Maths NCERT Class 8 EXPONENTS AND POWERS 104 MCQs with solutions

Maths NCERT Class 8

EXPONENTS AND POWERS 104 MCQs

EXPONENTS AND POWERS

1. The value of  a° is 

(a) 0

(b) 1

(c) 2

(d) a 

Answer:  (b) 1

Solution: 

(any number)°=1

(i.e.)  a°=1

2. The exponent of  aman is 

(a) amn

(b) am+n

(c) am-n

(d) amn

Answer:  (b) am+n

Solution: 

For any non-zero integer a as bases and exponents as m, n then  

aman= am+n

3. The exponent form of -2-5-2-6 is 

(a) 1-210

(b)  1-212

(c)  1-211

             (d)  1-28

Answer: (c)  1-211 

  Solution:

-2-5-2-6= -2-5+(-6)

                                       = -2-11=1-211

4. The exponent of 10410-12 is 

(a) 1106 

(b) 1104

(c) 1108

(d) 1105 

Answer: (c) 1108

Solution: 

10410-12=104+-12=104-12=10-8=1108

5. The exponential of 343-638 is

(a) 35

(b) 36

(c) 34

(d) 37

Answer: (b) 36

Solution: 

343-638=34+-6+8=36 

                                    =312-6=36          

6. For any non-zero integer a as aman is

(a)  am+n

(b) an-m

(c) am-n

(d) amn

Answer: (c) am-n

Solution: 

For any non-zero integer a as base and aman=ama-n=am+-n=am-n

7. The value of 4-2 is

(a) 818 

(b) 116

(c) 110

(d) 16

Answer: (b) 116

Solution: 

4-2=142=116

8. The simplified form of -5-4 is

(a) 625 

(b) 1625

(c) 1525

(d) 1425

Answer: (b) 1625

Solution: 

-5-4=1-54=1-5-5-5-5=1625

9. The value of 1n is

(a) n 

(b) 0

(c) 1

(d) -1n

Answer: (c) 1

Solution: 

11=12=13=1

1-1=1-2=1-3=1

               In general  1n=1 for any integer ‘n’ 

10. 12-5 has the simplified form as

(a) 30 

(b) 34

(c) 32

(d) 36

Answer: (c) 32

Solution: 

            12-5=1125=11212121212=1132=32                              

11. The correct form of amn is

(a) am+n 

(b) amn

(c) am-n

(d) amn

Answer: (b) amn

Solution: 

       For any non-zero integer ‘a’ as base and exponent as m and n  then

        amn=amn            

12. The exponent of ambm is

(a) am+n

(b) abm

(c) abm-n

(d) abmn

Answer: (b) abm

Solution: 

             For any non-zero integers a and b as bases and exponent as m and n  then

        ambn=abm             

13. The value of -54-56 is

(a) 125 

(b) 120

(c) 110

(d) 115

Answer: (a) 125

Solution: 

-54-56

=-54-56 

=-54-5-6

=-54+-6

=-5-2

=1-5-2=1-5-5=125

14. The result of 1234 in power notation with positive exponent is

(a) 1210 

(b) 1212

(c) 128

(d) 1214

Answer: (b) 1212

Solution: 

1234=1234=123×4=1212=1212 ∵amn=amn  

∵ 13=1

∴123=1323=123

15. The result of -32532 in power natation with positive exponent is

(a) 54 

(b) 52

(c) 56

(d) 53

Answer: (b) 52

Solution: 

-32532=-3532 ∵ ambm=abm

=-1×52=-1252

=1×52=52

16. The power natation with positive natation exponent of 4-64-84-5 is

(a) 142 

(b) 143

(c) 145

(d) 144 

Answer: (b) 143

Solution: 

4-64-84-5=4-64-84-5

        =4-6484-5 ∵ aman=am+n

        =4-6+84-5

                      =424-5

        =42+-5

        =4-3=143

17. The result of 3-3-6-3 in power notation with positive exponents is

(a) 1183 

(b) 1186

(c) 1-183

(d) 1-186

Answer: (c) 1-183

Solution: 

3-3-6-3=3×-6-3 ∵ ambm=abm

    =-18-3

    =1-183 

18. The exponent form of ambm is

(a) ab-m 

(b) abm

(c) ba2m

(d) ba-2m

Answer: (b) abm

Solution: 

  ambm=ab-m for any non-zero integers a and b and m as exponents  

19. The value of 40+9732 is

(a) 8 

(b) 10

(c) 6

(d) 12

Answer: (b) 10

Solution: 

40+9732=1+1932

=9+1932=109×9=109+19×9

= 10

20. The simplified form  3-1+9-13-2 is

(a) 12 

(b) 15

(c) 13

(d) 14

Answer: (c) 13

Solution: 

3-1+9-13-2=3×9-13-2

        =3132-13-2 

        =33-13-2 ∵ ambm=abm

        =3-33-2 

        =3-33-2=3-3 32=3-3+2=3-1=13

21. The value of 13-2+14-2+15-2 is

(a) 40 

(b) 50

(c) 70 

(d) 45

Answer: (b) 50

Solution: 

