Practice
Formulas
A quick-revision library of CAT formulas — search or filter by topic.
Topic
Simplification IdentitiesAlgebra
Square of a sum(a + b)2 = a2 + b2 + 2ab
Square of a difference(a − b)2 = a2 + b2 − 2ab
Sum/difference of squares link(a + b)2 = (a − b)2 + 4ab
Difference of squaresa2 − b2 = (a − b)(a + b)
Sum of cubesa3 + b3 = (a + b)(a2 − ab + b2)
Difference of cubesa3 − b3 = (a − b)(a2 + ab + b2)
Sum of squares (alternate form)a2 + b2 = 12[(a + b)2 + (a − b)2]
Quadratic EquationsAlgebra
General formax2 + bx + c = 0, a ≠ 0α, β are the two roots
Roots (quadratic formula)α, β = −b ± √(b2 − 4ac)2a
Sum of rootsα + β = −ba
Product of rootsα × β = ca
Reconstructing the equationx2 − (sum of roots)·x + (product of roots) = 0
Minimum value (a > 0)min = 4ac − b24a, at x = −b2a
Maximum value (a < 0)max = 4ac − b24a, at x = −b2a
Cubic & Higher-Degree EquationsAlgebra
Cubic — sum of rootsα + β + γ = −bafor ax³ + bx² + cx + d = 0
Cubic — sum of products, two at a timeαβ + βγ + γα = ca
Cubic — product of rootsαβγ = −da
n-degree — sum of rootsΣroots = −coeff. of xn−1coeff. of xnfor a degree-n polynomial equation
n-degree — sum of roots taken 2 at a time= coeff. of xn−2coeff. of xn
n-degree — sum of roots taken 3 at a time= −coeff. of xn−3coeff. of xn
n-degree — sum of roots taken r at a time= (−1)r coeff. of xn−rcoeff. of xn
n-degree — product of roots= (−1)n constant termcoeff. of xn
Remainder Theoremf(x) ÷ (x − a) leaves remainder f(a)
Factor Theorem(x − a) is a factor of f(x) ⇔ f(a) = 0
Linear Equations in Two VariablesAlgebra
System: ax+by=c, px+qy=ra/p = b/q = c/r → infinite solutions (same line)
Inconsistent systema/p = b/q ≠ c/r → no solution (parallel lines)
Unique-solution systema/p ≠ b/q → unique solution (intersecting lines)
Binomial TheoremAlgebra
Expansion(x + y)n = ΣKr xn−r yr, r = 0 to nK₀…Kₙ are the binomial coefficients
General (r+1)ᵗʰ termTr+1 = Kr xn−r yr
Binomial coefficientKr = nCr = n!r!(n − r)!
Exponent rulesum of exponents of x and y in any term = n
Sums of Counting NumbersNumber System
Sum of first n natural numbersn(n + 1)2
Sum of first n odd natural numbersn2
Sum of first n even natural numbersn(n + 1)
Sum of squares of first n natural numbersn(n + 1)(2n + 1)6
Sum of cubes of first n natural numbers[n(n + 1)2]2
LCM & HCFNumber System
LCM–HCF product ruleL × H = a × bfor two numbers a, b with LCM L and HCF H
LCM of fractionsLCM of numeratorsHCF of denominators
HCF of fractionsHCF of numeratorsLCM of denominatorsfractions in lowest form
Least number leaving remainder r(LCM of x, y, z) + r
Series of such numbers(LCM of x, y, z) × n + r
Factors & DivisorsNumber System
1 is not a prime number
Number of factors(m + 1)(n + 1)(p + 1)…N = aᵐ × bⁿ × cᵖ × … with a, b, c distinct primes
Ways as product of 2 factors (N not a perfect square)12{(p+1)(q+1)(r+1)…}p, q, r are the powers of N's prime factors
Ways as product of 2 different factors (N a perfect square)12{(p+1)(q+1)(r+1)… − 1}
Ways as product of 2 factors (N a perfect square)12{(p+1)(q+1)(r+1)… + 1}
Sum of the factors of N[ap+1 − 1a − 1][bq+1 − 1b − 1][cr+1 − 1c − 1]…N = aᵖ × bᵍ × cʳ × …
Totient function φ(N)N(1 − 1/a)(1 − 1/b)(1 − 1/c)…φ(N) = count of numbers ≤ N that are coprime to N; a, b, c are N's prime factors
Euler's theoremPφ(N) mod N = 1P is any natural number coprime to N
