AI Generated Notes (11/03/2026)

CAT Formula Sheet 2024

Number Systems

Covers formulas for sums of sequences and the division algorithm.

Sum of first n natural numbers = n(n+1)/2
Sum of squares of first n natural numbers = n(n+1)(2n+1)/6
Sum of cubes of first n natural numbers = [n(n+1)/2]^2
Sum of first n odd numbers = n^2
Divideknd = (Divisor x Quotient) + Remainder

Profit, Loss, and Discount

Defines key terms and formulas for calculating profit, loss, and discount percentages.

Profit Percentage = $(Profit/Cost Price) \times 100$
Loss Percentage = $(Loss/Cost Price) \times 100$
Selling Price (SP) = Cost Price (CP) + Profit OR CP – Loss
Cost Price (CP) = SP – Profit OR SP + Loss
Profit = SP – CP
Loss = CP – SP
/

Discount = Marked Price (MP) – SP
Discount Percentage = $(Discount/MP) \times 100$
$MP = SP/(1 – Discount Percentage/100)$
$MP = CP/(1 – Profit Percentage/100)$
$Profit Percent = (Profit \times 100)/C.P.$
$Loss Percent = (Loss \times 100)/C.P.$
$Selling Price = Cost Price \times \frac{(100 + Profit%)}{100}$ OR $Cost Price \times \frac{(100 - Loss%)}{100}$
$Loss% = (x/10)^2$
$Value\ of\ Loss = \frac{2x^2}{100^2 - x^2}$

HCF and LCM

Outlines methods for finding the Highest Common Factor and Lowest Common Multiple.

To find HCF: Break numbers into prime factors; HCF is the product of all common prime factors.
To find LCM: Break numbers into prime factors; LCM is the product of the highest power of all factors present.
HCF of A, B, C is the highest divisor that exactly divides all.
LCM of A, B, C is the lowest dividend exactly divisible by all.
Other number = $(LCM \times HCF) / 1st\ number$
$HCF\ of\ fractions = \frac{HCF\ of\ the\ numerators}{LCM\ of\ the\ Denominators}$
$LCM\ of\ fractions = \frac{LCM\ of\ the\ numerators}{HCF\ of\ the\ Denominators}$

Simple Interest

Defines simple interest and the formula for calculating the total amount.

Simple Interest (SI) = $Prt/100$ — Interest earned on principal P at rate r% over time t.
Amount (A) = $P + SI = P(1 + rt/100)$ — Total value after interest.

Compound Interest (CI)

Covers formulas for compound interest under different compounding periods and the difference between CI and SI.

$CI = A – P$ — Compound interest is the amount minus the principal.
Amount, $A = P(1+r/100)^t$ — Standard annual compounding.
For half-yearly compounding: $A = P(1+\frac{r/2}{100})^{2t}$
For quarterly compounding: (Formula implied as $P(1+\frac{r/4}{100})^{4t}$)
Difference between CI and SI for two years = $p(r/100)^2$
Difference between CI and SI for three years = $p(r/100)^2 (r/100+3)$
Difference for nth year: (Formula implied as interest on previous year's CI)

For depreciation: $V_f = V_i(1 - r/100)^t$, where $V_i$ is initial value, $V_f$ is final value, r is depreciation rate.

Time and Work

Relates work, time, and rate, and provides formulas for combined work rates.

Work Done = Time Taken × Rate of Work
Rate of Work = $1 / Time\ Taken$
Time Taken = $1 / Rate\ of\ Work$
If A does work in n days, A’s 1 day’s work = $1/n$ /
If A’s 1 day’s work = $1/n$, then A finishes work in n days.
If A (x days) and B (y days) work together, work done in one day = $1/x + 1/y$
Time taken by A and B together to complete work = $(xy) / (x + y)$

Speed, Time and Distance

Defines speed and its relationship with time and distance.

Speed = $Distance/Time$
Average Speed = $(Total\ distance\ travelled) / (Total\ time\ taken)$
Distance = Speed × Time
Time = Distance / Speed
Speed is inversely correlated with time and directly correlated with distance.

Averages

Covers the basic average formula and rules for changes to a data set.

Average = $(Sum\ of\ observations) / (Number\ of\ observations)$
If average of n numbers is A, sum = $nA$.
If m numbers (average B) are added, new average = $(nA + mB) / (n + m)$.
If each of n numbers (average A) is increased by x, new average = $(nA + nx) / n = A + x$.
If each of n numbers (average A) is decreased by x, new average = $(nA - nx) / n = A - x$.

Surds and Indices

Lists the fundamental laws of exponents and operations with surds.

