Basic Math Formulas and Algebraic Identities

(a+b)2=a2+2ab+b2

(a−b)2=a2−2ab+b2

(a+b)(a–b)=a2–b2

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

(x+a)(x–b)=x2+(a–b)x–ab

(x–a)(x+b)=x2+(b–a)x–ab

(x–a)(x–b)=x2–(a+b)x+ab

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

(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 

(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.

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−1${}^{-1}$ 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 = √ (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.

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.

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).

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).

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 . 

 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. 

● Binomial Theorem:

(i) Statement of Binomial Theorem : If n is a positive integer then

(a + x)ⁿ = aⁿ + ⁿC₁ a^{n - 1} x + ⁿC₂ a^{n - 2} x² + …………….. + ⁿC_r a^{n - r} x^r + ………….. + 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!] x^r+ ……………. ∞ (-1 < x < 1) ………….(2) 

(iii) The general term of the expansion (1) is (r+ 1)th term

= t_{r + 1} = ⁿC_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 + 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 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. 

Exponential Series:

(i) For all x, e^x = 1 + x/1! + x²/2! + x³/3! + …………… + 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!] ∙ x² + [(log_e a)³/3!] ∙ x³ + …………….. ∞.

● Logarithmic Series:

(i) log_e (1 + x) = x - x²/2 + x³/3 - ……………… ∞ (-1 < x ≤ 1). 

(ii) log_e (1 - x) = - x - x²/ 2 - x³/3 - ………….. ∞ (- 1 ≤ x < 1). 

(iii) ½ log_e [(1 + x)/(1 - x)] = x + x³/3 + x⁵/5 + ……………… ∞ (-1 < x < 1). 

(iv) log_e 2 = 1 - 1/2 + 1/3 - 1/4 + ………………… ∞. 

(v) log₁₀ m = µ log_e m where µ = 1/log_e 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.