13-2+14-2+15-2=9+16+25=50

∴13-2=1132=119=9

  14-2=1142=1116=16

  15-2=1152=1125=25

22. The simplified form of 2-1+3-1+6-10 is

(a) 1 

(b) 3

(c) 2

(d) 4

Answer: (a) 1 

Solution: 

2-1+3-1+6-10=12+13+160 ∵ x-m =1xm 

          =3+2+160

=660=10=1 ∵ a0=1

23. The value of -35-32 is

(a) 16625726 

(b) 16625729

(c) 16624726

(d) 16624723

Answer: (b) 16625729

Solution: 

-35-32=-35-3×2 ∵ amn=amn

      =35-6

      =-3-65-6 ∵ abm=ambm

      =56-36

      =5×5×5×5×5×5-3-3-3-3-3-3

      =15625759

24. The simplest of 8-1522-3 is

(a) 20 

(b) 40

(c) 30 

(d) 50

Answer: (d) 50

Solution: 

8-1522-3=8-1×5×52-4=1824×5×5

            =18×2×2×2×2×25=50

            = 50

25. The value of 4-13-15-1 is

(a) 160 

(b) 140

(c) 150

(d) 130 

Answer: (a) 160

Solution: 

4-13-15-1=4×3-15-1 ∵ ambm=abm 

=12-15-1

=12×5-1

=60-1

=160 ∵ a-m=1am

26. If the 6x6-4=65 then the value of x is

(a) 2 

(b) 1

(c) 4

(d) 6

Answer: (b) 1

Solution: 

6x6-4=65

6x6-4=65

6x 64=65

6x+4=65 

The exponents are equal since the bases are equal

∴    x+4=5      ⇒x=5-4

(i.e.) x=1 ∴  x=1

27. The simplification of 15-116-1-1 is

(a) -1 

(b) 1

(c) 2

(d) 3

Answer: (a) -1 

Solution: 

15-116-1-1=115-115-1 ∵ a-m=1am

=5-6-1

=-1-1

=-1

28. The value of 35-553-3 is

(a) 235

(b)  214

(c) 259

(d) 223

Answer: (c) 259

Solution: 

35-553-3=35-5353 ∵ a-m=1am

=35-5+3 53-3=153-3=353 

=35-2

=53-2

=5232 ∵ abm=ambm

=5×53×3=259

29. The rational number of 32-3 is

(a) 523 

(b) 827

(c) 625

(d) 324

Answer: (b) 827

Solution: 

32-3=1323=13323=2333=2×2×23×3×3=827

∵ a-m=1am,  ambm=abm 

30. If -25-3 is the given form then the rational number is

(a) -1258 

(b) 12516

(c) 12315

(d) 12512

Answer: (a) -1258

Solution: 

-25-3=1-253=1-2353=53-23=5×5×5-2×-2×-2

=-12516=-1258

=12516=-1258

31. The rational form of -23-1-352 is

(a) 2534 

(b) 2712

(c) 2750

(d) 2516

Answer: (c) 2750

Solution: 

-23-1-352=1-231-352 

  =321-352  

  =3121-3252 ∵ abm=ambm

=32-3-35×5

=2750

32. The power of a rational number with positive exponent of 5-25-4 is

(a) 153 

(b) 156

(c) 154

(d) 152

Answer: (b) 156

Solution: 

5-25-4=152154 ∵ a-m=1am

      =1×15254

      = 152+4 ∵ aman=am+n

=156=1656=156 ∵ abm=ambm

33. The positive exponent of -12-2-12-3 is

(a) 25 

(b) 23

(c) -25

(d) -32

Answer: (c) -25

Solution: 

  -12-2-12-3=-12-2+-3

=-12-5

=1-125

=1-1525=25-15=25-1=-25

34. The value of 3-14-12-45-1 is

(a) 209 

(b) -209

(c) 103

(d) -103

Answer: (b) -209

Solution: 

3-14-12-45-1=131421-45

  =134125-4

  =4325-4 

  =42325-4

  =49-51

  =-209

35. The simplified form of 3-1-4-1-1+2-1-3-1-1 is

(a) 20 

(b) 16

(c) 18

(d) 14

Answer: (c) 18

Solution: 

3-1-4-1-1+2-1-3-1-1=13-14-1+12-13-1     ∵a-m=1am

=4-312-1+3-26-1

=112-1+16-1

=1112+116

=12+6=18

36. The value of 3-1-4-1-16-1 is

(a) 74 

(b) 72

(c) 76 

(d) 70

Answer: (b) 72

Solution: 

3-1-4-1-16-1=13-14-116 

=112-161

=111261 ∵a-m=1an 

=12×6=72

37. The simplified value of 3-1+6-123-1 is

(a) 13 

(b) 12

(c) 14

(d) 15

Answer: (a) 13

Solution: 

   3-1+6-123-1=13+161231

=2+1632

=3623=13

38. The value of 4-1+23-1-1 is

(a) 45 

(b) 47

(c) 49

(d) 411

Answer: (b) 47

Solution: 

