Basic Math Formulas and Algebraic Identities
(a+b)2=a2+2ab+b2 |
(a–b)3=a3–b3–3ab(a–b) |
Laws of Indices:
(i) aᵐ ∙ aⁿ = aᵐ + ⁿ
(ii) aᵐ/aⁿ = aᵐ - ⁿ
(iii) (aᵐ)ⁿ = aᵐⁿ
(iv) a = 1 (a ≠ 0).
(v) a-ⁿ = 1/aⁿ
(vi) ⁿ√aᵐ = aᵐ/ⁿ
(iii) (aᵐ)ⁿ = aᵐⁿ
(iv) a = 1 (a ≠ 0).
(v) a-ⁿ = 1/aⁿ
(vi) ⁿ√aᵐ = aᵐ/ⁿ
(vii) (ab)ᵐ = aᵐ ∙ bⁿ.
(viii) (a/b)ᵐ = aᵐ/bⁿ
(ix) If aᵐ = bᵐ (m ≠ 0), then a = b.
(x) If aᵐ = aⁿ then m = n.
(viii) (a/b)ᵐ = aᵐ/bⁿ
(ix) If aᵐ = bᵐ (m ≠ 0), then a = b.
(x) If aᵐ = aⁿ then m = n.
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 | ) = + √(x² + y²) and
(b) amp. z (or, arg. z) = Ф = tan 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 = √ (x² + y²).
(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, i² = -1, i³ = -i, and i⁴ = 1.
(ix) | z₁ + z₂| ≤ | z₁ | + | z₂ |.
(x) | z₁ z₂ | = | z₁ | ∙ | z₂ |.
(xi) | z₁/z₂| = | z₁ |/| z₂ |.
(xii) (a) arg. (z₁ z₂) = arg. z₁ + arg. z₂ + m
(b) arg. (z₁/z₂) = arg. z₁ - arg. z₂ + 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) ω³ = 1 and 1 + ω + ω² = 0
(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 | ) = + √(x² + y²) and
(b) amp. z (or, arg. z) = Ф = tan 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 = √ (x² + y²).
(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, i² = -1, i³ = -i, and i⁴ = 1.
(ix) | z₁ + z₂| ≤ | z₁ | + | z₂ |.
(x) | z₁ z₂ | = | z₁ | ∙ | z₂ |.
(xi) | z₁/z₂| = | z₁ |/| z₂ |.
(xii) (a) arg. (z₁ z₂) = arg. z₁ + arg. z₂ + m
(b) arg. (z₁/z₂) = arg. z₁ - arg. z₂ + 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) ω³ = 1 and 1 + ω + ω² = 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.
(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 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.
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.
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.r .
(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).
(ii) The n th term of the above G.P. is t₀ = a.r .
(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).
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).
(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).
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 .
(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 .
Combination:
(i) Number of combinations of n different things taken r at a time = ⁿCr =
(ii) ⁿP₀ = r!∙ ⁿC₀.
(iii) ⁿC₀ = ⁿCn = 1.
(iv) ⁿCr = ⁿCn - r.
(v) ⁿCr + ⁿCn - 1 = C
(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.
(ii) ⁿP₀ = r!∙ ⁿC₀.
(iii) ⁿC₀ = ⁿCn = 1.
(iv) ⁿCr = ⁿCn - r.
(v) ⁿCr + ⁿCn - 1 = C
(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.
● Binomial Theorem:
(i) Statement of Binomial Theorem : If n is a positive integer then
(a + x)n = an + nC1 an - 1 x + nC2 an - 2 x2 + …………….. + nCr an - r xr + ………….. + xn …….. (1)
(ii) If n is not a positive integer then
(1 + x)n = 1 + nx + [n(n - 1)/2!] x2 + [n(n - 1)(n - 2)/3!] x3 + ………… + [{n(n-1)(n-2)………..(n-r+1)}/r!] xr+ ……………. ∞ (-1 < x < 1) ………….(2)
(iii) The general term of the expansion (1) is (r+ 1)th term
= tr + 1 = nCr an - r xr
(iv) The general term of the expansion (2) is (r + 1) th term
= tr + 1 = [{n(n - 1)(n - 2)....(n - r + l)}/r!] ∙ xr.
(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 + x2 + x3 + ………………….∞.
