Number Systems Test

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Number Systems Test - 30

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Weekly Quiz Competition

Question 1
1 / -0

The value of $$5\sqrt{3}-3\sqrt{12}+2\sqrt{75}$$ on simplifying is

A
$$5\sqrt{3}$$

B
$$6\sqrt{3}$$

C
$$\sqrt{3}$$

D
$$9\sqrt{3}$$

Solution

$$5\sqrt { 3 } -3\sqrt { 12 } +2\sqrt { 75 } =5\sqrt { 3 } -6\sqrt { 3 } +10\sqrt { 3 } =9\sqrt { 3 } \\ \\ \quad $$
Question 2
1 / -0

Which of the following is not a rational number?

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

B
$$\displaystyle \frac{-4}{19}$$

C
$$\displaystyle \frac{0}{8}$$

D
$$\displaystyle \frac{3}{0}$$

Solution

We know that a number that can be made by dividing two integers. (An integer is a number with no fractional part.)
The word comes from "ratio".
For example:
$$\dfrac {1}{2}$$ is a rational number. ($$1$$ divided by $$2$$, or the ratio of $$1$$ to $$2$$)

But $$\dfrac {1}{0}$$ is not a rational number because the result of this fraction is not defined. Similarly, $$\dfrac {3}{0}$$ is not a rational number.

Hence, $$\dfrac {3}{0}$$ is not a rational number.
Question 3
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Choose the rational number which does not lie between rational numbers $$ \displaystyle \frac{3}{5} $$ and $$ \displaystyle \frac{2}{3} $$ :

A
$$ \displaystyle \frac{46}{75} $$

B
$$ \displaystyle \frac{47}{75} $$

C
$$ \displaystyle \frac{49}{75} $$

D
$$ \displaystyle \frac{50}{75} $$

Solution

For $$\dfrac{3}{5}$$ multiply numerator and denominator by $$15$$ to make denominator $$75$$ that comes into $$\dfrac{45}{75}.$$
Similarly doing for second then we have $$\dfrac{50}{75}.$$
Now question is asking about rational lying between them.
So, we need to check the numerator only that lies in between $$45$$ and $$50$$ or not.
Clearly $$D$$ is correct.
Question 4
1 / -0

By how much does $$\displaystyle \sqrt{12}+\sqrt{18}$$ exceed $$\displaystyle \sqrt{3}+\sqrt{2}$$ ?

A
$$\displaystyle \sqrt{3}+2\sqrt{2}$$

B
$$\displaystyle \sqrt{2}+2\sqrt{3}$$

C
$$\displaystyle \sqrt{3}+\sqrt{2}$$

D
$$\displaystyle \sqrt{3}-2\sqrt{2}$$

Solution

Required difference $$\displaystyle =(\sqrt{12}+\sqrt{18})-(\sqrt{3}+\sqrt{2})$$
$$\displaystyle =(\sqrt{4\times 3}+\sqrt{9\times 2})-(\sqrt{3}+\sqrt{2})=2\sqrt{3}+3\sqrt{2}-\sqrt{3}-\sqrt{2}$$
$$\displaystyle =(2\sqrt{3}-\sqrt{3})+(3\sqrt{2}-\sqrt{2})=\sqrt{3}+2\sqrt{2}$$
Question 5
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$$\sqrt 5$$ is

A
an integer

B
a rational number

C
an irrational number

D
none of these

Solution

An integer is a whole number (not a fractional number) that can be positive, negative, or zero and no digits after the decimal point.
Examples of integers are: -5, 1, 5, 8, 97, and 3,043.

A Rational Number can be made by dividing two integers. (An integer is a number with no fractional part.)

$$\sqrt{5}$$ is an irrational number as it cant be represented as a fraction.

Answer is Option C
Question 6
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Find five rational numbers between $$\displaystyle\frac{-3}{2}$$ and $$\displaystyle\frac{5}{3}$$.

A
$$\displaystyle\frac{-8}{6},\,\displaystyle\frac{-13}{6},\,\displaystyle\frac{0}{6},\,\displaystyle\frac{1}{6}$$ and $$\displaystyle\frac{2}{6}$$

B
$$\displaystyle\frac{-8}{6},\,\displaystyle\frac{-7}{6},\,\displaystyle\frac{0}{6},\,\displaystyle\frac{1}{6}$$ and $$\displaystyle\frac{13}{6}$$

C
$$\displaystyle\frac{-8}{6},\,\displaystyle\frac{-7}{6},\,\displaystyle\frac{0}{6},\,\displaystyle\frac{1}{6}$$ and $$\displaystyle\frac{11}{6}$$

D
$$\displaystyle\frac{-8}{6},\,\displaystyle\frac{-7}{6},\,\displaystyle\frac{0}{6},\,\displaystyle\frac{1}{6}$$ and $$\displaystyle\frac{2}{6}$$