4-1+23-1-1=14+123-1

    =14+32-1

    =1+64-1

=74-1

=174=47

39. Given 3-1+4-12, the rational number of the form pq is

(a) 49142 

(b) 49140

(c) 49144

(d) 49146

Answer: (c) 49144

Solution: 

3-1+4-12=13+142 ∵a-m=1an

=4+3122

=7122

=72122 ∵ ambm=abm

=49144

40. The rational form 3-1-6-12 is

(a) 135 

(b) 136

(c) 132

(d) 134

Answer: (b) 136

Solution: 

3-1-6-12=13-162=2-162=162=1262=136 ∵abm= ambm

41. The value of 13-1-23-1-1 is

(a) 23

(b) 25

(c) 27

(d) 211

Answer: (a) 23

Solution: 

13-1-23-1-1=113-123-1

        =31-32-1

        =6-32-1

        =32-1=132=23

42. The rational form of -4-3 is

(a) -154 

(b) 164

(c) -164

(d) 134

Answer: (c) -164

Solution: 

-4-3=1-43=1-4×-4×-4=-164

43. 23-2 has the rational form as

(a) 49 

(b) 94 

(c) 23

(d) 45

Answer: (b) 94

Solution: 

23-2=1232=12232=3222=94 ∵abm= ambm

44. The value of 13-2 is

(a) 8 

(b) 10

(c) 9

(d) 7

Answer: (c) 9

Solution: 

13-2=32=9

45. The simplified form of 12-5

(a) 32 

(b) 34

(c) 30

(d) 36

Answer: (a) 32 

Solution: 

12-5=1125=11525=2515-321=32

            = 32

46. The value of 3-1+4-1 is

(a) 710 

(b) 78

(c) 79

(d) 712

Answer: (d) 712

Solution: 

3-1+4-1=13+14=4+312=112

47. The simplified form of 20+4-132 is

(a) 434 

(b) 454

(c) 452

(d) 435

Answer:  (b) 454

Solution: 

20+4-132=1+14×9

  =54×9

  =454

48. The value of 13-1+14-1+15-1 is

(a) 10 

(b) 14

(c) 12

(d) 8

Answer: (c) 12

Solution: 

13-1+14-1+15-1=113+114+115

=31+41+51=121=12

49. The simplified form of 3-16-13-2 is

(a) 13 

(b) 14

(c) 12

(d) 15

Answer: (c) 12

Solution: 

3-16-13-2=1316132

        =11819

        =11891=12

50. The simplified form of 5-14-12 is

(a) 1400 

(b) 1169

(c) 1196

(d) 1144

Answer: (a) 1400

Solution: 

5-14-12=15142=1202=12202=1400 ∵abm= ambm

51. The value of 6-17-13 is

(a) 342213

(b) 343216

(c) 341212

(d) 340213

Answer: (b) 343216

Solution: 

6-17-13=16173=16×73=763=7363=343216

52. The simplest form of 4-1+5-1-1 is

(a) 209 

(b) 109

(c) 119

(d) 89

Answer: (a) 209

Solution: 

4-1+5-1-1=14+15-1 ∵ a-m=1am

=5+420-1

=920-1=1920=209=209

53. The value of 4-15-1-16-1 is

(a) 53

(b) 107

(c) 103

(d) 52

Answer: (c) 103

Solution: 

4-15-1-16-1=1415-116

=120-116

=112016

=20×16

=103

54. The value after simplification of 42+32133 is

(a) 2527

(b) 2325

(c) 2521

(d) 2119

Answer: (a) 2527

Solution: 

42+32133=16+91333 ∵abm= ambm

      =25×127=2527

55. The value of 42-3234-3 is

(a) 44627

(b) 44827

(c) 44027 

(d) 44227

Answer: (b) 44827

Solution: 

42-3234-3=16-91343  

=7×13343 ∵abm= ambm

=7×4333

=7×6427=44827

56. The simplest form of 14-3-13-315-3 is

(a) 35124

(b) 36125

(c) 37125

(d) 35122

Answer: (c) 37125

Solution: 

14-3-13-315-3=1143-11331153

=11343-1133311353

=43-3353

=64-27125

=37125

57. The value of 52+42-32432 is

(a) 10 

(b) 12

(c) 16

(d) 18

Answer: (d) 18

Solution: 

52+42-32432 =25+16-9169

=32÷169

=32916

=18 

58. The number of be multiplied with 5-1 so that the product is equal to -7-1 is

(a) 57

(b) -57

(c) 37

(d) 47

Answer: (b) -57

Solution: 

Let ‘x’ be the number multiplied with 5-1 to get -7-1

(i.e.)     x=5-1=-7-1 

x=-7-15-1=1-715=-1751=-57 

  =-57

59. -47-1 is the product of two numbers.  If one number is 12-1 then the other number is

(a) 78

(b) 57

(c) -78

(d) 35

Answer: (c) -78

Solution: 

