ApexGovtPrep PRO • 37-Year Formula Vault & Revision Compendium
Verified Quantitative Formulas, Golden Grammar Rules & Static GK Master Points (1990 — 2026 TCS Shift Archive)
Formula Vault & Concept Master
31 Quantitative LaTeX Formulas, 30 Golden Rules of English Grammar, and 30 Static GK & Indian Polity modules verified against 37-Year TCS official question papers.
Quant Formulas
Grammar Rules
Static GK Modules
Direct Common Tangent (DCT) & Transverse Common Tangent (TCT)
Where d is the distance between centres, and r₁ and r₂ are the radii of the two circles. DCT always exists if circles do not lie inside one another. TCT exists only if d ≥ (r₁ + r₂).
- •DCT is always LONGER than or equal to TCT.
- •When circles touch externally (d = r₁ + r₂), DCT = 2√(r₁ · r₂) and TCT = 0.
- •Length of common chord when two identical circles pass through each other's centres = √3 · r.
Using (r₁ - r₂) instead of (r₁ + r₂) for Transverse Tangents causes instant -0.5 negative penalty.
Conditional Identity when a + b + c = 0
If (a + b + c) = 0, then a³ + b³ + c³ = 3abc. Also, if a² + b² + c² = ab + bc + ca, then a = b = c.
- •If (x - a)³ + (x - b)³ + (x - c)³ = 3(x - a)(x - b)(x - c), immediately set (x - a) + (x - b) + (x - c) = 0 => 3x = a + b + c.
- •If x + 1/x = k, then x³ + 1/x³ = k³ - 3k, and x² + 1/x² = k² - 2.
- •If x - 1/x = k, then x³ - 1/x³ = k³ + 3k, and x² + 1/x² = k² + 2.
Forgetting the minus sign: (x - 1/x = k) leads to (k³ + 3k), NOT (k³ - 3k).
Maximum and Minimum Values of Trigonometric Expressions
For expressions like a·sin²θ + b·cosec²θ, a·cos²θ + b·sec²θ, or a·tan²θ + b·cot²θ, the Minimum value is always 2√(a·b) when a > 0, b > 0.
- •sin²ⁿθ + cos²ⁿθ always has Max = 1 (at θ = 0° or 90°), and Min occurs at θ = 45°.
- •If sin θ + cos θ = √2 cos θ, then cos θ - sin θ = √2 sin θ.
- •If sec θ + tan θ = x, then sec θ - tan θ = 1/x, giving sec θ = (x² + 1)/(2x) and tan θ = (x² - 1)/(2x).
For a·sin²θ + b·cosec²θ when a < b, the minimum is (a + b), NOT 2√(ab).
Tangent-Secant Theorem & Intersecting Chord Power
If chords AB and CD intersect at P (internally or externally), PA · PB = PC · PD. If PT is a tangent from external point P and PAB is a secant, PT² = PA · PB.
- •Always measure from the intersection point P to both ends of the chord: PA and PB.
- •If P is midpoint of AB in secant, PA · PB = PA².
Writing PA · AB instead of PA · PB is the #1 mistake candidates make.
Incentre, Circumcentre, Orthocentre Angles & Centroid
Where I = Incentre (intersection of angle bisectors), O = Circumcentre (perpendicular bisectors), and H = Orthocentre (altitudes).
- •In a right triangle, Orthocentre is at the 90° vertex, Circumcentre is at the midpoint of hypotenuse (R = Hypotenuse / 2).
- •Euler Line: Orthocentre (H), Centroid (G), and Circumcentre (O) are collinear with HG : GO = 2 : 1.
- •Inradius of right triangle: r = (a + b - c) / 2.
Confusing Incentre angle (90° + A/2) with Circumcentre angle (2A).
Inradius, Circumradius & Heron's Area Formula
Where s = (a + b + c)/2 is semi-perimeter, r is inradius, R is circumradius, and Δ is the area of the triangle.
- •For Equilateral Triangle of side a: Area = (√3/4)a², Inradius r = a/(2√3), Circumradius R = a/√3, R : r = 2 : 1.
- •Distance between Incentre and Circumcentre: d = √(R² - 2Rr) (Euler's Formula).
Forgetting that r = Δ/s applies to ANY triangle (scalene, isosceles, equilateral).
