RD Sharma 2020 solution class 9 chapter 3 Rationalisation MCQS

MCQS

Page-3.16

Question 1:

$\sqrt{10}\times \sqrt{15}$ is equal to

(a) 5$\sqrt{6}$

(b) 6$\sqrt{5}$

(c) $\sqrt{30}$

(d) $\sqrt{25}$

Answer 1:

Given that, it can be simplified as 

Therefore given expression is simplified and correct choice is

Question 2:

$\sqrt[5]{6}\times \sqrt[5]{6}$is equal to

(a) $\sqrt[5]{36}$

(b) $\sqrt[5]{6\times 0}$

(c) $\sqrt[5]{6}$

(d) $\sqrt[5]{12}$

Answer 2:

Given that, it can be simplified as 

Therefore given expression is simplified and correct choice is.

Question 3:

The rationalisation factor of $\sqrt{3}$ is

(a) $-\sqrt{3}$

(b) $\frac{1}{\sqrt{3}}$

(c) $2\sqrt{3}$

(d) $-2\sqrt{3}$

Answer 3:

We know that rationalization factor for is. Hence rationalization factor of is.Hence the correct option is.

Question 4:

The rationalisation factor of $2+\sqrt{3}$ is

(a) $2-\sqrt{3}$

(b) $2+\sqrt{3}$

(c) $\sqrt{2}-3$

(d) $\sqrt{3}-2$

Answer 4:

We know that rationalization factor for is. Hence rationalization factor of is.Hence correct option is

Question 5:

If x = $\sqrt{5}+2$, then $x-\frac{1}{x}$ equals

(a) $2\sqrt{5}$

(b) 4

(c) 2

(d) $\sqrt{5}$

Answer 5:

Given that.Hence is given as
.We need to find
We know that rationalization factor for is. We will multiply numerator and denominator of the given expression by, to get

Therefore,

Hence the correct option is.

Question 6:

If $\frac{\sqrt{3-1}}{\sqrt{3}+1}$ = $a-b\sqrt{3}$, then

(a) a = 2, b =1

(b) a = 2, b =−1

(c) a = −2, b = 1

(d) a = b = 1

Answer 6:

Given that:
We are asked to find a and b
We know that rationalization factor for is . We will multiply numerator and denominator of the given expression by, to get

On equating rational and irrational terms, we get 

Comparing rational and irrational part we get

Hence, the correct choice is.

Question 7:

The simplest rationalising factor of $\sqrt[3]{500}$ is

(a) $\sqrt[3]{2}$

(b) $\sqrt[3]{5}$

(c) $\sqrt{3}$

(d) none of these

Answer 7:

Given that:.To find simplest rationalizing factor of the given expression we will factorize it as 

The rationalizing factor of is, since when we multiply given expression with this factor we get rid of irrational term.
Therefore, rationalizing factor of the given expression is
Hence correct option is.

Question 8:

The simplest rationalising factor of $\sqrt{3}+\sqrt{5}$ is

(a) $\sqrt{3}-5$

(b) $3-\sqrt{5}$

(c) $\sqrt{3}-\sqrt{5}$

(d) $\sqrt{3}+\sqrt{5}$

Answer 8:

We know that rationalization factor for is. Hence rationalization factor of is.

Question 9:

The simplest rationalising factor of $2\sqrt{5}$−$\sqrt{3}$ is

(a) $2\sqrt{5}+3$

(b) $2\sqrt{5}+\sqrt{3}$

(c) $\sqrt{5}+\sqrt{3}$

(d) $\sqrt{5}-\sqrt{3}$

Answer 9:

We know that rationalization factor for is. Hence rationalization factor of is.

Question 10:

If x =$\frac{2}{3+\sqrt{7}}$, then (x−3)² =

(a) 1
(b) 3
(c) 6
(d) 7

Answer 10:

Given that:
We know that rationalization factor for is. We will multiply numerator and denominator of the given expression by, to get

Therefore,

On squaring both sides, we get

Hence the value of the given expression is.

Question 11:

If $x=7+4\sqrt{3}$ and xy =1, then $\frac{1}{x^2}+\frac{1}{y^2}=$

(a) 64

(b) 134

(c) 194

(d) 1/49

Answer 11:

Given that,
Hence is given as 


We need to find
We know that rationalization factor for is. We will multiply numerator and denominator of the given expression by, to get

Since so we have 

Therefore,

Hence the value of the given expression is.

Question 12:

If $x+\sqrt{15}=4,$then $x+\frac{1}{x}$=

(a) 2
(b) 4
(c) 8
(d) 1

Answer 12:

Given that .It can be simplified as 

We need to find
We know that rationalization factor for is. We will multiply numerator and denominator of the given expression by, to get

Therefore,

Hence the value of the given expression is 8.Hence correct option is .

Page-3.17

Question 13:

If $x=\frac{\sqrt{5}+\sqrt{3}}{\sqrt{5}-\sqrt{3}}$ and $y=\frac{\sqrt{5}-\sqrt{3}}{\sqrt{5}+\sqrt{3}}$, then x + y +xy=

(a) 9
(b) 5
(c) 17
(d) 7

Answer 13:

Given that and.
We are asked to find
Now we will rationalize x. We know that rationalization factor for is. We will multiply numerator and denominator of the given expression by, to get

Similarly, we can rationalize y. We know that rationalization factor for is. We will multiply numerator and denominator of the given expression by, to get

Therefore,

Hence the value of the given expression is.

