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−n if m>n =1 ifm=n = 1an −m if m<n;a2R;a 6=0
13. (am)n = amn =(an)m
14. (ab)n = an:bn
15.abn = anbn
16. a0 =1wherea2R;a 6=0
17. a−n = 1an;an= 1
18. ap=q = qpap a−n
19. If am = an and a 6= 1;a6=0thenm=n 20. If an = bn where n 6=0,thena=b
21. If px;py are quadratic surds and if a+px = py,thena= 0 and x = y
22. If px;py are quadratic surds and if a+px = b+py then a = b and x = y
23. If a;m;n arepositivereal numbersanda 6=1,thenlogamn =logam+logan 24.Ifa;m;narepositiverealnumbers,a6=1, thenlogam n=logam−logan
25. If a and m are positive real numbers, a 6=1thenlogamn=nlogam
26. If a;b and k are positive real numbers, b 6=1;k6=1,thenlogba=logk a logk b
27. logb a = 1 loga b where a;b are positive real numbers, a 6=1;b6=1
28. if a;m;n are positive real numbers, a 6= 1 and if logam =logan,then m=n Typeset by AMS-TEX2
29. if a+ib =0 wherei=p−1, then a =b =0 30. if a+ib = x+iy,wher ei=p−1,then a = x and b=y
31. Theroots ofthe quadratic equation ax2+bx+c =0;a6= 0are −bpb2 −4ac ( The solution set of the equation is 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. p −b+p 2a ; −b− 2a 2a )
35. If and are the roots of the equation ax2 +bx+c =0;a6=0then i) +=−ba =−coe. of x coe. of x2 ii) = ca = constant term coe. of x2
36. The quadratic equation whose roots are and is (x−)(x −)=0 i.e. x2 − (+)x+=0 i.e. x2 − Sx+P =0whereS=Sum of the roots and P =Product of the roots.
37. For an arithmetic progression (A.P.) whose rst term is (a) and the common dierence is (d). i) nth term= tn = a+(n −1)d ii) The sum of the rst (n)terms=Sn=n where l =last term= a+(n −1)d. 2(a +l)=n2f2a+(n−1)dg
38. For a geometric progression (G.P.) whose rst term is (a) and common ratio is (γ), i) nth term= tn = aγn−1. ii) The sum of the rst (n)terms: Sn =a(1−γn) 1 −γ =a(γn−1) γ −1 =na ifγ<1 : if γ>1 if γ =1 39. For any sequence ftng;Sn−Sn−1 = tn where Sn =Sum of the rst (n) terms. n
40. 41. P γ=1 n P γ =1+2+3++n=n2(n+1). γ2 =12+22+32++n2=n6(n+1)(2n+1).
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−n if m>n =1 ifm=n = 1an −m if m<n;a2R;a 6=0
13. (am)n = amn =(an)m
14. (ab)n = an:bn
15.abn = anbn
16. a0 =1wherea2R;a 6=0
17. a−n = 1an;an= 1
18. ap=q = qpap a−n
19. If am = an and a 6= 1;a6=0thenm=n 20. If an = bn where n 6=0,thena=b
21. If px;py are quadratic surds and if a+px = py,thena= 0 and x = y
22. If px;py are quadratic surds and if a+px = b+py then a = b and x = y
23. If a;m;n arepositivereal numbersanda 6=1,thenlogamn =logam+logan 24.Ifa;m;narepositiverealnumbers,a6=1, thenlogam n=logam−logan
25. If a and m are positive real numbers, a 6=1thenlogamn=nlogam
26. If a;b and k are positive real numbers, b 6=1;k6=1,thenlogba=logk a logk b
27. logb a = 1 loga b where a;b are positive real numbers, a 6=1;b6=1
28. if a;m;n are positive real numbers, a 6= 1 and if logam =logan,then m=n Typeset by AMS-TEX2
29. if a+ib =0 wherei=p−1, then a =b =0 30. if a+ib = x+iy,wher ei=p−1,then a = x and b=y
31. Theroots ofthe quadratic equation ax2+bx+c =0;a6= 0are −bpb2 −4ac ( The solution set of the equation is 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. p −b+p 2a ; −b− 2a 2a )
35. If and are the roots of the equation ax2 +bx+c =0;a6=0then i) +=−ba =−coe. of x coe. of x2 ii) = ca = constant term coe. of x2
36. The quadratic equation whose roots are and is (x−)(x −)=0 i.e. x2 − (+)x+=0 i.e. x2 − Sx+P =0whereS=Sum of the roots and P =Product of the roots.
37. For an arithmetic progression (A.P.) whose rst term is (a) and the common dierence is (d). i) nth term= tn = a+(n −1)d ii) The sum of the rst (n)terms=Sn=n where l =last term= a+(n −1)d. 2(a +l)=n2f2a+(n−1)dg
38. For a geometric progression (G.P.) whose rst term is (a) and common ratio is (γ), i) nth term= tn = aγn−1. ii) The sum of the rst (n)terms: Sn =a(1−γn) 1 −γ =a(γn−1) γ −1 =na ifγ<1 : if γ>1 if γ =1 39. For any sequence ftng;Sn−Sn−1 = tn where Sn =Sum of the rst (n) terms. n
40. 41. P γ=1 n P γ =1+2+3++n=n2(n+1). γ2 =12+22+32++n2=n6(n+1)(2n+1).
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