Number Systems Test

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Number Systems Test - 35

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Weekly Quiz Competition

Question 1
1 / -0

The value of $$[2-3(2-3)^{-1}]^{-1}$$ is __________.

A
$$\displaystyle\frac{-1}{5}$$

B
$$\displaystyle\frac{1}{5}$$

C
$$-5$$

D
$$5$$

Solution

$$=$$$$[2-3(2-3)^{-1}]^{-1}$$
$$=$$$$[2-3(-1)^{-1}]^{-1}$$
$$=$$$$[2-3(-1)]^{-1}$$
$$=$$$$[2+3]^{-1}$$
$$=$$$$\dfrac{1}{5}$$
Question 2
1 / -0

Which one of the following is a rational number?

A
$${ \left( \sqrt { 2 } \right) }^{ 2 }$$

B
$$2\sqrt { 2 } $$

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

D
$$\cfrac { \sqrt { 2 } }{ 2 } $$

Solution

$$\textbf{Step -1: Check each option and find the correct one.}$$

$$\text{A rational number is a number that can be written in the form }\dfrac{p}{q}\text{ where }q\neq0.$$

$$(\sqrt2)^2=2$$

$$\text{and }2\text{ can be written as }\dfrac{2}{1}.$$

$$\therefore (\sqrt2)^2\text{ is a rational number.}$$

$$\text{Since, }\sqrt2\text{ is an irrational number so, }$$

$$2\sqrt2,2+\sqrt2,\dfrac{\sqrt2}{2}\text{ are also irrational numbers.}$$

$$\textbf{ Therefore, option A is correct.}$$
Question 3
1 / -0

The conjugate of the binomial surd $$\dfrac {1}{2}x + \dfrac {1}{2}\sqrt {y}$$ is?

A
$$\dfrac {1}{2}x + \dfrac {1}{3}\sqrt {y}$$

B
$$\dfrac {1}{3}x + \dfrac {1}{2}\sqrt {y}$$

C
$$\dfrac {1}{2}x - \dfrac {1}{2}\sqrt {y}$$

D
$$\dfrac {1}{2}y + \dfrac {1}{2}\sqrt {x}$$

Solution

The sum and difference of two simple quadratic surds are said to be conjugate of each other.
The conjugate of $$\dfrac{1}{2}x+\dfrac{1}{2}\sqrt y$$ is $$\dfrac{1}{2}x-\dfrac{1}{2}\sqrt y$$.
Question 4
1 / -0

Simplify:-
$$(2\sqrt 2\times 3\sqrt 5)\div 2$$ .

A
$$3\sqrt 5$$

B
$$15$$

C
$$12\sqrt 10$$

D
$$3\sqrt 10$$

Solution

$$\dfrac{(2\sqrt2\times3\sqrt5)}{2}=\dfrac{(6\sqrt{2\times5})}{2}=\dfrac{6\sqrt10}{2}\\ =3\sqrt{10}$$
Question 5
1 / -0

From the pairs of number given below, whose product is a natural number?

A
$$\sqrt12 , \sqrt3$$

B
$$\sqrt4 , \sqrt3$$

C
$$\sqrt10 , \sqrt3$$

D
$$\sqrt2 , \sqrt3$$

Solution

Option A: $$ \sqrt12\times\sqrt3=\sqrt{36}=6\Rightarrow $$natural number
Option B: $$\sqrt4\times\sqrt3=\sqrt{12}\Rightarrow$$ irrational number
Option C: $$\sqrt{10}\times\sqrt3=\sqrt{30}\Rightarrow $$ irrational number
Option D: $$\sqrt2\times\sqrt3=\sqrt6\Rightarrow $$ irrational number
Question 6
1 / -0

Which of the following numbers has the terminal decimal representation?

A
$$\dfrac{1}{7}$$

B
$$\dfrac{1}{3}$$

C
$$\dfrac{3}{5}$$

D
$$\dfrac{17}{3}$$

Solution

We know that a terminating decimal is a decimal that ends. It's a decimal with a finite number of digits. For example $$\dfrac {1}{4}=0.25$$, it has only two decimal digits.

Then,
Option $$A$$: $$\dfrac {1}{7}=0.1428..$$, non-terminating decimal number.
Option $$B$$: $$\dfrac {1}{3}=0.3333....$$, non-terminating decimal number.
Option $$C$$: $$\dfrac{3}{5}=0.6\Rightarrow $$ terminating decimal number.
Option $$D$$: $$\dfrac {17}{3}=5.6666...$$, non-terminating decimal number.
Hence, option $$C$$ is correct.
Question 7
1 / -0

Find two rational numbers lying between $$\dfrac{-1}{3}$$ and $$\dfrac{1}{2}$$.

