QA Formula Sheet

Number System

Indices & Exponents

$a m \times a n = a m+n$
$a m \div a n = a m-n$
$(a m) n = a mn$
$a 1/m = m \sqrta$
$a -m = 1 a m$

Funda Trick

If a^m = b^m (m ≠ 0):
If m is odd → a = b
If m is even → a = ±b

Factor Theory

Let N = x^a · y^b · z^c (where x, y, z are prime factors)
Total Factors (P): $(a+1)(b+1)(c+1)$
Sum of Factors: $x a+1 - 1 x - 1 \cdot y b+1 - 1 y - 1 \cdot z c+1 - 1 z - 1$
Ways as product of two factors:
P/2 (if N is not a perfect square)
(P+1)/2 (if N is a perfect square)

Legendre's Formula

Highest power of prime p in n!:

E p (n!) = ⌊n/p⌋ + ⌊n/p 2 ⌋ + ⌊n/p 3 ⌋ + \dots

Remainders & Modularity

Euler's Theorem: If HCF(M, N) = 1,
$Rem[M φ(N) / N] = 1$ where φ(N) = N(1 − 1/a)(1 − 1/b)…
Fermat's Theorem: If N is prime & HCF(M, N) = 1,
$Rem[M N-1 / N] = 1$
Wilson's Theorem: If N is prime,
$Rem[(N-1)! / N] = N-1$

Base Systems & Digits

Convert Base n to Decimal:
$(pqrst) n = p\cdotn 4 + q\cdotn 3 + r\cdotn 2 + s\cdotn + t$
LCM of Fractions: $LCM of Numerators HCF of Denominators$
HCF of Fractions: $HCF of Numerators LCM of Denominators$

Cyclicity & Last 2 Digits

The 5th power of any number has the same unit digit as the number itself.
Last 2 digits: 24 ^odd = 24 | 24 ^even = 76

Arithmetic

Averages

Weighted Average: $A w = W 1 X 1 + W 2 X 2 + \dots + W n X n W 1 + W 2 + \dots + W n$
AM: (a+b)/2 | GM: √(ab) | HM: 2ab/(a+b)

Funda Trick

AM ≥ GM ≥ HM (equal only if all elements are identical).
For two numbers: GM ² = AM × HM

Percentages & Profit/Loss

Successive Change (a% then b%): $Net change = a + b + ab/100 %$
False Weights: $% Profit = (Claimed Wt. Actual Wt. - 1) \times 100$
Discount: $MP - SP MP \times 100$

Funda Trick

An r% increase can be nullified by a 100r100+r% decrease in the other parameter.

Mixtures & Alligation

Alligation Rule: $Qty 1 Qty 2 = X 2 - X avg X avg - X 1$
Successive Replacement (replaced n times): $Final Qty = Initial Qty \times (1 - Replaced Vol Total Vol) n$

Time, Speed & Distance

Avg Speed (Equal Distance): $2S 1 S 2 / (S 1 + S 2)$
Avg Speed (Equal Time): $(S 1 + S 2) / 2$
Boats: Upstream = B − R | Downstream = B + R

Races & Circular Tracks

Meet at start (speeds a, b): LCM(L/a, L/b)
1st meet anywhere (opposite dir): L / (a+b)
1st meet anywhere (same dir): L / |a−b|

Time & Work

A takes a hrs, B takes b hrs → together: $ab a+b$
A, B, C together: $abc ab+bc+ca$

Pipes & Cisterns

Inlets do positive work (+1/t), Outlets do negative work (−1/t).

Algebra

Algebraic Identities

$a 3 \pm b 3 = (a \pm b)(a 2 \mp ab + b 2)$
$a n - b n = (a-b)(a n-1 + a n-2 b + \dots)$
aⁿ − bⁿ is divisible by (a+b) for even n.
aⁿ + bⁿ is divisible by (a+b) for odd n.
If a+b+c = 0, then: $a 3 + b 3 + c 3 = 3abc$

Quadratic & Cubic Equations

Quadratic (ax²+bx+c = 0):
Sum of roots = −b/a | Product = c/a
Cubic (ax³+bx²+cx+d = 0):
Σα = −b/a
Σαβ = c/a
αβγ = −d/a

Funda Trick

Max/Min of ax ²+bx+c occurs at x = −b/2a.
The extreme value = −Δ/4a (where Δ = b ²−4ac)

Inequalities & Modulus

$|x+y| \leq |x| + |y|$
$|x+y| \geq |x| - |y|$
If (x−a)(x−b) < 0 → a < x < b
If (x−a)(x−b) > 0 → x < a or x > b

Max/Min Logic

If sum x+y = k, product xy is maximum when x = y.
If product xy = k, sum x+y is minimum when x = y.