Question 14:

If x=$\frac{\sqrt{3}-\sqrt{2}}{\sqrt{3}+\sqrt{2}}$ and y = $\frac{\sqrt{3}+\sqrt{2}}{\sqrt{3}-\sqrt{2}}$, then x² + y +y² =

(a) 101
(b) 99
(c) 98
(d) 102

Answer 14:

Given that and.
We need to find
Now we will rationalize x. We know that rationalization factor for is. We will multiply numerator and denominator of the given expression by, to get

Similarly, we can rationalize y. We know that rationalization factor for is. We will multiply numerator and denominator of the given expression by, to get

Therefore,

Hence the value of the given expression is.

Question 15:

$\frac{1}{\sqrt{9}-\sqrt{8}}$ is equal to

(a) $3+2\sqrt{2}$

(b) $\frac{1}{3+2\sqrt{2}}$

(c) $3-2\sqrt{2}$

(d) $\frac{3}{2}-\sqrt{2}$

Answer 15:

Given that
We know that rationalization factor for is. We will multiply numerator and denominator of the given expression by, to get

Hence the correct option is.

Question 16:

The value of $\frac{\sqrt{48}+\sqrt{32}}{\sqrt{27}+\sqrt{18}}$ is

(a) $\frac{4}{3}$

(b) 4

(c) 3

(d) $\frac{3}{4}$

Answer 16:

Given that
We know that rationalization factor for is. We will multiply numerator and denominator of the given expression by, to get

We can factor irrational terms as 

Hence the value of given expression is.

Question 17:

If $\frac{5-\sqrt{3}}{2+\sqrt{3}}=x+y\sqrt{3}$, then

(a) x = 13, y = −7

(b) x = −13, y = 7

(c) x = −13, y = −7

(d) x = 13, y = 7

Answer 17:

Given that: .We need to find x and y
We know that rationalization factor for is . We will multiply numerator and denominator of the given expression   by, to get

Since
On equating rational and irrational terms, we get
Hence, the correct choice is.

Question 18:

If x = $\sqrt[3]{2+\sqrt{3}}$, then $x^3+\frac{1}{x^3}=$

(a) 2

(b) 4

(c) 8

(d) 9

Answer 18:

Given that .It can be simplified as 

We know that rationalization factor for is. We will multiply numerator and denominator of the given expression by, to get

Therefore,

Hence the value of the given expression is.

Question 19:

The value of $\sqrt{3-2\sqrt{2}}$ is

(a) $\sqrt{2}-1$

(b) $\sqrt{2}+1$

(c) $\sqrt{3}-\sqrt{2}$

(d) $\sqrt{3}+\sqrt{2}$

Answer 19:

Given that:.It can be written in the form as

Hence the value of the given expression is.

Question 20:

The value of $\sqrt{5+2\sqrt{6} }$ is

(a) $\sqrt{3}-\sqrt{2}$

(b) $\sqrt{3}+\sqrt{2}$

(c) $\sqrt{5}+\sqrt{6}$

(d) none of these

Answer 20:

Given that:.It can be written in the form as

Hence the value of the given expression is.

Question 21:

If $\sqrt{2}=1.4142$ then $\sqrt{\frac{\sqrt{2}-1}{\sqrt{2}+1}}$ is equal to

(a) 0.1718

(b) 5.8282

(c) 0.4142

(d) 2.4142

Answer 21:

Given that , we need to find the value of .
We can rationalize the denominator of the given expression. We know that rationalization factor for is. We will multiply numerator and denominator of the given expression by, to get

$\sqrt{\frac{\sqrt{2}-1}{\sqrt{2}+1}}=\frac{\sqrt{2}-1}{1}$
Putting the value of , we get 

Hence the value of the given expression is 0.14142 and correct choice is.

Question 22:

If $\sqrt{2}=1.414,$then the value of $\sqrt{6}-\sqrt{3}$ upto three places of decimal is

(a) 0.235

(b) 0.707

(c) 1.414

(d) 0.471

Answer 22:

Given that.We need to find.
We can factor out from the given expression, to get

Putting the value of, we get

Hence the value of expression must closely resemble be
The correct option is.

Question 23:

The positive square root of $7+\sqrt{48}$ is

(a) $7+2\sqrt{3}$

(b) $7+\sqrt{3}$

(c) $2+\sqrt{3}$

(d) $3+\sqrt{2}$

Answer 23:

Given that:.To find square root of the given expression we need to rewrite the expression in the form

Hence the square root of the given expression is
Hence the correct option is.

Question 24:

If $x=\sqrt{6}+\sqrt{5}$, then $x^2+\frac{1}{x^2}-2=$

(a) $2\sqrt{6}$

(b) $2\sqrt{5}$

(c) 24

(d) 20

Answer 24:

Given that.Hence is given as
We need to find
We know that rationalization factor for   is. We will multiply numerator and denominator of the given expression   by, to get

We know that  therefore,

Hence the value of the given expression is 20 and correct option is (d).

Question 25:

If $\sqrt{13-a\sqrt{10}}=\sqrt{8}+\sqrt{5}, \mathrm{then} a=$

(a) −5

(b) −6

(c) −4

(d) −2

Answer 25:

Given that:
We need to find a
The given expression can be simplified by taking square on both sides 

The irrational terms on right side can be factorized such that it of the same form as left side terms. 
Hence,

On comparing rational and irrational terms, we get.Therefore, correct choice is .