Let ‘x’ be then number to be multiplied with 12-1

(i.e.) x×12-1=-47-1

(i.e.) x=-47-112-1=12-47=12-74=-78

60. The number to be divided by -36-1 so that the quotient is equal to -6-1

(a) 16

(b) 15

(c) 13

(d) 12

Answer: (a) 16

Solution:  

Let ‘x’ be the number divides -36-1 to get -6-1

(i.e.) -36-1÷x=-6-1

-36-11x=-6-1 

(i.e.) 1-36x=-6-1 

(i.e.) 1-36x=1-6

(i.e.) x=636=16

61. The square of -45 is

(a) 165  

(b) 1625

(c) -1625

(d) 425

Answer: (b) 1625

Solution: 

-452=-4252=-4-45×5=1625

62. The cube of -13 is

(a) 125

(b) -125

(c) 127

(d) -127

Answer: (d) -127

Solution: 

-133=-1333=-1-1-13×3×3=-127

63. The following number which is not same as -454

(a) -4544

(b) -4544

(c) 44-54

(d) -45-45-45-45 

Answer: (b) -4544

Solution: 

-454=-4454=44-54=-45-45-45-45

-454-4454 (∵ Even power does not end up with negative value)

64. The following number which is not reciprocal of 563 

(a) 65-3

(b) 653

(c) 6353

(d) 56-3

Answer: (a) 65-3

Solution: 

Reciprocal of 563 is 653=6353 

(i.e.)   653=165-3=16-35-3=5-36-3=56-3

563=156-3=15-36-3=6-35-3=65-3

Clearly  65-3 is not reciprocal of 563

65. The following number which is not equal to -2764 is

(a) -34-3

(b) -343

(c) 34-3

(d) -34-34-34

Answer: (c) 34-3

Solution: 

(a) -34-3=-3343=3×3×34×4×4=-2764

(b)    -343=-34-34-34=-2764 

(c)    34-3=3-34-3=4333=433=6427

(d)    -34-34-34=-2764 ∴  34-3 is not equal is -2764

66. The exponential form of 43-143-143-143-1 is

(a) 81256 

(b) 80253

(c) 81251 

(d) 81254

Answer: (a) 81256

Solution: 

43-143-143-143-1=143143143143 ∵ a-m=1am

=34343434

=81256

67. 35-235-235-2 has the exponents form as 

(a) 15628725 

(b) 15627724

(c) 15626723

(d) 15625729

Answer: (d) 15625729

Solution: 

35-235-235-2=135213521352

=132521325213252

=523252325232

=259259259

=15625729

68. The rational number of the form pq of the number 57-173-1 is

(a) 25 

(b) 15

(c) 35

(d) 45

Answer: (c) 35

Solution: 

  57-173-1=157173

=7537

=35

69. The rational number of -5-1-43-1 is

(a) 310 

(b) 720

(c) 320 

(d) 910

Answer: (c) 320

Solution: 

-5-1-45-1=1-51-43

      =-15-34

      =320

70. The value of 4-15-13 is

(a) 12564

(b) 12362

(c) 12063

(d) 12461

Answer: (a) 12564

Solution: 

4-15-13=14153

=14×53

=543=5343 ∵abm= ambm

=12564

71. The negative exponents of the rational number 534-2 is

(a) 53-2

(b) 53-4

(c) 53-6

(d) 53-8

Answer: (d) 53-8

Solution: 

534-2=53-8 

72. The negative exponents of the rational number 574 is

(a) 75-2  

(b) 57-4

(c) 75-4

(d) 57-2

Answer: (c) 75-4

Solution: 

574=157-4=15-47-4=7-45-4=75-4 

73. The negative exponents of 57 is

(a) 5-7

(b) 15-7

(c) 17-5

(d) 7-5

Answer: (b) 15-7

Solution: 

57=15-7=1-75-7=15-7 ∵ a-m=1am

74. The rational number 153 has the negative exponents as

(a) 5-3

(b) 3-5

(c) 15-3

(d) 13-5

Answer: (a) 5-3

Solution: 

153=1353=5-3 ∵abm= ambm

75. The positive exponents of the rational number 54-4-3 is

(a) 547

(b) 5412

(c) 543

(d) 4512

Answer: (b) 5412

Solution: 

54-4-3=54-4×-3=5412 

76. The positive exponents of the rational number 45-3 is

(a) 453

(b) 153

(c) 543

(d) 143

Answer: (c) 543

Solution: 

45-3=1453=14353=5343=543

77. The positive exponent of 525-4 is

(a) 153

(b) 152

(c) 154

(d) 151

Answer: (b) 152

Solution: 

525-4=5254=5×55×5×5×5=152=152 

78. The simplified form of -122-2-1 is

(a) 14

(b) 116

(c) 112

(d) 110

Answer: (b) 116

Solution: 

-122-2-1=-122-2=-124=-1424=12×2×2×2

=116

79. The simplest form by 14-7-8-1-1 is

(a) 2 

(b) -3

(c) -2

(d) 3

Answer: (c) -2

Solution: 