Solution

Converting the given rational numbers with the same denominators
$$\cfrac{-3}{2}=\cfrac{-3\times3}{2\times3}=\cfrac{-9}{6}$$ and $$\cfrac{5}{3}=\cfrac{5\times2}{3\times2}=\cfrac{10}{6}$$

We know that $$-9\,<\,-8\,<\,-7\,<\,-6\,<\,\dots\,<\,0\,<\,1\,<\,2\,<\,8\,<\,9\,<\,10$$
$$\Rightarrow\cfrac{-9}{6}\,<\,\cfrac{-8}{6}\,<\,\cfrac{-7}{6}\,<\,\cfrac{-6}{6}\,<\,\dots\,<\,\cfrac{0}{6}\,<\,\cfrac{1}{6}\,<\,\cfrac{2}{6}\,<\,\dots\,<\,\cfrac{8}{6}\,<\,\cfrac{9}{6}\,<\,\cfrac{10}{6}$$.

Thus, we have the following five rational numbers between $$\cfrac{-3}{2}$$ and $$\cfrac{5}{3}$$
$$\Rightarrow \cfrac{-8}{6},\,\cfrac{-7}{6},\,\cfrac{0}{6},\,\cfrac{1}{6}and\cfrac{2}{6}$$
Question 7
1 / -0

Find $$9$$ rational numbers between $$-\displaystyle\frac{1}{9}\;and\;\displaystyle\frac{1}{5}$$.

A
$$\displaystyle\frac{-7}{45},\,\displaystyle\frac{-3}{45},\,\displaystyle\frac{-2}{45},\,\displaystyle\frac{-1}{45},\,\displaystyle\frac{2}{45},\,\displaystyle\frac{3}{45},\,\displaystyle\frac{4}{45},\,\displaystyle\frac{5}{45},\,\displaystyle\frac{8}{45}$$

B
$$\displaystyle\frac{-4}{45},\,\displaystyle\frac{-3}{45},\,\displaystyle\frac{-2}{45},\,\displaystyle\frac{-1}{45},\,\displaystyle\frac{10}{45},\,\displaystyle\frac{3}{45},\,\displaystyle\frac{4}{45},\,\displaystyle\frac{5}{45},\,\displaystyle\frac{8}{45}$$

C
$$\displaystyle\frac{-4}{45},\,\displaystyle\frac{-3}{45},\,\displaystyle\frac{-2}{45},\,\displaystyle\frac{-1}{45},\,\displaystyle\frac{2}{45},\,\displaystyle\frac{9}{45},\,\displaystyle\frac{4}{45},\,\displaystyle\frac{5}{45},\,\displaystyle\frac{8}{45}$$

D
$$\displaystyle\frac{-4}{45},\,\displaystyle\frac{-3}{45},\,\displaystyle\frac{-2}{45},\,\displaystyle\frac{-1}{45},\,\displaystyle\frac{2}{45},\,\displaystyle\frac{3}{45},\,\displaystyle\frac{4}{45},\,\displaystyle\frac{5}{45},\,\displaystyle\frac{8}{45}$$

Solution

Convert the rational numbers into equivalent rational numbers with the same denominator.

LCM of $$9$$ and $$5$$ is $$45$$.

$$-\displaystyle\frac{1}{9}=\displaystyle\frac{-1\times5}{9\times5}=\displaystyle\frac{-5}{45}$$ and $$\displaystyle\frac{1}{5}=\displaystyle\frac{1\times9}{5\times9}=\displaystyle\frac{9}{45}$$

The integers between $$-5$$ and $$9$$ are
$$-4,\,-3,\,-2,\,-1,\,0,\,1,\,2,\,3,\,4,\,5,\,6,\,7,\,8$$.

The corresponding rational numbers are $$\displaystyle\frac{-4}{45},\,\displaystyle\frac{-3}{45},\,\displaystyle\frac{-2}{45},\,\displaystyle\frac{-1}{45},\,\displaystyle\frac{0}{45},\,\displaystyle\frac{1}{45},\,\displaystyle\frac{2}{45},\,\displaystyle\frac{3}{45},\,\displaystyle\frac{4}{45},\,\displaystyle\frac{5}{45},\,\displaystyle\frac{6}{45},\,\displaystyle\frac{7}{45},\,\displaystyle\frac{8}{45}$$

On selecting any $$9$$ of them, we get $$9$$ rational numbers between $$-\displaystyle\frac{1}{9}$$ and $$\displaystyle\frac{1}{5}$$ as

$$\displaystyle\frac{-4}{45},\,\displaystyle\frac{-3}{45},\,\displaystyle\frac{-2}{45},\,\displaystyle\frac{-1}{45},\,\displaystyle\frac{2}{45},\,\displaystyle\frac{3}{45},\,\displaystyle\frac{4}{45},\,\displaystyle\frac{5}{45},\,\displaystyle\frac{8}{45}$$
Question 8
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Write any $$10$$ rational numbers between $$0\;and\;2$$.

