Every rule, shortcut and identity for the units we've covered. No worked examples — pure reference for last-minute revision.
a mod n = remainder when a is divided by n
0 ≤ (a mod n) < n (always non-negative, < n)
17 mod 5 = 2 (17 = 3·5 + 2)
38 mod 7 = 3 (38 = 5·7 + 3)
(a + b) mod n = ((a mod n) + (b mod n)) mod n
(a − b) mod n = ((a mod n) − (b mod n) + n) mod n
(a · b) mod n = ((a mod n) · (b mod n)) mod n
(−13) mod 5
= (−13 + 5·3) mod 5 ← add 5 until non-negative
= 2
a ≡ b (mod n) ⟺ n divides (a − b)
⟺ a and b have the SAME remainder when divided by n
17 ≡ 2 (mod 5) because 5 | (17 − 2) = 15
24 ≡ 3 (mod 7) because 7 | (24 − 3) = 21
Reflexive: a ≡ a (mod n)
Symmetric: a ≡ b ⟹ b ≡ a (mod n)
Transitive: a ≡ b and b ≡ c ⟹ a ≡ c (mod n)
If a ≡ b (mod n) and c ≡ d (mod n):
a + c ≡ b + d (mod n)
a − c ≡ b − d (mod n)
a · c ≡ b · d (mod n)
aᵏ ≡ bᵏ (mod n) for any k ≥ 0
Ratio of cheaper to dearer
= (dearer price − mean price)
─────────────────────────────
(mean price − cheaper price)
Cheaper (c) Dearer (d)
\ /
Mean (m)
/ \
(d − m) (m − c)
Ratio of cheaper : dearer = (d − m) : (m − c)
After n equal withdrawals & top-ups with water:
pure_left = initial · (1 − x/V)ⁿ
x = volume removed and replaced each time
V = total volume of the container
n = number of withdrawals
Downstream speed = boat speed + stream speed = b + s
Upstream speed = boat speed − stream speed = b − s
b = (downstream + upstream) ÷ 2
s = (downstream − upstream) ÷ 2
distance = speed × time
time = distance ÷ speed
speed = distance ÷ time
If a pipe fills a tank in t hours,
its rate = 1 / t (tank per hour).
Together, time to fill = (a · b) / (a + b)
(where a, b are the individual times — same shortcut
as the "two people working together" formula.)
Net rate = (1/a) − (1/b)
Time = 1 / net rate = (a · b) / (b − a)
(a = filling time, b = draining time; assumes b > a.)
In the time A finishes the race,
B covers (race length − x) metres.
⟹ speed_A : speed_B = L : (L − x)
where L = race length.
Time taken by A : T_A
Time taken by B : T_A + t
Speed ratio (A : B) = (T_A + t) : T_A
A runs L metres while B runs (L − x) metres
in the same time.
speed_A : speed_B = L : (L − x)
Profit-share ratio = ratio of investments
If A invests Rs P_A and B invests Rs P_B for the SAME duration,
profit(A) : profit(B) = P_A : P_B
Profit-share ratio = ratio of (investment × time)
Equivalent capital = Rs invested × months in business
Use these "capital · months" products as the ratio.
Working partner: takes a fixed salary OR % of profit FIRST,
then the remainder is split in capital ratio.
Sleeping partner: invests only — share is purely by capital ratio.
Add or subtract the same number ─ direction stays the same
Multiply / divide by POSITIVE ─ direction stays the same
Multiply / divide by NEGATIVE ─ FLIP the direction
(< becomes >, ≤ becomes ≥)
Solve 2x − 5 < 7
2x < 12
x < 6 → solution: (−∞, 6)
If a < x < b,
apply ANY allowed operation to ALL THREE parts at once.
−3 < 2x + 1 ≤ 7
−4 < 2x ≤ 6 (subtract 1 from each part)
−2 < x ≤ 3 (divide each part by +2)
A binary operation ∗ on a set S
is a rule that assigns to every ordered pair
(a, b) with a, b ∈ S a UNIQUE element a ∗ b ∈ S.
Commutative: a ∗ b = b ∗ a for all a, b ∈ S
Associative: (a ∗ b) ∗ c = a ∗ (b ∗ c)
Identity e : a ∗ e = e ∗ a = a (unique if it exists)
Inverse a⁻¹: a ∗ a⁻¹ = a⁻¹ ∗ a = e
Addition on ℕ: closed, commutative, associative; identity 0 (not in ℕ⁺); no inverses in ℕ.