Cyclic Quadrilateral: Ptolemy's & Brahmagupta's Theorems
For a cyclic quadrilateral where all 4 vertices lie on a circle, opposite angles sum to 180°. Product of diagonals equals sum of products of opposite sides.
- •If a cyclic quadrilateral is also circumscribed around a circle, Area Δ = √(abcd).
- •Opposite angles: ∠A + ∠C = 180° and ∠B + ∠D = 180°.
Applying Brahmagupta's formula to non-cyclic quadrilaterals produces incorrect areas.
Internal and External Angle Bisector Theorem & Stewart's Theorem
An angle bisector of a triangle divides the opposite side into segments proportional to the adjacent sides. AD is the internal bisector of ∠A.
- •For external angle bisector meeting BC extended at E: AB / AC = BE / CE.
- •Apollonius Theorem (Median Length): AB² + AC² = 2(AD² + BD²) where AD is the median.
Assuming angle bisector is perpendicular to opposite side (only true in isosceles/equilateral).
Frustum of a Right Circular Cone (Bucket)
Where R is base radius, r is top radius, h is vertical height, and l is slant height.
- •When a cone of height H is cut parallel to base at height h from apex: Volume ratio = (h / H)³.
- •Area of cross-section ratio = (h / H)².
Using (R² + r²) instead of (R² + r² + Rr) in volume loses 2.5 marks.
Sphere, Hemisphere & Solid Melting Conservation
Total surface area of solid hemisphere includes the circular top (2πr² + πr² = 3πr²). When n smaller spheres of radius r are melted to form a sphere of radius R: R³ = n · r³.
- •If radius increases by x%, Volume increases by [3x + 3x²/100 + x³/10000]%.
- •Surface area of sphere = 4πr² (equals CSA of cylinder of same radius & height 2r).
Using 2πr² instead of 3πr² for TOTAL surface area of a solid hemisphere.
Symmetric Quartic & Power Expansions
This identity is heavily tested in TCS SSC Tier 1 and Tier 2 papers. Given (x² + xy + y²) and (x² - xy + y²), adding them gives 2(x² + y²), subtracting gives 2xy.
- •If x⁴ + x²y² + y⁴ = 21 and x² + xy + y² = 7, then x² - xy + y² = 3.
- •Then x² + y² = (7 + 3)/2 = 5, and xy = (7 - 3)/2 = 2.
Forgetting that (x² + y²) is the arithmetic mean of the two factors.
Compound Angle & Product-to-Sum Formulas
If A + B = 45° or 225°, then (1 + tan A)(1 + tan B) = 2 and (cot A - 1)(cot B - 1) = 2.
- •If A + B + C = 180°, tan A + tan B + tan C = tan A · tan B · tan C.
- •sin θ · sin(60° - θ) · sin(60° + θ) = (1/4) sin 3θ.
- •cos θ · cos(60° - θ) · cos(60° + θ) = (1/4) cos 3θ.
- •tan θ · tan(60° - θ) · tan(60° + θ) = tan 3θ.
Forgetting the 1/4 factor in the sin and cos triple-angle product formulas.
Height and Distance Standard Triangle Ratios
Standard ratio multipliers eliminate the need to write full tan θ equations. If angles of elevation of top of tower from two points at distances a and b are complementary, Height h = √(ab).
- •Complementary Angles: h = √(a · b).
- •When moving distance d towards tower, angle changes from 30° to 60°: Height h = (d · √3) / 2.
Measuring distances a and b from each other instead of from the base of the tower.
Euler's Totient Theorem & Fermat's Little Theorem
Where φ(n) = n(1 - 1/p₁)(1 - 1/p₂) is Euler's totient function. Wilson's Theorem: (p - 1)! + 1 is divisible by prime p, i.e., (p - 1)! ≡ -1 ≡ (p - 1) (mod p).
- •To find remainder of a^k ÷ p (where p is prime): Divide power k by (p - 1) and find remainder.
- •Unit digit cycle of 2, 3, 7, 8 is 4. Unit digit cycle of 4, 9 is 2. 0, 1, 5, 6 always stay same.
Applying Fermat's theorem when the base and divisor are not coprime.
Composite Divisibility Rules for 72, 88, 99 & 7-11-13 Oscillators
To test divisibility by 72: Last 3 digits must be divisible by 8, and sum of all digits must be divisible by 9. For 88: Last 3 digits div by 8, and (Sum of odd places - Sum of even places) div by 11.
- •Divisibility by 8: If hundreds digit is EVEN, check last 2 digits. If hundreds digit is ODD, add 4 to last 2 digits and check.
- •A 6-digit number formed by repeating 3 digits (e.g. xyzxyz) is always divisible by 7, 11, 13, and 1001.
- •A 6-digit number formed by repeating 1 digit (e.g. 666666) is divisible by 3, 7, 11, 13, 37.
Finding multiple values of y in divisibility by 8 and picking the minimum when question asks for maximum.
Total, Even, Odd Factors and Sum of Factors of N
Where p, q, r are prime factors. Odd factors = product of (power + 1) of odd primes only. Even factors = Total - Odd factors = a(b+1)(c+1) where p=2.
- •Product of all factors of N = N^(Total Factors / 2).
- •Number of ways to express N as product of two coprime factors = 2^(k-1) where k is number of distinct prime factors.
Counting 1 as a prime factor when prime factorizing.
Trailing Zeroes in n! (Legendre's Formula)
Trailing zeroes are created by pairs of (2 × 5). In factorials, 2s are abundant, so the number of trailing zeroes is determined entirely by the highest power of 5 in n!.
- •Keep dividing n by 5, then divide quotient by 5, and sum all quotients.
- •Highest power of any prime p in n! = ⌊n/p⌋ + ⌊n/p²⌋ + ⌊n/p³⌋ + ...
Adding the original number n into the sum instead of only summing the quotients.
Difference Between CI and SI for 2 Years and 3 Years
Where D₂ is difference between CI and SI for 2 years, D₃ is difference for 3 years at annual rate R% on Principal P.
- •Ratio of D₃ to D₂: D₃ / D₂ = (300 + R) / 100. If D₃/D₂ is given as 31/10, R = 10% immediately!
- •When interest is compounded half-yearly: Rate becomes R/2%, Time becomes 2n.
Using D₃ = 3 · D₂ (only true if R=0, actual has extra PR³/100³ term).
Compound Interest Successive Effective Rate Formula
For standard annual rates: 5% for 2 yrs = 10.25%, 10% for 2 yrs = 21%, 10% for 3 yrs = 33.1%, 20% for 2 yrs = 44%, 20% for 3 yrs = 72.8%.
- •If sum doubles in n years at CI: It becomes 2^k times in (k · n) years.
- •If sum doubles in n years at SI: It becomes k times in (k - 1) · n years.
Applying SI linear multiplication to CI exponential growth.
Dishonest Dealer & Faulty Weight Formula
When a shopkeeper sells at cost price but uses a faulty weight (e.g. 900g instead of 1000g), profit is earned purely on the goods given to the customer.
- •If dealer marks up by m% and uses g grams instead of G grams: Net Multiplier = (1 + m/100) × (G / g).
Dividing the error by 1000g instead of 900g (actual weight given).
Successive Discounts & Marked Price to Cost Price Ratio
The MP/CP ratio formula solves 80% of SSC Tier 1 discount problems instantly without setting variable equations.
- •"Buy x, Get y Free" Equivalent Discount % = [y / (x + y)] · 100%. E.g. Buy 4 Get 1 Free = (1/5) · 100 = 20%.
- •For 3 successive discounts: D_net = 100 - 100(1 - d₁/100)(1 - d₂/100)(1 - d₃/100).
Writing [y / x] instead of [y / (x + y)] in "Buy x Get y Free".
Mean, Third, and Fourth Proportional
In continued proportion a:b = b:c, b is the mean proportional. In a:b = c:d, d is the fourth proportional.
- •If (a + x) / (b + x) = (c + x) / (d + x), then x = (ad - bc) / [(a + d) - (b + c)].
- •Componendo and Dividendo: If a/b = c/d, then (a + b)/(a - b) = (c + d)/(c - d).
Mixing up Third Proportional (needs 2 terms) with Fourth Proportional (needs 3 terms).
Rule of Alligation & Repeated Replacement Formula
Where x is total capacity of vessel, y is quantity drawn out and replaced with water each time, and n is number of replacement operations.
- •Alligation can be applied to Average Speeds, Interest Rates, Profit/Loss % on items, and Class Test Averages.
