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Page No 3.14:

Question 1:

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

(i) 35

(ii) 325  

(iii) 112

(iv) 25

(v) 3+12

(vi) 2+53

(vii) 325

Answer:

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

Hence the given expression is simplified to.

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

Hence the given expression is simplified to.

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

Hence the given expression is simplified to.

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

Hence the given expression is simplified to.

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

Hence the given expression is simplified to.

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

 

Hence the given expression is simplified to.

Page No 3.14:

Question 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) 23

(ii) 310

(iii) 5+12

(iv) 10+152

(v) 2+33

(vi) 2-15

Answer:

(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 expression can be round off to three decimal places as.

Hence the given expression is simplified to.

(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 can be round off to three decimal places as.

Hence the given expression is simplified to.

Page No 3.14:

Question 3:

Express each one of the following with rational denominator:

(i) 13+2

(ii) 16-5

(iii) 1641-5

(iv) 3053-35

(v) 125-3

(vi) 3+122-3

(vii) 6-426+42

(viii) 32+125-3

(ix) b2a2+b2+a

Answer:

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

Hence the given expression is simplified with rational denominator to.

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

Hence the given expression is simplified with rational denominator to.

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

Hence the given expression is simplified with rational denominator to.

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

Hence the given expression is simplified with rational denominator to.

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

Hence the given expression is simplified with rational denominator to.

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

Hence the given expression is simplified with rational denominator to.

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

Hence the given expression is simplified with rational denominator to.

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

Hence the given expression is simplified with rational denominator to.

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

Hence the given expression is simplified with rational denominator to .

Page No 3.14:

Question 4:

Rationales the denominator and simplify:

(i) 3-23+2

(ii) 5+237+43

(iii) 1+23-22

(iv) 26-535-26

(v) 43+5248+18

(vi) 23-522+33

Answer:

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

Hence the given expression is simplified to.

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

Hence the given expression is simplified to.

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

Hence the given expression is simplified to.

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

Hence the given expression is simplified to.

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

Hence the given expression is simplified to.

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

Hence the given expression is simplified to.

Page No 3.14:

Question 5:

Simplify:

(i) 5+35-3+5-35+3

(ii) 12+3+25-3+12-5

(iii) 25+3+13+2+35+2

Answer:

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

Hence the given expression is simplified to.

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

Hence the given expression is simplified to.

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

Hence the given expression is simplified to.

Page No 3.14:

Question 6:

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

(i) 3-13+1=a-b3

(ii) 4+22+2=n-b

(iii) 3+23-2=a+b2

(iv) 5+337+43=a+b3

(v) 11-711+7=a-b77

(vi)    4+354-35=a+b5

Answer:

(i) 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 

Hence, we get.

(ii) 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 

Hence, we get.

(iii) 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 

Hence, we get

(iv) 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 

Hence, we get.

(v) 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 

Hence, we get.

(vi) 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 

Hence, we get.

Page No 3.14:

Question 7:

Find the value of 65-3, it being given that 3=1.732 and 5=2.236

Answer:

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

Putting the values of and, we get

Hence value of the given expression is.



Page No 3.15:

Question 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) 3-53+25

(ii) 1+23-22

Answer:

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

Putting the values of, we get 

Hence the given expression is simplified to.

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

Putting the value of, we get 

Hence the given expression is simplified to.

Page No 3.15:

Question 9:

Simplify:

(i) 32-2332+23+123-2
(ii) 7+353+5-7-353-5

Answer:

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

Hence the given expression is simplified to.

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

Hence the given expression is simplified to.

Page No 3.15:

Question 10:

If x = 2+3, find the value of x3+1x3

Answer:

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

As therefore, 

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

Putting the value of and , we get

Hence the value of the given expression

Page No 3.15:

Question 11:

If x = 3+8, find the value of x2+1x2

Answer:

We know that. We have to find the value of . As therefore,

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

Putting the value of x and , we get

Hence the given expression is simplified to.

Page No 3.15:

Question 12:

If x=3+12, find the value of 4x3+2x2-8x+7.