Multiplication on ℤ: closed, commutative, associative; identity 1; inverses only for ±1.
Subtraction on ℤ: closed, NOT commutative, NOT associative.
aᵐ · aⁿ = aᵐ⁺ⁿ
aᵐ ÷ aⁿ = aᵐ⁻ⁿ (a ≠ 0)
(aᵐ)ⁿ = aᵐⁿ
(a · b)ⁿ = aⁿ · bⁿ
(a / b)ⁿ = aⁿ / bⁿ (b ≠ 0)
a⁰ = 1 (a ≠ 0)
a⁻ⁿ = 1 / aⁿ (a ≠ 0)
a^(1/n) = ⁿ√a
a^(m/n) = ⁿ√(aᵐ) = (ⁿ√a)ᵐ
√(a · b) = √a · √b
√(a / b) = √a / √b (b > 0)
Rationalise: 1 / (√a + √b) · (√a − √b) / (√a − √b)
= (√a − √b) / (a − b)
log_b x = y ⟺ bʸ = x
(b > 0, b ≠ 1, x > 0)
log₁₀ 1000 = 3 (because 10³ = 1000)
log₂ 32 = 5 (because 2⁵ = 32)
log_b (x · y) = log_b x + log_b y
log_b (x / y) = log_b x − log_b y
log_b (xⁿ) = n · log_b x
log_b 1 = 0
log_b b = 1
log_b x = (log_c x) / (log_c b)
Common bases:
log x = log₁₀ x (common log)
ln x = log_e x (natural log, e ≈ 2.71828)
aₙ = a + (n − 1) · d
a = first term, d = common difference, n = term number.
Sₙ = n/2 · [ 2a + (n − 1) · d ]
= n/2 · ( a + aₙ ) (when last term known)
If a, b, c are in AP, then b = (a + c) / 2.
Sum of first n natural numbers: 1 + 2 + … + n = n(n+1)/2
aₙ = a · rⁿ⁻¹
a = first term, r = common ratio (r ≠ 0).
Sₙ = a · (rⁿ − 1) / (r − 1) if r > 1
Sₙ = a · (1 − rⁿ) / (1 − r) if r < 1
Sₙ = n · a if r = 1
S∞ = a / (1 − r) valid only when |r| < 1
Geometric mean of a and c: b = √(a · c)
(a, b, c in GP ⟺ b² = a · c)
A STATEMENT is a sentence that is either TRUE or FALSE,
but not both.
"7 is a prime number." statement (true)
"Close the door." NOT a statement (command)
"x + 2 = 5" NOT a statement (depends on x)
Negation ¬p "not p"
Conjunction p ∧ q "p and q"
Disjunction p ∨ q "p or q"
Implication p → q "if p then q"
Biconditional p ↔ q "p if and only if q"
p q | p∧q | p∨q | p→q | p↔q
T T | T | T | T | T
T F | F | T | F | F
F T | F | T | T | F
F F | F | F | T | T
∀x P(x) "for ALL x, P(x) is true" (universal)
∃x P(x) "there EXISTS x such that P(x)" (existential)
Negation rule:
¬(∀x P(x)) ≡ ∃x ¬P(x)
¬(∃x P(x)) ≡ ∀x ¬P(x)
Original : p → q "if p then q"
Converse : q → p "if q then p"
Inverse : ¬p → ¬q "if not p then not q"
Contrapositive : ¬q → ¬p "if not q then not p"
¬(p ∧ q) ≡ ¬p ∨ ¬q
¬(p ∨ q) ≡ ¬p ∧ ¬q
¬(p → q) ≡ p ∧ ¬q
¬(p ↔ q) ≡ (p ∧ ¬q) ∨ (¬p ∧ q)
p → q : "If it rains, the match is cancelled."
Converse : If the match is cancelled, it rained.
Inverse : If it does not rain, the match is not cancelled.
Contrapositive : If the match is not cancelled, it did not rain.
Tautology : compound statement ALWAYS true e.g. p ∨ ¬p
Contradiction : compound statement ALWAYS false e.g. p ∧ ¬p
Contingency : sometimes true, sometimes false
Modus ponens : p → q, p ⟹ q
Modus tollens : p → q, ¬q ⟹ ¬p
Hypothetical : p → q, q → r ⟹ p → r
Disjunctive : p ∨ q, ¬p ⟹ q
Direct proof : assume p, deduce q, conclude p → q.