A
-1/6,1 /6

B
-2/3, 2/3

C
none

D
both

Solution
Question 8
1 / -0

The rational number that lies between $$\dfrac{3}{7}$$ and $$\dfrac{2}{3}$$ is

A
$$\dfrac{2}{5}$$

B
$$\dfrac{4}{7}$$

C
$$\dfrac{3}{7}$$

D
$$\dfrac{2}{3}$$

Solution

$$ \dfrac{3}{7}\times \dfrac{3}{3} = \dfrac{9}{21},$$

$$\dfrac{2}{3}\times \dfrac{7}{7} = \dfrac{14}{21} $$

Rational numbers between $$ \dfrac{3}{7} $$ & $$ \dfrac{2}{3} $$ are $$ \dfrac{10}{21},\dfrac{11}{21},\dfrac{12}{21},\dfrac{13}{21} $$

$$ \dfrac{12}{21}= \dfrac{4}{7} $$

Hence, $$ \dfrac{4}{7} $$ lies between $$\dfrac37$$ and $$\dfrac23$$.
Question 9
1 / -0

Which of the following is irrational

A
$$\sqrt {\dfrac {4}{9}} $$

B
$$\dfrac {4}{5}$$

C
$$\sqrt {7}$$

D
$$\sqrt {81}$$

Solution

$${\textbf{Step-1: Identify rational and irrational}}$$
$$\text{A = }\sqrt{4/9}$$
$$\text{=}\sqrt{\text2/3}$$ $${\text{Rational}}$$
$${\text{B = 4/5}}$$ $${\text{Rational}}$$
  $$\text{C = }\sqrt {\text{7}} $$    $${\text{Irrational}}$$
  $$\text{D = }\sqrt {{\text{81}}} $$ = 9 $${\text{Rational}}$$
$${\textbf{Hence option C is correct}}$$
Question 10
1 / -0

Simplify $$2\sqrt{5}$$ + $$10\sqrt{5}$$ + $$25\sqrt{5}$$+ $$31\sqrt{5}$$.

A
$$68\sqrt{5}$$

B
$$28\sqrt{5}$$

C
$$48\sqrt{5}$$

D
$$\sqrt{5}$$

Solution

$$2\sqrt{5}+10\sqrt{5}+25\sqrt{5}+31\sqrt{5}$$

$$=(2+10+25+31)\sqrt5$$

$$=68\sqrt{5}$$

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Re-Attempt Weekly Quiz Competition

Number Systems Test - 35

Result

Number Systems Test - 35

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SHARING IS CARING

Question 1

$$\displaystyle\frac{-1}{5}$$

$$\displaystyle\frac{1}{5}$$

$$-5$$

$$5$$

Solution

Question 2

$${ \left( \sqrt { 2 } \right) }^{ 2 }$$

$$2\sqrt { 2 } $$

$$2+\sqrt { 2 } $$

$$\cfrac { \sqrt { 2 } }{ 2 } $$

Solution

Question 3

$$\dfrac {1}{2}x + \dfrac {1}{3}\sqrt {y}$$

$$\dfrac {1}{3}x + \dfrac {1}{2}\sqrt {y}$$

$$\dfrac {1}{2}x - \dfrac {1}{2}\sqrt {y}$$

$$\dfrac {1}{2}y + \dfrac {1}{2}\sqrt {x}$$

Solution

Question 4

$$3\sqrt 5$$

$$15$$

$$12\sqrt 10$$

$$3\sqrt 10$$

Solution

Question 5

$$\sqrt12 , \sqrt3$$

$$\sqrt4 , \sqrt3$$

$$\sqrt10 , \sqrt3$$

$$\sqrt2 , \sqrt3$$

Solution

Question 6

$$\dfrac{1}{7}$$

$$\dfrac{1}{3}$$

$$\dfrac{3}{5}$$

$$\dfrac{17}{3}$$

Solution

Question 7

-1/6,1 /6

-2/3, 2/3

none

both

Solution

Question 8

$$\dfrac{2}{5}$$

$$\dfrac{4}{7}$$

$$\dfrac{3}{7}$$

$$\dfrac{2}{3}$$

Solution

Question 9

$$\sqrt {\dfrac {4}{9}} $$

$$\dfrac {4}{5}$$

$$\sqrt {7}$$

$$\sqrt {81}$$

Solution

Question 10

$$68\sqrt{5}$$

$$28\sqrt{5}$$

$$48\sqrt{5}$$

$$\sqrt{5}$$

Solution

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