Sum of numbers less than & coprime to Nφ(N) × N2
Ways to write N as product of 2 coprime numbers2n − 1n = number of distinct prime factors of N
Product of all factors of NN(number of factors)/2
PercentagesArithmetic
Successive percentage change(a + b + ab100)%two successive changes of a% and b%
A is r% more/less than BB is 100r100 ± r% less/more than A
Simple & Compound InterestArithmetic
Simple InterestSI = P × r × n100P = principal, r% = annual rate, n = years
Compound Interest amountA = P(1 + r100)n
AmountA = Principal + Interest
Population growthP′ = P(1 + r100)nP = original population, P′ = population after n years
CI − SI, 2 years(CI)2 − (SI)2 = Pa2a = r/100
CI − SI, 3 years(CI)3 − (SI)3 = Pa2(a + 3)
Time to become Y times at SIyears = Y − 1X − 1 × Tgiven it becomes X times in T years
Time to become Y times at CIyears = T × n, where Xn = Y
Profit, Loss & DiscountArithmetic
Profit percentageProfit % = ProfitCP × 100
Selling priceSP = CP + P% of CP = CP(1 + P100)
DiscountDiscount = Marked Price − Selling Price
Discount percentageDiscount % = DiscountMarked Price × 100
Equal SP, one at +X% one at −X%net loss = X2100%
Ratio & ProportionArithmetic
Proportiona : b :: c : d ⇒ ad = bc
a < b, x positivea/b < (a+x)/(b+x) and a/b > (a−x)/(b−x)
a > b, x positivea/b > (a+x)/(b+x) and a/b < (a−x)/(b−x)
Componendo Lawif a/b = c/d, then (a+b)/b = (c+d)/d
Dividendo Law(a−b)/b = (c−d)/d
Componendo & Dividendo Law(a+b)/(a−b) = (c+d)/(c−d)
±c/±d ratio identity(a ± c)/(b ± d) = a/b
Chained equal ratios (=K)if a/b = c/d = e/f = K, then (a+c+e)/(b+d+f) = K
Weighted chained ratios(pa + qc + re)/(pb + qd + rf) = Kp, q, r are any weights, not all zero
Alligation, Mixtures & MeansArithmetic
Alligation rulew1 : w2 = (x2 − x) : (x − x1)weights inversely proportional to distance from the mean
Arithmetic MeanAM = x1 + x2 + … + xnn
Geometric MeanGM = ⁿ√(x1 × x2 × … × xn)
Harmonic MeanHM = n1/x1 + 1/x2 + … + 1/xn
Repeated dilutionK = K0(V − xV)nK₀ initial conc., V volume, x volume replaced each time
Time, Speed & DistanceArithmetic
Unit conversion1 km/hr = 5/18 m/s, 1 m/s = 18/5 km/hr
Average speed= Total DistanceTotal Time
Average speed, constant distanceSavg = 2S1S2S1 + S2harmonic mean of the two speeds
Average speed, constant timeSavg = S1 + S22arithmetic mean of the two speeds
Boats & streamsD = B + R, U = B − RD = downstream speed, U = upstream speed, B = boat's speed in still water, R = stream speed
Boat speed / stream speedB = D + U2, R = D − U2
Equal downstream/upstream distanceTimedown/Timeup = Upstream speed / Downstream speed
Clock angleθ = |30H − 112M|H = hours, M = minutes
Triangle PropertiesGeometry & Mensuration
Area — base & heightArea = 12 × b × h
Heron's FormulaArea = √(s(s − a)(s − b)(s − c))s = (a+b+c)/2
Area — included angleArea = 12ab·sin θ
Area — inradiusArea = r × s
Cosine Rulecos θ = a2 + b2 − c22ab
Law of SinesArea = abc4RR = circumradius
Angle Bisector TheoremBD/DC = AB/AC
Apollonius Theoremb2 + c2 = 2m2 + 12a2m = median to side a
Apollonius Theorem (isosceles case)m2 = b2 − a24
Acute triangle relationAC2 = AB2 + BC2 − 2 × BC × BD
Obtuse triangle relationAC2 = AB2 + BC2 + 2 × BC × BD
Right triangle, altitude to hypotenuseAD2 = BD × DC
Right triangle, leg-projection (AB)AB2 = BD × BC
Right triangle, leg-projection (AC)AC2 = CD × BC
PolygonsGeometry & Mensuration
Number of diagonalsn(n − 3)2
