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Page No 9:

Question 1:

What are rational numbers? Give ten examples of rational numbers.

Answer:

The numbers that can be written in the pq form, where p and q are integers and q ≠ 0 are known as rational numbers.

Examples of rational numbers:
(1)  45      (2) 15     (3) 54     (4) 41 = 4       (5) 52     (6) 17      (7) 01 = 0        (8) 95        (9) 55 = 1       (10) 49

Page No 9:

Question 2:

Represent each of the following rational numbers on the number line:
(i) 5
(ii) −3
(iii) 57
(iv) 83
(v) 1.3
(vi) −2.4
(vii) 236

Answer:

(i)


(ii)


(iii)


(iv) 83=223


(v) 1.3=1310=1310


(vi) -2.4=-2410=-125=-225


(vii) 236=356

Page No 9:

Question 3:

Find a rational number lying between
(i) 14 and13
(ii) 38 and25
(iii) 1.3 and 1.4
(iv) 0.75 and 1.2
(v) -1and12
(vi) -34and-25

Answer:

(i) Let:
x = 14 and y = 13
Rational number lying between x and y:
12x + y = 1214 + 13
= 724

(ii) Let:
x = 38 and y = 25
Rational number lying between x and y:
12x + y = 1238 + 25
= 1215+1640 = 3180

(iii) Let:
x = 1.3 and y = 1.4
Rational number lying between x and y:
12x + y = 121.3+1.4
= 122.7= 1.35

(iv) Let:
x = 0.75 and y = 1.2
Rational number lying between x and y:
12x + y = 120.75+1.2
= 121.95= 0.975

(v) Let:
x = -1 and y = 12
Rational number lying between x and y:
12x + y = 12-1 + 12
= -14

(vi) Let:
x-34 and y = -25
Rational number lying between x and y:
12x + y = 12-34 - 25
= 12-15-820 = -2340

Page No 9:

Question 4:

Find three rational numbers lying between 15and14.

Answer:

Let:
x
= 15, y = 14 and n = 3
We know:
d = y-xn+1 = 14-153+1 = 1204 = 180

So, three rational numbers lying between x and y are:

(x + d), (x + 2d) and (x + 3d)

= 15+180, 15+280 and 15+380

= 1780,1880 and 1980

Page No 9:

Question 5:

Find five rational numbers lying between 25and34.

Answer:

Let:
x = 25 , y = 34 and n = 5
We know:
d = y-xn+1 = 34-255+1 = 7206 = 7120

So, five rational numbers between x and y are:
(x + d), (x + 2d), (x + 3d), (x + 4d) and (x + 5d)

= 25+7120, 25+14120 , 25+21120,25+28120 and 25+ 35120

= 55120,62120,69120,76120 and 83120



Page No 10:

Question 6:

Insert six rational numbers between 3 and 4.

Answer:

Let:
x = 3, y = 4 and n = 6
We know:
d = y-xn+1 = 4-36+1 = 17 
So, six rational numbers between x and y are:
(x + d), (x + 2d), (x + 3d), (x + 4d), (x + 5d) and (x + 6d)
= 3+17, 3+27 , 3+37,3+47 , 3+ 57 and 3+67

= 227,237,247,257, 267 and 277

Page No 10:

Question 7:

Insert 16 rational numbers between 2.1 and 2.2.

Answer:

Let:
x = 2.1, y = 2.2 and n = 16

We know:
d = y-xn+1=2.2-2.116+1=0.117=1170= 0.005 (approx.)
So, 16 rational numbers between 2.1 and 2.2 are:
(x + d), (x + 2d), ...(x + 16d)
= [2.1 + 0.005], [2.1 + 2(0.005)],...[2.1 + 16(0.005)]
= 2.105, 2.11, 2.115, 2.12, 2.125, 2.13, 2.135, 2.14, 2.145, 2.15, 2.155, 2.16, 2.165, 2.17, 2.175 and 2.18



Page No 15:

Question 1:

Without actual division, find which of the following rationals are terminating decimals.
(i) 1380
(ii) 724
(iii) 512
(iv) 835
(v) 16125

Answer:

(i) Denominator of 1380 is 80.
And,
80 = 24×5
Therefore, 80 has no other factors than 2 and 5.
Thus, 1380 is a terminating decimal.

(ii) Denominator of 724 is 24.
And,
24 = 23×3
So, 24 has a prime factor 3, which is other than 2 and 5.
Thus, 724 is not a terminating decimal.

(iii)  Denominator of 512 is 12.
And,
12 = 22×3
So, 12 has a prime factor 3, which is other than 2 and 5.
Thus, 512 is not a terminating decimal.

(iv) Denominator of 835 is 35.
And,
35 = 7×5
So, 35 has a prime factor 7, which is other than 2 and 5.
Thus, 835 is not a terminating decimal.

(v) Denominator of 16125 is 125.
And,
125 = 53
Therefore, 125 has no other factors than 2 and 5.
Thus, 16125 is a terminating decimal.

Page No 15:

Question 2:

Convert each of the following into a decimal.
(i) 58
(ii) 916
(iii) 725
(iv) 1124
(v) 2512

Answer:

(i) 58 = 0.625
By actual division, we have:


(ii) 916 = 0.5625
By actual division, we have:


(iii) 725 = 0.28
By actual division, we have:
 

(iv) 1124 =
By actual division, we have:
 

(v)  2512 = 2912 =
By actual division, we have:

Page No 15:

Question 3:

Express each of the following as a fraction in simplest form.
(i) 0.3
(ii) 1.3
(iii) 0.34
(iv) 3.14
(v) 0.324
(vi) 0.17
(vii) 0.54
(viii) 0.163

Answer:

(i) 0.3
Let x=0.3¯
∴ x = 0.3333...                           ...(i)
10x = 3.3333...                           ...(ii)
On subtracting (i) from (ii), we get:
9x = 3
= x = 13

∴ 0.3  = 13

(ii) 1.3
Let x=1.3¯
∴ x = 1.3333...                    ...(i)
10x = 13.3333...                  ...(ii)
On subtracting (i) from (ii), we get:
9x = 12
= x = 43

1.3=43

(iii) 0.34
Let x0.34
x = 0.3434...                    ...(i)
100x = 34.3434...                ...(ii)
On subtracting (i) from (ii), we get:
99x = 34
= x = 3499

0.34=3499

(iv) 3.14
Let x = 3.14
x = 3.1414...                    ...(i)
100x = 314.1414...              ...(ii)
On subtracting (i) from (ii), we get:
99x = 311
= x = 31199

3.14 = 31199

(v) 0.324
Let x = 0.324
x = 0.324324...                    ...(i)
1000x = 324.324324...            ...(ii)
On subtracting (i) from (ii), we get:
999x = 324
= x = 324999=1237

0.324= 1237

(vi) 0.17
Let x = 0.17
x = 0.1777...                 
10x = 1.777...              ...(i)
100x = 17.777....         ...(ii)
On subtracting (i) from (ii). we get:
90x = 16
= x = 845
0.17 = 845

(vii) 0.54
Let x = 0.54
x = 0.5444...                 
10x = 5.4444...               ...(i)
100x = 54.4444...           ...(ii)
On subtracting (i) from (ii), we get:
90x = 49
= x = 4990
0.54= 4990

(viii) 0.163
Let x0.163
x = 0.16363...                 
10x = 1.6363...               ...(i)
1000x = 163.6363...       ...(ii)
On subtracting (i) from (ii), we get:
990x = 162
= x = 162990=955

0.163 = 955

Page No 15:

Question 4:

Write, whether the given statement is true or false. Give reasons.
(i) Every natural number is a whole number.
(ii) Every whole number is a natural number.
(iii) Every integer is a rational number.
(iv) Every rational number is a whole number.
(v) Every terminating decimal is a rational number.
(vi) 0 is a rational number.

Answer:

(i) True
Natural numbers start from 1 to infinity and whole numbers start from 0 to infinity; hence, every natural number is a whole number.

(ii) False
0 is a whole number but not a natural number, so every whole number is not a natural number.

(iii) True
Every integer can be expressed in the pq form.

(iv) False
Because whole numbers consist only of numbers of the form p1, where p is a positive number. On the other hand, rational numbers are the numbers whose denominator can be anything except 0.

(v) True
Every terminating decimal can be easily expressed in the pq form.

(vi) True
Every terminating decimal can be easily expressed in the pq form.

(vii) True
0 can be expressed in the form pq, so it is a rational number.



Page No 20:

Question 1:

What are irrationl numbers? How do they differ from rational numbers? Give examples.

Answer:

A number that can neither be expressed as a terminating decimal nor be expressed as a repeating decimal is called an irrational number. A rational number, on the other hand, is always a terminating decimal, and if not, it is a repeating decimal.
Examples of irrational numbers:
0.101001000...
0.232332333...

Page No 20:

Question 2:

Classify the following numbers as rational or irrational. Give reasons to support your answer:
(i) 4
(ii) 196
(iii) 21
(iv) 43
(v) 3+3
(vi) 7-2
(vii) 236
(viii) 0.6
(ix) 1.232332333...
(x) 3.040040004....
(xi) 3.2576
(xii) 2.356565656...
(xiii) π
(xiv) 227

Answer:

(i) 4=2 It is a rational number.(ii) 196=14 It is a rational number.(iii) 21=3×7=4.58257...It is an irrational number.(iv) 43If a is a positive integer, which is not a perfect square, thena is an irrational number.Here, 43 is not a perfect square, so it is irrational.(v) 3+3 The sum of a rational number and an irrational number is an irrational number. So, it is an irrational number.(vi) 7-2 The difference of an irrational number and a rational number is an irrational number, so it is an irrational number.(vii) 236The product of a rational number and an irrational number is an irrational number, so it is an irrational number.(viii) .666666 is a rational number because it is a repeating decimal.(ix) 1.232332333... is an irrational number because it is a non-terminating, non-repeating decimal.(x) 3.040040004... is an irrational number because it is a non-terminating, non-repeating decimal.(xi) 3.2576 is a rational number because it is a terminating decimal.(xii) 2.356565656... is a rational number because it is repeating.(xiii) π= 3.14285... is an irrational number because it is a non-terminating, non-repeating decimal.(xiv) 227is a rational number because it can be expressed in the pqform.



Page No 21:

Question 3:

Represent 2, 3 and5on the real line.

Answer:


Let X'OX be a horizontal line taken as the x-axis and O be the origin representing 0.
Take OA = 1 unit and AB ⊥ OA such that AB = 1 unit.
Join OB.
Now,
OB = OA2+ AB2 = 12 + 12 = 2 units
Taking O as the centre and OB as the radius, draw an arc, meeting OX at P.
We have:
OP = OB = 2 units
Thus, point P represents 2 on the number line.
Now, draw BC ⊥ OB such that BC = 1 unit.
Join OC.
We have:
OC = OB2+BC2= 22+12 =3 units
Taking O as the centre and OC as the radius, draw an arc, meeting OX at Q.
We have:
OQ = OC = 3units
Thus, point Q represents 3 on the number line.
Now, draw CD ⊥OC such that CD = 1 unit.
Join OD.
We have:
OD = OC2+OD2 =32+12 = 4= 2 units
Now, draw DE ⊥OD such that DE = 1 unit. 
Join OE.
We have:
OE = OD2+DE2= 22+12=5 units
Taking O as the centre and OE as the radius, draw an arc, meeting OX at R.
We have:
OR = OE = 5 units
Thus, point R represents 5 on the number line.

Page No 21:

Question 4:

Represent 6and7on the real line.

Answer:


Let X'OX be a horizontal line taken as the x-axis and O be the origin representing 0.
Take OA = 2 units and AB ⊥ OA such that AB = 1 unit.
Now, join OB.
We have: OB = OA2+ AB2 = 22 + 12 = 5 units
Taking O as the centre and OB as the radius, draw an arc, meeting OX at P.
Thus, we have:
OP = OB = 5 units
Here, point P represents 5 on the number line.
Now, draw BC ⊥ OB such that BC = 1 unit.
Join OC.
We have: OC = OB2+BC2= 52+12 =6 units
Taking O as the centre and OC as the radius, draw an arc, meeting OX at Q.
Thus, we have:
OQ = OC = 6units
Here, point Q represents 6 on the number line.
Now, draw CD ⊥OC such that CD =1 unit.
Join OD.
We have: OD = OC2+OD2 =62+12 = 7units
Taking O as the centre and OE as the radius, draw an arc, meeting OX at R.
Now,
OR = OD = 7 units
Thus, point R represents 7 on the number line.

Page No 21:

Question 5:

Giving reason in each case, show that each of the following numbers is irrational.
(i) 4+5
(ii) -3+6
(iii) 57
(iv) -38
(v) 25
(vi) 43

Answer:

(i) 4+5 because the sum of a rational number and an irrational number is an irrational number.(ii) -3+6 because the difference of a rational number and an irrational number is an irrational number.(iii) 57 because the product of a rational number and an irrational number is an irrational number.(iv) -38 because the product of a rational number and an irrational number is an irrational number.(v) 25 because the quotient of a rational number and an irrational number is an irrational number.(vi) 43 because the quotient of a rational number and an irrational number is an irrational number.

