Number Systems Test

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Number Systems Test - 28

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Weekly Quiz Competition

Question 1
1 / -0

The number obtained on rationalising the denominator of $$\dfrac{1}{\sqrt{9}-\sqrt{8}}$$ is ?

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

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

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

D
$$\dfrac{1}{2}\left ( 3-2\sqrt{2} \right )$$

Solution

We use the identity $$\left( \sqrt { a } +\sqrt { b } \right) \left( \sqrt { a } -\sqrt { b } \right) =a-b$$.

So, $$\dfrac { 1 }{ \sqrt { 9 } -\sqrt { 8 } } =\dfrac { 1 }{ \sqrt { 9 } -\sqrt { 8 } } \times \left( \dfrac { \sqrt { 9 } +\sqrt { 8 } }{ \sqrt { 9 } +\sqrt { 8 } } \right) \\$$

$$ =\dfrac { \sqrt { 9 } +\sqrt { 8 } }{ 9-8 }$$

$$=\dfrac { \sqrt { 9 } +\sqrt { 8 } }{ 1 } $$

$$=\sqrt { 3^2 } +\sqrt { 4\times 2 } $$

$$=3+2\sqrt { 2 }$$
Question 2
1 / -0

The number obtained on rationalizing the denominator of $$\dfrac {1}{\sqrt {7} - 2}$$ is

A
$$\dfrac {\sqrt {7} + 2}{3}$$

B
$$\dfrac {\sqrt {7} - 2}{3}$$

C
$$\dfrac {\sqrt {7} + 2}{5}$$

D
$$\dfrac {\sqrt {7} + 2}{45}$$

Solution

We use the identity $$\left( a+\sqrt { b } \right) \left( a-\sqrt { b } \right) ={ a }^{ 2 }-b$$.
Multiply and divide $$\dfrac { 1 }{ \sqrt { 7 } -2 }$$ by $$\sqrt { 7 } +2$$ to get
$$\dfrac { 1 }{ \sqrt { 7 } -2 } \times \dfrac { \sqrt { 7 } +2 }{ \sqrt { 7 } +2 } =\dfrac { \sqrt { 7 } +2 }{ 7-4 } =\dfrac { \sqrt { 7 } +2 }{ 3 }$$

Hence, option $$A.$$
Question 3
1 / -0

Which of the following is an irrational number ?

A
$$\sqrt{23}$$

B
$$\sqrt{225}$$

C
$$0.3796$$

D
$$7.478$$

Solution

In the given options,
$$\sqrt { 225 }=15$$ so, it is not an irrational number.

Options $$C$$ and $$D$$ are terminating decimals. So, they are also rational numbers.

$$\sqrt{23}4.7958\dots$$ which is a non-terminating non-repeating decimal hence, it is an irrational number.

So, option $$A$$ is the correct answer.
Question 4
1 / -0

$$\pi$$ is a/an _______ .

A
Rational number

B
Integer

C
Irrational number

D
Whole number

Solution

The value of $$\pi$$ is equal to $$ 3.14159265358\dots$$ which is a non-terminating and non-repeating decimal hence, $$\pi$$ is an irrational number.
Question 5
1 / -0

$$\pi$$ is an ________ while $$\dfrac{22}{7}$$ is rational.

A
Integer

B
Whole Number

C
Rational Number

D
Irrational Number

Solution

The value $$\dfrac{22}7$$ is a rational number, as it can be expressed in the form $$\dfrac pq$$.
We consider it as an approximate value of $$\pi$$ because $$\pi$$ is close to $$\dfrac{22}7$$.
But actually its value is $$3.14159....$$, which is neither terminating nor repeating.
Thus, $$\pi $$ is irrational, but $$\dfrac{22}7$$ is rational.
Question 6
1 / -0

Which one of the following is an irrational number ?

A
0.14

B
0.1416

C
0.14169452

D
0.4014001400014543.....

Solution

$$ {\textbf{Step -1: Observe the options given and find the non-terminating and non-repeating number.}} $$

$$ {\text{As }}0.4014001400014.....{\text{is non - terminating and non-repeating so it is a irrational number}}{\text{.}} $$

$$ {\textbf{Final Answer: Hence, the correct answer is option D}}{\text{.}} $$
Question 7
1 / -0

If $$\displaystyle \frac{\sqrt{3}-1}{\sqrt{3}+1} = a+b \sqrt{3}$$, find the values of $$a$$ and $$b$$.

