Important Mathematical Formula

☞.  Algebra Formulas :- The following are some of the important algebraic formulas.

1. (a + b)² = a² + 2ab + b²

2.  (a - b)²  = a² - 2ab + b²

       3.  a² - b²  = (a + b) (a - b) 

       4.  a² - b²  =  (a+b)² - 2ab or (a-b)² + 2ab

       5. 2(a²+b²)  =  (a + b)² + (a - b)²

       6. (a + b)² - (a - b)²  =  4ab

     7. a⁴ + b⁴ = (a+b) (a-b)[(a+b)²-2ab]

     8.  (a + b)² = (a - b)²  + 4ab

       9.  (a - b)²  =  (a + b)² - 4ab

      10. a⁴ + b⁴ = [(a+b)²-2ab]²-2ab]²-2(ab)²

      11. (a + b + c)² = a² + b² + c² + 2ab + 2bc + 2ac

     12. (a + b - c)² = a² + b² + c² + 2ab - 2bc - 2ac

      13. (a - b - c)² = a² + b² + c² - 2ab +2bc - 2ac)

      14. (a - b + c)²  = a² + b² + c² - 2ab - 2bc + 2ac

     15. a⁴ + a²b² + b⁴ = (a² + ab + b²)( a² – ab + b²)

      

      16.  (a + b)³  = a³ + b³ + 3ab(a + b)

      17.  (a - b)³  =  a³ - b³ - 3ab(a - b)

      18.  a³ + b³ = (a + b)(a² - ab + b²) 

     19. a³ + b³ = (a+b)³-3ab(a+b)

       20.  a³ - b³  = (a - b)(a² + ab + b²)

     21. a³ - b³  = (a-b)³+3ab(a-b) 

      22. (a + b + c + d)² = a² + b² + c² + d² + 2(ab + ac + ad + bc + bd + cd)

    23. a³ + b³ + c³ – 3abc = (a + b + c)(a² + b² + c² -ab – bc – ca) 

    24. a³ + b³ + c³ – 3abc = 1/2 (a + b + c)[(a – b)² + (b – c)² + (c – a)²]

    25. If a + b + c = 0 then a³ + b³ + c³ = 3abc

    26. (x + a)(x + b) (x + c) = x³ + (a + b + c) x² + (ab + bc + ac)x + abc

    27. (x – a)(x – b) (x – c) = x³ – (a + b + c) x² + (ab + bc + ac)x – abc

    28.  (x + a)(x + b)  = x² + (a + b)x + ab

     29 .  (x + a)(x - b)  = x² + (a - b)x - ab

     30.  (x - a)(x + b)  = x² + (b - a)x - ab

     31.  (x - a)(x - b)  = x²  - (a + b)x + ab

      

     32.  (x + a) (x + b) (x + c) = x³ + (a + b +c)x² + (ab + bc + ca)x + abc

     33.  a³ + b³ + c³ - 3abc  = (a + b + c)(a² + b² + c² - ab - bc -ac)

    
    34. a⁴ + a² + 1 = (a²+a+1)(a²-a+1)

    

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☞.  Laws of Indices :-  The laws of indices are:-

     (i) a^mxaⁿ  = a^m+n

     (ii) aman=am−n$\frac{{\mathrm{<br>}}^{}}{}$
$\frac{{\mathrm{<br>}}^{}}{}$      (iii)  (a^m)ⁿ  =  a^mn

    (iv) (a^mbⁿ)^p =   a^mp b^np

    (v) a^-m = 1am

   (vi)  a = 1 (a ≠ 0)

  (vii)  ⁿ√aᵐ = aᵐ/ⁿ 

  (viii) (ab)ᵐ = aᵐ x bⁿ

  (ix) (a/b)ᵐ = aᵐ/bⁿ

  (x) If aᵐ = bᵐ (m ≠ 0), then a = b

  (xi) If aᵐ = aⁿ then m = n

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☞. Rules of Zero:-The law of Zero are:-

   (1)  a¹  = 1

   (2) a⁰ = 1

   (3) a*0 = 0

   (4) a0a⁰  is undefined

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☞. Surds :-

(i) The surd conjugate of √a + √b (or a + √b) is √a - √b (or a - √b) and conversely.

(ii) If a is rational, √b is a surd and a + √b (or, a - √b) = 0 then a = 0 and b = 0.

(iii) If a and x are rational, √b and √y are surds and a + √b = x + √y then a = x and b = y

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☞. Arithmetical Progression (A.P.):-

(i) The general form of an A. P. is a, a + d, a + 2d, a + 3d,..... where a is the first term and d, the common difference of the A.P.

(ii) The nth term of the above A.P. is t₀ = a + (n - 1)d.

(iii) The sum of first n terns of the above A.P. is s = n/2 (a + l) = (No. of terms/2)[1st term + last term] or, S = ⁿ/₂ [2a + (n - 1) d]

(iv) The arithmetic mean between two given numbers a and b is (a + b)/2.

