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Basic Math Formulas
LIST OF IMPORTANT MATH FORMULAS AND RESULTSAlgebra:● Laws of Indices:(i) a^{m} ∙ a^{n} = a^{m + n}(ii) a^{m}/a^{n} (iii) (a^{m})^{n} = a^{mn} (iv) a^{0} = 1 (a ≠ 0). (v) a^{ n} = 1/a^{n} (vi) ^{n}√a^{m} = a^{m/n} (vii) (ab)^{m} = a^{m} ∙ b^{n}. (viii) (a/b) ^{m} = a^{m}/b^{n} (ix) If a^{m} = b^{m} (m ≠ 0), then a = b. (x) If a^{m} = a^{n} then m = n. ● 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. ● Complex Numbers:(i) The symbol z = (x, y) = x + iy where x, y are real and i = √1, is called a complex (or, imaginary) quantity;x is called the real part and y, the imaginary part of the complex number z = x + iy.(ii) If z = x + iy then z = x  iy and conversely; here, z is the complex conjugate of z. (iii) If z = x+ iy then (a) mod. z (or,  z  or,  x + iy  ) = + √(x^{2} + y^{2}) and (b) amp. z (or, arg. z) = Ф = tan^{1} y/x (π < Ф ≤ π). (iv) The modulus  amplitude form of a complex quantity z is z = r (cosф + i sinф); here, r =  z  and ф = arg. z (π < Ф <= π). (v)  z  =  z  = z ∙ z = √ (x^{2} + y^{2}). (vi) If x + iy= 0 then x = 0 and y = 0(x,y are real). (vii) If x + iy = p + iq then x = p and y = q(x, y, p and q all are real). (viii) i = √1, i^{2} = 1, i^{3} = i, and i^{4} = 1. (ix)  z_{1} + z_{2} ≤  z_{1}  +  z_{2} . (x)  z_{1} z_{2}  =  z_{1}  ∙  z_{2} . (xi)  z_{1}/z_{2} =  z_{1} / z_{2} . (xii) (a) arg. (z_{1} z_{2}) = arg. z_{1} + arg. z_{2} + m (b) arg. (z_{1}/z_{2}) = arg. z_{1}  arg. z_{2} + m where m = 0 or, 2π or, ( 2π). (xiii) If ω be the imaginary cube root of unity then ω = ½ ( 1 + √3i) or, ω = ½ (1  √3i) (xiv) ω^{3} = 1 and 1 + ω + ω^{2} = 0 ● Variation:(i) If x varies directly as y, we write x ∝ y or, x = ky where k is a constant of variation.(ii) If x varies inversely as y, we write x ∝ 1/y or, x = m ∙ (1/y) where m is a constant of variation. (iii) If x ∝ y when z is constant and x ∝ z when y is constant then x ∝ yz when both y and z vary. ● 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 t_{n} = 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) 1^{2} + 2^{2} + 3^{2} +……………. + n^{2} = [n(n+ 1)(2n+ 1)]/6. (vii) 1^{3} + 2^{3} + 3^{3} + . . . . + n^{3} = [{n(n + 1)}/2 ]^{2}. ● Geometrical Progression (G.P.) :(i) The general form of a G.P. is a, ar, ar^{2}, ar^{3}, . . . . . 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 t_{n} = a.r^{n  1} . (iii) The sum of first n terms of the above G.P. is S = a ∙ [(1  r^{n})/(1 – r)] when 1 < r < 1 or, S = a ∙ [(r^{n} – 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 + ar^{2} + ……………. ∞ = a/(1 – r) where (1 < r < 1). ● Theory of Quadratic Equation :ax^{2} + bx + c = 0 ... (1)(i) Roots of the equation (1) are x = {b ± √(b^{2} – 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 x^{2} ); and product of its roots = αβ = c/a = (Constant term /(Coefficient of x^{2}). (iii) The quadratic equation whose roots are α and β is x^{2}  (α + β)x + αβ = 0 i.e. , x^{2}  (sum of the roots) x + product of the roots = 0. (iv) The expression (b^{2}  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 b^{2}  4ac > 0; (b) real and equal when b^{2}  4ac = 0; (c) imaginary when b^{2}  4ac < 0; (d) rational when b^{2} 4ac is a perfect square and (e) irrational when b^{2}  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). ● 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 ^{n}P_{r} = n!