D
None

Solution

The reciprocal of a positive rational number is positive.
Question 2
1 / -0

The rational number between $$\cfrac{1}{2}$$ and $$\cfrac{6}{10}$$ is

A
$$\cfrac{1}{4}$$

B
$$\cfrac{3}{4}$$

C
$$\cfrac{21}{40}$$

D
$$\cfrac{33}{100}$$

Solution

$$\dfrac{1}{2}$$ =$$\dfrac{5}{10}$$
$$\dfrac{5}{10}$$ and $$\dfrac{6}{10}$$
Multiplying the denominator and numerator by $$4$$ we get,
$$\dfrac{5}{10}\times 4=\dfrac{20}{40}$$ and
$$\dfrac{6}{10}\times 4=\dfrac{24}{40}$$
Number between $$\dfrac{20}{40}$$ and $$\dfrac{24}{40}$$ is $$\dfrac{21}{40}$$
Question 3
1 / -0

If p: every fraction is a rational number
q: every rational number is a fraction
then which of the following is correct?

A
p is true and q is false.

B
p is false and q is true.

C
Both p and q are true.

D
Both p and q are false.

Solution

Fraction is defined as a part of a whole thing. Every fraction is of the form $$\dfrac{m}{n}$$ where $$m$$ is a whole number and $$n $$ is a natural number.

Rational number is defined as the number of the form $$\dfrac{a}{b}$$ where $$a$$ and $$b$$ are integers and $$b\neq 0$$.

Since all whole numbers and natural numbers are present in set of integers every fraction is a rational number. Examples : $$\dfrac{4}{5}, \dfrac{4}{7}$$

So, the statement $$p$$ is true.

But every rational number is not a fraction. Examples : $$\dfrac{3}{-4}, \dfrac{17}{-16}$$. Here the denominators are not natural numbers and hence they are not fractions.

So, the statement $$q$$ is false.
Question 4
1 / -0

$$\displaystyle \frac{-2}{-19}$$ is a

A
Positive rational number.

B
Negative rational number.

C
Either positive or negative number.

D
Has neither a positive numerator nor a negative number

Solution

$$\because$$ Both numerator and denominator are negative (i.e., same sign)
Question 5
1 / -0

$$(2+\sqrt 3)$$ is

A
rational

B
irrational

C
an integer

D
not real

Solution

$$\sqrt{3}$$ is an irrational number. So any irrational number added to a rational number gives an irrational number.
So, $$2+\sqrt{3}$$ is an irrational number
Answer is Option B
Question 6
1 / -0

$$\displaystyle \frac{-5}{0}$$ is a ........

A
Positive rational number.

B
Negative rational number.

C
Either positive or negative rational number.

D
Neither positive nor negative rational number.

Solution

$$\because$$ Denominator is '0', it is not a rational number as it makes number undefined.
Question 7
1 / -0

Write five rational numbers which are smaller than $$2$$.

A
$$1,\,\displaystyle\frac{1}{2},\,0,\,-1,\,-\displaystyle\frac{1}{2}$$

B
$$0, 1 , 1.414, \sqrt3, -1$$

C
$$0, 1 , \sqrt2, \sqrt3, -1$$

D
$$0, 1 , 1.732, \sqrt2, -1$$

Solution

Five rational numbers less than $$2$$ may be taken $$1,\,\displaystyle\frac{1}{2},\,0,\,-1,\,-\displaystyle\frac{1}{2}$$
(There can be many more such rational numbers).
Question 8
1 / -0

Which of the following is a rational number (s)?

A
$$\displaystyle \frac{-2}{9}$$

B
$$\displaystyle \frac{4}{-7}$$

C
$$\displaystyle \frac{-3}{17}$$

D
All the three given numbers

Solution

A rational number is a number that can be expressed as a fraction $$\dfrac{p}{q} $$ where $$p$$ and $$q$$ are integers and $$q\neq0$$.
A rational number $$\dfrac{p}{q}$$ is said to have numerator $$p$$ and denominator $$q.$$

In the given options, both numerator and denominator are integers.
Hence, all the three given numbers are rational number.
Question 9
1 / -0

The rationalising factor of $$ \displaystyle \left ( a+\sqrt{b} \right ) $$ is

A
$$ \displaystyle a-\sqrt{b} $$

B
$$ \displaystyle \sqrt{a}-b $$

C
$$ \displaystyle \sqrt{a}-\sqrt{b} $$

D
None of this

Solution

Rationalizing factor of $$a+\sqrt{b}$$ is $$a-\sqrt{b}$$
$$(a+\sqrt{b})\times (a-\sqrt{b})=a^2-b^2$$
Question 10
1 / -0

Which of the following expresses zero law of exponents?

A
$$0^{x} = 0$$

B
$$x^{0} = 1$$

C
$$x^{1} = x$$

D
None of the above

Solution

According to the zero law of exponents,
any non - zero number raised to the power of zero is equal to $$1$$.
$$\therefore x^{0} = 1$$, where $$x\neq 0$$
So, option $$B$$ is correct.

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Number Systems Test - 21

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Number Systems Test - 21

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Question 1

True

False

Cannot be determined

None

Solution

Question 2

$$\cfrac{1}{4}$$

$$\cfrac{3}{4}$$

$$\cfrac{21}{40}$$

$$\cfrac{33}{100}$$

Solution

Question 3

p is true and q is false.

p is false and q is true.

Both p and q are true.

Both p and q are false.

Solution

Question 4

Positive rational number.

Negative rational number.

Either positive or negative number.

Has neither a positive numerator nor a negative number

Solution

Question 5

rational

irrational

an integer

not real

Solution

Question 6

Positive rational number.

Negative rational number.

Either positive or negative rational number.

Neither positive nor negative rational number.

Solution

Question 7

$$1,\,\displaystyle\frac{1}{2},\,0,\,-1,\,-\displaystyle\frac{1}{2}$$

$$0, 1 , 1.414, \sqrt3, -1$$

$$0, 1 , \sqrt2, \sqrt3, -1$$

$$0, 1 , 1.732, \sqrt2, -1$$

Solution

Question 8

$$\displaystyle \frac{-2}{9}$$

$$\displaystyle \frac{4}{-7}$$

$$\displaystyle \frac{-3}{17}$$

All the three given numbers

Solution

Question 9

$$ \displaystyle a-\sqrt{b} $$

$$ \displaystyle \sqrt{a}-b $$

$$ \displaystyle \sqrt{a}-\sqrt{b} $$

None of this

Solution

Question 10

$$0^{x} = 0$$

$$x^{0} = 1$$

$$x^{1} = x$$

None of the above

Solution

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