The document presents fundamental algebra questions and solutions tailored for entrepreneurs, covering various algebraic concepts, such as factoring, divisibility, and exponents. It includes problem-solving examples and links to additional resources for enhancing numerical and analytical skills in business contexts. The author emphasizes the importance of supporting social entrepreneurship and disseminating knowledge globally.

1. BASIC ALGEBRA FOR ENTREPRENEURS by : DR. T.K. JAIN AFTERSCHO ☺ OL centre for social entrepreneurship sivakamu veterinary hospital road bikaner 334001 rajasthan, india FOR – PGPSE / CSE PARTICIPANTS [email_address] mobile : 91+9414430763

2. My words..... My purpose here is to give a few questions on fundamentals of Algebra . I welcome your suggestions. I also request you to help me in spreading social entrepreneurship across the globe – for which I need support of you people – not of any VIP. With your help, I can spread the ideas – for which we stand....

3. If (x^3 – 1/x^3)= 14, what is (x-1/x) ? Options : 5,4,3,2 let us assume 1/x = y x^3 -y^3 = 14 (x-y)^3 = x^3 -3x^2y+3xy^2-y^3 (x-y)^3 =14 -3xy(x-y) (x-y)^3 +3xy(x-y)=14 (x-Y)^3 +3(x-y) – 14=0 (we can cancel xy, as y is 1/x) at this stage put the options, when you put x-y = 2, you are able to solve it so answer is 2

4. What must be added to 1/x to make it equal to x ? X – 1/x = (x^2 -1) / x answer

5. Is (x^3 – 5^3) divisible by (x-3) ? Yes

6. Is (x^3 – 5^3) divisible by (x+3) ? No it will be divisible only if the power is even example : (x^4 – 5^4) is divisible by (x+3)

7. Is (x^3+ 5^3) divisible by (x+3) ? Yes, but only when power is odd

8. Is (x^3 + 5^3) divisible by (x-3) ? Never

9. X^3 -3x^2+4x+k contains (x+6) as factor, what is the value of k ? X+6 = 0, so x = -6 Take x=-6 -216 -108 -24 +k = 0 k = 348 answer

10. (x+2) and (x-1) are factors of (x^3+10x^2+mx+n), what is the value of m and n ? Options : 5,-3 17,-8, 23,-19, 7,-18 solution : take x = 1 1+10+mx+n or m+n = -11 take x = -2 -8+40-2m+n or -2m+n = -32 or 3n=-54 or n=-18 so last option is correct

11. If 5^x /125 = 1, what is X ? We know that 5^3 = 125, 125/125=1, so x=3 answer

12. If 3^x – 3^(x-1) = 18, what is x^x Options : 3,8,27,216 try with 27, x should be 3 let us put the value of x in the equation we find : 27 – 9 = 18 so X = 3 and x^x = 27 answer

13. If 2^x – 2^(x-1) = 16, then what is the value of x^4 ? Options : 16, 81, 256, 625 let us try with option 625, here x = 5 so put the value of X in the equation, we get : 32 – 16 = 16 so x= 5, and x^4 = 625 answer

14. If (a+b+c) = 0, then a^3+b^3+c^3 is equal to what ? Options : 0, 3abc, abc, (ab+bc+ac) answer : = 3abc formula : (a+b+c)=0 then a^3+b^3+c^3 = 3abc

15. a+b = 5, and ab = 6, what is (a^3+b^3) ? (a-b)^2 = (a+b)^2 – 4ab ab =6, a+b = 5 (a-b)^2=25-24 = 1 (a+b)^3 = a^3 +b^3 +3a^2b+3ab^2 or : (a+b)^3 = a^3 +b^3 +3ab(a+b) (5)^3 = a^3+b^3 +3*6(5) 125 – 90 = a^3+b^3 a^3+b^3 = 35 ans

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17. What will you get from (.4)^(1/2) ? Options : 1. .2 2. .-2 3. .02 4 none of these answer : square root of .4 will be close to .6, so answer is none of these

18. What will you get from (.8)^(1/3) ? a. .2 b .02 c .04 d none of these answer : answer will be more than .9, so answer is none of these

19. What will you get from the following : (.9)^(1/2) a. .3 b. .03 c. between .8 and .9 d. none of these answer : none of these

20. Due to clerical mistake , + is written as *, / is written as - , * is written as +, - is written as =. what will you get from this : 2 of 2 =4 * 6- 3 (2+4-2) Solution : first remove clerical mistake : 2 of 2-4 + 6 / 3 (2*4/2) first solve bracket : 2 of 2-4+ 6 /3 * 4 solve of 4 -4+2 * 4 now solve multiply 4 – 4 + 8 = 8 answer

