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This section includes 7 InterviewSolutions, each offering curated multiple-choice questions to sharpen your Current Affairs knowledge and support exam preparation. Choose a topic below to get started.
| 1. |
Express each of the following as a product of prime factors only in exponential form: (i) `108 xx 192` (ii) `1270` (iii) `729 xx 64` (iv) `768` |
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Answer» `108=2^2xx3^3` `192=2^6xx3` |
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| 2. |
Express the following numbers in standard form.(i) 0.0000000000085 (ii) 0.00000000000942(iii) 6020000000000000 (iv) 0.00000000837(v) 31860000000 |
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Answer» (i) `8.5 * 10^-2` (ii) `9.42 * 10^-2` (iii) 6.02*10^15` (iv) `6.37*10^-9` (v) `3.186 * 10^10` answer |
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| 3. |
Express 256 as a power 2. |
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Answer» `256= 2*2*2*2*2*2*2*2` `256=2^8` answer |
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| 4. |
Express the following numbers in usual form. (i) `3.02 xx 10^-6` (ii) `4.5 xx 10^4` (iii) `3 xx 10^(-8)` (iv) `1.0001 xx 10^9` (v) `5.8 xx 10^12` (vi) `3.61492 xx 10^6` |
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Answer» (i) `0.00000302` (ii) `45000` (iii) `0.00000003` (iv) `1000100000` (v) `5800000000000` (vi) `3614920` answer |
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| 5. |
Express `4^(-3)` as a power with the base 2 |
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Answer» `(4)^-3` `= (2*2)^-3` `= (2^2)^-3` `= 2^-6` answer |
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| 6. |
Identify the greater number, wherever possible, in each of the following ? (i) `4^3 or 3^4` (ii) `5^3 or 3^5` (iii) `2^8 or 8^2` (iv) `100^2 or 2^100` |
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Answer» (i) `4^3 or 3^4` `4^3 = 4*4*4 = 64` `3^4 = 3*3*3*3 = 81` `:. 3^4> 4^3` (ii) `5^3 or 3^5` `5^3= 125` `3^5= 243` `:. 3^5> 5^3` (iii) `2^8 or 8^2` `2^8 = (2*2*2*2)^2 = 16^2= 96` `8^2 = 64` `:. 2^8 > 8^2` (iv)`100^2 or 2^100` `100^2 = 10000` `2^100= (2^10)^10 = 1024^10` `:. 2^100 > 100^2` answers |
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| 7. |
If `a^x=b, b^y=c and c^z=a` then find the value of xyz |
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Answer» `a^x = b; b^y = c ; c^z = a` `a^x = b` `log a^x = log b` `x log a = log b` `x = log b/log a = log_a b` now , `b^y = c` `log b^y = log c` `y log b = log c` `y = log_b c` `c^z = a` `log c^z = log a` `z logc = log a` `z = log_c a` now, `xyz = log_a b xx log_b c xx log_c a` `xyz = log b/ log a xx log c/ log b xx log a/log c` `xyz = 1` Answer |
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| 8. |
Find value of `(81)^0.16xx81^(0.09)` |
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Answer» `(81)^0.16xx(81)^0.09` `=(3^4)^0.16xx(3^4)^0.09` `=3^(0.64) xx 3^(0.36)` `=3^1 = 3` `:. (81)^0.16xx(81)^0.09 = 3` |
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| 9. |
Find the number from each of the following expanded forms : (a) `8 xx 10^4 + 6 xx 10^3 + 0 xx 10^2 + 4 xx 10^1 + 5 xx 10^0` (b) `4 xx 10^5 + 5 xx 10^3 + 3 xx 10^2 + 2 xx 10^0` (c) `3 xx 10^4 + 7 xx 10^2 + 5 xx 10^0` |
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Answer» a) `86045` b) `405302` c) `30705` d) `900230` answers |