(vii) (1 + x)-1 = I - x + x2 - x3 + ……………∞.
(viii) (1 - x)-2 = 1 + 2x + 3x2 + 4x3 + . . . . ∞ .
(ix) (1 + x)-2 = 1 - 2x + 3x2 - 4x3 + . . . . ∞ .
(a + x)n = an + nC1 an - 1 x + nC2 an - 2 x2 + …………….. + nCr an - r xr + ………….. + xn …….. (1)
(ii) If n is not a positive integer then
(1 + x)n = 1 + nx + [n(n - 1)/2!] x2 + [n(n - 1)(n - 2)/3!] x3 + ………… + [{n(n-1)(n-2)………..(n-r+1)}/r!] xr+ ……………. ∞ (-1 < x < 1) ………….(2)
(iii) The general term of the expansion (1) is (r+ 1)th term
= tr + 1 = nCr an - r xr
(iv) The general term of the expansion (2) is (r + 1) th term
= tr + 1 = [{n(n - 1)(n - 2)....(n - r + l)}/r!] ∙ xr.
(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 + x2 + x3 + ………………….∞.
(vii) (1 + x)-1 = I - x + x2 - x3 + ……………∞.
(viii) (1 - x)-2 = 1 + 2x + 3x2 + 4x3 + . . . . ∞ .
(ix) (1 + x)-2 = 1 - 2x + 3x2 - 4x3 + . . . . ∞ .
● Logarithm:
(i) If ax = M then loga M = x and conversely.
(ii) loga 1 = 0.
(iii) loga a = 1.
(iv) a logam = M.
(v) loga MN = loga M + loga N.
(vi) loga (M/N) = loga M - loga N.
(vii) loga Mn = n loga M.
(viii) loga M = logb M x loga b.
(ix) logb a x 1oga b = 1.
(x) logb a = 1/logb a.
(xi) logb M = logb M/loga b.
(ii) loga 1 = 0.
(iii) loga a = 1.
(iv) a logam = M.
(v) loga MN = loga M + loga N.
(vi) loga (M/N) = loga M - loga N.
(vii) loga Mn = n loga M.
(viii) loga M = logb M x loga b.
(ix) logb a x 1oga b = 1.
(x) logb a = 1/logb a.
(xi) logb M = logb M/loga b.
Exponential Series:
(i) For all x, ex = 1 + x/1! + x2/2! + x3/3! + …………… + xr/r! + ………….. ∞.
(ii) e = 1 + 1/1! + 1/2! + 1/3! + ………………….. ∞.
(iii) 2 < e < 3; e = 2.718282 (correct to six decimal places).
(iv) ax = 1 + (loge a) x + [(loge a)2/2!] ∙ x2 + [(loge a)3/3!] ∙ x3 + …………….. ∞.
(ii) e = 1 + 1/1! + 1/2! + 1/3! + ………………….. ∞.
(iii) 2 < e < 3; e = 2.718282 (correct to six decimal places).
(iv) ax = 1 + (loge a) x + [(loge a)2/2!] ∙ x2 + [(loge a)3/3!] ∙ x3 + …………….. ∞.
● Logarithmic Series:
(i) loge (1 + x) = x - x2/2 + x3/3 - ……………… ∞ (-1 < x ≤ 1).
(ii) loge (1 - x) = - x - x2/ 2 - x3/3 - ………….. ∞ (- 1 ≤ x < 1).
(iii) ½ loge [(1 + x)/(1 - x)] = x + x3/3 + x5/5 + ……………… ∞ (-1 < x < 1).
(iv) loge 2 = 1 - 1/2 + 1/3 - 1/4 + ………………… ∞.
(v) log10 m = µ loge m where µ = 1/loge 10 = 0.4342945 and m is a positive number.
(ii) loge (1 - x) = - x - x2/ 2 - x3/3 - ………….. ∞ (- 1 ≤ x < 1).
(iii) ½ loge [(1 + x)/(1 - x)] = x + x3/3 + x5/5 + ……………… ∞ (-1 < x < 1).
(iv) loge 2 = 1 - 1/2 + 1/3 - 1/4 + ………………… ∞.
(v) log10 m = µ loge m where µ = 1/loge 10 = 0.4342945 and m is a positive number.
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.
(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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