What topics are covered in the GRADSKOOL CAT Quant tool?

The CAT Quant Revision tool covers all 34 topics tested in CAT QA — Number Theory, Algebra, Geometry, Mensuration, Arithmetic, Modern Maths (P&C, Probability, Sets), and Data Interpretation. Each topic includes concept notes, formulas, and practice questions.

Is the GRADSKOOL CAT Maths tool free?

Yes — the CAT CAT Quantitative Aptitude Tool tool is completely free. No login, no paywall, no time limit. All 34 topics are accessible immediately.

How is this tool different from CAT mock tests?

Mock tests test your speed and strategy under exam conditions. This tool is for concept revision and targeted practice — use it to strengthen individual topics before applying them in mocks. It is most effective when used alongside GRADSKOOL's CAT mock series.

Can I use this tool for XAT and GMAT preparation as well?

Yes. The QA topics covered — Number Theory, Algebra, Geometry, and Arithmetic — overlap significantly with XAT and GMAT Quantitative sections. The tool is labelled for CAT but the content is directly applicable to all MBA entrance quant sections.

How many questions are in the CAT Quant Revision tool?

The tool covers all 34 QA topics with multiple questions per topic. Total question count exceeds 500, ranging from concept-level to CAT-difficulty application questions.

CAT Quantitative Aptitude Tool — All 34 Topics

Classification of Numbers ▾

Natural Numbers

1, 2, 3, …

Counting numbers. Zero is NOT included.

Whole Numbers

0, 1, 2, 3, …

Natural numbers plus zero.

Integers

…, −2, −1, 0, 1, 2, …

All positives, negatives, and zero. No fractions.

Rational

p/q, q ≠ 0

Decimal either terminates or repeats. e.g. 1/3 = 0.333…

Irrational

Non-terminating, non-repeating

Cannot be p/q. e.g. √2, π, e, φ

Prime

Exactly 2 factors: 1 and itself

2 is the ONLY even prime. 1 is NOT prime.

Composite

More than 2 factors

Smallest composite = 4. 1 is neither prime nor composite.

Co-prime

GCD(a, b) = 1

They share no common factor other than 1. Need not be prime themselves.

Perfect Number

Sum of proper divisors = number

6 = 1+2+3. 28 = 1+2+4+7+14. Next: 496.

Factor Count

N = aᵖbᵍcʳ → (p+1)(q+1)(r+1)

e.g. 12 = 2²×3¹ → (2+1)(1+1) = 6 factors

Sum of Factors

∏ (pᵢᵃⁱ⁺¹ − 1)/(pᵢ − 1)

Geometric series product per prime factor.

Sum of N Naturals

n(n+1)/2

1+2+…+100 = 5050

Sum of N Odds

n²

1+3+5+…+(2n−1) = n²

Sum of N Evens

n(n+1)

2+4+6+…+2n = n(n+1)

Trailing Zeros in n!

⌊n/5⌋ + ⌊n/25⌋ + ⌊n/125⌋ + …

Count factors of 5. e.g. 100!: 20+4 = 24 zeros.

Highest Power of p in n!

⌊n/p⌋ + ⌊n/p²⌋ + …

Legendre's formula. e.g. power of 2 in 32! = 31.

Unit Digit Cyclicity

2,3,7,8 → period 4 | 4,9 → period 2 | 0,1,5,6 → period 1

e.g. 7¹=7, 7²=9, 7³=3, 7⁴=1, then repeats.

Terminating Decimal

p/q terminates iff q = 2ᵃ × 5ᵇ only

e.g. 3/8 = 3/2³ → 0.375 ✓. 1/3 ✗

0.999… = 1

10x − x = 9 ⟹ x = 1

Exact equality, not approximation.

Subset Chain

ℕ ⊂ 𝕎 ⊂ ℤ ⊂ ℚ ⊂ ℝ ⊂ ℂ

Every natural is whole, every whole is integer, etc.

TRICK: n² − m² = prime p ⟹ n−m=1 and n+m=p. So n = (p+1)/2. Works only when p is odd prime.

TRICK: Any number n where n(n+1) is always even — product of consecutive integers is ALWAYS even. Use this to eliminate options instantly.

TRICK: For unit digit of aᵇ — only the unit digit of a matters, and b mod cyclicity gives position. Cyclicity of 2,3,7,8 = 4. Cyclicity of 4,9 = 2. Cyclicity of 0,1,5,6 = 1.

TRICK: n³ − n = n(n−1)(n+1) = product of 3 consecutive integers → always divisible by 6.

TRICK: For any prime p > 3, p² − 1 is always divisible by 24. Memorise this cold.