14-7-8-1-1=1141-8-1

=4-8-1

=-12-1

=1-12=-2

80. The simplified form of 232313-43-16-1 is

(a) 3281

(b) 3180

(c) 3285

(d) 3185

Answer: (a) 3281

Solution: 

232313-43-16-1=23611341316

=2636114341316

=263634141316  

=2634-61316

=26321316

=64613×32

=323133=323×3×3×3

=3281

81. If 12-212-4=12-2x then the value of x is

(a) 3

(b) 2

(c) 4

(d) 5

Answer: (a) 3

Solution: 

12-212-4=12-2x

12-2-4=126=12-2x  

Since bases are equal 

∴ Exponents are equal

(i.e.)  -6=-2x

x=-6-2=3

82. If 13213-4=132m-1 then the value of ‘m’ is

(a) 12 

(b) 13

(c) -12

(d) -13

Answer: (c) -12

Solution: 

13213-4=132m-1

132-4=132m-1-2=132m-1

Since bases are equal therefore it exponents are equal

-2=2m-1

(i.e.) -2+1=2m

(i.e.) -1=2m

(i.e.) m=-12

83. If x=32223-4 then the value of x-1 is

(a) 326

(b) 236

(c) 325

(d) 235

Answer: (b) 236

Solution: 

x=32223-4

  =32221234

  =9412434

  =943424

  =3×32×23×3×3×32×2×2×2=3626=326

  x=326

  ∴ x-1=32-6=236

84. If x=45-2142 then the value of x-2 is

(a) 5-2 

(b) 5-4

(c) 4-3

(d) 4-2

Answer: (b) 5-4 

Solution: 

x=45-2142

  =45-2412

  =4121452

  =4214252

x=52 

∴x-2= 52-2=5-4    ⇒ x-2=5-4 

85. The value of  ‘x’ for which 32x3-4=36 is

(a) 2

(b) 4

(c) 1

(d) 5

Answer: (c) 1

Solution: 

32x3-4=36    ⇒ 32x3-4=36  ⇒32x34=36 

32x34=36

32x+4=36 ∵ aman=am+n

Since bases are equal therefore its exponents is equal

2x+4=6

2x=2

x=1

86. The suitable number of the given exponent                                                                     1×103+4×101+5×100+6×10-1+2×10-2 is

(a) 1025.63

(b) 1045.62

(c) 1035.64

(d) 1065.67

Answer: (b) 1045.62

Solution: 

1×103+4×101+5×100+6×10-1+2=1×1000+0×100+4×10+5×1+610+2100

=1000+0+40+5+0.6+0.002

=1045.62

87. The number which corresponds to given exponents 

1×103+3×102+2×101+7×100+2×10-1+6×10-2+8×10-3 is

          

(a) 1347.286 

(b) 1327.628

(c) 1237.628

(d) 1327.268

Answer: (d) 1327.268

Solution: 

1×103+3×102+2×101+7×100+2×10-1+6×10-2+8×10-3

=1×1000+3×100+2×10+7×1+210+6100+81000

=1000+300+20+7+.2+.06+.008

= 1327.268

88. The standard form of the given number 0.000000465 is 

(a) 4.65×10-5

(b) 4.65×10-6

(c) 4.65×10-7

(d) 4.65×10-4

Answer: (c) 4.65×10-7

Solution: 

0.000000465=4651,000,000,000

=4.65109102

=4.65107

=4.65×10-7

89. The standard form of 62100000 is

(a) 6.21×107 

(b) 6.21×105

(c) 6.21×104

(d) 6.21×103

Answer: (a) 6.21×107

Solution: 

62100000=621×100000

      =6.21×100×100000

      =6.21×102105

      =6.21×107

90. The standard form of 0.0000051 is

(a) 5.1×10-5

(b) 5.1×10-6

(c) 5.1×10-4

(d) 5.1×10-3

Answer: (b) 5.1×10-6

Solution: 

0.0000051=5110000000

        =5.1×1010000000

        =5.11000000=5.1106=5.1×10-6 

91. The standard form of the number 26350000 is

(a) 2.635×105

(b) 2.635×103

(c) 2.635×104

(d) 2.635×107

Answer: (d) 2.635×107

Solution: 

26350000=2635×10000

      =2.635×1000×10000

      =-2.635×103104

      =2.635×107

92. The usual form of the given number 5.03×10-6  is

(a) .0000503 

(b) .000503

(c) .00000503

(d) .000000503

Answer: (c) .00000503

Solution: 

5.03×10-6=503×10-210-6

          =503×10-8

          =503100,000,000

          =  .00000503

93. The usual form of the number 4.7×1012 is

(a) 4700000000

(b) 4700000000000

(c) 47000000000

(d) 47000000 

Answer: (b) 4700000000000

Solution: 

4.7×1012=4710×1,000,000,000,000

=47×100,000,000,000

= 4700000000000

94. The usual form of 5×10-8 is

(a) .0000005

(b) .000005

(c) .000005

(d) .00000005

Answer: (d) .00000005

Solution: 