Product rule: $a^m × a^n = a^{(m+n)}$
Quotient rule: $a^m / a^n = a^{(m-n)}$
Power rule: $(a^m)^n = a^{(m×n)}$
Negative exponent rule: $a^{-m} = 1 / a^m$
Rational exponent rule: $a^{m/n} = \sqrt[n]{a^m}$
Fractional exponent rule: $a^{p/q} = \sqrt[q]{a^p}$
Surds multiplication: $\sqrt{a} × \sqrt{b} = \sqrt{ab}$
Surds division: $\sqrt{a} / \sqrt{b} = \sqrt{a/b}$
Surds addition: $\sqrt{a} + \sqrt{b} ≠ \sqrt{a+b}$
Surds subtraction: $\sqrt{a} – \sqrt{b} ≠ \sqrt{a-b}$

Logarithm

Defines logarithms and lists key identities and rules.

Definition: If $x>0$ and $b≠1$, then $y = \log_b x$ if and only if $x = b^y$.
Logarithmic identities:
- $\log_b(xy) = \log_b x + \log_b y$
- $\log_b(x/y) = \log_b x – \log_b y$
- $\log_b(x^p) = p \log_b x$
- $\log_b 1 = 0$
- $\log_b b = 1$
- $\log_b(x) = 1 / \log_x(b)$
Change of base: $\log_b(x) = \log_a(x) / \log_a(b)$
Common logarithm: $\log_{10}(x) = \log(x)$
Natural logarithm: $\log_e(x) = \ln(x)$

Set Theory & Function

Covers set operations, cardinality, De Morgan's laws, and key set formulas.

De Morgan’s Law: $(A ∩ B)’ = A’ ∪ B’$ and $(A ∪ B)’ = A’ ∩ B’$
Cardinality: The number of elements in a set.
Union: $A ∪ B = {x: x ∈ A \text{ or } x ∈ B}$
Intersection: $A ∩ B = {x: x ∈ A \text{ and } x ∈ B}$
Complement: $A’ = {x: x ∉ A}$
Difference: $A – B = {x: x ∈ A \text{ and } x ∉ B}$
Cartesian product: $A × B = {(a, b): a ∈ A \text{ and } b ∈ B}$
$n( A ∪ B ) = n(A) + n(B) – n (A ∩ B)$
$n(A∪B)=n(A)+n(B)$ {when A and B are disjoint sets}
$n(U)=n(A)+n(B)–n(A∩B)+n((A∪B)^c)$
$n(A∪B)=n(A−B)+n(B−A)+n(A∩B)$
$n(A−B)=n(A∩B)−n(B)$
$n(A−B)=n(A)−n(A∩B)$
$n(A^c)=n(U)−n(A)$
$n(P∪Q∪R)=n(P)+n(Q)+n(R)–n(P⋂Q)–n(Q⋂R)–n(R⋂P)+n(P⋂Q⋂R)$

COMMUTATIVE LAWS

$A ∪ B = B ∪ A$
$A ∩ B = B ∩ A$

ASSOCIATIVE LAWS

$(A ∪ B) ∪ C = A ∪ (B ∪ C)$
$(A ∩ B) ∩ C = A ∩ (B ∩ C)$

DISTRIBUTIVE LAWS

$A ∪ (B ∩ C) = (A ∪ B) ∩ (A ∪ C)$
$A ∩ (B ∪ C) = (A ∩ B) ∪ (A ∩ C)$

IDENTITY LAWS

$A ∪ ∅ = A$
$A ∩ U = A$

COMPLEMENT LAWS

$A ∪ A^c = U$
$A ∩ A^c = ∅$

DE MORGAN’S LAW

$(A ∪ B)^c = A^c ∩ B^c$
$(A ∩ B)^c = A^c ∪ B^c$

Trigonometry

Defines trigonometric ratios, sum/difference formulas, and reciprocal identities.

Sine = Opposite/ Hypotenuse
Cosine = Adjacent/ Hypotenuse
Tangent = Opposite/ Adjacent
Secant = Hypotenuse/ Adjacent
Cosecant = Hypotenuse/ Opposite
Cotangent = Adjacent/ Opposite
Sum and Difference Formulas:
- $\sin(u + v) = \sin(u)\cos(v) + \cos(u)\sin(v)$
- $\cos(u + v) = \cos(u)\cos(v) – \sin(u)\sin(v)$
- $\sin(u – v) = \sin(u)\cos(v) – \cos(u)\sin(v)$
- $\cos(u – v) = \cos(u)\cos(v) + \sin(u)\sin(v)$
Reciprocal Identities:
- $\csc θ = 1/\sin θ$
- $\sec θ = 1/ \cos θ$
- $\cot θ = 1/\tan θ$
- $\sin θ = 1/\csc θ$
- $\cos θ = 1/\sec θ$
- $\tan θ = 1/\cot θ$

Mensuration

Provides area and perimeter formulas for various 2D shapes.