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| 10. |
Write exponential form for `8 xx 8 xx 8 xx 8` taking base as `2.` |
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Answer» `= 8*8*8*8` `= 2^3*2^3*2^3*2^3` `= 2^(3+3+3+3)` using `a^x*a^y*a^z = a^(x+y+z) ` `2^12` answer |
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| 11. |
Express each of the following as rational number : `(a) (-5/7)^(-4)` `(b) (2/3)^(-2)` |
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Answer» (a) `(-5/7)^-4 = (-7/5)^4 = 2401/625` (b) `(2/3)^-2 = (3/2)^2 = 9/4` |
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| 12. |
Express the following numbers in the standard form:(i) 5985.3 (ii) 65,950 (iii) 3,430,000 (iv) 70,040,000,000 |
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Answer» 1.)`5985.3=5.9835xx10^3` 2.)`65950=6.595xx10^4` 3.)`3430000=3.43xx10^6` 4.)`70040000000=7.04xx10^4` |
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| 13. |
Express the number appearing in the following statements in standard form.(a) The distance between Earth and Moon is 384,000,000 m.(b) Speed of light in vacuum is 300,000,000 m/s.(c) Diameter of the Earth is 1,27,56,000 m.(d) Diameter of the Sun is 1,400,000,000 m.(e) In a galaxy there are on an average 100,000,000,000 stars.(f) The universe is estimated to be about 12,000,000,000 years old.(g) The distance of the Sun from the centre of the Milky Way Galaxy is estimated to be 300,000,000,000,000,000,000 m.(h) 60,230,000,000,000,000,000,000 molecules are contained in a drop of water weighing 1.8 gm.(i) The earth has 1,353,000,000 cubic km of sea water.(j) The population of India was about 1,027,000,000 in March, 2001. |
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Answer» (a) `384,000,000m = 3.84 xx 10^8m` (b) `300,000,000 = 3 xx 10^8m/s` (c) `1,27,56,000= 1.2756 xx 10^7m` (d) `1,400,000,000m= 1.4 xx 10^9m` (e) `1,00,000,000,000= 1 xx 10^11` (f) `12,000,000,000 = 1.2 xx 10^10` (g) `300,000,000,000,000,000,000 = 3 xx 10^20` (h) `60,230,000,000,000,000,000,000= 6.023 xx 10^22` (i) `1,353,000,000 = 1.353 xx 10^9` (j) `1,027,000,000 = 1.027 xx 10^9` answer |
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| 14. |
Express the following numbers in standard form:(i) 5,00,00,000 (ii) 70,00,000 (iii) 3,18,65,00,000(iv) 3,90,878 (v) 39087.8 (vi) 3908.78 |
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Answer» (i) `5,00,00,000 = 5 xx 10^7` (ii) `70,00,000 = 7 xx 10^6` (iii) `3,18,65,00,000 = 3.1865 xx 10^9` (iv) `390878 = 3.90878 xx 10^5` (v) `39087.8 = 3.90878 xx 10^4` (vi) `3908.78 = 3.90878 xx 10^3` answers |
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| 15. |
Simplify and write the answer in the exponential form. (i) `(3^7/3^2) xx 3^5` (ii) `2^3 xx 2^2 xx 5^5` |
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Answer» (i) `(3^7)/(3^2)*3^5 = (3^7*3^5)/(3^2)` `= 3^(7+5)/2^2` `= 3^12/3^2` `= 3^(12-2) = 3^10` (ii) `2^3*2^2*5^5` `= 2^5*5^5` `= (2*5)^5= 10^5` (iii) `(6^2*6^4) -: 6^3` `= 6^(2+4-3) = 6^3` (iv) `[(2)^3 *3^6] * 5^6` `= 2^(2*3) *3^6*5^6` `= (2*3*6)^6` `= 30^6` answer |
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| 16. |
Simplify : (i) `(12^4+9^3 xx 4)/(6^3 xx 8^2 xx 27)` (ii) `2^3 xx a^3 xx 5a^4` |
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Answer» (i) `(12^4*9^3*4)/(6^3*8^2*27)` `= ((2^2*3)^4*(3^2)^3 *2^2)/((2*3)^3 *(2^3)^2 * 3^3)` `= (2^8*3^4*3^6*2^2)/(2^3*3^3*2^6*3^3)` `= 2^(8+2-3-6)*3^(4+6-3-3)` `= 2*3^4 = 162` (ii) `2^3*a^3*5a^4` `= 2^3*a^3*a^4*5` `= 2^3*5*a^(3+4)` `= 8*5*a^7 = 40a^7` answer |