Answer:

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.



Page No 3.16:

Question 1:

Write the value of 2+3 2-3. 

Answer:

Given that 

It can be simplified as 

Hence the value of the given expression is.

Page No 3.16:

Question 2:

Write the reciprocal of 5+2.

Answer:

Given that, it’s reciprocal is given as 

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

Hence reciprocal of the given expression is.

Page No 3.16:

Question 3:

Write the rationalisation factor of 7-35.

Answer:

The rationalizing factor of is. Hence the rationalizing factor of is .

Page No 3.16:

Question 4:

If 3-13+1=x+y3, find the values of x and y.

Answer:

It is 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

On equating rational and irrational terms, we get 

Hence, we get.

Page No 3.16:

Question 5:

If x=2-1, then write the value of 1x.

Answer:

Given that.Hence is given as

We know that rationalization factor for is . We will multiply each side of the given expression by, to get

Hence the value of the given expression is.

Page No 3.16:

Question 6:

If a=2+1, then find the value of a-1a.

Answer:

Given that, hence is given as

.we are asked to find

We know that rationalization factor for is . We will multiply each side of the given expression by, to get

Therefore,

Hence value of the given expression is.

Page No 3.16:

Question 7:

If x=2+3,  find the value of x+1x.

Answer:

Given that, hence 1xis given as

.We are asked to find

We know that rationalization factor for is . We will multiply each side of the given expression by, to get

Therefore,

Hence value of the given expression is.

Page No 3.16:

Question 8:

Write the rationalisation factor of 5-2.

Answer:

Given that, we know that rationalization factor of is

So the rationalization factor of is.

Page No 3.16:

Question 9:

Simplify 3+22.

Answer:

We are asked to simplify. It can be written in the form as

Hence the value of given expression is.

Page No 3.16:

Question 10:

Simplify 3-22.

Answer:

We are asked to simplify. It can be written in the form as

Hence the value of the given expression is.

Page No 3.16:

Question 11:

If x= 3+22, then find the value of x-1x.

Answer:

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

Therefore,

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

Hence,

Therefore, value of the given expression is.

Page No 3.16:

Question 1:

10×15 is equal to

(a) 56

(b) 65

(c) 30

(d) 25

Answer:

Given that, it can be simplified as 

Therefore given expression is simplified and correct choice is

Page No 3.16:

Question 2:

65×65 is equal to

(a) 365

(b) 6×05

(c) 65

(d) 125

Answer:

Given that, it can be simplified as 

Therefore given expression is simplified and correct choice is.



Page No 3.17:

Question 3:

The rationalisation factor of 3 is

(a) -3

(b) 13

(c) 23

(d) -23

Answer:

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

Page No 3.17:

Question 4:

The rationalisation factor of 2+3 is

(a) 2-3

(b) 2+3

(c) 2-3

(d) 3-2

Answer:

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

Page No 3.17:

Question 5:

If x = 5+2, then x-1x equals

(a) 25

(b) 4

(c) 2

(d) 5

Answer:

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.

Page No 3.17:

Question 6:

If 3-13+1 = a-b3, then

(a) a = 2, b =1

(b) a = 2, b =−1

(c) a = −2, b = 1

(d) a = b = 1

Answer:

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.

Page No 3.17:

Question 7:

The simplest rationalising factor of 5003 is

(a) 23

(b) 53

(c) 3

(d) none of these

Answer:

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.

Page No 3.17:

Question 8:

The simplest rationalising factor of 3+5 is

(a) 3-5

(b) 3-5

(c) 3-5

(d) 3+5

Answer:

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

Page No 3.17:

Question 9:

The simplest rationalising factor of 253 is

(a) 25+3

(b) 25+3

(c) 5+3

(d) 5-3

Answer:

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

Page No 3.17:

Question 10:

If x =23+7, then (x−3)2 =

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

Answer:

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.

Page No 3.17:

Question 11:

If x=7+43 and xy =1, then 1x2+1y2=

(a) 64

(b) 134

(c) 194

(d) 1/49

Answer:

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.