TRICK: LCM of fractions = LCM of numerators / GCD of denominators. GCD of fractions = GCD of numerators / LCM of denominators.

TRICK: Perfect square ↔ odd number of total factors. Use this to check if a number is a perfect square without computing the root.

TRICK: If GCD(a,b) = G, then a = Gx, b = Gy where GCD(x,y)=1. LCM = Gxy. So LCM × GCD = a × b (only for 2 numbers).

TRAP: 22/7 is RATIONAL (it's p/q). π is IRRATIONAL. Students confuse these — 22/7 is only an approximation of π.

TRICK: To find max power of prime p in n!, use ⌊n/p⌋ + ⌊n/p²⌋ + … Stop when pᵏ > n. This is also the exact number of trailing zeros when p=5 (combined with p=2, but 5 is always the constraint).

10 questions · 1 min 30 sec each · Instant feedback

Divisibility Rules ▾

Div by 2

Last digit ∈ {0,2,4,6,8}

Even last digit. e.g. 4736 ✓

Div by 3

Sum of all digits divisible by 3

e.g. 123 → 1+2+3=6 ✓

Div by 4

Last 2 digits divisible by 4

e.g. 1732 → 32÷4=8 ✓

Div by 5

Last digit is 0 or 5

e.g. 145, 230 ✓

Div by 6

Divisible by BOTH 2 AND 3

Both tests must pass simultaneously.

Div by 7

Double last digit, subtract from rest. Repeat.

e.g. 343 → 34−6=28 → 2−16=−14 ✓

Div by 8

Last 3 digits divisible by 8

e.g. 1256 → 256÷8=32 ✓

Div by 9

Sum of all digits divisible by 9

e.g. 729 → 7+2+9=18 ✓

Div by 10

Last digit is 0

e.g. 230, 1000 ✓

Div by 11

|Alternating digit sum| = 0 or multiple of 11

e.g. 2728 → 2−7+2−8 = −11 ✓

Div by 12

Divisible by BOTH 3 AND 4

Both tests simultaneously.

Div by 25

Last 2 digits divisible by 25

e.g. 1375 → 75÷25=3 ✓

Div by 2ⁿ

Last n digits divisible by 2ⁿ

General rule for all powers of 2.

abba palindrome

Always divisible by 11

a−b+b−a = 0 always.

Fermat's Little Theorem

aᵖ⁻¹ ≡ 1 (mod p), gcd(a,p)=1

p prime. e.g. 2⁶ ≡ 1 (mod 7).

Wilson's Theorem

(p−1)! ≡ −1 (mod p)

For any prime p. e.g. 6! ≡ −1 (mod 7).

Trailing zeros in n!

⌊n/5⌋ + ⌊n/25⌋ + …

Count of 5s in prime factorisation of n!

Remainder shortcut

(a×b) mod m = [(a mod m)(b mod m)] mod m

Break large products into smaller remainders.

TRICK: Divisibility by 7 is slow — use remainder cycling instead. 10 ≡ 3, 100 ≡ 2, 1000 ≡ 6 (mod 7). Multiply digits by these weights from right.

TRICK: 4-digit palindrome abba is ALWAYS divisible by 11 without checking. a−b+b−a = 0. Save time in MCQs.

TRICK: For remainder of 2ⁿ ÷ 7 — cycle is 2,4,1 (period 3). For 2ⁿ ÷ 9 — cycle is 2,4,8,7,5,1 (period 6). Memorise these.

TRICK: If a number is divisible by both 3 and 4, it is divisible by 12. But if divisible by 2 and 6, it is STILL only divisible by 6, not necessarily 12. The factors must be co-prime.

TRICK: n! is divisible by all primes ≤ n. So 7! is NOT divisible by 11 (11 > 7). Instant answer for many CAT questions.

TRICK: Wilson's theorem shortcut — 6! mod 7 = 720 mod 7 = 6 = 7−1. Verify: 6!=720, 720/7=102r6. ✓

TRAP: 23 trailing zeros in n! is IMPOSSIBLE. Zeros jump from 22 (at n=94) to 24 (at n=100) because 100=4×25 adds two 5s at once. Check: any multiple of 25 skips a count.

TRICK: For (a+b) mod m — reduce a and b individually first, then add. For large power expressions, find the pattern after 2-3 terms.