A
$$a =2, b= -1$$

B
$$a =2, b= 1$$

C
$$a =-2, b= -1$$

D
$$a =-2, b= 1$$

Solution

$$\displaystyle \frac{\sqrt{3}-1}{\sqrt{3}+1}$$

$$=\displaystyle \frac{\sqrt{3}-1}{\sqrt{3}+1} \times \frac{\sqrt{3}-1}{\sqrt{3}-1}$$

$$=\displaystyle \frac{(\sqrt{3}-1)^2}{(\sqrt{3})^2-1^2}$$

$$=\displaystyle \frac{3+1-2\sqrt{3}}{3-1}$$

$$=\displaystyle \frac{4-2 \sqrt{3}}{2}$$

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

$$= a+b\sqrt{3}$$

$$a=2,b=-1$$
Question 8
1 / -0

Simplify:
$$5\sqrt{2} -\sqrt{98}+\sqrt{162}$$

A
$$7\sqrt{2}$$

B
$$6\sqrt{2}$$

C
$$3\sqrt{2}$$

D
none

Solution

$$5\sqrt{2} -\sqrt{98}+\sqrt{162}$$
$$=\sqrt{50} -\sqrt{98}+\sqrt{162}$$
$$=5\sqrt{2} -7\sqrt{2}+9\sqrt{2}$$
$$=\sqrt{2}(5-7+9)=7\sqrt{2}$$
Question 9
1 / -0

Simplify:
$$4\sqrt{8} + 4\sqrt{2} - 3\sqrt{2}$$

A
$$6\sqrt{2}$$

B
$$4\sqrt{2}$$

C
$$9\sqrt{2}$$

D
$$5\sqrt{2}$$

Solution

$$4\sqrt{8} + 4\sqrt{2} - 3\sqrt{2}$$
$$=4\sqrt{2^2 \times 2} + \sqrt{2^4 \times 2} - 3\sqrt{2}$$
$$=((4 \times 2)+4 -3) \sqrt{2}$$
$$=9 \sqrt{2}$$
Question 10
1 / -0

Simplify:
$$7\sqrt{5} - 4\sqrt{5}+\sqrt{125}$$

A
$$3\sqrt{5}$$

B
$$4\sqrt{5}$$

C
$$5\sqrt{5}$$

D
$$8\sqrt{5}$$

Solution

$$7\sqrt{5} - 4\sqrt{5}+\sqrt{125}$$
$$=7\sqrt{5} - 4\sqrt{5}+5\sqrt{5}$$
$$=(7-4+5)\sqrt{5}=8\sqrt{5}$$

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Re-Attempt Weekly Quiz Competition

Number Systems Test - 28

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Number Systems Test - 28

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SHARING IS CARING

Question 1

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

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

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

$$\dfrac{1}{2}\left ( 3-2\sqrt{2} \right )$$

Solution

Question 2

$$\dfrac {\sqrt {7} + 2}{3}$$

$$\dfrac {\sqrt {7} - 2}{3}$$

$$\dfrac {\sqrt {7} + 2}{5}$$

$$\dfrac {\sqrt {7} + 2}{45}$$

Solution

Question 3

$$\sqrt{23}$$

$$\sqrt{225}$$

$$0.3796$$

$$7.478$$

Solution

Question 4

Rational number

Integer

Irrational number

Whole number

Solution

Question 5

Integer

Whole Number

Rational Number

Irrational Number

Solution

Question 6

0.14

0.1416

0.14169452

0.4014001400014543.....

Solution

Question 7

$$a =2, b= -1$$

$$a =2, b= 1$$

$$a =-2, b= -1$$

$$a =-2, b= 1$$

Solution

Question 8

$$7\sqrt{2}$$

$$6\sqrt{2}$$

$$3\sqrt{2}$$

none

Solution

Question 9

$$6\sqrt{2}$$

$$4\sqrt{2}$$

$$9\sqrt{2}$$

$$5\sqrt{2}$$

Solution

Question 10

$$3\sqrt{5}$$

$$4\sqrt{5}$$

$$5\sqrt{5}$$

$$8\sqrt{5}$$

Solution

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