5×10-8=51,00,000,000=.00000005

95. The number 2.0002109 has the usual form as

(a) 2002000

(b) 200200

(c) 20020000

(d) 200200000

Answer: (d) 200200000

Solution: 

2.0002109=2000210000×1,000,000,000

=2002×100000

=200,200,000

96. The usual form of 5.42639×106 is

(a) 5426930 

(b) 5426390

(c) 5426039

(d) 5426309

Answer: (b) 5426390

Solution: 

5.42639×106=542639100,000×1,000,000

  =542639×10

  = 5426390

97. The standard form of 0.0000000000058 is

(a) 5.8×10-8

(b) 5.8×10-11

(c) 5.8×10-12

(d) 5.8×10-7

Answer: (c) 5.8×10-12

Solution: 

0.0000000000058=5810000000000000

          =581013

          =5.8×101013

            =5.81012=5.8×10-12

98. The standard form of 7030000000000000 is

(a) 7.03×1013

(b) 7.03×1012

(c) 7.03×1015

(d) 7.03×1014

Answer: (c) 7.03×1015

Solution: 

7030000000000000 =703×10,000,000,000,0000

=703×1013

=7.03×1021013

=7.03×1015

99. The standard form of 1 micron is equal to 11,000,000  m is

(a) 1×10-5

(b) 1×10-7

(c) 1×10-6

(d) 1×10-8

Answer: (c) 1×10-6

Solution: 

Given    11000000=1106=1×10-6    m

(i.e.) 1 micron  =1×10-6    m

100. The standard form of size of a bacteria. 0000005m is

(a) 5×10-6

(b) 5×10-5

(c) 5×10-7

(d) 5×10-4

Answer: (c) 5×10-7

Solution: 

Given    0.0000005=510,000,000=5107=5×10-7

101. In a stack if there are 5 books each of thickness 10mm and 4 paper sheets each of thickness 0.015mm then the total thickness of the stack is

(a) 5.06×101mm

(b) 5.06×102mm

(c) 5.006×101mm

(d) 5.006×102mm

Answer: (c) 5.006×101mm

Solution: 

Thickness of one book = 10mm

∴ Thickness of 5 books =5×10mm=50mm

Thickness of 1 paper sheet = 0.015mm

∴ Thickness of 4 paper sheet =4×0.015mm

= 0.06mm

Total thickness = 50mm + 0.06mm

  = 50.06mm

  = 5.006×101mm

102. The standard form of size of a plant cell 0.00001275m is

(a) 1.275×10-3

(b) 1.275×10-2

(c) 1.275×10-4

(d) 1.275×10-5

Answer: (d) 1.275×10-5

Solution: 

Given   0.00001275=1275100000000

=1275108=1.275×103108

=1.275105

=1.275×10-5 

103. A number if it is expressed as the product of a number between 1 and 10 and integral power of 10 then it is

(a) usual form

(b) standard form

(c) normal form

(d) general form

Answer: (b) standard form

Solution: 

A number is said to be in the standard form if it is expressed as the product of a number between 1 and 10 and integral power of 10.

104. A number is said to be in the standard form when it is written as k×10n where

(a) 1<k<10 

(b) 1<k≤10

(c) 1≤k<10

(d) 1≤k≤10

Answer: (c) 1≤k<10

Solution: 

A number is said to be in the standard form when it is written as k×10n where 1≤k<10

105. The standard form of thickness of a thick paper 0.07mm is 

(a) 7×10-2mm

(b) 7×10-3mm

(c) 7×10-4mm

(d) 7×10-6mm

Answer: (a) 7×10-2mm

Solution: 

Given

0.07=7100=7102=7×10-2

∴ Thickness of a thick paper is 7×10-2mm

106. A number is said to be in the standard form when it in written as k×10n where 1≤k<10 and ‘n’ is

(a) whole number  

(b) an integer

(c) natural number

(d) positive integer

Answer: (b) an integer

Solution: 

A number is said to be in the standard form when it is written as k10n where 1≤k<10 and ‘n’ is an integer.

 

 

 

                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                             

 

QUIZ

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UG TRB MATHEMATICS SYLLABUS

 

UG TRB MATHEMATICS SYLLABUS

UNIT-1 ALGEBRA and TRIGONOMETRY

Polynomial Equations — Imaginary and Irrational Roots Relation between Roots and Coef1iclents symmetric function of Roots in terms of coefficient- Transformation of equation — Reciprocal equation - Increase or Decrease the roots of given equation - Removal of terms — Descartes's rule of signs — Approxmate solution of roots of polynomial by Homer's Method Cardan's method of solution of cubic polynomial — Summation of series using Binomial — Exponentialand Logarithmic series.

Symmetric — Skew symmetilc, Hermitian — Skew Hermitian, Orthogonal Matrices, Unitary Matrices Eigen Values — Eigen Vectors — Cayley-HamiIton Theorem — Similar Matrices — Diagonalizalion of Matrices.

Prime Number, Composite Number, Decomposition of a Composite Number as a Product of primes uniquely — Divisor of a positive Integer — Euler Function. Congruence Modulo n, Highest power of prime number p Contained in n! — Application of Maxima and Minima — Prime and Composite numbers — Euler's function Φ(N) — Congruences — Fermat‘s, Wilson's and Lagrange's theorems.

