Rationalisation- 2 RD Sharma Solutions | Mathematics (Maths) Class 9 PDF Download

RD Sharma Solutions: Exercise 3.2 - Rationalisation

Q.1. Rationalise the denominator of each of the following (i-vii):

(i)

(ii) 

(iii)

(iv)

(v)

(vi)

(vii)

Proof: (i) We know that rationalization factor foris.We will multiply numerator and denominator of the given expressionby, to get

Hence the given expression is simplified to.

(ii) We know that rationalization factor foris.We will multiply numerator and denominator of the given expressionby, to get

Hence the given expression is simplified to.

(iii)We know that rationalization factor foris. We will multiply numerator and denominator of the given expressionby,to get

Hence the given expression is simplified to.

(iv) We know that rationalization factor foris. We will multiply numerator and denominator of the given expressionby , to get

Hence the given expression is simplified to.

(v) We know that rationalization factor foris. We will multiply numerator and denominator of the given expressionby to get

Hence the given expression is simplified to.

(vii) We know that rationalization factor foris. We will multiply numerator and denominator of the given expression by , to get

Hence the given expression is simplified to.

Q.2. Find the value to three places of decimals of each of the following. It is given that √2 = 1.414, √3 = 1.732, √5 = 2.236  and √10 = 3.162.

(i)

(ii)

(iii)

(iv)

(v)

(vi)

Proof: (i) We know that rationalization factor of the denominator is. We will multiply numerator and denominator of the given expression  by, to get

The value of expression can be round off to three decimal places as.

Hence the given expression is simplified to.

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

The value of expression can be round off to three decimal places as.

Hence the given expression is simplified to

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

The value of expression can be round off to three decimal places as.

Hence the given expression is simplified to.

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

The value of expression can be round off to three decimal places as.

Hence the given expression is simplified to.

(v) Given that

Putting the value of, we get

The value of expression1.24401 can be round off to three decimal places as 1.244.

Hence the given expression is simplified to 1.244.

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

Putting the value of and, we get

The value of expression 0.1852 can be round off to three decimal places as 0.185

Hence the given expression is simplified to 0.185.

Q.3. Express each one of the following with rational denominator:

(i)

(ii)

(iii)

(iv)

(v)

(vi)

(vii)

(viii)

(ix)

Proof: 

(i) We know that rationalization factor foris. We will multiply numerator and denominator of the given expressionby, to get

Hence the given expression is simplified with rational denominator to.

(ii) We know that rationalization factor foris. We will multiply numerator and denominator of the given expressionby, to get

Hence the given expression is simplified with rational denominator to.

(iii) We know that rationalization factor foris. We will multiply numerator and denominator of the given expressionby, to get

Hence the given expression is simplified with rational denominator to.

(iv) We know that rationalization factor foris . We will multiply numerator and denominator of the given expressionby, to get

Hence the given expression is simplified with rational denominator to.

(v) We know that rationalization factor foris.We will multiply numerator and denominator of the given expressionby, to get

Hence the given expression is simplified with rational denominator to.

(vi) We know that rationalization factor foris. We will multiply numerator and denominator of the given expressionby, to get

Hence the given expression is simplified with rational denominator to.

(vii) We know that rationalization factor foris. We will multiply numerator and denominator of the given expressionby, to get

Hence the given expression is simplified with rational denominator to.

(viii) We know that rationalization factor foris. We will multiply numerator and denominator of the given expressionby, to get

Hence the given expression is simplified with rational denominator to.

(ix) We know that rationalization factor foris. We will multiply numerator and denominator of the given expressionby, to get

Hence the given expression is simplified with rational denominator to .

Q.4. Rationales the denominator and simplify:

(i)

(ii)

(iii)

(iv)

(v)

(vi)

Proof: 

(i) We know that rationalization factor foris. We will multiply numerator and denominator of the given expressionby, to get

Hence the given expression is simplified to.

(ii) We know that rationalization factor foris. We will multiply numerator and denominator of the given expressionby, to get

Hence the given expression is simplified to.

(iii) We know that rationalization factor foris. We will multiply numerator and denominator of the given expressionby, to get

Hence the given expression is simplified to.

