MATHEMATICAL FORMULAE Algebra 1. (a + b)2= a2+ 2ab + b2; a2+ b2= (a + b)2− 2ab 2. (a − b)2= a2− 2ab + b2; a2+ b2= (a − b)2+ 2ab 3. (a + b + c)2= a2+ b2+ c2+ 2(ab + bc + ca) 4. (a + b)3= a3+ b3+ 3ab(a + b); a3+ b3= (a + b)3− 3ab(a + b) 5. (a − b)3= a3− b3− 3ab(a − b); a3− b3= (a − b)3+ 3ab(a − b) 6. a2− b2= (a + b)(a − b) 7. a3− b3= (a − b)(a2+ ab + b2) 8. a3+ b3= (a + b)(a2− ab + b2) 9. an− bn= (a − b)(an−1+ an−2b + an−3b2+ ··· + bn−1) 10. an= a.a.a... n times 11. am.an= am+n 12.am an= am−nif m > n = 1 if m = n = 1 an−mif m < n;a ∈ R,a 6= 0 13. (am)n= amn= (an)m 14. (ab)n= an.bn 15. ?a b ?n =an bn 16. a0= 1 where a ∈ R,a 6= 0 17. a−n= 1 an,an= 1 a−n 18. ap/q= q√ap 19. If am= anand a 6= ±1,a 6= 0 then m = n 20. If an= bnwhere n 6= 0, then a = ±b 21. If√x,√y are quadratic surds and if a +√x =√y, then a = 0 and x = y 22. If√x,√y are quadratic surds and if a+√x = b+√y then a = b and x = y 23. If a,m,n are positivereal numbersand a 6= 1, then logamn = logam+logan 24. If a,m,n are positive real numbers, a 6= 1, then loga ?m n ? = logam−logan 25. If a and m are positive real numbers, a 6= 1 then logamn= nlogam 26. If a,b and k are positive real numbers, b 6= 1,k 6= 1, then logba =logka logkb 27. logba = 1 logabwhere a,b are positive real numbers, a 6= 1,b 6= 1 28. if a,m,n are positive real numbers, a 6= 1 and if logam = logan, then m = n Typeset by AMS-TEX 2 29. if a + ib = 0 where i =√−1, then a = b = 0 30. if a + ib = x + iy, where i =√−1, then a = x and b = y 31. The roots of the quadratic equation ax2+bx+c = 0; a 6= 0 are−b ±√b2− 4ac 2a The solution set of the equation is ( −b +√∆ 2a ,−b −√∆ 2a ) where ∆ = discriminant = b2− 4ac 32. The roots are real and distinct if ∆ > 0. 33. The roots are real and coincident if ∆ = 0. 34. The roots are non-real if ∆ < 0. 35. If α and β are the roots of the equation ax2+ bx + c = 0,a 6= 0 then i) α + β =−b a = −coeff. of x coeff. of x2 ii) α · β =c a=constant term coeff. of x2 36. The quadratic equation whose roots are α and β is (x − α)(x − β) = 0 i.e. x2− (α + β)x + αβ = 0 i.e. x2− Sx + P = 0 where S =Sum of the roots and P =Product of the roots. 37. For an arithmetic progression (A.P.) whose first term is (a) and the common difference is (d). i) nthterm= tn= a + (n − 1)d ii) The sum of the first (n) terms = Sn=n 2(a + l) =n 2{2a + (n − 1)d} where l =last term= a + (n − 1)d. 38. For a geometric progression (G.P.) whose first term is (a) and common ratio is (γ), i) nthterm= tn= aγn−1. ii) The sum of the first (n) terms: Sn =a(1 − γn) 1 − γ ifγ < 1 =a(γn− 1) γ − 1 if γ > 1 = na if γ = 1 . 39. For any sequence {tn},Sn− Sn−1 = tn where Sn =Sum of the first (n) terms. 40. n P γ=1γ = 1 + 2 + 3 + ··· + n =n 2(n + 1). 41. n P γ=1γ2= 12+ 22+ 32+ ··· + n2=n 6(n + 1)(2n + 1). 3 42. n P γ=1γ3= 13+ 23+ 33+ 43+ ··· + n3=n2 4(n + 1)2. 43. n! = (1).(2).(3).... .(n − 1).n. 44. n! = n(n − 1)! = n(n − 1)(n − 2)! = ..... 45. 0! = 1. 46. (a +b)n= an+ nan−1b+n(n − 1) 2! an−2b2+n(n − 1)(n − 2) 3! an−3b3+ ··· + bn,n > 1.
Posted on: Tue, 10 Sep 2013 07:59:13 +0000
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