21. What should we add or subtract or multiply or divide from 13333 to make it a perfect square ? Find minimum digit If we add 125, we get : 13458, which is square of 116 if we deduct 108, we get 13225 so we should deduct 108

22. (149)^2 – (78)^2 – (737) – (98)^2 = ? 22201 – 6084 – 737 – 9604 = 5776 answer

23. Which of these is highest : 2/5, 15/18 , 223/226, 1202/1205 ? Here difference between denominator and numerator is same in each case, so the answer will be 1202 / 1205 answer

24. What is the .the sum of first 20 terms of series is 1/5*6 +1/6*7+1/7*8----- We can write it as : 1/30, 1/42, 1/56 we can write it as (1/5-1/6)+(1/6 – 1/7)+ (1/7-1/8) . . . . here you can see that in first bracket we have -1/6 and in 2 nd bracket we have + 1/6 and so on.. when we open bracket, we get : 1/5+1/25 we get 6/25 answer

25. The least among the following :- a. 0.2 b.1/0.2 c. 0.3^2 d. 0.22 a. 0.2 b.1/0.2 = 5 c. 0.3^2 =.09 d. 0.22 so the answer = C

26. If 1/3.718 =0.2689,then find value of 1/0.0003718 ? =2689 just shift the decimals as per the question

27. 4.036 divided by 0.04 = ? 100.9 answer

28. (0.96)3 – (0.1)3/ (0.96)2 +0.096 +(0.1)2 =? (a^3 – b^3) / (a+b)^2 = a-b a = .96 b = .1 answer = a-b = .86 answer

29. (a+b) * (a-b) =? = a^2 – b^2

30. (a+b)^3 = ? A^3 + 3a^2b + 3ab^2 +b^3

31. (a-b)^3 =? A^3 - 3a^2b + 3ab^2 -b^3

32. What is .000000001 in scientific notation ? = .000000001*10^9 / 10^9 =1*10^(-9)

33. What is .000000001234 in scientific notation ? = 1.234* 10^(-9)

34. What will you get from : .3879999999.......? Here 9 is repeating and 387 is non-repeating deduct 387 from the number and divide by 9000 we use as many 9 as there are repeating digits we use as many 0 as we have non-repeatig digits. =(3879-387)/(9000) =3492 / 9000 answer

35. What will you get from .358585858.....? Here 58 are being repeated we will deduct 3 (non repeating) from 358 and divide it by 990 (as many 9 as there are repeating digits and as many 0 as there are non repeating digits) = (358 – 3) / (990) =355/990 =71/198 answer

36. Put these in ascending order : 2/3, 25/39, 18/-19, 55/70, .04555..? Ascending means from small to big the smallest number is 18/-19 (as it is in negative). Then we have .04555..... now compare 2/3 and 25/39, in order to compare these, let us multiply the numerators of these by denominators of the other numbers, we get : 78 and 75, the bigger number is 78 so 2/3 is bigger. Now compare it with 55/70. We get : 140 and 165, 165 is bigger so final order : -18/19, .45555.., 25/39, 2/3, 55/70 answer

37. What will you get from (91.9432)^2 – (41.9432)^2 ? (a^2 – b^2) = (a-b) * ( a+b) = (91.9432 – 41.9432) * ( 91.9432+41.9432) (50) * (133.8864) =6694.32 answer

38. .49*.98*2+.76*.38*2-.98*1.52 = ? Simplify it : .98*.98+.76*.76-2*.98*.76 (a-b)^2 = a^2 +b^2-2ab =(.98-.76)^2 =(.22)^2 =.0484 answer

39. What is the remainder when (2x^3 +5x^2 -4x -6) is divided by (2x -1) ? 2x-1 = 0 so x = ½ or .5 put this value in the equation .25+1.25-2-6 =1.5 – 8 = -6.5 answer

40. Which of these is / are factor of (x^2 +4) ? (x+2) , (x-2) , (x^2 -2), (none of these ) answer = none of these

41. Which of these is a factor of (X^3 -5x^2+8x-4) ? (x-1), (x-2)^2, (x+2), (x+1) , (none) answer = (x-1)

42. What is the remainder when (x^11 +1) is divided by (x+1) ? Answer = zero

43. (x+1/x) = 3, what is the value of (x^6 +1/x^6) ? Options : 927, 414, 364, 322 let us assume y = 1/x find its cube : x^3+3x^2y+3xy^2+y^3 = 27 =x^3+y^3 + 9 or x^3+y^3 = 18 now square it : x^6+2x^3y^3+y^6 = 324 (we can cancel x^3 and y^3) x^6+y^6 = 324-2 = 322 answer

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