(iv) We know that rationalization factor foris. We will multiply numerator and denominator of the given expressionby, to get

Hence the given expression is simplified to.

(v) We know that rationalization factor foris. We will multiply numerator and denominator of the given expressionby, to get

Hence the given expression is simplified to.

(vi) We know that rationalization factor foris  We will multiply numerator and denominator of the given expressionby, to get

Hence the given expression is simplified to.

Q.5. Simplify:

(i)

(ii)

(iii)

Proof:(i) We know that rationalization factor forandareandrespectively. We will multiply numerator and denominator of the given expressionandbyandrespectively, to get

Hence the given expression is simplified to 8.

(ii) We know that rationalization factor forandareandrespectively. We will multiply numerator and denominator of the given expression and by and respectively, to get

Hence the given expression is simplified to 0.

(iii) We know that rationalization factor forandareandrespectively. We will multiply numerator and denominator of the given expression and by and respectively, to get

Hence the given expression is simplified to 0.

Q.6. In each of the following determine rational numbers a and b:

(i)

(ii)

(iii)

(iv)

(v)

(vi)

Proof: 

(i) We know that rationalization factor foris. We will multiply numerator and denominator of the given expressionby, to get

On equating rational and irrational terms, we get 

Hence, we get.

(ii) We know that rationalization factor foris. We will multiply numerator and denominator of the given expressionby, to get

On equating rational and irrational terms, we get 

Hence, we get.

(iii) We know that rationalization factor foris. We will multiply numerator and denominator of the given expressionby, to get

On equating rational and irrational terms, we get 

Hence, we get. 

(iv) We know that rationalization factor foris. We will multiply numerator and denominator of the given expressionby, to get

On equating rational and irrational terms, we get 

Hence, we get.

(v) We know that rationalization factor foris. We will multiply numerator and denominator of the given expressionby, to get

On equating rational and irrational terms, we get 

Hence, we get.

(vi) We know that rationalization factor foris. We will multiply numerator and denominator of the given expressionby, to get

On equating rational and irrational terms, we get 

Hence, we get.

Q.7. Find the value of, it being given that √3=1.732 and √5 = 2.236

Proof: 

We know that rationalization factor foris. We will multiply denominator and numerator of the given expressionby, to get

Putting the values of √3 and √5, we get

Hence value of the given expression is.

Q.8. Find the values of each of the following correct to three places of decimals, it being given that √2=1.4142,√3=1.732, √5=2.2360, √6=2.4495 and √10=3.162,

(i)

(ii)

Proof: (i) We know that rationalization factor foris. We will multiply numerator and denominator of the given expressionby, to get

Putting the values of, we get 

Hence the given expression is simplified to.

(ii) We know that rationalization factor foris. We will multiply numerator and denominator of the given expressionby, to get

Putting the value of, we get 

Hence the given expression is simplified to.

Q.9. Simplify:

(i)

(ii)

Proof: (i) We know that rationalization factor forand areandrespectively. We will multiply numerator and denominator of the given expressionandbyandrespectively, to get

Hence the given expression is simplified to.

(ii) We know that rationalization factor forandareandrespectively. We will multiply numerator and denominator of the given expressionandbyandrespectively, to get

Hence the given expression is simplified to.

Q.10. If x = 2 + √3, find the value of x³+1/x³

Proof: We know that. We have to find the value of.

Astherefore, 

We know that rationalization factor foris. We will multiply numerator and denominator of the given expressionby, to get

Putting the value of  and  , we get

Hence the value of the given expression

Q.11. If x = 3+√8, find the value of x²+1/x²

Proof: We know that. We have to find the value of. 

As therefore,

We know that rationalization factor foris. We will multiply numerator and denominator of the given expressionby, to get

Putting the value of x and  , we get

Hence the given expression is simplified to.

Q.12. If x = ,  find the value of 4x³+2x²−8^x+7

Proof: We have, 

It can be simplified as 

On squaring both sides, we get 

The given equation can be rewritten as.

Therefore, we have 

Hence, the value of given expression is.