/(n  1)! = n (n – 1)(n  2) ∙∙∙∙∙∙∙∙ (n  r + 1). (iv) Number of permutations of n different things taken all at a time = ^{n}P_{n} = 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!}/_{(p!q!r!)} (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 n^{r} . ● Combination:(i) Number of combinations of n different things taken r at a time = ^{n}C_{r} = ^{n!}/_{(r!(n – r)!)}.(ii) ^{n}P_{r} = r!∙ ^{n}C_{r}. (iii) ^{n}C_{0} = ^{n}C_{n} = 1. (iv) ^{n}C_{r} = ^{n}C_{n  r}. (v) ^{n}C_{r} + ^{n}C_{n  1} = ^{n + 1}C_{r} (vi) If p ≠ q and ^{n}C_{p} = ^{n}C_{p} then p + q = n. (vii) ^{n}C_{r}/^{n}C_{r  1}= (n  r + 1)/r. (viii) The total number of combinations of n different things taken any number at a time = ^{n}C_{1} + ^{n}C_{2} + ^{n}C_{3} + …………. + ^{n}C_{n} = 2^{n} – 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. ● Binomial Theorem:(i) Statement of Binomial Theorem : If n is a positive integer then(a + x)^{n} = a^{n} + ^{n}C_{1} a^{n  1} x + ^{n}C_{2} a^{n  2} x^{2} + …………….. + ^{n}C_{r} a^{n  r} x^{r} + ………….. + x^{n} …….. (1) (ii) If n is not a positive integer then (1 + x)^{n} = 1 + nx + [n(n  1)/2!] x^{2} + [n(n  1)(n  2)/3!] x^{3} + ………… + [{n(n1)(n2)………..(nr+1)}/r!] x^{r}+ ……………. ∞ (1 < x < 1) ………….(2) (iii) The general term of the expansion (1) is (r+ 1)th term = t_{r + 1} = ^{n}C_{r} a^{n  r} x^{r} (iv) The general term of the expansion (2) is (r + 1) th term = t_{r + 1} = [{n(n  1)(n  2)....(n  r + l)}/r!] ∙ x^{r}. (v) There is one middle term is the expansion ( 1 ) when n is even and it is (n/2 + 1)th term ; the expansion ( I ) will have two middle terms when n is odd and they are the {(n  1)/2 + 1} th and {(n  1)/2 + 1} th terms. (vi) (1  x)^{1} = 1 + x + x^{2} + x^{3} + ………………….∞. (vii) (1 + x)^{1} = I  x + x^{2}  x^{3} + ……………∞. (viii) (1  x)^{2} = 1 + 2x + 3x^{2} + 4x^{3} + . . . . ∞ . (ix) (1 + x)^{2} = 1  2x + 3x^{2}  4x^{3} + . . . . ∞ . ● 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 ^{logam} = 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} M^{n} = 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. ● Exponential Series:(i) For all x, e^{x} = 1 + x/1! + x^{2}/2! + x^{3}/3! + …………… + x^{r}/r! + ………….. ∞.(ii) e = 1 + 1/1! + 1/2! + 1/3! + ………………….. ∞. (iii) 2 < e < 3; e = 2.718282 (correct to six decimal places). (iv) a^{x} = 1 + (log_{e} a) x + [(log_{e} a)^{2}/2!] ∙ x^{2} + [(log_{e} a)^{3}/3!] ∙ x^{3} + …………….. ∞. ● Logarithmic Series:(i) log_{e} (1 + x) = x  x^{2}/2 + x^{3}/3  ……………… ∞ (1 < x ≤ 1).(ii) log_{e} (1  x) =  x  x^{2}/ 2  x^{3}/3  ………….. ∞ ( 1 ≤ x < 1). (iii) ½ log_{e} [(1 + x)/(1  x)] = x + x^{3}/3 + x^{5}/5 + ……………… ∞ (1 < x < 1). (iv) log_{e} 2 = 1  1/2 + 1/3  1/4 + ………………… ∞. (v) log_{10} m = µ log_{e} m where µ = 1/log_{e} 10 = 0.4342945 and m is a positive number. ● Formula