Mensuration Formulas for Circles

Circumference of circle= $2πr$
Area of circle= $πr^2$
Length of arc= $θ/360×2πr$
Area of Sector= $θ/360×πr^2$
Area of Segment= $θ/360×πr^2–1/2×r^2×\sinθ$

Mensuration Formulas for Triangles

Perimeter of triangle (P) = $a + b + c$
Semi Perimeter (s) = $\frac{a+b+c}{2}$
Area of triangle = $1/2×b×h$ (base & height)
Area of triangle (Heron's) = $\sqrt{s(s–a)(s–b)(s–c)}$

Mensuration Formulas for Quadrilaterals

Area of any Quadrilateral = $\frac{1}{2}×one\ diagonal×(h1+h2)$
Area of any Quadrilateral = $\frac{1}{2}×d1d2×\sinθ$ (d1, d2 are diagonals, θ angle between them)
Area of any Quadrilateral = $\sqrt{(s–a)(s–b)(s–c)(s–d)–abcd×\cos^2α}$, where $α$ = average of a pair of opposite angles, $s=\frac{a+b+c+d}{2}$
Perimeter of Rectangle = $2(length+breadth)$
Area of Rectangle= $length \times breadth$
Perimeter of Square= $4a$
Area of Square = $a^2$
Perimeter of Parallelogram = $2(a+b)$
Area of Parallelogram= $base \times height$
Perimeter of Rhombus = $4a$
Area of Rhombus = $\frac{1}{2}×d1×d2$
Area of Trapezium= $\frac{1}{2}×(a+b)×h$ (a, b parallel sides)

Mensuration Formulas for Polygons

Area of a Regular Polygon= $1/2 (apothem)(perimeter)$
Sum of interior angles = $(n – 2)·180°$
Interior angle (regular polygon) =$\frac{n−2}{n}×180^o$
Exterior angle (regular polygon) =$360/n$

Coordinate Geometry

Lists key formulas for points, lines, distances, and slopes in the coordinate plane.

Distance Formula: Distance between $P(x_1, y_1)$ and $Q(x_2, y_2)$ is $\sqrt{(x_2 – x_1)^2 + (y_2 – y_1)^2}$.
Slope Formula: Slope through $P(x_1, y_1)$ and $Q(x_2, y_2)$ is $m = (y_2 – y_1) / (x_2 – x_1)$.
Equation of a Line: Through $(x_1, y_1)$ with slope m: $y – y_1 = m(x – x_1)$ or $y = mx + (y_1 – mx_1)$.
Midpoint Formula: Midpoint of segment PQ is $[ (x_1 + x_2) / 2, (y_1 + y_2) / 2 ]$.
Section Formula: Point dividing PQ in ratio m:n: $x = (mx_2 + nx_1)/(m+n)$, $y = (my_2 + ny_1)/(m+n)$.
Slope of a Parallel Line: Same as original slope m.
Slope of a Perpendicular Line: $-1/m$.

Permutation & Combination

Provides formulas for counting permutations and combinations under various conditions.

Permutations of n Different Things

Number of permutations of n different things taken all at a time = $^nP_n = n!$
Taken r at a time: $^nP_r = \frac{n!}{(n−r)!}$
Taken at most r at a time: $^nP_1 + ^nP_2 + ... + ^nP_r$
Taken at least r at a time: $^nP_r + ^nP_{r+1} + ... + ^nP_n$
When one particular thing always occurs: $r ⋅ ( ^{n-1}P_{r-1} )$
When one particular thing never occurs: $^{n-1}P_r$
When k particular things always occur: $( ^rP_k )( ^{n-k}P_{r-k} )$
When k particular things never occur: $^{n-k}P_r$
All n things, m specified always together: $m!(n−m+1)!$
All n things, m specified never together: $n! – [m!(n − m +1)!]$

Combination of n Different Things

n things taken r, x things always happen: $^{n-x}C_{r-x}$
n things taken r, x things never happen: $^{n-x}C_r$nn
n things taken r, x always & y never happen: $^{n-x-y}C_{r-x}$
n things taken r, x things cannot be together in any pick: $^{n-x}C_r - x$
Selections of zero or more from n distinct things: $^nC_0 + ^nC_1 + ... + ^nC_n = 2^n$
Selections of one or more from n distinct things: $^nC_1 + ^nC_2 + ... + ^nC_n = 2^n − 1$k

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