Multiple Choice Questions(MCQs)

Q.1. √10 × √15  is equal to

(a) 5√6

(b) 6√5

(c) √30

(d) √25

Proof: 

Given that, it can be simplified as 

Therefore given expression is simplified and correct choice is

Q.2. is equal to

(a) 5√36

(b) 5√6×0

(c) 5√6

(d) 5√12

Proof: Given that, it can be simplified as 

Therefore given expression is simplified and correct choice is.

Q.3. The rationalisation factor of √3 is

(a) −√3

(b) 1/√3

(c) 2√3

(d) −2√3

Proof: We know that rationalization factor foris. Hence rationalization factor of is.Hence the correct option is.

Q.4. The rationalisation factor of 2+√3 is

(a) 2−√3

(b) 2+√3

(c) √2–3

(d) √3−2

Proof: We know that rationalization factor foris. Hence rationalization factor of is. Hence correct option is 

Q.5. If x = √5+2, then x−1/x equals

(a) 2√5

(b) 4

(c) 2

(d) √5

Proof: 

Given that.Henceis given as.

We need to find

We know that rationalization factor foris. We will multiply numerator and denominator of the given expressionby, to get

Therefore,

Hence the correct option is.

Q.6. If = a − b√3, then

(a) a = 2, b =1

(b) a = 2, b =−1

(c) a = −2, b = 1

(d) a = b = 1

Proof: Given that:

We are asked to find a and b

We know that rationalization factor foris. We will multiply numerator and denominator of the given expressionby, to get

On equating rational and irrational terms, we get 

Comparing rational and irrational part we get

a = 2, b = 1

Hence, the correct choice is.

Q.7. The simplest rationalising factor of  is

(a)

(b)

(c) √3

(d) none of these

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

The rationalizing factor ofis, 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.

Q.8. The simplest rationalising factor of √3 + √5 is

(a) √3 − 5

(b) 3 − √5

(c) √3 − √5

(d) √3 + √5

Proof: We know that rationalization factor foris. Hence rationalization factor ofis.

Q.9. The simplest rationalising factor of 2√5 − √3 is

a) 2√5 + 3

(b) 2√5 + √3

(c) √5 + √3

(d) √5 - √3

Proof: We know that rationalization factor foris. Hence rationalization factor ofis.

Q.10. if x = then (x−3)² =

(a) 1

(b) 3

(c) 6

(d) 7

Proof: Given that:

We know that rationalization factor foris. We will multiply numerator and denominator of the given expressionby, to get

Therefore,

On squaring both sides, we get

Hence the value of the given expression is.

Q.11. If x = 7 + 4√3 and xy = 1, then 1/x²+1/y² =

(a) 64

(b) 134

(c) 194

(d) 1/49

Proof: Given that,

Hence y is given as 

We need to find 

We know that rationalization factor foris. We will multiply numerator and denominator of the given expressionby, to get

Since so we have 

Therefore,

Hence the value of the given expression is.

Q.12. If x + √15 = 4, then x + 1/x=

(a) 2

(b) 4

(c) 8

(d) 1

Proof: Given that.It can be simplified as 

We need to find

We know that rationalization factor foris. We will multiply numerator and denominator of the given expressionby, to get

Therefore,

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

Q.13. If x = and y =, then x + y + xy =

(a) 9

(b) 5

(c) 17

(d) 7

Proof: Given that and.

We are asked to find x + y + xy

Now we will rationalize x. We know that rationalization factor foris. We will multiply numerator and denominator of the given expressionby, to get

Similarly, we can rationalize y. We know that rationalization factor foris. We will multiply numerator and denominator of the given expressionby, to get

Therefore,

Hence the value of the given expression is.

Q.14. if x = and y = , then x² + y + y² =

(a) 101

(b) 99

(c) 98

(d) 102

Proof: Given that and.

We need to find 

x² + y + y²

Now we will rationalize x. We know that rationalization factor foris. We will multiply numerator and denominator of the given expressionby, to get

Similarly, we can rationalize y. We know that rationalization factor foris. We will multiply numerator and denominator of the given expressionby, to get

Therefore,

Hence the value of the given expression is.

Q.15.  is equal to

(a)

(b)

(c)

(d)

Proof: Given that

We know that rationalization factor foris. We will multiply numerator and denominator of the given expressionby, to get

Hence the correct option is.