- •When using alligation on Profit %, the resulting ratio is always the ratio of COST PRICES.
Thinking alligation gives ratio of Selling Prices instead of Cost Prices.
Men-Days-Hours Formula (MDH / W Rule)
Where M = Number of workers, D = Days, H = Hours per day, E = Efficiency, W = Work done (or wages earned, food eaten, trenches dug).
- •If A is x% more efficient than B: Ratio of Efficiency = (100 + x) : 100, Ratio of Days taken = 100 : (100 + x).
- •If A and B can do a work in x days, A alone takes (x + a) days, and B alone takes (x + b) days, then x = √(a · b).
Putting wages/work done in the numerator instead of denominator.
Alternate Day Work & Leakage Pipes
When Inlet Pipe A fills in a hours and Outlet/Leak B empties in b hours (where b > a), net filling time = (ab) / (b - a).
- •Always convert into Total LCM Units of tank capacity.
- •In alternate-cycle problems: Stop the cycle when remaining capacity is LESS THAN or equal to the 1st person's daily output.
Letting the cycle overshoot full tank capacity and then subtracting overfilled water.
Average Speed & Distance When Time Difference is Given
Where s₁ and s₂ are two speeds, and Δt is the difference in arrival times (e.g. 10 min late vs 5 min early = 15 min difference).
- •For 3 equal distances at speeds x, y, z: Average Speed = (3xyz) / (xy + yz + zx).
- •If late by t₁ and early by t₂: Total time diff Δt = (t₁ + t₂)/60 hours.
- •If late by t₁ and late by t₂: Total time diff Δt = |t₁ - t₂|/60 hours.
Forgetting to convert time in minutes to hours (divide by 60) when speeds are in km/h.
Trains Crossing Objects & Relative Speeds
Use (+) in relative speed when trains move in OPPOSITE directions. Use (-) when moving in the SAME direction. Speed conversion: 1 km/h = 5/18 m/s, 1 m/s = 18/5 km/h.
- •Two trains start at same time from A and B towards each other. After meeting, they reach destinations in T₁ and T₂ hours: Speed Ratio S₁ / S₂ = √(T₂ / T₁).
Multiplying by 18/5 instead of 5/18 when converting km/h to m/s.
Boats and Streams Upstream & Downstream Velocity
Where D = Downstream Speed (u + v), and U = Upstream Speed (u - v). Total round trip time for distance d: T = d/(u+v) + d/(u-v) = (2du) / (u² - v²).
- •If boat takes k times as long to row upstream as to row downstream: u / v = (k + 1) / (k - 1).
Adding stream speed in upstream instead of subtracting.
Distance, Section Formula & Perpendicular Lines
Slope of line ax + by + c = 0 is m = -a/b. Two lines are parallel if m₁ = m₂. Two lines are perpendicular if m₁ · m₂ = -1.
- •Area of triangle with vertices (x₁,y₁), (x₂,y₂), (x₃,y₃): Area = (1/2)|x₁(y₂ - y₃) + x₂(y₃ - y₁) + x₃(y₁ - y₂)|.
- •Centroid of triangle: G = ((x₁+x₂+x₃)/3, (y₁+y₂+y₃)/3).
Forgetting the negative sign in the slope formula of general line m = -a/b.
Empirical Relationship: Mode, Median & Mean
Karl Pearson's empirical formula relates Mode, Median, and Mean for moderately skewed distributions. Standard Deviation σ = √(Variance).
- •Coefficient of Variation (CV) = (Standard Deviation / Mean) × 100%.
- •If every value in dataset is multiplied by k: Mean, Median, Mode and Standard Deviation are all multiplied by k. Variance is multiplied by k².
Writing Mode = 2Median - 3Mean instead of 3Median - 2Mean.
Arithmetic (AP) & Geometric (GP) Progression Sums
Nth term of AP: T_n = a + (n-1)d. Nth term of GP: T_n = a · r^(n-1). 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.
- •Sum of first n ODD numbers = n².
- •Sum of first n EVEN numbers = n(n + 1).
- •Arithmetic Mean (AM) ≥ Geometric Mean (GM) ≥ Harmonic Mean (HM) with GM² = AM · HM.
Applying S_∞ formula when common ratio r ≥ 1 (series diverges).