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Number-System

Most important question

Q1. 

Solution

Q2. When 15 is divided by , the Quotient is:

• 1) 
• 2) 
• 3) 
• 4) ₅

Solution

Q3. If x = 2 + , then eQuals to:

• 1) 2
• 2) 
• 3) 
• 4) 4

Solution

Q4. If x = 3 + 2, find the value of .

Solution

Q5. If x =  and y =  then find the value of x + y.

Solution

 Therefore, x + y = 9 + 

Q6. 

Solution

Q7. Express  with rational denominator.

Solution

Q8. Represent the irrational number  on the number line.

Solution

Q9. Two rational numbers between are:

• 1) 
• 2) 
• 3) 
• 4) 

Solution

  _{Therefore, two possible rational numbers are:} 

Q10. Find the value of: 

Solution

  

Q11. Which of the following number is an irrational number?

• 1) 
• 2) 
• 3) 
• 4) 

Solution

 _{is an irrational number.}

Q12. An irrational number between is:

• 1) 0.75
• 2) 0.70
• 3) 0.750750075000…
• 4) 0.7512

Solution

Let,   We observe that in the first decimal place a has digit 7 and b has digit 7. In the second decimal place a has digit 1 and b has digit 7. So, irrational between them is 0.750750075000…

Q13. 

Solution

The commutative and associative do not hold for subtraction or division as: let a= 2, b = 4, c = 6 a - b is not eQual to b - a :  a - b =2 - 4 = -2 ,  b-a = 4 - 2 = 2 a  b is not eQual to b  a :  a- (b-c) is not eQual to (a-b) - c : a -(b-c) = 2 -(4 -6)= 4  ;   (a-b) - c = (2-4) - 6 = -8  is not eQual to  :  a (b c) = 3 ;  (ab) c =

Q14. 

Solution

Q15. Simplify the following by rationalizing the denominators. 

Solution

Q16. when simplified is:

• 1) Positive and rational
• 2) Negative and irrational
• 3) Negative and rational
• 4) Positive and irrational

Solution

 Positive and rational

Q17. Which of the following is rational number?

• 1) 
• 2) 0
• 3) 1 + 
• 4) 2 

Solution

0 is a rational number.

Q18. Express  in the form of  where p and Q are integers and Q  0.

Solution

Let x =  So x = 3.424242... Multiply by 100, we get 100x = 342.4242... Subtracting, we get 99x = 339 

Q19. Add .

• 1) 
• 2) 
• 3) 
• 4) 

Solution

Q20. Write the following in the ascending order of their magnitude. 

Solution

 _{Converting the powers with powers of same denominator, we get: In ascending order, numbers are:}  ie = 

Q21. Simplify the following into a fraction with rational denominator. 

Solution

Q22. The product or Quotient of a non-zero rational number with an irrational number is:

• 1) Natural number
• 2) Rational number
• 3) Whole number
• 4) Irrational number

Solution

The product or Quotient of a non-zero rational number with an irrational number is irrational. Example: 

Q23. The value of  is:

• 1) 2
• 2) 0
• 3) 1
• 4) -1

Solution

Q24. Which of the following is an irrational number?

• 1) 
• 2) 
• 3) 0.3796
• 4) 

Solution

 _{is an irrational number.}

Q25. Convert  in the form of decimals.

• 1) 0.35
• 2) 0.27
• 3) 6.26
• 4) 0.28

Solution

Q26. 

Solution

Q27. Simplified value of  is:

• 1) 5
• 2) 3
• 3) 25
• 4) 1

Solution

Q28. Prove that 

Solution

 =  =  =  =  = RHS

Q29. Simplify: 

• 1) 
• 2) 
• 3) 
• 4) 

Solution

Q30. Simplify 

Solution

Q31. Simplified value of  is:

• 1) 4
• 2) 0
• 3) 16
• 4) 1

Solution

Q32.   

Solution

Q33. Simplify 

Solution

Q34. If x = 3 +  then 

Solution

x = 3 +  = 1 + 2 +  x =     = -  

Q35. Which of the following is the value of 

• 1) 
• 2) -4
• 3) 4
• 4) 

Solution

Q36. 

Solution

Q37. Find three rational numbers between . How many rational numbers can be determined lying between these numbers?

Solution

Here  So three numbers are:  There are infinite rational numbers lying between any two rational numbers.

Q38. Prove that 

Solution

Q39. Show that OC =  

Solution

OA = 2 units on the number line. AB is the of unit length perpendicular to OA. OB is the hypotenuse of the right triangle OAB   

Q40. Write in decimal form and say what kind of decimal expansion it has?

Solution

  is a non-terminating recurring decimal.

Q41. Write the following in the descending order of their magnitude. 

Solution

Q42. Evaluate

Solution

Q43. If , find the value of .

Solution

Q44. Express 5. in the form of , where p and Q are integers and Q  0.

Solution

Let  …..(i) Multiplying both sides by 10,  …(ii) Sub (i) from (ii) 

Q45. 

Solution

Q46. Find four rational numbers between.

Solution

We have to find four rational numbers between. Multiplying numerator and denominator of the rational numbers by 5 we get,  Thus, four rational numbers betweenare: 

Q47. Find the value of 

Solution

Q48. Simplify: 

Solution

Q49. The value of  is:

• 1) 1
• 2) 0
• 3) 
• 4) 

Solution

= 

Q50. Find 4 rational numbers between 

Solution

LCM of 3 and 5 is 15.  _{Rational numbers between}  Any four rational numbers between 

Q51. 

Solution

Q52. The simplest rationalization factor of is:

• 1) 
• 2) 
• 3) 50
• 4) 

Solution

since =

Q53. The number  expressed as a vulgar fraction is

• 1) 
• 2) 
• 3) 
• 4) 

Solution

Let x =  ….(i) 100x =  ….(ii) Subtracting (i) from (i) ∴ 99x = 413 ∴ x = 

Q54. Simplify the following by rationalizing the denominator. 

Solution

 =  =  = = 

Q55. If = 1.732 and = 1.414, the value of is:

• 1) 0.318
• 2) 
• 3) 3.146
• 4) 

Solution

Q56. Simplify the following by rationalizing the denominator. .

Solution

                                                                                     

Q57.  is:

• 1) 
• 2) 
• 3) 
• 4) 

Solution

Q58. Express  in the form  where p and Q are integers and Q ≠ 0.

Solution

Let x = … (i) Multiply eQ. (i) by 10, we get 10 x = 157.1212    

Q59. if x = 3 - 2, find the value of  - 

Solution

Q60. Visualise the representation of  on the number line upto 4 decimal places i.e. 4.6777.

Solution

Q61. If (5)^{x - 3}  (3)^{2x - 8} = 225 then find the value of x

Solution

Q62. A number is irrational if and only if its decimal representation is:

• 1) non- terminating
• 2) non- terminating and repeating
• 3) non- terminating and non-repeating
• 4) Terminating

Solution

A number is irrational if and only if its decimal representation is: non- terminating and non-repeating

Q63. 

Solution

Q64. 

Solution

Q65. Value of  is

• 1) -3
• 2) 9
• 3) 
• 4) 

Solution

Q66. 

Solution

 By comparing, we get a = 5 and b = 2.

Q67. 

Solution

Q68. Find the value of x if 2⁴ x 2⁵ = (2³)^x

Solution

  2⁹ = 2^3x 9 = 3x x = 3

Q69. If x =  find the value of x² + xy + y².

Solution

X =  =  = X² =  =  y =  =  y² = xy =  = 1 x² + xy + y² =  =  = 

Q70. Represent  on the number line.

Solution

Q71. Simplify: 

Solution

 

Q72. Represent  on the number line:

Solution

 BE = BD = 1. Draw a line segment AB = 4.2 unit and BC = 1 unit 2. From midpoint O of AC and OA =r draw semi circle 3. Draw  _{4. Draw} 

Q73. Simplify: 

Solution

  =2 + 1 + 2³ =2 + 1 + 8 = 11

Q74. If x = 3+2, check whether  is rational or irrational.

Solution

   rational.