(v) 1 + 2 + 3 + ...... + n = [n(n + 1)]/2.

(vi) 1² + 2² + 3² +……………. + n² = [n(n+ 1)(2n+ 1)]/6.

(vii) 1³ + 2³ + 3³ + . . . . + n³ = [{n(n + 1)}/2 ]².

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☞. Geometrical Progression (G.P.) :-

(i) The general form of a G.P. is a, ar, ar², ar³, . . . . . where a is the first term and r, the common ratio of the G.P.

(ii) The n th term of the above G.P. is t₀ = a.rn−1${}^{n-1}$ .

(iii) The sum of first n terms of the above G.P. is S = a ∙ [(1 - rⁿ)/(1 – r)] when -1 < r < 1
or, S = a ∙ [(rⁿ – 1)/(r – 1) ]when r > 1 or r < -1.

(iv) The geometric mean of two positive numbers a and b is √(ab) or, -√(ab).

(v) a + ar + ar² + ……………. ∞ = a/(1 – r) where (-1 < r < 1).

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☞.Theory of Quadratic Equation:-

     ax² + bx + c = 0 ... (1)

(i) Roots of the equation (1) are x = {-b ± √(b² – 4ac)}/2a.

(ii) If α and β be the roots of the equation (1) then,sum of its roots = α + β = - b/a = - (coefficient of x)/(coefficient of x² );and product of its roots = αβ = c/a = (Constant term /(Coefficient of x²).

(iii) The quadratic equation whose roots are α and β is x² - (α + β)x + αβ = 0 i.e. , x² - (sum of the roots) x + product of the roots = 0.

(iv) The expression (b² - 4ac) is called the discriminant of equation (1).

(v) If a, b, c are real and rational then the roots of equation (1) are

(a) real and distinct when b² - 4ac > 0;

(b) real and equal when b² - 4ac = 0;

(c) imaginary when b² - 4ac < 0;

(d) rational when b²- 4ac is a perfect square and

(e) irrational when b² - 4ac is not a perfect square.

(vi) If α + iβ be one root of equation (1) then its other root will be conjugate complex quantity α - iβ and conversely (a, b, c are real).

(vii) If α + √β be one root of equation (1) then its other root will be conjugate irrational quantity α - √β (a, b, c are rational).

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☞. Permutation:-

(i) ⌊n (or, n!) = n (n – 1) (n – 2) ∙∙∙∙∙∙∙∙∙ 3∙2∙1.

(ii) 0! = 1.

(iii) Number of permutations of n different things taken r ( ≤ n) at a time ⁿP₀ = n!/(n - 1)! = n (n – 1)(n - 2) ∙∙∙∙∙∙∙∙ (n - r + 1).

(iv) Number of permutations of n different things taken all at a time = ⁿP₀ = n!.

(v) Number of permutations of n things taken all at a time in which p things are alike of a first kind, q things are alike of a second kind, r things are alike of a third kind and the rest are all different, is ⁿ<span style='font-size: 50%'>!/₀

(vi) Number of permutations of n different things taken r at a time when each thing may be repeated upto r times in any permutation, is nʳ .

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☞. Combination:-

(i) Number of combinations of n different things taken r at a time = ⁿCr = n!r!(n−r)!$\frac{n!}{r!(n-r)!}$

(ii) ⁿP₀ = r!∙ ⁿC₀.

(iii) ⁿC₀ = ⁿCn = 1.

(iv) ⁿCr = ⁿCn - r.

(v) ⁿCr + ⁿCn - 1 = n+1${}^{n+1}$Cr${}_r$

(vi) If p ≠ q and ⁿCp = ⁿCq then p + q = n.

(vii) ⁿCr/ⁿCr - 1 = (n - r + 1)/r.

(viii) The total number of combinations of n different things taken any number at a time = ⁿC₁ + ⁿC₂ + ⁿC₃ + …………. + ⁿC₀ = 2ⁿ – 1.

(ix) The total number of combinations of (p + q + r + . . . .) things of which p things are alike of a first kind, q things are alike of a second kind r things are alike of a third kind and so on, taken any number at a time is [(p + 1) (q + 1) (r + 1) . . . . ] - 1. 

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☞. Logarithm:-

(i) If a^x = M then log_a M = x and conversely. 

(ii) log_a 1 = 0. 

(iii) log_a a = 1. 

(iv) a ^log_am = M. 

(v) log_a MN = log_a M + log_a N. 

(vi) log_a (M/N) = log_a M - log_a N. 

(vii) log_a Mⁿ = n log_a M.

(viii) log_a M = log_b M x log_a b. 

(ix) log_b a x 1og_a b = 1. 

(x) log_b a = 1/log_b a. 

(xi) log_b M = log_b M/log_a b. 

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