Page No 21:

Question 6:

State in each case, whether the given statement is true of false.
(i) The sum of two rational numbers is rational.
(ii) The sum of two irrational numbers is irrational.
(iii) The product of two rational numbers is rational.
(iv) The product of two irrational number is irrational.
(v) The sum of a rational number and an irrational number is irrational.
(vi) The product of a nonzero rational number and an irrational number is a rational number.
(vii) Every real number is rational.
(viii) Every real number is either rational or irrational.
(ix) πis irrational and227is rational.

Answer:

(i) True

(ii) False
Example: 2+3+2-3=4Here, 4 is a rational number.

(iii) True

(iv) False
Example: 3×3=3Here, 3 is a rational number.

(v) True

(vi) False
Example: 4×5=45 Here, 45 is an irrational number.

(vii) False 
Real numbers can be divided into rational and irrational numbers.

(viii) True

(ix) True



Page No 25:

Question 1:

Add:
(i) 23-52and3+22
(ii) 22+53-75and33-2+5
(iii) 237-12+611and137+322-11

Answer:

(i) 23-52+3+22=23+3+22-52=33-32(ii) 22+53-75+33-2+5=22-2+53+33+5-75=2+83-65(iii) 237-122+611+137+322-11=237+137-11+611+322-122=7+511+2



Page No 26:

Question 2:

Multiply:
(i) 35 by 25
(ii) 615 by 43
(iii) 26 by 33
(iv) 38 by 32
(v) 10 by 40
(vi) 328 by 27

Answer:

(i) 35×25=3×2×5×5=6×5=30(ii) 615×43=6×4×5×3×3=24×3×5=725(iii) 26×33=2×3×2×3×3=6×3×2=182(iv) 38×32=3×3×2×2×2×2=9×4=36   (v) 10×40=2×5×2×2×2×5=2×2×2×2×5×5=2×2×5=20(vi) 328×27=67×4×7=6×7×4=42×2=84

Page No 26:

Question 3:

Divide:
(i) 166 by 42
(ii) 125 by 43
(iii) 1821 by 67

Answer:

(i) 16642=162342=43(ii) 121543=125×343=35(iii) 182167=187367=33

Page No 26:

Question 4:

Simplify:
(i) 4+24-2
(ii) 5+35-3
(iii) 6-66+6
(iv) 5-22-3
(v) 5-32
(vi) 3-32

Answer:

(i) 4+24-2=42-22            [a+ba-b=a2-b2]=16-2=14    (ii) 5+35-3=52-32=5-3=2(iii) 6-66+6=62-62=36-6=30(iv) 5-22-3=5×2-5×3-2×2+2×3    =10-15-2+6(v) 5-32=52+32-2×53      [a-b2=a2+b2-2ab]=5+3-215=8-215(vi) 3-32=32+32-2×3×3             [a-b2=a2+b2-2ab]=9+3-63=12-63

Page No 26:

Question 5:

Represent 3.2geometrically on the number line.

Answer:

Draw a line segment AB = 3.2 units and extend it to C such that BC = 1 unit.
Now, find the midpoint O of AC.
Taking O as the centre and OA as the radius, draw a semicircle.
Now, draw BD ⊥ AC, intersecting the semicircle at D.
Here,
BD = 3.2 units
Taking B as the centre and BD as the radius, draw an arc meeting AC produced at E. 
Thus, we have:
BE = BD = 3.2 units

Page No 26:

Question 6:

Represent 7.28 geometrically on the number line.

Answer:

Draw a line segment AB = 7.28 units and extend it to C such that BC = 1 unit.
Now, find the midpoint O of AC.
Taking O as the centre and OA as the radius, draw a semicircle.
Now, draw BD ⊥ AC, intersecting the semicircle at D.
We have:
BD = 7.28 units
Now, taking B as the centre and BD as the radius, draw an arc meeting AC produced at E. 
Thus, we have:
BE = BD =7.28 units  

Page No 26:

Question 7:

Mention the closure property, associative law, commutative law, existance of identity, existance of inverse of each real number for each of the operations (i) addition (ii) multiplication on real numbers.

Answer:

ADDITION PROPERTIES OF REAL NUMBERS

(i) Closure property: The sum of two real numbers is always a real number.  
(ii) Associative law: (a + b) + c = a + (b + c) for all real numbers a, b and c.
(iii) Commutative law: a + b = b + a for all real numbers a and b.
(iv) Existence of additive identity: 0 is called the additive identity for real numbers.
      As, for every real number a ,  0 + a = a + 0 = a
(v) Existence of additive inverse: For each real number a, there exists  a real number (-a) such that  a + (-a) = 0 = (-a) + a. Here, a and (-a) are the additive inverse of each other.
 

MULTIPLICATION PROPERTIES OF REAL NUMBERS

(i) Closure property: The product of two real numbers is always a real number.  
(ii) Associative law: (ab)c = a(bc)  for all real numbers a, b and c.
(iii) Commutative law: a ×b = b ×a for all real numbers a and b.
(iv) Existence of multiplicative identity: 1 is called the multiplicative identity for real numbers.
        As, for every real number a ,  1 ×a = a × 1 = a

(v) Existence of multiplicative inverse: For each real number a, there exists  a real number 1a such that  a 1a = 1 = 1aa. Here, a and 1a  are the multiplicative inverse of each other



Page No 30:

Question 1:

Rationalise the denominator of each of the following:
17

Answer:

On multiplying the numerator and denominator of the given number by 7, we get:

 17 = 17×77 = 77

Page No 30:

Question 2:

Rationalise the denominator of each of the following:
523

Answer:

On multiplying the numerator and denominator of the given number by 3, we get:

 523 = 523×33 = 156

Page No 30:

Question 3:

Rationalise the denominator of each of the following:
12+3

Answer:

On multiplying the numerator and denominator of the given number by 2-3, we get:
 12+3 = 12+3×2-32-3 =2-322-32= 2-34-3=2-31 = 2-3

Page No 30:

Question 4:

Rationalise the denominator of each of the following:
15-2

Answer:

On multiplying the numerator and denominator of the given number by 5+2, we get:
 15-2 = 15-2×5+25+2 =5+252-22= 5+25-4=5+21 = 5+2

Page No 30:

Question 5:

Rationalise the denominator of each of the following:
15+32

Answer:

On multiplying the numerator and denominator of the given number by 5-32, we get:
 15+32 = 15+32×5-325-32 =5-3252-322= 5-3225-18=5-327 

Page No 30:

Question 6:

Rationalise the denominator of each of the following:
16-5

Answer:

On multiplying the numerator and denominator of the given number by 6+5, we get:
 16-5 = 16-5×6+56+5 =6+562-52= 6+56-5=6+5 

Page No 30:

Question 7:

Rationalise the denominator of each of the following:
47+3

Answer:

On multiplying the numerator and denominator of the given number by 7-3, we get:
47+3=47+3×7-37-3=47-372-32= 47-37-3= 47-34= 7-3

Page No 30:

Question 8:

Rationalise the denominator of each of the following:
3-13+1

Answer:

On multiplying the numerator and denominator of the given number by 3-1, we get:
 3-13+1 = 3-13+1×3-13-1 =3-1232-12= 3+1-233-1=4-232= 2(2-3)2=2-3

Page No 30:

Question 9:

Rationalise the denominator of each of the following:
3-223+22

Answer:

On multiplying the numerator and denominator of the given number by 3-22, we get:
 3-223+22 = 3-223+22 ×3-223-22 =3-22232-222= 9+8-1229-8=17-122

Page No 30:

Question 10:

Find the values of a and b in each of the following.
3+13-1=a+b3

Answer:

We have:
3+13-1

Now, rationalising the denominator of the given number by multiplying both the numerator and denominator with 3+1, we get:
3+13-1=3+13-1×3+13+1=3+1232-12=3+1+233-1=4+232=22+32= 2+3
∴ 2+3= a + b3
So, on comparing the LHS and the RHS, we get:
a = 2 and b = 1

Page No 30:

Question 11:

Find the values of a and b in each of the following.
3+23-2=a+b2

Answer:

We have:
3+23-2

Now, rationalising the denominator of the given number by multiplying both the numerator and denominator with 3+2, we get:
3+23-2=3+23-2×3+23+2=3+2232-22=9+2+629-2=11+627=117+672

117 + 672= a + b2
So, on comparing the LHS and the RHS, we get:
a117 and b = 67

Page No 30:

Question 12:

Find the values of a and b in each of the following.
5-65+6=a-b6

Answer:

We have:
5-65+6

Now, rationalising the denominator of the given number by multiplying both the numerator and denominator with 5-6, we get:
5-65+6=5-65+6×5-65-6=5-6252-62=25+6-10625-6=31-10619=3119-10196

3119 - 10196 = a -b6
So, on comparing the LHS and the RHS, we get:
a3119 and b = 1019

Page No 30:

Question 13:

Find the values of a and b in each of the following.
5+237+43=a-b3

Answer:

We have :
5+237+43

Now, rationalising the denominator of the given number by multiplying both the numerator and denominator with 7-43, we get:
5+237+43=5+237+43×7-437-43=35+143-203-2472-432=11-6349-48=11-63

11-63 = a - b3
So, on comparing the LHS and the RHS, we get:
a = 11 and b = 6

Page No 30:

Question 14:

Simplify:5-15+1+5+15-1

Answer:

 5-15+1+5+15-1Taking the LCM of both the fractions and adding them, we get:5-12+5+125+15-1= 5+1-25+5+1+2552-12= 125-1= 124 = 3

Page No 30:

Question 15:

Simplify: 4+54-5+4-54+5.

Answer:

4+54-5+4-54+5=4+52+4-524-54+5=42+52+2×4×5+42+52-2×4×516-5=16+5+85+16+5-8516-5=4211



Page No 31:

Question 16:

If x=4-15, find the value of x+1x.

Answer:

We have:x=4-151x=14-15×4+154+15=4+1516-15=4+15

 x+1x=4-15+4+15=8

Page No 31:

Question 17:

If x=2+3, find the value of x2+1x2.

Answer:

We have:x=2+3Now,1x=12+3×2-32+3=2-322-32=2-3 x+1x=2+3+2-3=4 x+1x=4Squaring both sides, we get: x+1x2=42 x2+1x2+2×x×1x=16x2+1x2=14

Page No 31:

Question 18:

Show that 13-8-18-7+17-6-16-5+15-2=5.

Answer:

LHS= 13-8-18-7+17-6-16-5+15-2

=13-8×3+83+8-18-7×8+78+7+17-6×7+67+6-16-5×6+56+5+15-2×5+25+2=3+89-8-8+78-7+7+67-6-6+56-5+5+25-4=3+8-8-7+7+6-6-5+5+2=3+2=5

So, LHS = RHS



Page No 32:

Question 1:

Simplify:
(i) 62/5×63/5
(ii) 31/2×31/3
(iii) 75/6×72/3

Answer:

(i) 625×635= 625+35= 655= 6       (am×an=am+n)(ii) 312×313=312+13=33+26=356(iii) 756×723=756+23=75+46=796=732

Page No 32:

Question 2:

Simplify:
(i) 61/461/5
(ii) 81/282/3
(iii) 56/752/3

Answer:

(i) 614615=614-15=65-420= 6120         aman=am-n(ii) 812823=812-23=83-46=8-16(iii) 567523=567-23=518-1421=5421

Page No 32:

Question 3:

Simplify:
(i) 31/4×51/4
(ii) 25/8×35/8
(iii) 61/2×71/2

Answer:

(i) 314×514=1514                            (am×bm)=abm(ii) 258×358=658(iii) 612×712=4212

Page No 32:

Question 4:

Simplify:
(i) (34)1/4
(ii) (31/3)4
(iii) 134

Answer:

(i)  3414=34×14=3                           (a)mn=amn(ii)  3134=313×4=343(iii)  13412=134×12=132=19=3-2

Page No 32:

Question 5:

Evaluate:
(i) (49)1/2
(ii) (125)1/3
(iii) (64)1/6

Answer:

(i) 4912=7212=72×12=7(ii) 12513=5313=53×13=5(iii) 6416=2616=26×16=2

Page No 32:

Question 6:

Evaluate:
(i) (25)3/2
(ii) (32)2/5
(iii) (81)3/4

Answer:

(i) 2532= 52×32= 53 = 125               (am)n= amn(ii) 3225= 2525= 25×25= 22= 4(iii) 8134= 3434= 34×34= 33= 27

Page No 32:

Question 7:

Evaluate:
(i) (64)−1/2
(ii) (8)−1/3
(iii) (81)−1/4

Answer:

(i) 64-12=82-12=82×-12=8-1=18(ii) 8-13=23-13=23×-13=2-1=12(iii) 81-14=34-14=34×-14=3-1=13



Page No 33:

Question 1:

Which of the following is an irrational number?
(a) 3.14
(b) 3.14
(c) 3.14
(d) 3.141141114...