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| 17. |
Express the following numbers in standard form.(i) 0.000035 (ii) 4050000 |
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Answer» 1.)`3.5xx10^(-5)` 2.)`4.05xx10^6` |
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| 18. |
`27^x=9/(3^x)`, Find the value of x |
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Answer» `27^x = 9/3^x` `=>(3^3)^x = 3^2/3^x` `=>3^(3x) = 3^(2-x)` `=>3x = 2-x` `=>4x = 2` `=> x = 1/2`, is the required solution. |
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| 19. |
Express the following numbers in usual form. (i) `3.52 xx 10^2` (ii) `7.54 xx 10^-4` (iii) `3 xx 10^(05)` |
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Answer» (i) `3.52 * 10^5 = 352000` (ii) `7.54 * 10^-4 = 0.000754` (iii) `3*10^-5 = 0.00003` answers |
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| 20. |
If `(9^(n+2) xx (3^(-n/2))^(-2)-27^n)/(3^(3m)xx2^3xx10)=1/27` prove that m-n=1 |
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Answer» `(9^(n+2) xx (3^(-n/2))^-2 - 27^n)/(3^(3m)xx2^3xx10) = 1/27` `=>((3^2)^(n+2) xx 3^n - (3^3)^n)/(3^(3m)xx2^3xx10) = 1/3^3` `=>(3^(2n+4) xx 3^n - 3^(3n))/(3^(3m)xx2^3xx10) = 1/3^3` `=>(3^(3n+4)*3^3 - 3^(3n)*3^3) = (3^(3m)xx2^3xx10)` `=>3^(3n+7) - 3^(3n+3) = 3^(3m)xx2^3xx10` `=>3^(3n+3)(3^4 - 1) = 3^(3m)xx2^3xx10` `=>3^(3n+3)(80) = 3^(3m)xx2^3xx10` `=>3^(3n+3)(2^3 xx 10) = 3^(3m)xx2^3xx10` `=>3n+3 = 3m` `=>n+1 = m` `=>m - n = 1` |
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| 21. |
`( 3125^x xx 5^9)/ 625 = (125)^-1` |
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Answer» `(3125^x xx5^9)/625 = 125^-1` `=>(3125^x xx5^9)/5^4 = 1/5^3` `=>(3125^x xx5^8) = 1` `=>((5^5)^x xx 5^8) = 5^0` `=>5^(5x+8) = 5^0` `5x+8 = 0=> x = -8/5` |
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| 22. |
Simplify. (i) `(25 xx t^(-4))/(5^3 xx 10 xx t^(-8)) (t != 0)` (ii) `(3^(-5) xx 10^(-5) xx 125)/(5^(-7) xx 6^(-5))` |
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Answer» (i) `(25 * t^-4)/(5^-3*10*t^-8)` ` = 5^(2+3)/(t^(4-8)*10)` `= (625t^4)/2` (ii) `(3^-5 * 10^-5* 5^3)/(5^-7 * 6^-5)` `= 3^-5*10^-5*5^10*6^5` `= (5^10*6^5)/(3^5*10^5)` `= 5^5` `= 3125` answer |
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| 23. |
Simplify : (i) `2 xx 10^3` (ii) `7^2 xx 2^2` (iii) `2^3 xx 5` (iv) `3 xx 4^4` |
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Answer» (i) `2*10^3= 2*10*10*10= 2000` (ii) `7^2*2^2 = 7*7*2*2 = 49*4 = 196` (iii)`2^3*5= 2*2*2*5= 40` (iv)`3^2*10^4= 3^2*2^4*5^4` `= 9*10000` `=90000` (iv) `5^2*3^3` `= 5*5*3*3*3= 25*27` `= 25*(25+2)= 625+50 = 675` |
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| 24. |
Simplify and express the result in power notation with positive exponent. (i) `(-4)^5 -: (-4)^8` (ii) `(1/(2^3))^2` (iii) `(-3)^4 xx (5/3)^4` (iv) `(3^(-7) -: 3^(-10)) xx 3^(-5)` (v) `2^(-3) xx (-7)^(-3)` |
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Answer» (i) `(-4 ) -: (-4)^8` `(-4)^(5-8)` `(-4)^-3` `= 1/-4^3` (ii) `(1/2^3)^2` `= (2^-3)^2 = 2^-6` `= 1/2^6` (iii) `(-3)^4 xx (5/3)^4` `= (-1)^4*(3)^(4-4)* 5^4` `= 5^4` (iv) `(3^-7 -: 3^-10) xx 3^-5` `= 3^(-7 + 10) xx 3^-5 ` `= 3^3 xx 3^-5` `= 3^(-5+3) = 3^-2` `= 1/3^2` (v) `2^-3 xx (-7)^-3` `(2 xx -7)^-3= -14^-3` `1/14^3` answers |
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