10 questions · 1 min 30 sec each · Instant feedback

Factors and Multiples ▾

Number of Factors

N = p₁ᵃ¹ × p₂ᵃ² × … → (a₁+1)(a₂+1)…

e.g. 360 = 2³×3²×5 → 4×3×2 = 24 factors

Sum of Factors

∏ (pᵢᵃⁱ⁺¹ − 1)/(pᵢ − 1)

e.g. Sum of factors of 12 = (2³−1)/(2−1) × (3²−1)/(3−1) = 7×4 = 28

Product of All Factors

N^(d(N)/2)

d(N) = number of factors. Product of all factor pairs.

HCF × LCM

= a × b (for exactly 2 numbers only)

Does NOT extend to 3 numbers.

LCM of Fractions

LCM(numerators) / GCD(denominators)

GCD of Fractions

GCD(numerators) / LCM(denominators)

Perfect Square ↔ Odd Factors

N is perfect square iff d(N) is odd

Because one factor pairs with itself (the square root).

Co-prime Factor Pairs

2^(number of distinct prime factors)

e.g. 12 = 2²×3 → k=2 → 2²=4 co-prime pairs.

Euler's Totient

φ(N) = N × ∏(1 − 1/p)

Count of integers ≤ N that are co-prime to N. φ(36)=12.

Legendre's Formula

vₚ(n!) = ⌊n/p⌋ + ⌊n/p²⌋ + …

Highest power of prime p dividing n!

Divisor Sum Formula

σ(N) = ∏ (pᵢᵃⁱ⁺¹−1)/(pᵢ−1)

Sum includes N itself.

Perfect Number Condition

σ(N) = 2N

Sum of ALL divisors (including N) = 2N.

TRICK: Number of ODD factors — only use the odd prime factors. Ignore all powers of 2. e.g. Odd factors of 72 = 72÷8=9. 9=3². Odd factors = (2+1)=3.

TRICK: Number of EVEN factors = Total factors − Odd factors. Always faster than recounting.

TRICK: For HCF × LCM = a × b, rearrange as: LCM = a×b/HCF. If HCF and LCM are given and one number, find the other in one step.

TRICK: Two numbers are given in ratio a:b with HCF=H. Then numbers are Ha and Hb. LCM = H×a×b (since gcd(a,b)=1).

TRICK: N has exactly 3 factors ⟺ N = p² for some prime p. The 3 factors are 1, p, p².

TRICK: N has exactly 4 factors ⟺ N = p³ OR N = p×q (distinct primes). Knowing this saves time on factor-count questions.

TRICK: Sum of all factors of pᵃ = (pᵃ⁺¹ − 1)/(p − 1). This is a geometric series 1+p+p²+…+pᵃ.

TRAP: LCM × HCF = a × b is ONLY valid for 2 numbers. For 3 numbers a,b,c: LCM × HCF ≠ a×b×c in general.

TRICK: φ(p) = p−1 for any prime p. φ(pⁿ) = pⁿ − pⁿ⁻¹. Use this for quick totient calculation.

TRICK: Product of all factors of N = N^(d(N)/2). If N=36, d(36)=9. Product = 36^(9/2) = 36^4.5.

10 questions · 1 min 30 sec each · Instant feedback

Percentages ▾

x% of y

xy/100

e.g. 15% of 200 = 30

% Change

(New − Old)/Old × 100

Positive = increase. Negative = decrease.

Successive %

Net = a + b + ab/100

Two consecutive changes a% and b%. e.g. 10%+10% = 21%.

Multiplying Factor

(100+r)/100 for increase | (100−r)/100 for decrease

1.2 = 20% up. 0.8 = 20% down.

Reverse %

Original = Final × 100/(100+r)

Find original before r% increase.

Price↑r% → Qty must ↓

100r/(100+r) %

To keep expenditure constant.

False Weight Profit

(True − False)/False × 100

800g instead of 1kg → 200/800 = 25%.

Same SP Both Items

Net loss = (x/10)² %

One item x% gain, other x% loss at same SP → always net loss.

% to Key Fractions

25%=1/4, 33.33%=1/3, 16.67%=1/6, 12.5%=1/8, 11.11%=1/9

Memorise for CAT speed.

Expenditure = Price × Qty

If P↑a% and Q↓b%, Net = a+b+ab/100 change

Use successive % formula.

TRICK: Two successive discounts of a% and b% → net discount = a+b−ab/100. Same formula as successive %. Remember the sign is MINUS for discounts.

TRICK: If price increases by r%, consumption must decrease by 100r/(100+r)% to keep expenditure same. Quick: 25% rise → 20% cut. 50% rise → 33.33% cut.

TRAP: When two items are sold at the SAME selling price, one at x% profit and one at x% loss, there is ALWAYS a net loss of (x/10)²%. The losses are NOT symmetric.