Answer:

(d) 3.141141114...

Because 3.141141114... is neither a repeating decimal nor a terminating decimal, it is an irrational number.

Page No 33:

Question 2:

Which of the following is an irrational number?
(a) 49
(b) 916
(c) 5
(d) 205

Answer:

(c) 5

Because 5 cannot be expressed in the pq form where p and q are integers (q0), it is an irrational number. The remaining options have a perfect root; they can be expressed in the pq form, so they are rational numbers.

Page No 33:

Question 3:

Which of the following is an irrational number?
(a) 0.32
(b) 0.321
(c) 0.321
(d) 0.3232232223...

Answer:

(d) 0.3232232223...

Because 0.3232232223... is neither a repeating decimal nor a terminating decimal, it is an irrational number.

Page No 33:

Question 4:

Which of the following is a rational number?
(a) 2
(b) 23
(c) 225
(d) 0.1010010001....

Answer:

(c) 225

Because 225 is a square of 15, i.e., 225 = 15, and it can be expressed in the pq form, it is a rational number.

Page No 33:

Question 5:

Every rational number is
(a) a natural number
(b) a whole number
(c) an integer
(d) a real number

Answer:

(d) a real number

Every rational number is a real number, as every rational number can be easily expressed on the real number line.

Page No 33:

Question 6:

Between any two rational numbers there
(a) is no rational number
(b) is exactly one rational numbers
(c) are infinitely many rational numbers
(d) is no irrational number

Answer:

(c) are infinitely many rational numbers

Because the range between any two rational numbers can be easily divided into any number of divisions, there can be an infinite number of rational numbers between any two rational numbers.

Page No 33:

Question 7:

The decimal representation of a rational number is
(a) always terminating
(b) either terminating or repeating
(c) either terminating or non-repeating
(d) neither terminating nor repeating

Answer:

(b) either terminating or repeating

As per the definition of rational numbers, they are either repeating or terminating decimals.

Page No 33:

Question 8:

The decimal representation of an irrational number is
(a) always terminating
(b) either terminating or repeating
(c) either terminating or non-repeating
(d) neither terminating nor repeating

Answer:

(d) neither terminating nor repeating

As per the definition of irrational numbers, these are neither terminating nor repeating decimals.



Page No 34:

Question 9:

Decimal expansion of 2is
(a) a finite decimal
(b) a terminating or repeating decimal
(c) a non-terminating and non-repeating decimal
(d) none of these

Answer:

(c) a non-terminating and non-repeating decimal

Because 2 is an irrational number, its decimal expansion is non-terminating and non-repeating.

Page No 34:

Question 10:

The product of two irrational number is
(a) always irrational
(b) always rational
(c) always an integer
(d) sometimes rational and sometimes irrational

Answer:

(d) sometimes rational and sometimes irrational

For example:
2 is an irrational number, when it is multiplied with itself  it results into 2, which is a rational number.
2  when multiplied with 3, which is also an irrational number, results into 6, which is an irrational number.

Page No 34:

Question 11:

Which of the following is a true statment?
(a) The sum of two irrational numbers is an irrational number
(b) The product of two irrational numbers is an irrational number
(c) Every real number is always rational
(d) Every real number is either rational or irrational

Answer:

(d) Every real number is either rational or irrational.

Because a real number can be further categorised into either a rational number or an irrational number, every real number is either rational or irrational.

Page No 34:

Question 12:

Which of the following is a true statment?
(a) π and 227are both rationals
(b) π and 227are both irrationals
(c) π is rational and 227is irrational
(d) π is irrational and 227is rational

Answer:

(d) π is irrational and 227 is rational.
Because the value of π is neither repeating nor terminating, it is an irrational number. 227, on the other hand, is of the form pq, so it is a rational number.

Page No 34:

Question 13:

A rational number between 2 and 3 is
(a) 122+3
(b) 123-2
(c) 2.5
(d) 1.5

Answer:

(d) 1.5
Because 2 = 1.414... and 3 = 1.732..., a rational number between these two values is 1.5

Page No 34:

Question 14:

(125)−1/3 = ?
(a) 5
(b) −5
(c) 15
(d) -15

Answer:

(c) 15
We have:
125-13 =1 12513 =1 (5)313 = 15

Page No 34:

Question 15:

32+488+12=?
(a) 2
(b) 2
(c) 4
(d 8

Answer:

(b) 2

We have:
32 + 488+12 = 16×2+16×34×2+4×3 = 42+4322+23 =42+322+3 =2



Page No 35:

Question 16:

23×24×3212=?
(a) 2
(b) 2
(c) 22
(d) 42

Answer:

(a) 2

We have:
23×24×3212= 213×214×25112= 213×214×2512= 213+14+512= 24+3+512 = 21=2

Page No 35:

Question 17:

8116-3/4=?
(a) 49
(b) 94
(c) 278
(d) 827

Answer:

(d) 827

8116-34168134=1681143=234143=233=827

Page No 35:

Question 18:

64-24=?
(a) 4
(b) 14
(c) 8
(d) 18

Answer:

(d) 18

We have:
 64-24=26-24= 2-124= 2-3= 123 = 18

Page No 35:

Question 19:

14-3=?
(a) 2+3
(b) 2-3
(c) 1
(d) none of these

Answer:

(a) (2 + 3)
On rationalising the denominator, we get:
14-3= 12-3×2+32+3= 2+322-32= 2+34-3= 2+3

Page No 35:

Question 20:

13+22=?
(a) 3-2217
(b) 3-2213
(c) 3-22
(d) none of these

Answer:

(c) 3-22
On rationalising, we get:
13+22=13+22×3-223-22=3-22(3)2-222=3-229-8=3-22

Page No 35:

Question 21:

If x=7+43, then x+1x=?
(a) 83
(b) 14
(c) 49
(d) 48

Answer:

(b) 14
x = 7+43
Thus, we have:
1x=17+43=17+43×17+43=7-4372-432=7-43
∴ x + 1x = 7+43 + 7-43 = 7+7 = 14

Page No 35:

Question 22:

If 2=1.41, then 12=?
(a) 0.075
(b) 0.75
(c) 0.705
(d) 7.05

Answer:

(c) 0.705
∵2 = 1.41
We have:
12=12×22 = 22 = 1.412  = 0.705

Page No 35:

Question 23:

If 7=2.646, then17=?
(a) 0.375
(b) 0.378
(c) 0.441
(d) none of these

Answer:

(b) 0.378
∵7 = 2.646
We have:
17 = 17×77 = 77 = 2.6467 = 0.378

Page No 35:

Question 24:

10×15=?
(a) 25
(b) 56
(c) 65
(d) none of these

Answer:

(b) 56

10×15 5×2×5×3 =5×2×5×3= 56

Page No 35:

Question 25:

(625)0.16 × (625)0.09 =?
(a) 5
(b) 25
(c) 125
(d) 625.25

Answer:

(a) 5

We have:
6250.16×6250.09=6250.16+0.09=6250.25=62525100=62514=(5)414=5

Page No 35:

Question 26:

If 2=1.414, then 2-12+1=?
(a) 0.207
(b) 2.414
(c) 0.414
(d) 0.621

Answer:

(c) 0.414

On rationalising  2-12+1 by multiplying it by 2-1 both in the numerator and denominator, we get:

 2-12+1×2-12-1= 2-1222-12= 2-12-1 = 2-1 2 = 1.414 2-1 = 1.414 - 1 = 0.414

Page No 35:

Question 27:

The simplest for of 1.6 is
(a) 833500
(b) 85
(c) 53
(d) none of these

Answer:

(c) 53
Let x = 1.6666666...       ...(i)
Multiplying by 10 on both sides, we get:
10x = 16.6666666...       ...(ii)
Subtracting (i) from (ii), we get:
9x = 15
x = 159 =53



Page No 36:

Question 28:

The simplest form of 0.54 is
(a) 2750
(b) 611
(c) 47
(d) none of these

Answer:

(b) 611
Let x = 0.545454...               ...(i)
Multiplying both sides by 100, we get:
100x = 54.5454545...           ...(ii)
Subtracting (i) from (ii), we get:
99x = 540
x5499 = 611

Page No 36:

Question 29:

The simplest form of 0.32is
(a) 1645
(b) 3299
(c) 41333
(d) none of these

Answer:

(c) 2990
Let x = 0.3222222222...          ...(i)
Multiplying by 10 on both sides, we get:
10x = 3.222222222...              ...(ii)
Again, multiplying by 10 on both sides, we get:
100x = 32.222222222...          ...(iii)
On subtracting (ii) from (iii), we get:
90x = 29
x = 2990

Page No 36:

Question 30:

The simplest form of 0.123 is
(a) 41330
(b) 37330
(c) 41333
(d) none of these

Answer:

(d) none of these
Let x = 0.12333333333...         ...(i)
Multiplying by 100 on both sides, we get:
100x = 12.33333333...             ...(ii)
Multiplying by 10 on both sides, we get:
1000x = 123.33333333...         ...(iii)
Subtracting (ii) from (iii), we get:
900x = 111

x = 111900

Page No 36:

Question 31:

An irrational number between 5 and 6 is
(a) 125+6
(b) 5+6
(c) 5×6
(d) none of these

Answer:

(c) 5×6

An irrational number between a and b is given as ab.

Page No 36:

Question 32:

An irrational number between 2 and 3is
(a) 2+3
(b) 2×3
(c) 51/4
(d) 61/4

Answer:

(d) 61/4
An irrational number between 2and 3:2×3=614

Page No 36:

Question 33:

An irrational number between 17and27is
(a) 1217+27
(b) 17×27
(c) 17×27
(d) none of these

Answer:

(c) 17×27

An irrational number between a and b is given as ab.

Page No 36:

Question 34:

Assertion: Three rational numbers between 25 and 35are920, 1020and 1120.
Reason: A rational number between two rational numbers p and q is 12p+q.
(a) Both Assertion and Reason are true and Reasom is a correct explanation of Assertion.
(b) Both Assertion and Reason and Reasom are true but Reasom is not a correct explanation of Assertion.
(c) Assertion is true and Reasom is false.
(d) Assertion is false and Reasom is true.

Answer:

(a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.
Rational number between 25 and 35: 25+352=12=1020Rational number between 25 and 1020: 25+10202=1840=920Rational number between 35 and 1020: 35+10202=2240=1120

So, Assertion and Reason are correct (property of rational numbers). Also, Reason is the correct explanation of Assertion.



Page No 37:

Question 35:

Assertion: 3is an irrational number.
Reason: Square root a positive integer which is not a perfect square is an irrational number.
(a) Both Assertion and Reason are true and Reasom is a correct explanation of Assertion.
(b) Both Assertion and Reason and Reasom are true but Reasom is not a correct explanation of Assertion.
(c) Assertion is true and Reasom is false.
(d) Assertion is false and Reasom is true.

Answer:

(a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.

3is not a perfect square; hence, it is irrational. 3 33 is not a perfect square and hence is irrational and the reason is correct explanation for the assertion thus (a) is correct

Page No 37:

Question 36:

Assertion: e is irrational number.
Reason: π is an irrational number.
(a) Both Assertion and Reason are true and Reasom is a correct explanation of Assertion.
(b) Both Assertion and Reason and Reasom are true but Reasom is not a correct explanation of Assertion.
(c) Assertion is true and Reasom is false.
(d) Assertion is false and Reasom is true.

Answer:

(b) Both Assertion and Reason are true, but Reason is not a correct explanation of Assertion.

It is known that e and π are irrational numbers, but Reason is not the correct explanation.

Page No 37:

Question 37:

Assertion: 3is an irrational number.
Reason: The sum of rational number and an irrational number is an irrational number.
(a) Both Assertion and Reason are true and Reasom is a correct explanation of Assertion.
(b) Both Assertion and Reason and Reasom are true but Reasom is not a correct explanation of Assertion.
(c) Assertion is true and Reasom is false.
(d) Assertion is false and Reasom is true.

Answer:

(b) Both Assertion and Reason are true, but Reason is not a correct explanation of Assertion.
3 is not a perfect square and is irrational.Reason: Let the sum of a rational number a and an irrational number b be a rational number c.Thus, we have: a +b=c b=c-aNow, c-a is rational because both c and a are rational, but b is irrational; thus, we arrive at a contradiction.Hence, the sum of a rational number and an irrational number is an irrational number.Thus, Reason R is not a correct explanation.

Page No 37:

Question 38:

Match the following columns:

Column I Column II
(a) 6.54 is ....... . (p) 14
(b) π is ...... . (q) 6
(c) The length of period of 17=...... . (r) a rational number
(d) If x=2-3, then x2+1x2....... . (s) an irrational number
(a) ........
(b) ........
(c) ........
(d) ........

Answer:

(a) Because it is a non-terminating and repeating decimal, it is a rational number.

(b) π is an irrational number.