Exterior angle (regular polygon)360° / n
Interior angle (convex polygon)180° − 360°/n
Sum of exterior angles360°
Sum of interior angles(n − 2) × 180°
Circle TheoremsGeometry & Mensuration
Intersecting chordsPA × PB = PC × PDP = point where chords AB and CD meet, inside or outside the circle
Tangent–secant relationPA × PB = PT2AB extended to P, PT the tangent from P
Angle at centre vs. circumference∠AOB = 2 × ∠AXBO = centre, X = any point on the major arc
Exterior angle of cyclic quadrilateral∠BCE = ∠DABE is on side DC extended beyond C
Cyclic parallelograma parallelogram inscribed in a circle is always a rectangle/square
Inscribed square/rectangle diagonaldiagonal = diameter of the circle
Common tangent length (direct)√((distance between centres)2 − (r1 − r2)2)r₁, r₂ are the two circles' radii
Common tangent length (transverse)√((distance between centres)2 − (r1 + r2)2)
Alternate Segment Theoremangle between chord & tangent = angle in alternate segment
Ptolemy's TheoremAB × CD + BC × DA = AC × BD
2D MensurationGeometry & Mensuration
RectanglePerimeter = 2(a + b), Area = ab
SquarePerimeter = 4a, Area = a2
TrianglePerimeter = a+b+c, Area = 12bh
Right-angled trianglePerimeter = b+h+d, Area = 12bh, d = √(b2+h2)
Equilateral trianglePerimeter = 3a, Area = √34a2, h = √32a
Isosceles right-angled trianglePerimeter = 2a+d, Area = 12a2, d = a√2
ParallelogramPerimeter = 2(a+b), Area = ah
RhombusPerimeter = 4a, Area = 12d1 × d2d₁, d₂ are the two diagonals
QuadrilateralArea = 12(AC)(h1 + h2)AC = one diagonal; h₁, h₂ = altitudes to AC from the other two vertices
TrapeziumArea = 12h(a + b)
CircleCircumference = 2πr, Area = πr2
SemicirclePerimeter = πr + 2r, Area = 12πr2
Ring (shaded region)Perimeter = 2π(R+r), Area = π(R2 − r2)R = outer radius, r = inner radius
Sector of a circleArc length = (θ/360°) × 2πr, Area = (θ/360°) × πr2θ = central angle of the sector
Segment of a circleArea = r2[πθ360° − sin θ2]θ = angle subtended at the centre by the arc
Pathway across the middleA = w(l + b − w)l = length, b = breadth of the rectangle, w = path width
Pathway outsidePerimeter = 2[l+b+4w], A = 2w(l+b+2w)
Pathway insidePerimeter = 2[l+b−4w], A = 2w(l+b−2w)
Solids (3D Mensuration)Geometry & Mensuration
CuboidTSA = 2(lb+bh+hl), Volume = lbhTSA = total surface area; l, b, h = length, breadth, height
CubeTSA = 6a2, Volume = a3a = edge length
Right prismLSA = perimeter(base) × height, Volume = area(base) × heightLSA = lateral (side) surface area
Right circular cylinderLSA = 2πrh, TSA = 2πr(r+h), Volume = πr2hr = base radius, h = height
Right pyramidLSA = 12 × perimeter(base) × slant height, Volume = 13 × area(base) × height
Right circular coneLSA = πrl, TSA = πr(l+r), Volume = 13πr2h, l = √(r2+h2)l = slant height
SphereTSA = 4πr2, Volume = 43πr3r = radius
HemisphereTSA = 3πr2, Volume = 23πr3
Spherical shellTSA = 4π(R2+r2), Volume = 43π(R3−r3)R = outer radius, r = inner radius
Trigonometric Ratios & IdentitiesTrigonometry
Angle conversionπ radians = 180°
sin θ= Height / Hypotenuse
cos θ= Base / Hypotenuse
tan θ= Height / Base = sin θ / cos θ
Pythagorean identitysin2θ + cos2θ = 1
Secant identity1 + tan2θ = sec2θ
Cosecant identity1 + cot2θ = cosec2θ
Standard Angle ValuesTrigonometry
0°, 30°, 45°, 60°, 90° — sin, cos, tan0°sin = 0, cos = 1, tan = 0
30°sin = 1/2, cos = √3/2, tan = 1/√3
45°sin = 1/√2, cos = 1/√2, tan = 1