Expansions of Power of sinnX, cosnX, tannx — Summation by C + i S method, Telescopic Summation - Expansion of sinx, cosx, tanx in lerms of x - Sum of Roots of Trigonometric Equation, Formation of Equation With Trigonometric Roots - Hyperbolic Functions — Relation Between Circular and Hyperbolic Function — Inverse Hyperbolic Function — Logarithm of a complex number — Principal Value and General Values,

UNIT II DIFFERENTIAL CALCULUS, INTEGRAL CALCULUS and ANALYTICAL GEOMETRY

 

n* derivatives —Trigonometrical Transformations — Leibnitz Theorem — Implicit functions - Partial Differentiation— Maxima / Minima of a function of Ma variables — Lagrangian multiplier method - Radius of curvature in Cartesian and Polar forms — Angle between radius vector and tangent — Slope of tangent of a polar curve — p-r equations — Center of Curvature — Evolutes, Envelopes —Asymptotes of Algebraic curves - Asymptotes by inspection — Intersection of a curve with asymptotes. 

Evaluation of Double and Triple integrals — Applications of Multiple lntegrals in finding volumes, surface areas of solids — Areas of curved surfaces — Jacobians — Transformation of fntegrals using Jacobians — Indefinite integrals - Beta and Gamma Functions and their properties — Evaluation of lntegrals using Beta and Gamma Functions.

 Pole and Polar — Conjugate points and Conjugate lines, Conjugate diameters - Polar Coordinates — General Polar Equation of a Straight line — General Polar Equation of a Conic

UNIT III DIFFERENTIAL EQUATIONS and LAPLACE TRANSFORMATIONS

 Ordinary Differential Equations - Homogeneous Equations - Exact equations - Integrating Factors - Linear equations - Reduction of order — Second order Linear differential equations — General soluton of homogeneous Equations — Homogeneous equation with constant coefficients — Method of undetermined coefficients - method of Variation of Parameters - System of first order equations — Linear systems - Homogeneous linear systems with constant coefficients.

Partial Differential Differential Equations - Formation of Partial Oifferential Differential Equations by eliminating arbitrary constants and arbitrary functions. Solving PDEs: Complele integral - Singular Integral - general integral - Lagrange's equation Pp+Qq=R - Charpit's mefhod and special types of first order equations. 

Laplace transform of elementary functions — Laplace transforms of special functions like unit step function. Dirac Delta function — Properties of Laplace Transformation and Laplace Transforms of derivatives and integrals — Evaluation of integrals using Laplace transform - Initial value theorem - Final value theorem — Laplace transform of periodic functions — Inverse Laplace transforms — Convolution theorem — Application of Laplace transformations in solving first and second order linear differential equations and simultaneous linear ordinary differential equations.

UNIT -IV   VECTOR CALCULUS and FOURIER SERIES, FOURIER TRANSFORMS

Vector Differentiation — Velocity and Acceleration — Vector valued functions and Scalar potentials — Gradient — Divergence — Curl — Directional Derivative — Unit normal to a surface — Laplacian double operator — Harmonic functions.

Vector Integration — Line integral — Conservative force field — Determining Scalar Potential from a conservative force field — Work done by a force — Surface Integral — Volume inlegral — Theorems of Gauss, Stokes, and Green.

 Fourier Series — Expansions of Periodic functions of period 2x - Expansion of even and odd functions — half range series — Evaluation of Infinite Series using Fourier Series expansions — Fourier Transforms — Infinite Fourier Transform — Fourier Sine and Cosine transforms — Simple properties of Fourier Transforms — Convolution Theorem — Parseval's identity.

UNIT -V ALGEBRAIC STRUCTURES

Groups — Subgroups, cyclic Groups and properties of cyclic groups, Lagrange's Theorem — Counting Principles — Normal subgroups, Quotient groups, Homomorphism, Automorphism, Cayley's theorem, Permutation groups — Rings — Some special classes of Rings — Integral domain, Homomorphism of rings — Ideal and Quotient rings — Prime ideal, ktaximum Ideals —the field and quotients of an integral domain — Euclidean rings — Algebra of Linear transformation, Characteristic roots, matrices, Canonical forms, Triangular Foims — Problems of converting Linear Transformation to Matrices and vice-versa — Vector Space — Definition and examples — Linear dependence — Independence, Sub spaces and Dual spaces — Inner product spaces.

UNIT-VI REAL ANALYSlS

Sets — Countable and Uncountabfe sets — Real Number system R — Functions — Real Valued functions, Equivalence and Countability — Infremum and Supremum of a subset of R — BoIzanO- Weierstrass Theorem — Sequences of real numbers — Convergent and Divergent Sequences — Monatane Sequences — Cauchy Sequences — Limit Superior and Limit Inferior of a sequence — Sub Sequences — Infinite series — Alternating Series — Conditional convergence and Absolute convergence — Tesls of Absolute convergence — Continuity and Uniform Continuity of a real valued function of a real variable — Limit of a function at a point — Coninuity and Differentiabllity of real valued functions — Rolle’s Theorem — Mean Value Theorems — Inverse function theorem, Taylor‘s Theorem with remainder forms — Pawer senes expansion — Riemann Integrability — Sequences and Series of Functions.