(c) 17=.142857142857...
Hence, its period is 6.
                                                     
(d)
x2+1x2=2-32+12-32=22+33-2×2×3+122+33-2×2×3=4+3-4×3+14+3-4×3=7-4×3+17-4×3=7-4×327-4×3+17-4×3=72+432-2×7×43+17-4×3=49+48-563+17-4×3=98 -5637-4×3=14×7-4×37-4×3=14

Page No 37:

Question 39:

Match the following columns:

Column I Column II
(a) 81-24=...... . (p) 4
(b) If abx-2=bax-4, then x = ........ . (q) 29
(c) If x=9+45, then x-1x= ...... . (r) 19
(d) 8116-3/4×6427-1/3=? (s) 3

(a) ......
(b) ......
(c) ......
(d) ......

Answer:

(a)
81-214=9-414=9-4×14=9-1=19     

(b)

abx-2=bax-4ba2-x=bax-42-x=x-42x=6  x=3

(c)

x=9+45and 1x=19+45×9-459-45=9-4581-80=9-45
Now,x+1x=9+45+9-45=18Thus, we have:x+1x=18We know:x-1x2=x+1x-2×x×1xx-1x2=18-2x-1x2=16Taking the square root of both sides, we get:x-1x=4

(d)

324×-34×433×-13=32-3×43-1=32-3×34=3-32-3×322=3-3×32-3×22=3-22-1=29                                                                                   



Page No 38:

Question 40:

Give an example of two irrational numbers whose sum as well as the product is rational.

Answer:

Consider 2+3 and 2-3.Their sum:2+3+2-3=4It is a rational number.Their product: 2+3×2-3=22-32=4-3=1It is also a rational number.

Page No 38:

Question 41:

If x is rational and y is irrational, then show that (x + y) is always irrational.

Answer:

Let x be a rational number and y be an irrational number.  Now, let (x+y) be a rational number c.So, we have:x+y=cy=c-xNow, c-x is a rational number because both x and c are rational, but y is irrational; thus, we arrive at a contradiction.Hence, the sum of a rational number and an irrational number is an irrational number.

Page No 38:

Question 42:

Is the product of a rational and irrational number always irrational? Give an example.

Answer:

No.Let x=0 and y=5 x is rational and y is irrational.Their product, xy=0×5=0 The product obtained is rational.

Page No 38:

Question 43:

Give an example of a number x such that x2 is an irrational number and x4 is a rational number.

Answer:

Let x =514  x2=5142=514×2=5125 is irrational. And, x4=51445 is rational.

Page No 38:

Question 44:

The number 4.17expressed as a vulgar fraction is
(a) 417100
(b) 41799
(c) 41399
(d) 41390

Answer:

(c)41399
Let x = 4.171717171...  ...(i)
Multiplying by 100 on both sides, we get:
100x = 417.171717...    ...(ii)
Subtracting (i) from (ii), we get:
99x = 413
x = 41399

Page No 38:

Question 45:

If x=2+3, find the value of x2+1x22.

Answer:

x=2+3Now, 1x=12+3×2-32-3=2-322-32=2-3So,x+1x=2+3+2-3=4 x+1x=4Squaring both sides, we get:x2+1x2+2=16x2+1x2=14Then, again squaring both sides, we get:x2+1x22=196

Page No 38:

Question 46:

If 3-13+1=a-b3, find the values of a and b.

Answer:

3-13+1=3-13+1×3-13-1=3-123-1=32-2×1×3+12=4-232=2-3We know:a-b3=2-3 a=2 and b=1

Page No 38:

Question 47:

If 4+54-5=a+b5, find the values of a and b.

Answer:

4+54-5=4+54-5×4+54+5=4+5216-5=16+2×4×5+511=21+8511We know:a+b5 =21+8511a+b5 =2111+8511Thus, we have:a=2111 and b=811

Page No 38:

Question 48:

If5-15+1+5+15-1=a+b5, find the values of a and b.

Answer:

We have:5-15+1+5+15-1=5-12+5+125+15-1=5+1-25+5+1+2552-1=124=3We know:a+b5=3Thus, we have:a=3 and b=0

Page No 38:

Question 49:

If 2+332-23=a+b6, find the values of a and b.

Answer:

2+332-23=2+332-23×32+2332+23=6+26+36+618-12=12+566
=2+566We know:a+b6 = 2+566 a = 2 and  b =56

Page No 38:

Question 50:

If x=3+23-2and y=3-23+2, find (x2 + y2).

Answer:

x =3+23-2×3+23+2=3+223-2=3+2+223=5+26Now,y =3-23+2×3-23-2=3-223-2=3+2-223=5-26Thus, we have:xy=5+265-26=25-24=1x+y=5+26+5-26x+y=10Squaring both sides, we get:x+y2=102x2+y2+2xy=100x2+y2+2×1=100x2+y2=100-2=98



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Question 51:

If x=12 -3, show that the value of (x3 − 2x2 − 7x + 5) is 3.

Answer:

x=12-3By rationalising, we get:x=12-3×2+32+3x=2+34-3x=2+3x-2=3Squaring both sides, we get:x-22=3x2+4-4x=3x2-4x+1=0We can write x3-2x2-7x+5 as x3-4x2+x+2x2-8x+2+3.Thus, we have:x3-2x2-7x+5=x3-4x2+x+2x2-8x+2+3=xx2-4x+1+2x2-4x+1+3=x0+20+3=3

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Question 52:

If x=3+8, show that x2+1x2=34.

Answer:

x=3+81x=13+8×3-83-8=3-89-8=3-8
And,x+1x=3+8+3-8=6Thus, we have:x+1x=6Squaring both sides, we get:x+1x2=62x2+1x2+2=36x2+1x2=36-2x2+1x2=34

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Question 53:

If x=2+3, show that x3+1x3=52.

Answer:

x=2+3Thus, we have:1x=12+3×2-32-3=2-34-3=2-3Now,x+1x=2+3+2-3=4x+1x3=43x3+1x3+3×x×1x×x+1x=64x3+1x3+3×4  =64     x+1x=4x3+1x3+12 =64x3+1x3=64-12x3+1x3=52

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Question 54:

If x=3-22, show that x-1x=±2.

Answer:

We have:x=3-22Now,1x=13-22×3+223+22=3+2232-222=3+22
x+1x=3-22+3+22=6We have:x-1x2=x+1x-2x-1x2=6-2x-1x2=4
Taking the square root of both sides, we get:x-1x=±2

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Question 55:

If x=5+26, show that x+1x=±23.