Q75. Express  as a rational number in the form where p and Q are integers and Q 0.

Solution

Let x = _{…(1) Multiply by 1000, we get} 1000x = _{…(2) Subtracting (1) from (2), we get 999x = 1 Or, x =} 

Q76. Simplify.

Solution

  

Q77. Express.

Solution

 x = 0.001001001…… ------------- (i) Multiplying by 1000 1000x = 1000(0.001001001…) 1000x = 1.001001001… ------------- (ii) Subtracting (i) from (ii) 999x = 1

Q78. Write  in decimal form and say what kind of decimal expansion it has?

Solution

  Thus, decimal expansion of  is non terminating repeating.

Q79. Show that .

Solution

Q80. Find an irrational number between  and 

Solution

We can see that    = , and we can easily calculate   To find an irrational between and, we find a number which is non-terminating non-recurring lying between them. There are infinitely many such numbers. An example of such a number is 0.150150015000150000...

Q81. Express  in the form of , where p and Q are integers. and Q 0.

Solution

x =  x = 3.575757... 100x = 357.5757... Subtracting, we get 99x = 354 

Q82. Express in the form  where p and Q are integers Q 0

Solution

Q83. Represent  on the number line.

Solution

Q84. Simplify: 

Solution

3  

Q85. Represent  on the number line.

Solution

Using Pythagoras Theorem: 5=2²+1² Taking positive sQuare root we get  (i) We mark a point 'A' representing 2 units on number line. (ii) Now we construct AB of unit length perpendicular to OA. Then we join OB (iii) Now taking O as centre and OB as radius we draw an arc, intersecting number line at point C. (iv) Point C represents  on number line.

Q86. Express  as a rational number in the form , where p and Q are integers, and Q  0.

Solution

Let x = 0.4777 ... Multiply by 10, we get 10x = 4.777... … (I) Again multiply by 10, we get 100x = 47.77... … (II) Subtracting (II) from (I), we get 90x = 43 x = 

Q87. If a = 2, b = 3 then find the values of the following: (i) (a^b + b^a)^-1 (ii) (a^a + b^b)^-1

Solution

(i) (a^b + b^a)^-1 = (2³ + 3²)^-1 = (8 + 9)^-1 =17^-1 =  _(ii) (a^a + b^b)^-1 = (2² + 3³)^-1 = (4 + 27)^-1 = 31^-1 = 

Q88. Simplify  and express the result in the exponential form of x.

Solution

 =  =  = 

Q89. Simplify the following: 

Solution

Q90. If xyz = 1, than show that 

Solution

Q91. Express  with rational denominator.

Solution

Q92. Simplify : 

Solution

Q93. Simplify the following by rationalizing the denominators: 

Solution

  

Q94. Evaluate : , given that  = 3.162.

Solution

Q95. Simplify : .

Solution

 =  =  = 

Q96. Prove that  is not a rational number on the basis of decimal representation.

Solution

Q97. Simplify: 

Solution

 =  ₌  = 

Q98. If x = 5 - 2 then find the value of x² + .

Solution

x = 5 - 2 

Q99. Simplify:

Solution

  ₌  

Q100. 

Solution

Q101. Simplify: 

Solution

 =    

Q102. Represent  on the number line.

Solution

 Here, we first draw OA=1 Unit. And AB=1 Unit perpendicular to OA. Then, by Pythagoras theorem, in  Using a compass with centre O and radius OB, Draw an arc intersecting the number line at point P. Then, P corresponds to _.

Q103. Represent  on the number line.

Solution

Q104. Simplify : 

Solution

Q105. Prove that 

Solution

Q106. Find the value of 'p' if = 225.

Solution

   (Comparing powers) p =5

Q107. Express 2.02 in the form of where p and Q are integers and Q 0.

Solution

If x = 2. 02 Multiply by 100, we get 100x = 202. _{… (1) Again, multiply by 10, we get} 1000x = 2025. _{… (2) Subtracting (1) from (2), we get} 900x = 1823 Or, x = 

Q108. Find the value of a and b if  = a - b.

Solution

Q109. 

Solution