Metric spaces — Limits of a function at a point in metric spaces — functions continuous on a metric space — various reformulations of continuity of a function in a metric space - open sets — closed sets — discontinuous functions on the real line.

UNIT VII COMPLEX ANALYSIS

Algebra of Complex Numbers — Function of Complex Variable — Mappings, Limits — Theorems on Limits, confinuity, differentiability — CauEhy-Riemann Equations - Analytic Functions — Harmonic Function — Conformal mapping

— Mobius Transformations — Elementary Transformation — Bilinear Transformations — Cross ratio — Fixed points of biJinear transformations — Special Bilinear transformations.

Contours — Contour Integrals - Anti Deñvatives — Cauchy-Goursat Theorem- Power Series — Complex Integration

— Cauchy*s theorem, Morera's theorem, Cauchy's Integral Formula — Liouville's Theorem — Maximum Modulus Principle

— Schwarz's Lemma — Taylor's series — Laurent's sefies — Calculus of Residues — Residue Theorem — Evaluation of Integrals - Definite integrafs of Trigonometrlc functions — Argument principle and Rouche's Theorem.

UNIT VIII MECHANICS

Statics: Farces on a rigid body —Moment of a force — General motion of a rigid body — Equivalent system of forces

— Parallel Forces — Forces along the sides of Triangle Couples.

Resultant of several coplanar forces — Equation of line of action of the resultant — Equilibrium of rigid body under three Coplanar forces — Reduction of Coplanar fDrces into single force and couples — Laws of friction, angle of friction. Equilibrium of a body on a rough inclined plane acted on by several forces — Equilibrium of a uniform Homogeneous string — Catenary — Suspension bridge — Centre of Gravity ol uniform rigid bodies,

Dynamics: Velocity and AcceJeration — Coplanar motion — Rectilinear motion under constant forces — Acceleration and retardation thrust on a plane — Motion atong a Vertical line under gravity — Motion along an inclinerl plane — motion of connected particles — Newton's Laws of mofion.

Work, Energy and power — Work — Conservative field of force — Power —Rectilinear motion under varying force Simple Harmonic Motion (S,H.M) — S.H.M along a horizontal line — S.H.M along a Vertical line — Motion under gravity in a resisting medium.

Path of a projectile - Particle projected on an inclined plane — Analysis of forces acting on particles and rigid bodies on static equilibrium, equivalent systems of forces, friction, centroids and moments of inertia — Elastic Medium, Impact - Impulsive force — Impact of sphere — Impact of two smooth spheres — Impact of two spheres of two smooth sphere on a plane — oblique impact of two smooth spheres.

Circular motion — Conical Pendulum motion of a cyclist on circular path — Circular motion on a vertical plane relative rest in revolving cone — simple pendulum — Central Orbits — Conic as Centered Orbit — Moment of inertia

UNIT IX OPERATIONS RESEARCH

Linear Programming — Formulation — Graphical Solution - Simplex Method — Big —M method — Two phase method - Duality — Primal dual relation — dual simplex method — revised simplex method — Sensitivity analysis — Transportation Problem — Assignment Problem — Queuing Theory — Basic Concepts — Steady State analysis of M/M/1 and M/M/Systems with infinite and finite capacities.

 PERT-and CPM — Project network diagram — Critical path — PERT computations-Inventory Models- Basic Concept -EOQ Models — uniform Demand rate infinite and finite protection rate with no shonage — Classical newspaper boy problem with discrete demand — purchase inventory model with one price brake — Game theory - Two person Zero — Sum game with saddle point — without saddle point — Dominance — Solving 2xn or mx2 game by graphical method — Integer programming — Branch and bound method

UNIT—X STATISTICS/PROBABILITY

Measures of central tendency - Measures of Dispersion — Moments - Skewness and Xurtosis — Correlation — Rank Correlation — Regression — Regression line of x on y and y on x — Index Numbers — Consumer Price Index numbers — Conversion of chain base Index Number into fixed base index numbers - Curve Fitting — Principle of Least Squares - Fitting a straight line — Fitting a second degree parabola — Fitting of power curves - Theory of Attributes — Attributes — Consistency of Data — Independence and Associate of data.

Theory of Probability — Sample Space - Axioms of Probability - Probability function — Laws of Addition - Conditional Probability — Law of multiplication - Independent — Boole's Inequality - Bayes’ Theorem — Random Variables Distribution function — Discrete and continuous random variables — Probability density functions — Mathematical Expectation — Moment Generating Functions — Cumutates - Characteristic functions — Theoretical distributions —

Binomial, Poisson, Normal distributions — Properties and conditions of a normal curve — Test of significance of sample and large samples — Z-test - Student's t-test — F-test - Chi square and contingency coefficient.