TRICK: A is r% more than B. B is r/(100+r) × 100 % less than A. e.g. A is 25% more than B → B is 20% less than A. Fraction: 1/4 more means 1/5 less.

TRICK: Candidate A gets p% votes and wins by d votes → total votes = d×100/(2p−100). Memorise this pattern — appears every CAT.

TRICK: Fresh fruit has x% water. After drying it has y% water. Dry weight = Fresh weight × (100−x)/(100−y).

TRICK: Population/value after n years at r% compounded = P×(1+r/100)ⁿ. Same formula as CI. Use for depreciation with minus sign.

10 questions · 1 min 30 sec each · Instant feedback

Ratio and Proportions ▾

Ratio

a : b = a/b

Compare quantities of same kind.

Proportion

a/b = c/d ⟹ ad = bc

Cross-multiplication rule.

Compounded Ratio

(a×c×e) : (b×d×f)

Multiply corresponding terms.

Duplicate Ratio

a² : b²

Square of a:b.

Sub-duplicate Ratio

√a : √b

Square root of a:b.

Triplicate Ratio

a³ : b³

Cube of a:b.

Mean Proportion

b² = ac

b is mean proportion of a and c. b = √(ac).

Third Proportion

c = b²/a

If a:b = b:c, then c = b²/a.

k-method

a/b = c/d = k ⟹ each ratio = k

All equal ratios share the same constant k.

Componendo–Dividendo

(a+b)/(a−b) = (c+d)/(c−d)

From a/b = c/d. Avoids solving for a,b,c,d individually.

If 4A = 5B = 7C

A:B:C = 1/4 : 1/5 : 1/7 = 35:28:20

Take reciprocals when equated to same constant.

TRICK: If a:b = m:n, and a+b = S, then a = Sm/(m+n) and b = Sn/(m+n). Don't cross-multiply — just use this directly.

TRICK: If xA = yB = zC, then A:B:C = 1/x : 1/y : 1/z. Take reciprocals of the coefficients. Fastest method for 4A=5B=7C type questions.

TRICK: Componendo-Dividendo — if a/b = c/d is given and the question asks for (a+b)/(a−b), apply C-D directly without solving for a and b. Saves 40 seconds.

TRICK: Alloy/Mixture ratio problems — use alligation cross method: (Higher−Mean):(Mean−Lower). Always draw the cross mentally.

TRICK: In a:b:c problems — if a:b is given and b:c is given, make b common by LCM, then read off the ratio directly.

TRAP: Ratios can be added only if they refer to the SAME total. a:b = 2:3 does NOT mean a = 2 and b = 3 unless the total is specified.

TRICK: Partnership profit ratio = Ratio of (Capital × Time). If capitals are same, ratio is time. If time is same, ratio is capital.

10 questions · 1 min 30 sec each · Instant feedback

Profit, Loss, Discounts ▾

Profit %

(SP − CP)/CP × 100

Always calculated on Cost Price.

Loss %

(CP − SP)/CP × 100

Always calculated on Cost Price.

SP given Profit %

SP = CP × (100+p)/100

p = profit percent.

CP from SP (profit)

CP = SP × 100/(100+p)

Find original cost.

Discount %

(Discount/MP) × 100

Discount always on Marked Price, NOT CP.

SP after Discount

SP = MP × (100−d)/100

d = discount percent.

Successive Discounts

1 − (1−d₁/100)(1−d₂/100)

Net single equivalent discount.

False Weight Profit

(True − False)/False × 100

Charge for True, deliver False weight.

Same SP Both Items

Net Loss = (x/10)² %

One x% gain + one x% loss at same SP = always loss.

Dishonest Buying + Selling

Profit = [(100+b)(100+s)/100²−1]×100

b% gain while buying, s% gain while selling.

MP from CP

MP = CP × (100+markup)/100

Mark up is always on CP.

Overall Profit in Chain

P_final = CP × ∏(1 + rᵢ/100)

r₁%, r₂%,… profits at each link of chain.

TRICK: If two items are sold at the SAME price — one at x% profit and one at x% loss — net result is ALWAYS a loss of (x/10)²%. No need to calculate. e.g. x=20 → loss = 4%.

TRICK: Profit on cost vs profit on sale. If 'profit is 25% of SP', convert: profit on CP = 25/(100−25) = 33.33%. Memorise: profit x% on SP → CP profit = x/(100−x)×100%.

TRICK: Trader uses false weight w grams instead of W grams and sells at CP. Profit% = (W−w)/w × 100. Note: denominator is the FALSE weight, not true.