A
$$\displaystyle\frac{1}{10},\,\displaystyle\frac{2}{10},\,\displaystyle\frac{3}{10},\,\displaystyle\frac{4}{10},\,\displaystyle\frac{5}{10},\,\displaystyle\frac{6}{10},\,\displaystyle\frac{7}{10},\,\displaystyle\frac{88}{10},\,\displaystyle\frac{9}{10},\,\displaystyle\frac{10}{10}$$

B
$$\displaystyle\frac{1}{10},\,\displaystyle\frac{2}{10},\,\displaystyle\frac{3}{10},\,\displaystyle\frac{4}{10},\,\displaystyle\frac{21}{10},\,\displaystyle\frac{6}{10},\,\displaystyle\frac{7}{10},\,\displaystyle\frac{8}{10},\,\displaystyle\frac{9}{10},\,\displaystyle\frac{10}{10}$$

C
$$\displaystyle\frac{1}{10},\,\displaystyle\frac{2}{10},\,\displaystyle\frac{3}{10},\,\displaystyle\frac{4}{10},\,\displaystyle\frac{35}{10},\,\displaystyle\frac{6}{10},\,\displaystyle\frac{7}{10},\,\displaystyle\frac{8}{10},\,\displaystyle\frac{9}{10},\,\displaystyle\frac{10}{10}$$

D
$$\displaystyle\frac{1}{10},\,\displaystyle\frac{2}{10},\,\displaystyle\frac{3}{10},\,\displaystyle\frac{4}{10},\,\displaystyle\frac{5}{10},\,\displaystyle\frac{6}{10},\,\displaystyle\frac{7}{10},\,\displaystyle\frac{8}{10},\,\displaystyle\frac{9}{10},\,\displaystyle\frac{10}{10}$$

Solution

Let us write $$0$$ as $$\dfrac{0}{10}$$ and $$2$$ as $$\dfrac{20}{10}.$$

So, ten rational numbers between these will be,
$$\dfrac{1}{10},\;\dfrac{2}{10},\;\dfrac{3}{10},\;\dfrac{4}{10},\;\dfrac{5}{10},\;\dfrac{6}{10},\;\dfrac{7}{10},\ \dfrac{8}{10},\ \dfrac{9}{10},\; 1$$

Hence, option $$D$$ is correct.
Question 9
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By how much does $$(\displaystyle 5\sqrt{7}-2\sqrt{5})$$ exceeds $$(\displaystyle 3\sqrt{7}-4\sqrt{5})$$?

A
$$\displaystyle 5(\sqrt{7}+\sqrt{5})$$

B
$$\displaystyle\sqrt{7}+\sqrt{5}$$

C
$$\displaystyle 2(\sqrt{7}+\sqrt{5})$$

D
$$\displaystyle 7(\sqrt{2}+\sqrt{5})$$

Solution

$$ (5\sqrt{7}-2\sqrt{5})-(3\sqrt{7}-4\sqrt{5})$$
$$ =(5\sqrt{7}-2\sqrt{5}-3\sqrt{7}+4\sqrt{5})$$
$$ =2\sqrt{7}+2\sqrt{5}$$
$$=2(\sqrt{7}+\sqrt{5})$$
Question 10
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The rational number $${\cfrac{0}{7}}$$:

A
Has a positive numerator

B
Has a negative numerator

C
Has either a positive numerator or a negative numerator

D
Has neither a positive numerator nor a negative numerator

Solution

In the given question numerator is $$0$$ and $$0$$ is neither positive nor negative. So the correct answer will be option D.

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Re-Attempt Weekly Quiz Competition

Number Systems Test - 30

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Number Systems Test - 30

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Question 1

$$5\sqrt{3}$$

$$6\sqrt{3}$$

$$\sqrt{3}$$

$$9\sqrt{3}$$

Solution

Question 2

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

$$\displaystyle \frac{-4}{19}$$

$$\displaystyle \frac{0}{8}$$

$$\displaystyle \frac{3}{0}$$

Solution

Question 3

$$ \displaystyle \frac{46}{75} $$

$$ \displaystyle \frac{47}{75} $$

$$ \displaystyle \frac{49}{75} $$

$$ \displaystyle \frac{50}{75} $$

Solution

Question 4

$$\displaystyle \sqrt{3}+2\sqrt{2}$$

$$\displaystyle \sqrt{2}+2\sqrt{3}$$

$$\displaystyle \sqrt{3}+\sqrt{2}$$

$$\displaystyle \sqrt{3}-2\sqrt{2}$$

Solution

Question 5

an integer

a rational number

an irrational number

none of these

Solution

Question 6

$$\displaystyle\frac{-8}{6},\,\displaystyle\frac{-13}{6},\,\displaystyle\frac{0}{6},\,\displaystyle\frac{1}{6}$$ and $$\displaystyle\frac{2}{6}$$

$$\displaystyle\frac{-8}{6},\,\displaystyle\frac{-7}{6},\,\displaystyle\frac{0}{6},\,\displaystyle\frac{1}{6}$$ and $$\displaystyle\frac{13}{6}$$

$$\displaystyle\frac{-8}{6},\,\displaystyle\frac{-7}{6},\,\displaystyle\frac{0}{6},\,\displaystyle\frac{1}{6}$$ and $$\displaystyle\frac{11}{6}$$

$$\displaystyle\frac{-8}{6},\,\displaystyle\frac{-7}{6},\,\displaystyle\frac{0}{6},\,\displaystyle\frac{1}{6}$$ and $$\displaystyle\frac{2}{6}$$

Solution

Question 7

$$\displaystyle\frac{-7}{45},\,\displaystyle\frac{-3}{45},\,\displaystyle\frac{-2}{45},\,\displaystyle\frac{-1}{45},\,\displaystyle\frac{2}{45},\,\displaystyle\frac{3}{45},\,\displaystyle\frac{4}{45},\,\displaystyle\frac{5}{45},\,\displaystyle\frac{8}{45}$$

$$\displaystyle\frac{-4}{45},\,\displaystyle\frac{-3}{45},\,\displaystyle\frac{-2}{45},\,\displaystyle\frac{-1}{45},\,\displaystyle\frac{10}{45},\,\displaystyle\frac{3}{45},\,\displaystyle\frac{4}{45},\,\displaystyle\frac{5}{45},\,\displaystyle\frac{8}{45}$$

$$\displaystyle\frac{-4}{45},\,\displaystyle\frac{-3}{45},\,\displaystyle\frac{-2}{45},\,\displaystyle\frac{-1}{45},\,\displaystyle\frac{2}{45},\,\displaystyle\frac{9}{45},\,\displaystyle\frac{4}{45},\,\displaystyle\frac{5}{45},\,\displaystyle\frac{8}{45}$$

$$\displaystyle\frac{-4}{45},\,\displaystyle\frac{-3}{45},\,\displaystyle\frac{-2}{45},\,\displaystyle\frac{-1}{45},\,\displaystyle\frac{2}{45},\,\displaystyle\frac{3}{45},\,\displaystyle\frac{4}{45},\,\displaystyle\frac{5}{45},\,\displaystyle\frac{8}{45}$$

Solution

Question 8

$$\displaystyle\frac{1}{10},\,\displaystyle\frac{2}{10},\,\displaystyle\frac{3}{10},\,\displaystyle\frac{4}{10},\,\displaystyle\frac{5}{10},\,\displaystyle\frac{6}{10},\,\displaystyle\frac{7}{10},\,\displaystyle\frac{88}{10},\,\displaystyle\frac{9}{10},\,\displaystyle\frac{10}{10}$$

$$\displaystyle\frac{1}{10},\,\displaystyle\frac{2}{10},\,\displaystyle\frac{3}{10},\,\displaystyle\frac{4}{10},\,\displaystyle\frac{21}{10},\,\displaystyle\frac{6}{10},\,\displaystyle\frac{7}{10},\,\displaystyle\frac{8}{10},\,\displaystyle\frac{9}{10},\,\displaystyle\frac{10}{10}$$

$$\displaystyle\frac{1}{10},\,\displaystyle\frac{2}{10},\,\displaystyle\frac{3}{10},\,\displaystyle\frac{4}{10},\,\displaystyle\frac{35}{10},\,\displaystyle\frac{6}{10},\,\displaystyle\frac{7}{10},\,\displaystyle\frac{8}{10},\,\displaystyle\frac{9}{10},\,\displaystyle\frac{10}{10}$$

$$\displaystyle\frac{1}{10},\,\displaystyle\frac{2}{10},\,\displaystyle\frac{3}{10},\,\displaystyle\frac{4}{10},\,\displaystyle\frac{5}{10},\,\displaystyle\frac{6}{10},\,\displaystyle\frac{7}{10},\,\displaystyle\frac{8}{10},\,\displaystyle\frac{9}{10},\,\displaystyle\frac{10}{10}$$

Solution

Question 9

$$\displaystyle 5(\sqrt{7}+\sqrt{5})$$

$$\displaystyle\sqrt{7}+\sqrt{5}$$

$$\displaystyle 2(\sqrt{7}+\sqrt{5})$$

$$\displaystyle 7(\sqrt{2}+\sqrt{5})$$

Solution

Question 10

Has a positive numerator

Has a negative numerator

Has either a positive numerator or a negative numerator

Has neither a positive numerator nor a negative numerator

Solution

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