Q.16. The value ofis

(a) 4/3

(b) 4

(c) 3

(d) 3/4

Proof: Given that

We know that rationalization factor foris. We will multiply numerator and denominator of the given expressionby, to get

We can factor irrational terms as 

Hence the value of given expression is.

Q.17. If=x + y√3, then

(a) x = 13, y = −7

(b) x = −13, y = 7

(c) x = −13, y = −7

(d) x = 13, y = 7

Proof: Given that: .We need to find x and y

We know that rationalization factor foris. We will multiply numerator and denominator of the given expressionby, to get

Since

On equating rational and irrational terms, we get x = 13 and y = -7

Hence, the correct choice is.

Q.18. If x =, then x³ + 1/x³=

(a) 2

(b) 4

(c) 8

(d) 9

Proof: Given that.It can be simplified as 

We know that rationalization factor foris. We will multiply numerator and denominator of the given expressionby, to get

Therefore,

Hence the value of the given expression is.

Q.19. The value ofis

(a) √2 − 1

(b) √2 + 1

(c) √3 − √2

(d) √3 + √2

Proof: Given that:.It can be written in the formas

Hence the value of the given expression is.

Q.20. The value ofis

(a) √3 − √2

(b) √3 + √2

(c) √5 + √6

(d) none of these

Proof: 

Given that:.It can be written in the formas

Hence the value of the given expression is.

Q.21. If √2=1.4142 thenis equal to

(a) 0.1718

(b) 5.8282

(c) 0.4142

(d) 2.4142

Proof: Given that , we need to find the value of .

We can rationalize the denominator of the given expression. We know that rationalization factor foris. We will multiply numerator and denominator of the given expressionby, to get

Putting the value of, we get 

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

Q.22. If √2=1.414, then the value of √6 −√3 upto three places of decimal is

(a) 0.235

(b) 0.707

(c) 1.414

(d) 0.471

Proof: 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 0.707

The correct option is.

Q.23. The positive square root of 7 + √48 is

(a) 7 + 2√3

(b) 7 + √3

(c) 2 + √3

(d) 3 + √2

Proof: Given that:7 + √48.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.

Q.24. If x = √6 + √5, then x² + 1/x² − 2 =

(a) 2√6

(b) 2√5

(c) 24

(d) 20

Proof: 

Given that.Hence is given as

We need to find

We know that rationalization factor foris. We will multiply numerator and denominator of the given expressionby, to get

We know thattherefore,

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

Q.25. If, then a = 

(a) −5

(b) −6

(c) −4

(d) −2

Proof: 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 a = -4 .Therefore, correct choice is.

Fill in the Blanks Type Questions(FBQs)

Q.1. The number obtained by rationalizing the denominator of  is __________.

Proof:

Multiply and divide by (√7−√2), we get

(Using the identity: (a + b)(a − b)=a² − b²)

Hence, the number obtained by rationalizing the denominator ofis.

Q.2. if, then A = ____________ and B = ____________.

Proof:

Multiply and divide by (√9 + √8), we get

(Using the identity: (a + b)(a − b)=a²−b²)

√9 + √8

=3 + 2√2

Now, it is given that

⇒A=3 and B=2

Hence, if = , then A = 3 and B = 2.

Q.3. After rationalizing the denominator of, we get the denominator as __________.

Proof:

Multiply and divide by, we get

(Using the identity: (a + b)(a − b)=a²−b²)

Hence, after rationalizing the denominator of, we get the denominator as.

Q.4. If a=2 + √3, then a − 1/a=___________.

Proof: Given: a = 2 + √3

Multiply and divide RHS by (2−√3), we get

(Using the identity: (a + b)(a − b)=a²−b²)

Thus,

Hence, if a=2 + √3, then a − 1/a = 2√3.

Q.5. If a=5 + 2√6, then a + 1/a = _________.

Proof: Given: a= 5 + 2√6

Now,

Multiply and divide RHS by (5−2√6), we get

(Using the identity: (a + b)(a − b)=a²−b²)

Thus,

= 10

Hence, if a = 5 + 2√6, then a + 1/a = 10.

Q.6. If x=√6 + √5, then x²+1/x²−2=_________.