Contrapositive proof : assume ¬q, deduce ¬p, conclude p → q.
Proof by contradiction: assume ¬(claim), derive a contradiction.
Counter-example : ONE case where the claim fails disproves "for all".
A function f : A → B assigns to each x ∈ A
exactly one element f(x) ∈ B.
A = domain (allowed inputs)
B = codomain
range(f) = { f(x) : x ∈ A } ⊆ B
Polynomial : f(x) = aₙxⁿ + … + a₁x + a₀
Rational : f(x) = p(x) / q(x) (q(x) ≠ 0)
Modulus : f(x) = |x|
Exponential : f(x) = aˣ (a > 0, a ≠ 1)
Logarithmic : f(x) = log_a x (a > 0, a ≠ 1, x > 0)
Denominator ≠ 0 (rule out zeros of q(x))
Even root ≥ 0 (radicand non-negative)
log_a x : x > 0
lim f(x) = L
x → a
means: f(x) can be made as close to L as we wish
by taking x close enough to a (but x ≠ a).
lim (f ± g) = lim f ± lim g
lim (f · g) = lim f · lim g
lim (f / g) = (lim f) / (lim g) (lim g ≠ 0)
lim (k · f) = k · lim f
lim (xⁿ − aⁿ) / (x − a) = n · aⁿ⁻¹
x→a
lim (eˣ − 1) / x = 1
x→0
lim (log(1 + x)) / x = 1
x→0
lim (1 + 1/x)ˣ = e
x→∞
f'(x) = lim ( f(x + h) − f(x) ) / h
h→0
Geometrically: slope of the tangent at (x, f(x)).
d/dx [ xⁿ ] = n · xⁿ⁻¹
d/dx [ k ] = 0 (constant)
d/dx [ eˣ ] = eˣ
d/dx [ aˣ ] = aˣ · ln a
d/dx [ ln x ] = 1 / x
d/dx [ log_a x ] = 1 / (x · ln a)
Sum / difference : (f ± g)' = f' ± g'
Constant multiple: (k · f)' = k · f'
Product : (f · g)' = f'·g + f·g'
Quotient : (f / g)' = (f'·g − f·g') / g²
Chain : d/dx [ f(g(x)) ] = f'(g(x)) · g'(x)
MC = dC/dx (rate of change of cost C wrt output x)
MR = dR/dx (rate of change of revenue R wrt output x)
MP = dP/dx (marginal profit, where P = R − C)
Average cost AC = C(x) / x
Average revenue AR = R(x) / x
f'(x) > 0 on an interval ⟹ f is INCREASING there
f'(x) < 0 on an interval ⟹ f is DECREASING there
f'(x) = 0 at x = c ⟹ c is a CRITICAL point
At a critical point c (where f'(c) = 0):
f''(c) > 0 ⟹ local MINIMUM at c
f''(c) < 0 ⟹ local MAXIMUM at c
f''(c) = 0 ⟹ test fails — use first-derivative sign change
Random experiment : an action whose outcome cannot be
predicted with certainty in advance.
Sample space S : the SET of all possible outcomes.
Outcome : a single element of S.
Toss a coin : S = { H, T }
Roll a die : S = { 1, 2, 3, 4, 5, 6 }
Two coins : S = { HH, HT, TH, TT }
n(S) = total number of equally likely outcomes
n(A) = number of outcomes favourable to event A
For two independent stages with m and n outcomes:
total outcomes = m · n (multiplication principle)
Sure event : A = S (P = 1)
Impossible : A = ∅ (P = 0)
Simple : A has exactly one outcome
Compound : A has more than one outcome
Complement A' : all outcomes in S that are NOT in A
P(A) = n(A) / n(S)
= favourable outcomes / total outcomes
(1) 0 ≤ P(A) ≤ 1
(2) P(S) = 1
(3) If A and B are MUTUALLY EXCLUSIVE (A ∩ B = ∅),
P(A ∪ B) = P(A) + P(B)
P(A ∪ B) = P(A) + P(B) − P(A ∩ B)
P(A') = 1 − P(A)
P(A − B) = P(A) − P(A ∩ B)
P(A | B) = P(A ∩ B) / P(B) (P(B) > 0)
Read "probability of A GIVEN B".