Functions & Logarithms

Even Function: f(−x) = f(x)
Odd Function: f(−x) = −f(x)
$log(ab) = log a + log b$
Change of base: $log b (a) = log a log b$

Geometry & Trigonometry

Triangle Properties

Area (Heron's): $\sqrt[s(s-a)(s-b)(s-c)]$ where s = (a+b+c)/2
Area (Inradius): r × s
Circumradius: $R = abc 4 \times Area$
Apollonius Theorem: $AB 2 + AC 2 = 2(AD 2 + BD 2)$

Special Triangles

30-60-90: Sides → x : √3·x : 2x
45-45-90: Sides → x : x : √2·x

Quadrilaterals & Polygons

Interior Angle Sum: (n−2) × 180°
Number of Diagonals: $n(n-3) / 2$
Rhombus Area: $(d 1 \times d 2) / 2$
Trapezium Area: $½ \times h \times (a+b)$
Sum of squares of diagonals of a parallelogram = Sum of squares of all four sides.

Circles

Intersecting Chords/Secants: $PA \times PB = PC \times PD$
Tangent-Secant: $PT 2 = PA \times PB$
Direct Common Tangent length: $\sqrt[d 2 - (r 1 -r 2) 2]$
Transverse Common Tangent length: $\sqrt[d 2 - (r 1 +r 2) 2]$

Mensuration (3D) & Coordinates

Cone Volume: ⅓πr²h
Sphere Volume: (4/3)πr³
Slope: $m = (y 2 -y 1) / (x 2 -x 1)$
Sine Rule: $a/sin A = b/sin B = c/sin C = 2R$
Cosine Rule: $cos A = (b 2 +c 2 -a 2) / 2bc$

Funda Trick

Largest sphere in a cube of side a → radius = a/2.
Cube inside a hemisphere of radius r → edge = r√(2/3)

Modern Math

Set Theory & Binomial

$n(A\cupB\cupC) = Σn(A) - Σn(A\capB) + n(A\capB\capC)$
Number of subsets of n elements: 2ⁿ
Binomial Expansion: $(x+1) n = x n + nx n-1 + \dots + 1$
Sum of binomial coefficients: $Σ n C r = 2 n$

Permutations & Combinations

Selection (ⁿC_r): $n! r!(n-r)!$
Arrangement (ⁿP_r): $n! (n-r)!$
Circular Arrangement: (n−1)!
Necklace: (n−1)! / 2
Identical items in r distinct groups: $n+r-1 C r-1$

Derangement Formula

No item occupies its original position:

D n = n! \times (1 - 1/1! + 1/2! - 1/3! + \dots + (-1) n /n!)

Probability

$P(A\cupB) = P(A) + P(B) - P(A\capB)$
Independent Events: $P(A\capB) = P(A) \times P(B)$
Binomial Probability (r successes in n trials): $P = n C r \times p r \times (1-p) n-r$
Odds in favour: Favourable / Unfavourable

Progressions & Series

AP Sum: $S n = (n/2)[2a + (n-1)d]$
GP Sum: $S n = a(1-r n) / (1-r)$
Infinite GP (|r| < 1): $S \infty = a / (1-r)$
Sum of first n natural numbers: $n(n+1) / 2$
Sum of squares: $n(n+1)(2n+1) / 6$
Sum of cubes: $[n(n+1)/2] 2$

You may also be interested in

Number System

Indices & Exponents

Factor Theory

Remainders & Modularity

Base Systems & Digits

Arithmetic

Averages

Percentages & Profit/Loss

Mixtures & Alligation

Time, Speed & Distance

Time & Work

Algebra

Algebraic Identities

Quadratic & Cubic Equations

Inequalities & Modulus

Functions & Logarithms

Geometry & Trigonometry

Triangle Properties

Quadrilaterals & Polygons

Circles

Mensuration (3D) & Coordinates

Modern Math

Set Theory & Binomial

Permutations & Combinations

Probability

Progressions & Series

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