Proof: Given: x = √6 +√5

Now,

Multiply and divide RHS by (√6 − √5), we get

(Using the identity: (a + b)(a − b)=a²−b²)

Thus,

= 20

Hence, if x = √6 +√5, then x²+ 1/x²−2 = 20.

Q.7. If x = 3 − √8, then (x − 1/x)² =  ___________.

Proof: Given: x = 3 − √8

Now,

Multiply and divide RHS by (3 + √8), we get

(Using the identity: (a + b)(a − b)=a²−b²)

Thus,

= 32

Hence, if x = 3 − √8, then (x − 1/x)² = 32.

Q.8. if x = and y= , then x + y = __________.

Proof: Given:

x= 

y= 

Now,

x =

Multiply and divide RHS by (√3+√5), we get

(Using the identity: (a + b)(a − b)=a²−b²)

Multiply and divide RHS by (√3-√5), we get

(Using the identity: (a + b)(a − b)=a²−b²)

Thus,

Hence, x + y = −2√3.

Q.9. if , then A=_________.

Proof: Given:

Now,

(Using the identity: (a + b)² = a² + b² + 2ab)

Hence, if , then A = √2.

Q.10. =__________.

Proof:

(Using the identity: (a + b)² = a² + b² + 2ab)

= √6 + 1     ....(1)

(Using the identity: (a - b)² = a² + b² - 2ab)

=√6 − 1     ....(2)

From (1) and (2)

= 2

Hence, = 2.

Q.11. If x=and y=, then (x−y)²=____________.

Proof: Given:

x= 

y=

Now,

x= 

Multiply and divide RHS by (√10+√8), we get

(Using the identity: (a + b)(a − b) = a²−b²)

y= 

Multiply and divide RHS by, we get

(Using the identity: (a + b)(a − b) = a²−b²)

Thus,

= 32

hence, x = and y = , then (x − y)² = 32.

Q.12. if, then x =__________.

Proof: Given:

Now,

(Using the identity: (a + b)² = a² + b² + 2ab)

⇒ −x = 4

⇒ x = −4

Hence, if then x = −4.

Very Short Answer Type Questions(VSAQs)

Q.1. Write the value of (2+√3) (2−√3).

Proof: Given that

It can be simplified as

Hence the value of the given expression is. 

Q.2. Write the reciprocal of 5 + √2.

Proof: 

Given that, it’s reciprocal is given as 

It can be simplified by rationalizing the denominator. The rationalizing factor ofis, we will multiply numerator and denominator of the given expressionby, to get

Hence reciprocal of the given expression is.

Q.3. Write the rationalisation factor of 7−3√5.

Proof: The rationalizing factor ofis. Hence the rationalizing factor of is.

Q.4. if= x + y√3,  find the values of x and y.

Proof: It is given that;

.we need to find x and y

We know that rationalization factor foris. We will multiply numerator and denominator of the given expressionby, to get

On equating rational and irrational terms, we get 

Hence, we get.

Q.5. If x = √2 −1, then write the value of 1/x.

Proof: Given that.Henceis given as

We know that rationalization factor foris. We will multiply each side of the given expressionby, to get

Hence the value of the given expression is.

Q.6. If a = √2 + 1, then find the value of a − 1/a.

Proof: Given that, hence is given as

.we are asked to find 

We know that rationalization factor foris. We will multiply each side of the given expressionby, to get

Therefore,

Hence value of the given expression is.

Q.7. If x = 2 + √3,  find the value of x + 1/x.

Proof: Given that, hence 1/x is given as

.We are asked to find 

We know that rationalization factor foris. We will multiply each side of the given expressionby, to get

Therefore,

Hence value of the given expression is.

Q.8. Write the rationalisation factor of √5 − 2.

Proof: Given that, we know that rationalization factor ofis.

So the rationalization factor ofis.

Q.9. Simplify

Proof: We are asked to simplify. It can be written in the formas

Hence the value of given expression is.

Q.10. Simplify

Proof: We are asked to simplify . It can be written in the formas

Hence the value of the given expression is.

Q.11. If x =then find the value of √x − 1/√x.

Proof: Given that:.It can be written in the formas

Therefore,

We know that rationalization factor foris. We will multiply numerator and denominator of the given expressionby, to get

Hence,

Therefore, value of the given expression is.