Page No 3.17:

Question 12:

If x+15=4, then x+1x=

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

Answer:

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 No 3.17:

Question 13:

If x=5+35-3 and y=5-35+3, then x + y +xy=

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

Answer:

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.

Page No 3.17:

Question 14:

If x=3-23+2  and y = 3+23-2 , then x2 + y +y2 =

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

Answer:

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.

Page No 3.17:

Question 15:

19-8 is equal to

(a) 3+22

(b) 13+22

(c) 3-22

(d) 32-2

Answer:

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.

Page No 3.17:

Question 16:

The value of 48+3227+18 is

(a) 43

(b) 4

(c) 3

(d) 34

Answer:

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.



Page No 3.18:

Question 17:

If 5-32+3=x+y3, then

(a) x = 13, y = −7

(b) x = −13, y = 7

(c) x = −13, y = −7

(d) x = 13, y = 7

Answer:

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.

Page No 3.18:

Question 18:

If x = 2+33, then x3+1x3=

(a) 2

(b) 4

(c) 8

(d) 9

Answer:

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.

Page No 3.18:

Question 19:

The value of 3-22 is

(a) 2-1

(b) 2+1

(c) 3-2

(d) 3+2

Answer:

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

Hence the value of the given expression is.

Page No 3.18:

Question 20:

The value of 5+26  is

(a) 3-2

(b) 3+2

(c) 5+6

(d) none of these

Answer:

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

Hence the value of the given expression is.

Page No 3.18:

Question 21:

If 2=1.4142 then 2-12+1 is equal to

(a) 0.1718

(b) 5.8282

(c) 0.4142

(d) 2.4142

Answer:

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


2-12+1=2-11

Putting the value of , we get 

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

Page No 3.18:

Question 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

Answer:

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.

Page No 3.18:

Question 23:

The positive square root of 7+48 is

(a) 7+23

(b) 7+3

(c) 2+3

(d) 3+2

Answer:

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.

Page No 3.18:

Question 24:

If x=6+5, then x2+1x2-2=

(a) 26

(b) 25

(c) 24

(d) 20

Answer:

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).

Page No 3.18:

Question 25:

If 13-a10=8+5, then a=

(a) −5

(b) −6

(c) −4

(d) −2

Answer:

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 .



Page No 3.2:

Question 1:

Simplify each of the following:

(i) 43×163

(ii) 1250424

Answer:

(i) We know that. We will use this property to simplify the expression.

Hence the value of the given expression is .

(ii) We know that. We will use this property to simplify the expression.

Hence the value of the given expression is.

Page No 3.2:

Question 2:

Simplify the following expressions:

(i) 4+7) (3+2

(ii) 3+3) (5-2

(iii) 5-2) (3-5

Answer:

(i) We can simplify the expression as 

Hence the value of the expression is

(ii) We can simplify the expression as 

Hence the value of the expression is

(iii) We can simplify the expression as 

Hence the value of the expression is .

Page No 3.2:

Question 3:

Simplify the following expressions:

(i) 11+11 11-11

(ii) 5+7 5-7 

(iii) 8-2 8+2 

(iv) 3+33-3

(v) 5-2 5+2 

Answer:

(i) We know that. We will use this property to simplify the expression.

Hence the value of expression is 110.

 

(ii) We know that. We will use this property to simplify the expression.

Hence the value of expression is 18.

(iii) We know that. We will use this property to simplify the expression.

Hence the value of expression is 6

(iv) We know that. We will use this property to simplify the expression.

Hence the value of expression is 6.

(v) We know that. We will use this property to simplify the expression.

Hence the value of expression is 3.



Page No 3.3:

Question 4:

Simplify the following expressions:

(i) 3+72

(ii) 5-32

(iii) 25+322

Answer:

(i) We know that. We will use this property to simplify the expression.

Hence the value of expression is

(ii) We know that. We will use this property to simplify the expression.

Hence the value of expression is

(iii) We know that. We will use this property to simplify the expression.

Hence the value of expression is.



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