Answer:

We have:x=5+261x=15+26×5-265-26=5-2652-262=5-26
Now, x+1x=5+26+5-26=10Thus, we have:x+1x2=x+1x+2x+1x2=10+2x+1x2=12
Taking the square root of both sides, we get:x+1x=±23



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Question 1:

Find two rational numbers lying between 13and12.

Answer:

Because13is less than 12, let x=13, y=12 and n= 2.Rational numbers between x and y are x+d and x+2d.Here,d=y-xn+1=12-132+1=16×3=118Required numbers: x+d=13+118=6+118=718 and x+2d=13+218=49

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Question 2:

Find four rational numbers between 35and45.

Answer:

Let:x=35, y=45 and n=4Thus, we have:d=y-xn+1=45-354+1=15×5=125The four rational numbers are x+d, x+2d, x+3d and x+4d.35+125, 35+225, 35+325 and 35+4251625, 1725, 1825 and 1925

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Question 3:

Write four irrational numbers between 0.1 and 0.2.

Answer:

Non-repeating and non-terminating decimals are irrational numbers; thus, any such numbers lying between 0.1 and 0.2 will be irrational.
These can be 0.125986634275... ,0.165323296581422..., 0.121221222212226... and 0.13133133331333311333...

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Question 4:

Express 12504in its simplest form.

Answer:

125014=2×62514=214×62514=214×5414                 amn=amn=214×54×14=5×24 

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Question 5:

Express 23. 18as a pure surd.

Answer:

23×18=232×3×3=23×2×9=23×2×3=22=2×2×2=8



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Question 6:

Divide 1675by512.

Answer:

1675512=16×5×5×352×2×3=16×25×35×4×3=16×5×35×2×3=8

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Question 7:

Express 0.123 as a rational number in the form pq, where p and q are integers and q ≠ 0.

Answer:

Let x=0.1232323           ...(i) 10x=1.23232323     ...(ii)Also,1000x=123.232323      ...(iii)Subtracting (ii) from (iii), we get:990x=122x=122990=61495

Thus, we have:
0.123=61495

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Question 8:

If 632-23=a2+b3, find the values of a and b.

Answer:

We have:632-23=a2+b3632-23×32+2332+23=6×32+23322-232=182+12318-12=6×32+236
32+23=a2+b3Thus, we have: a=3 and b = 2

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Question 9:

The simplest form of 64729-1/6is
(a) 23
(b) 32
(c) 43
(d) 34

Answer:

(b) 32
64729-16=7296416We know:26=64 and 36=729Thus, we have: 7296416=36×1626×16=32

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Question 10:

Which of the following is irrational?
(a) 0.14
(b) 0.1416
(c) 0.1416
(d) 0.1401401400014.....

Answer:

(d) 0.1401401400014...

Since, it is non terminating and non repeating
Therefore,  number is irrational.

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Question 11:

Between two rational numbers
(a) there is no rational number
(b) there is exactly one rational number
(c) there are infinitely many irrational numbers
(d) there is no irrational number

Answer:

(c) there are infinitely many irrational numbers

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Question 12:

Decimals representation of an irrational number is
(a) always a terminating decimal
(b) either a terminating or a repeating decimal
(c) either a terminating or a non-repeating decimal
(d) always non-terminating and non-repeating decimal

Answer:

(d) always a non-terminating and non-repeating decimal

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Question 13:

If x=7+52, then x2+1x2=?
(a) 160
(b) 198
(c) 189
(d) 156

Answer:

(b) 198

We have:x= 7+52Now,1x=17+52×7-527-52=7-5249-50=-7-52=52-7And, x+1x=7+52+52-7=102We know:x+1x2=x2+1x2+21022=x2+1x2+2x2+1x2=200-2=198

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Question 14:

Rationalise the denominator of 53-4243+32.

Answer:

53-4243+32=53-4243+32×43-3243-32=53-42×43-32432-322=53×43-53×32-42×43+42×32432-322=20×3-156-166+12×216×3-9×2=60-316+2448-18=84-31630

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Question 15:

Simplify: 127-1/3+1625-1/4.

Answer:

127-13+1625-14=2713+62514=3313+5414=3+5=8

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Question 16:

Find the smallest of the number 63, 246 and 84.

Answer:

We can write the given numbers in the following ways:

63=613246=241684=814

The powers of the given numbers are 13, 16 and 14.
The LCM of the denominators of these powers is 12.

Because on raising all three numbers by the same power, the smallest number will still remain the smallest, raising each number to the power of 12, we get:
613×12=64=12962416×12=242=576814×12=83=512

Of these three, 512 is the smallest number, which has been derived from 84.

Hence, 84 is the smallest of all three numbers.



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Question 17:

Match the following columns:

Column I Column II
(a) π is ...... . (p) a rational number
(b) 3.1416 is ...... . (q) an irrational number
(c) 0.23 = ....... . (r) 730
(d) 0.23 = ...... . (s) 2399

(a) ......
(b) ......
(c) ......
(d) ......

Answer:

(a) π is an irrational number.

(b) 3.141614161416... is a rational number because it is a non-terminating but repeating decimal.

(c) Let x = 0.23232323...    ...(i)
100x = 23.232323...       ...(ii)
Subtracting (i) from (ii), we get:
99x = 23
Or,
x = 2399

(d) Let x = 0.233333333..     ...(i)
10x = 2.33333333..          ...(ii)
We have:
100x = 23.333333333          ...(iii)
Subtracting (ii) from (iii), we get:
90x = 21
Or,
x = 2190=730

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Question 18:

If x=5+35-3 and y=5-35+3, find the value of (x2 + y2).

Answer:

x=5+35-3 and y=5-35+3x2=5+35-32=5+3+2155+3-215=8+2158-215y2=5-35+32=5+3-2155+3+215=8-2158+215
Now, x2+y2=8+2158-215+8-2158+215=8+2152+8-21528-2158+215=82+2152+2×8×215+82+2152-2×8×21582-2152=64+60+3215+64+60-321564-60=2484=62

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Question 19:

If 2=1.41 and 5=2.24, find the value of 382+55+282-55.

Answer:

382+55+282-55=382+55×82-5582-55+282-55×82+5582+55=3×82-5564×2-25×5+2×82+5564×2-25×5=3×82-553+2×82+553=82-55+2×82+553=82+2382+2355-55=821+23+5523-1=8253+55-13=4031.41-532.24=18.8-3.73=15.07

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Question 20:

Prove that 8116-3/4×259-3/2÷52-3=1.

Answer:

8116-34×259-32÷52-3=168134×92532×52--3=234×34×352×32×523=233×353×523=23×33×5333×53×23=1



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