Multiplication rule:
P(A ∩ B) = P(B) · P(A | B) = P(A) · P(B | A)
A and B are INDEPENDENT ⟺ P(A ∩ B) = P(A) · P(B)
Equivalently: P(A | B) = P(A) and P(B | A) = P(B)
If B₁, B₂, …, Bₙ partition the sample space S,
P(A) = P(A | B₁)·P(B₁) + P(A | B₂)·P(B₂) + …
+ P(A | Bₙ)·P(Bₙ)
Raw (ungrouped) : individual values x₁, x₂, …, xₙ
Discrete frequency : value xᵢ with frequency fᵢ
Continuous (grouped) : class intervals with class mark xᵢ
Total frequency : N = Σ fᵢ
Class mark : xᵢ = (lower + upper) / 2
Class width : h = upper − lower
cfᵢ = f₁ + f₂ + … + fᵢ
The cf of a class is the count of all observations
up to and including that class. Used in median,
quartile and percentile formulas.
Pie sector angle = (frequency / total) · 360°
Pie sector % = (frequency / total) · 100
Direct : x̄ = Σ(fᵢ · xᵢ) / N
Assumed mean : x̄ = A + Σ(fᵢ · dᵢ) / N dᵢ = xᵢ − A
Step-deviation : x̄ = A + h · ( Σ(fᵢ · uᵢ) / N ) uᵢ = (xᵢ − A)/h
Median = L + h · ( (N/2 − cf) / f )
L = lower limit of the median class
N = total frequency
cf = cumulative frequency BEFORE the median class
f = frequency of the median class
h = class width
Median class: the class containing the (N/2)ᵗʰ observation.
Mode = L + h · ( (f₁ − f₀) / (2f₁ − f₀ − f₂) )
L = lower limit of the modal class (highest frequency)
f₁ = frequency of the modal class
f₀ = frequency of the class BEFORE
f₂ = frequency of the class AFTER
h = class width
Empirical relation: Mode ≈ 3·Median − 2·Mean
Range = Maximum value − Minimum value
For grouped data: Range = upper limit of highest class
− lower limit of lowest class.
MD about mean : MD = Σ fᵢ · | xᵢ − x̄ | / N
MD about median : MD = Σ fᵢ · | xᵢ − M | / N
Coefficient of MD = MD / (mean or median used)
Variance : σ² = Σ fᵢ · ( xᵢ − x̄ )² / N
Std deviation : σ = √ σ²
Shortcut : σ² = Σ fᵢ · xᵢ² / N − x̄²
Coefficient of variation (CV) = ( σ / x̄ ) · 100 (%)
Lower CV ⟹ more consistent data.
Symmetric : Mean = Median = Mode skewness = 0
Positively skewed : Mean > Median > Mode tail on the RIGHT
Negatively skewed : Mean < Median < Mode tail on the LEFT
Sk = ( Mean − Mode ) / σ
= 3 · ( Mean − Median ) / σ (when mode is ill-defined)
Range : −3 ≤ Sk ≤ +3 (in practice usually −1 to +1).
Sk(Bowley) = ( Q₃ + Q₁ − 2·Q₂ ) / ( Q₃ − Q₁ )
Q₂ = median. Used when extreme values make Karl Pearson unreliable.
Range : −1 ≤ Sk(Bowley) ≤ +1.
Qₖ = L + h · ( ( k·N/4 − cf ) / f ) k = 1, 2, 3
Q₁ class: contains (N/4)ᵗʰ observation (lower quartile)
Q₂ class: contains (N/2)ᵗʰ observation (= median)
Q₃ class: contains (3N/4)ᵗʰ observation (upper quartile)
Pₖ = L + h · ( ( k·N/100 − cf ) / f ) k = 1 … 99
P₂₅ = Q₁, P₅₀ = Q₂ = median, P₇₅ = Q₃.
IQR = Q₃ − Q₁
QD = ( Q₃ − Q₁ ) / 2
Coefficient of QD = ( Q₃ − Q₁ ) / ( Q₃ + Q₁ )
SI = ( P · R · T ) / 100
P = principal (Rs)
R = annual rate of interest (%)
T = time (years)
A = P + SI
= P · ( 1 + R·T / 100 )
P = 100 · SI / ( R · T )
R = 100 · SI / ( P · T )
T = 100 · SI / ( P · R )
A = P · ( 1 + R/100 ) ᵀ
CI = A − P
P = principal, R = annual rate (%), T = time (years).
