Conditions for stuff:

Quadratic roots:

${(c_{1} a_{2} - a_{1} c_{2})}^{2} = (b_{1} a_{2} - a_{1} b_{2}) (c_{1} b_{2} - c_{2} b_{1}) for common root in quadratic$
$Roots of a quadratic are greater than a value ' k' if: a f (k) > 0, - \frac{b}{2 a} > k, D \geq 0$
$Roots of a quadratic are lesser than a value ' k' if: a f (k) > 0, - \frac{b}{2 a} < k, D \geq 0$
$Roots are on either side of a value ' k' if: a f (k) < 0$
$Exactly one root of a quadratic is between 2 values ' k_{1}',' k_{2}' if: f (k_{1}) f (k_{2}) < 0$
$Both roots of a quadratic are between 2 values ' k_{1}',' k_{2}' if: a f (k_{1}) > 0, a f (k_{2}) > 0, k_{1} < - \frac{b}{2 a} < k_{2}, D \geq 0$
$If one root is bigger than a smaller value ' k_{1}' and other root is larger than a bigger value ' k_{2}' then: a f (k_{1}) < 0, a f (k_{2}) < 0$
In this case, both 'k₁' and 'k₂' are between the roots

Loci in Argand plane:

1. $Point inside a line segment \equiv | z - z_{1} | + | z - z_{2} | = | z_{1} + z_{2} |$
2. $Ellipse \equiv | z - z_{1} | + | z - z_{2} | = k, k > | z_{1} - z_{2} |, e = \frac{| z_{1} - z_{2} |}{k}$
3. $Ray emitting from one side of a line segment \equiv | z - z_{1} | - | z - z_{2} | = | z_{1} - z_{2} |$
4. $Point on the line but not on the line segment \equiv | (| z - z_{1} | - | z - z_{2} |) | = | z_{1} - z_{2} |$
5. $Hyperbola \equiv | (| z - z_{1} | - | z - z_{2} |) | = k, k < | z_{1} - z_{2} |, e = \frac{| z_{1} - z_{2} |}{k}$
1. $Apollonius circle \equiv \frac{| z - z_{2} |}{| z - z_{1} |} = k, k \neq 0, 1$
2. $Circle \equiv | z - z_{1} | = r$
3. $Circle with diametrically opposite points given \equiv {| z - z_{1} |}^{2} + {| z - z_{2} |}^{2} = {| z_{1} - z_{2} |}^{2}$
4. $Circle with diametrically opposite points given \equiv (z - z_{1}) (z - z_{2}) + (z - z_{2}) (z - z_{1}) = 0$
1. $Line \equiv \arg (z - z_{1}) = θ$
2. $Line with 2 points given \equiv a z + z a + b = 0, a = i (z_{2} - z_{1}), b = i (z_{1} z_{2} - z_{1} z_{2}), slope = \frac{- re (a)}{im (a)}$
$Perpendicular bisector \equiv | z - z_{1} | = | z - z_{2} |$
$Arc \equiv \arg (\frac{z - z_{1}}{z - z_{2}}) = α$
$Concyclic points \equiv [\frac{z_{4} - z_{1}}{z_{2} - z_{1}}] [\frac{z_{2} - z_{3}}{z_{4} - z_{3}}] is real$
$Equilateral triangle \equiv \frac{1}{z_{1} - z_{2}} + \frac{1}{z_{2} - z_{3}} + \frac{1}{z_{3} - z_{1}} = 0$
$Section formula for complex number \equiv \frac{m z_{2} + n z_{1}}{m + n} for internal and \frac{m z_{2} - n z_{1}}{m - n} for external$

Means of m^th power:

If m < 0 or m > 1:

\frac{a_{1}^{m} + a_{2}^{m} + a_{3}^{m} + ... + a_{n}^{m}}{n} > {(\frac{a_{1} + a_{2} + a_{3} + ... + a_{n}}{n})}^{m}

If 0 < m < 1:

\frac{a_{1}^{m} + a_{2}^{m} + a_{3}^{m} + ... + a_{n}^{m}}{n} < {(\frac{a_{1} + a_{2} + a_{3} + ... + a_{n}}{n})}^{m}

If m = 0 or m = 1 or a₁ = a₂ = a₃ = ... = a_n:

\frac{a_{1}^{m} + a_{2}^{m} + a_{3}^{m} + ... + a_{n}^{m}}{n} = {(\frac{a_{1} + a_{2} + a_{3} + ... + a_{n}}{n})}^{m}

Permutations & combinations under restrictions:

Particular objects are separated:

n objects in total arranged such that r objects must be separated number of permutations = {}^{n - r + 1}P_{r} * (n - r + 1)!

Particular objects are together:

n objects in total arranged so that a_{1} objects of 1^{st} type, a_{2} objects of 2^{nd} type, ..., a_{k} objects of k^{th} type are together number of permutations = (n - [a_{1} + a_{2} + a_{3} + ... + a_{k}] + k)! * (a_{1})! * (a_{2})! * ... * (a_{k})!

Circular permutations:

Number of circular permutations = \frac{number of linear permutations}{n}, n is number of seats Number of clockwise permutations = number of anti-clockwise permutations = \frac{number of circular permutations}{2}

Selection from identical objects:

Number of ways of selecting r objects from n identical objects = 1 Number of ways of selecting any number of objects from n identical objects = n + 1 Number of ways of selecting at least 1 object from n identical objects = n

Divisors of a number:

Prime factorization of N = \prod [{p_{i}}^{a^{i}}] Number of divisors of N = \prod (a_{i} + 1) Sum of divisors of N = \prod (\frac{{p_{i}}^{a_{i} + 1} - 1}{p_{1} - 1}) Number of ways of writing N as a product of 2 natural numbers = \frac{\prod [a_{i} + 1]}{2} if N is not a perfect square, \frac{\prod [a_{i}]}{2} if N is a perfect square

Division and distribution of distinct objects:

Second degree equation in 2 variables:

{a x}^{2} + 2 h x y + {b y}^{2} + 2 g x + 2 f y + c = 0, represents 2 lines if:

a b c + 2 f g h - {a f}^{2} - {b g}^{2} - {c h}^{2} = 0, h^{2} \geq a b, g^{2} \geq a c, f^{2} \geq b c

Conditions for stuff:

Quadratic roots:

Loci in Argand plane:

Means of mth power:

If m < 0 or m > 1:

If 0 < m < 1:

If m = 0 or m = 1 or a1 = a2 = a3 = ... = an:

Permutations & combinations under restrictions:

Particular objects are separated:

Particular objects are together:

Circular permutations:

Selection from identical objects:

Divisors of a number:

Division and distribution of distinct objects:

Second degree equation in 2 variables:

Means of m^th power:

If m = 0 or m = 1 or a₁ = a₂ = a₃ = ... = a_n: