Points:

• Set difference is not commutative
• Union,intersection,symmetric difference are commutative and associative
• Union and intersection are distributive over each other
• Compliment is a involution
• f(x)/(x-k) has remainder of f(k)
• If a polynomial appear to have more roots than its degree then all the coefficients are 0
• Squaring can lead to extraneous roots, dividing can lead to loss of roots
• Domain of a function can help to eliminate extraneous roots
• If a root is common then eliminate the x² term to find it
• Roots with multiplicity = n are also roots of the first n-1 derivative
• Between 2 roots of a polynomial, there exists at least 1 root of the derivative of the polynomial
• The term between any 2 terms in a AP/GP/HP is the mean(AM/GM/HM respectively) of the 2 terms
• Sigma(∑) is a linear operator, it is not distributive over multiplication or division
• When differences of consecutive terms form an AP (in a QP) then the general term is given by T_n = an²+bn+c
• When differences of consecutive terms form a GP then the general term is given by T_n=arⁿ+b
• If i,j are independent variables in ∑∑(T_i*T_j) then, ∑∑(T_i*T_j) = (∑T_i)(∑T_j), this can be generalized to any number of independent variables
• If the general term can be expressed as T_n = ∓(V(n)-V(n-1)) then the series is telescopic i.e., ∑T_n = ∓(V(n)-V(0))
• For V(n)-V(n-a), ∑T_n = V(n)+V(n-1)+...+V(n-a)-[V(0)+V(-1)+...+V(-a+1)]

Common non-formulae:

• |(|x|-|y|)| ≤ |x±y| ≤ |x| + |y|
• Sum of k^th(k≠pn,p is an integer) power of n^th roots of unity = 0
• Product of n^th roots of unity = 1 if n is odd , -1 is n is even
• 1. n^th term of an AP = a + (n-1)d
  2. n^th term of a GP = ar^n-1
  3. n^th term of a HP = 1/(a+(n-1)d)
  4. n^th term of a AGP = [a+(n-1)d]r^(n-1)
• HM ≤ GM ≤ AGM ≤ AM ≤ QM {QM = √(Σx²)/n}
• 1. Sum of first n natural numbers = n(n+1)/2
  2. Sum of squares of the first n natural numbers = n(n+1)(2n+1)/6
  3. Sum of cubes of the first n natural numbers = [n(n+1)/2]²
• Sum of first n powers of a = (a-aⁿ⁺¹)/(1-a)

Conditions for stuff:

Quadratic roots:

• (c₁a₂-a₁c₂)² = (b₁a₂-a₂b₁)(c₁b₂-b₁c₂) for common root in quadratic
• Roots of a quadratic are greater than a value if af(k) > 0 , -b/2a > k , D ≥ 0
• Roots of a quadratic are lesser than a value if af(k) > 0 , -b/2a < k , D ≥ 0
• Roots are on either side of a value if af(k) < 0
• Exactly one root of a quadratic is between 2 values if f(k₁)f(k₂) < 0
• Both roots of a quadratic are between 2 values if af(k₁) > 0 , af(k₂) > 0 , k₁ < -b/2a < k₂ , D ≥ 0
• If one root is bigger than the smaller value and other root is larger than the bigger value then af(k₁) < 0 , af(k₂) < 0

Loci in Argand plane:

• 1. Point inside a line segment ≡ |z-z₁| + |z-z₂| = |z₁+z₂|
  2. Ellipse ≡ |z-z₁| + |z-z₂| = k , k>|z₁-z₂| , e = |z₁-z₂|/k
  3. Ray emitting from one side of a line segment ≡ |z-z₁|-|z-z₂| = |z₁-z₂|
  4. Point on the line but not on the line segment ≡ |(|z-z₁|-|z-z₂|)| = |z₁-z₂|
  5. Hyperbola ≡ |(|z-z₁|-|z-z₂|)| = k , k<|z₁-z₂| , e = |z₁-z₂|/k
• 1. Apollonius circle ≡ |z-z₁|/|z-z₂| = k , k ≠ 0,1
  2. Circle ≡ |z-z₁| = r
  3. Circle with diameter points ≡ |z-z₁|² + |z-z₂|² = |z₁-z₂|²
  4. Circle with diameter points ≡ (z-z₁)(z̅-z̅₂)+(z-z₂)(z̅-z̅₁) = 0
• 1. Line ≡ arg(z-z₁) = θ
  2. Line with 2 points ≡ a̅z + z̅a + b = 0 , a = i(z₂-z₁) , b = i(z₁z̅₂-z̅₁z₂) , slope = -re(a)/im(a)
• Perpendicular bisector ≡ |z-z₁| = |z-z₂|
• Arc ≡ arg([z-z₁]/[z-z₂]) = α
• Concyclic points ≡ [(z₄-z₁)/(z₂-z₁)] *[(z₂-z₃)/(z₄-z₃)] is real
• Equilateral triangle ≡ 1/(z₁-z₂) + 1/(z₂-z₃) + 1/(z₃-z₁) = 0
• Section formula for complex number ≡ (mz₂+nz₁)/(m+n) for internal and (mz₂-nz₁)/(m-n) for external

Formulae:

• Number of power sets = 2ⁿ
• 1. A-B = B'-A'
  2. A⊆B = B'⊆A'
  3. (A∪B)' = A'∩B'
  4. (A∩B)' = A'∪B'
  5. A-(B∪C) = (A-B)∩(A-C)
  6. n(A∪B) = n(A) + n(B) - n(A∩B)
  7. n(A∪B∪C) = n(A) + n(B) + n(C) - n(A∩B) - n(B∩C) - n(C∩A) + n(A∩B∩C)
• √a+bi = ±sqrt[{sqrt(a² + b²)+a}/2] + sgn(b)*isqrt[{sqrt(a² + b²)-a}/2]
• |z₁ ∓ z₂| = |z₁|² + |z₂|² ∓Re(z₁z̅₂)
• Distance = |z₁-z₂|
• 1. Insertion of n AMs , d = (b-a)/n+1 , A_k = a + kd
  2. Insertion of n GMs , r = (b/a)^(1/(n+1)) , G_k = ar^k
  3. Insertion of n HMs , D = (a-b)/[(n+1)ab] , 1/H_k = (1/a) + kD
• 1. Sum of n terms in AP , S = (n/2)[2a+(n-1)d] = (n/2)[a+a_n]
  2. Sum of n terms in GP , S = a(1-rⁿ)/(1-r)
  3. Sum of infinite terms in GP {when |r|<1}, S_∞ = a/(1-r)
  4. Sum of n terms of in AGP , S(1-r) = a + dr(1-r^n-1)/(1-r) - [a+(n-1)d]rⁿ
• ∑_{1≤i≤j≤n}[f(i)*f(j)] = [∑∑(f(i)*f(j))+∑_i=j(f(i)*f(j))]/2

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