Half-yearly : A = P · ( 1 + R/200 ) ^(2T)
Quarterly : A = P · ( 1 + R/400 ) ^(4T)
Monthly : A = P · ( 1 + R/1200 ) ^(12T)
General m times a year for T years:
A = P · ( 1 + R/(100·m) ) ^(m·T)
If rate changes year-on-year (R₁, R₂, R₃ …):
A = P · (1 + R₁/100) · (1 + R₂/100) · (1 + R₃/100) · …
Population / depreciation:
Growth (rate R%/yr) : A = P · (1 + R/100)ᵀ
Depreciation (rate R%/yr) : A = P · (1 − R/100)ᵀ
Nominal rate r : the stated annual rate, ignoring
how often compounding actually happens.
Per-period rate i = r / m (m = periods per year)
Effective rate = ( 1 + r/m ) ᵐ − 1
r = nominal annual rate (decimal)
m = number of compounding periods per year
Nominal 12% per annum, compounded quarterly:
EAR = ( 1 + 0.12 / 4 )⁴ − 1
= ( 1.03 )⁴ − 1
≈ 0.1255
≈ 12.55 % per annum.
Annuity : a sequence of EQUAL payments PMT made
at EQUAL intervals.
Ordinary annuity : payments at the END of each period.
Annuity due : payments at the BEGINNING of each period.
PMT = periodic payment, i = rate per period,
n = number of periods.
FV = PMT · ( (1 + i)ⁿ − 1 ) / i
Use when saving regularly toward a target (e.g. SIP, recurring deposit).
Each payment grows by compound interest until the end of period n.
PV = PMT · ( 1 − (1 + i)⁻ⁿ ) / i
Use to value a stream of equal future payments today
(e.g. loan EMI calculation, pension valuation).
Annuity due: multiply the FV or PV formula by (1 + i).
Distance between P(x₁, y₁) and Q(x₂, y₂):
d = √( (x₂ − x₁)² + (y₂ − y₁)² )
Section formula — point dividing PQ in ratio m : n
internally : ( (m·x₂ + n·x₁)/(m+n), (m·y₂ + n·y₁)/(m+n) )
midpoint : ( (x₁ + x₂)/2 , (y₁ + y₂)/2 )
Slope m = (y₂ − y₁) / (x₂ − x₁) (x₁ ≠ x₂)
Slope-intercept : y = m·x + c
Point-slope : y − y₁ = m · (x − x₁)
Two-point : (y − y₁) / (y₂ − y₁) = (x − x₁) / (x₂ − x₁)
Intercept form : x/a + y/b = 1
General form : A·x + B·y + C = 0
Parallel : m₁ = m₂
Perpendicular : m₁ · m₂ = −1
Distance from point (x₀, y₀) to line A·x + B·y + C = 0:
d = | A·x₀ + B·y₀ + C | / √( A² + B² )
Distance between parallel lines A·x + B·y + C₁ = 0
and A·x + B·y + C₂ = 0:
d = | C₁ − C₂ | / √( A² + B² )
Centre (h, k), radius r :
(x − h)² + (y − k)² = r²
Centre at origin :
x² + y² = r²
x² + y² + 2·g·x + 2·f·y + c = 0
Centre = ( −g , −f )
Radius = √( g² + f² − c )
Real circle exists only when g² + f² − c > 0.
Let S(x, y) = (x − h)² + (y − k)² − r².
S(x₀, y₀) < 0 ⟹ point is INSIDE the circle
S(x₀, y₀) = 0 ⟹ point is ON the circle
S(x₀, y₀) > 0 ⟹ point is OUTSIDE the circle
Vertex : (0, 0)
Focus : (a, 0)
Directrix : x = −a
Axis : y = 0 (the x-axis)
Latus rectum : 4·a (chord through focus ⟂ axis)
y² = −4·a·x opens LEFT focus (−a, 0), directrix x = a
x² = 4·a·y opens UP focus (0, a), directrix y = −a
x² = −4·a·y opens DOWN focus (0, −a), directrix y = a
In every case the axis passes through the vertex and the focus,
and the latus rectum has length 4·a.
For a point P(x, y) on y² = 4·a·x :
focal distance = x + a
Defining property of a parabola:
distance(P, focus) = distance(P, directrix).