60°sin = √3/2, cos = 1/2, tan = √3
90°sin = 1, cos = 0, tan = ∞
Quadrant signs ("School After To College")QI: all + · QII: sin,cosec + · QIII: tan,cot + · QIV: cos,sec +
Arithmetic Progression (AP)Progressions
nth termTn = a + (n − 1)da = first term, d = common difference
Sum of n termsSn = n2[2a + (n − 1)d]
Sum of n terms (first & last)Sn = a + l2 × nl = last term
nth term from sumsTn = Sn − Sn−1
Geometric Progression (GP)Progressions
nth termTn = arn−1a = first term, r = common ratio
Sum of n terms, r < 1Sn = a 1 − rn1 − r
Sum of n terms, r > 1Sn = a rn − 1r − 1
Sum to infinityS∞ = a1 − r
Harmonic Progression & MeansProgressions
nth term of HPTn = 1a + (n − 1)d
Harmonic Mean of two numbersHM = 2aba + b
AM–GM–HM inequalityAM ≥ GM ≥ HM
GM as mean of AM & HM(GM)2 = AM × HM
LogarithmsModern Maths
Product ruleloga(XY) = logaX + logaYa = base, X and Y = the values being logged
Quotient ruleloga(X/Y) = logaX − logaY
Power ruleloga(Xk) = k·logaX
Root in baselogakX = 1klogaX
Root of argumentloga(ᵏ√X) = 1klogaX
Fractional-power baseloga1/kX = k·logaX
Identity of 1loga1 = 0
Same base identitylogxX = 1
Reciprocal base swaplogaX = 1logXa
Change of baselogaX = logbXlogba
Exponent inversea(logaX) = X
Default base conventionunmentioned base is taken as 10
Permutations & CombinationsModern Maths
PermutationsnPr = n!(n − r)!
CombinationsnCr = n!(n − r)! r!
Combination–permutation linknCr = nPrr!
Symmetry of combinationsnCr = nCn−r
Sum of all combinationsnC0 + nC1 + … + nCn = 2n
Distributing n identical items among r personsn+r−1Cr−1
Permutations with repeated itemsn!p! q! r!p, q, r items of each repeated kind
Selections from grouped items{(p+1)(q+1)(r+1)…} − 1selecting some or all from p items of one type, q of another, r of another…
Circular arrangements(n − 1)!
Dividing (m+n) things into groups of m, n(m + n)!m! n!
Dividing 2m things into two equal groups(2m)!2!(m!)2
ProbabilityModern Maths
Probability of an eventP(E) = favourable outcomesall possible outcomesP(E) = probability that event E occurs
Complement ruleP(not E) = 1 − P(E)
Odds in favour= favourable cases / unfavourable cases
Odds against= unfavourable cases / favourable cases
Addition Law (general)P(A or B) = P(A) + P(B) − P(A ∩ B)
Addition Law (mutually exclusive)P(A ∩ B) = 0
Independent eventsP(A ∩ B) = P(A) × P(B)
Multiplication LawP(E and F) = P(E) × P(F)E, F independent
FundamentalsCoordinate Geometry
Distance formula√((x2 − x1)2 + (y2 − y1)2)between points (x₁,y₁) and (x₂,y₂)
Area of a triangle (vertices)12|x1(y2−y3) + x2(y3−y1) + x3(y1−y2)|(x₁,y₁), (x₂,y₂), (x₃,y₃) are the triangle's vertices
Section formula (internal/external)(mx2 ± nx1m ± n, my2 ± ny1m ± n)point dividing (x₁,y₁)–(x₂,y₂) in the ratio m:n; + for internal, − for external division
Mid-point(x1+x22, y1+y22)
Centroid of a triangle(x1+x2+x33, y1+y2+y33)
Slope of a linem = y2 − y1x2 − x1
Parallel linesm1 = m2
Perpendicular linesm1m2 = −1
Equations of a LineCoordinate Geometry
Slope–intercept formy = mx + c
Point–slope formy − y1 = m(x − x1)
Two-point formy − y1 = y2 − y1x2 − x1(x − x1)
Length of perpendicular from a pointp = |ax1 + by1 + c| / √(a2 + b2)from point (x₁,y₁) to the line ax + by + c = 0
Perpendicular distance from origin= |c| / √(a2 + b2)
Distance between two parallel lines= |c2 − c1| / √(a2 + b2)lines ax + by + c₁ = 0 and ax + by + c₂ = 0