MATHEMATICAL FORMULAE Algebra a— 1. 2. 3. 4. 5. 6. 7. 9. 10. 11. 12. 13. 14. 15. 16. 17. 18. 19. 20. 21. 22. 23. 24. 25. 26. 27. 28. = (12 + 2ab 1)2; (12 b2 = 2 — 2(Jb 1)2; (12 + b2 _ — 2ab (J — b) 2 2ab (a + b + = a2 + b2 + c2 + 2(ab + bc + ca) = a3 + b3 + 3ab(a + b); (13 + b3 - 3ab(a + b) (a — b)3 2 — b2 a 3 — b3 a 3 3 — b3 — 3ab(a — b); a3 — b3 (a + b)(a — b) (a - + 3ab(a - b) a a m a a a b + an—3b2 + = a.a.a. .. n times n .a m n if m > n if m = n 1 a (am n mn (ab)n = an.bn bn 0 if m < n; a e R, a (an m a = 1 where a e R, a # 0 1 a a 1 a— ap/q — = an and a # ± 1, a # 0 then m = n = where n # 0, then a = ±b If v,'fi, are quadratic surds and if a + V"Ü, then a = 0 and x = y If A/G, are quadratic surds and if a + = b + v/'Ü then a = b and x = y If a, m, n are positive real numbers and a # 1, then loga mn = loga m+loga n If a, m, n are positive real numbers, a # 1, then log = loga m — loga n If a and m are positive real numbers, a # 1 then loga mn = n loga m logk a If a, b and k are positive real numbers, b # 1, k # 1, then logb a logk b 1 where a, b are positive real numbers, a # 1, b # 1 logb a loga b if a, m, n are positive real numbers, a # 1 and if log am = loga n, then Typeset by AMS-TEX
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2 29. 30. 31. 32. 33. 34. 35. 36. 37. 38. - sn-l 39. 40. 41. if a + 0 where i — —1, then a = b = 0 if a + ib = x + iy, where i — —1, then a = x and b = y The roots of the quadratic equation ax2+bx+c = 0; a # O are The solution set of the equation is where A = discriminant = b2 The roots are real and distinct if A > 0. The roots are real and coincident if A = 0. The roots are non-real if A < 0. If a and [3 are the roots of the equation ax2 + bm + c — 0, a # 0 then —b coeff. of x coeff. of x2 a constant term ii) a — coeff. of eT2 a The quadratic equation whose roots are a and [3 is @ — a)@ — [3) = 0 2 — (a + + CN/3 = 0 i.e. x i.e. x2 — sx + P = 0 where S =Sum of the roots and P =Product of the roots. For an arithmetic progression (A.P.) whose first term is (a) and the common difference is (d). i) nth term= tn — ii) The sum of the first (n) terms = Sn = where I =last term= a + (n — l)d. 2 2 For a geometric progression (G.P.) whose first term is (a) and common ratio is i) nth term= tn — any n I ii) The sum of the first (n) terms: if7 < 1 if > 1 if = 1 where S For any sequence {tn terms. _1+2+3+. — 12 22 32 a(l a(7n — na n 2 1) n =Sum of the first (n) n 6
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