TRICK: Shopkeeper gives x articles free with every y purchased. Effective discount% = x/(x+y) × 100.

TRICK: MP is marked r% above CP. Discount d% is offered. Profit = (100+r)(100−d)/100 − 100 %. Quick: if markup = discount = 20%, profit = 0.8×1.2−1 = −4% (loss).

TRAP: Discount is ALWAYS on Marked Price. Profit/Loss is ALWAYS on Cost Price. Mixing these up is the #1 error in this chapter.

TRICK: CP = x, SP = y. Profit% = (y−x)/x×100. If you only know profit% and SP: CP = SP×100/(100+profit%). If you know profit% and CP: SP = CP×(100+profit%)/100.

10 questions · 1 min 30 sec each · Instant feedback

Surds and Indices ▾

Product Rule

aᵐ × aⁿ = aᵐ⁺ⁿ

Same base: add exponents.

Quotient Rule

aᵐ ÷ aⁿ = aᵐ⁻ⁿ

Same base: subtract exponents.

Power Rule

(aᵐ)ⁿ = aᵐⁿ

Power of a power: multiply.

Negative Index

a⁻ⁿ = 1/aⁿ

Negative exponent = reciprocal.

Fractional Index

a^(1/n) = ⁿ√a

nth root expressed as exponent.

Zero Index

a⁰ = 1, a ≠ 0

Any non-zero number to power 0 = 1.

Rationalising

1/(√a+√b) = (√a−√b)/(a−b)

Multiply by conjugate.

Surd Addition

a√c + b√c = (a+b)√c

Only like surds can be added.

(√a + √b)(√a − √b)

= a − b

Difference of surds. Key identity.

√(a/b)

= √a / √b

Square root distributes over division.

If aˣ = bʸ = cᶻ and b² = ac

then 2/y = 1/x + 1/z

Important exam identity linking index and proportion.

Surds on Number Line

√n is between ⌊√n⌋ and ⌈√n⌉

e.g. √50 is between 7 and 8.

TRICK: To simplify √(a ± 2√b), try writing as (√x ± √y)² = x+y ± 2√(xy). So find x+y = a and xy = b. e.g. √(7+2√12) = √(4+3+2√12) = √4 + √3 = 2+√3.

TRICK: 2^(4n+3) always ends in 8. 2^(4n+1) ends in 2. 2^(4n+2) ends in 4. 2^(4n) ends in 6. Memorise the pattern once.

TRICK: If x + 1/x = k, then x² + 1/x² = k²−2 and x³ + 1/x³ = k³−3k. These telescoping identities appear repeatedly in CAT.

TRICK: (a^m)^n = a^(mn) but a^(m^n) ≠ a^(mn). Be careful about bracket placement in tower exponents — evaluate from the top down.

TRICK: Comparing surds — to compare √a + √b vs √c + √d, square both sides. e.g. √3 + √5 vs √2 + √6. Square: 8+2√15 vs 8+2√12. 2√15 > 2√12. So √3+√5 > √2+√6.

TRICK: a^(log_a x) = x. This identity instantly simplifies expressions of the form 2^(log₂ 5) = 5.

10 questions · 1 min 30 sec each · Instant feedback

Linear Equations ▾

Standard Form

ax + b = 0 → x = −b/a

Single variable.

Two Variable System

a₁x+b₁y=c₁ and a₂x+b₂y=c₂

Solve by elimination or substitution.

Unique Solution

a₁/a₂ ≠ b₁/b₂

Lines intersect at one point.

Infinite Solutions

a₁/a₂ = b₁/b₂ = c₁/c₂

Lines are identical (coincident).

No Solution

a₁/a₂ = b₁/b₂ ≠ c₁/c₂

Lines are parallel, never meet.

Cramer's Rule

x = (c₁b₂−c₂b₁)/D, y = (a₁c₂−a₂c₁)/D

D = a₁b₂ − a₂b₁

Age Problems

Present + n = future | Present − n = past

Set up as two equations from two conditions.

Number Problems

10a + b = 2-digit number

Reverse = 10b + a. Difference = 9(a−b).

Speed-Work Framework

Rate × Time = Work

Rate = 1/Time. Combined rates add.

TRICK: For 2-digit number problems — digit sum and difference trick. If sum of digits=S and difference of digits=D: the two digits are (S+D)/2 and (S−D)/2.

TRICK: Reverse of a 2-digit number ab = 10a+b. Reversed = 10b+a. Difference = 9(a−b). Sum = 11(a+b). Memorise both.

TRICK: For age problems — maintain a constant: (Age difference) never changes. A is always 'd' years older than B. Use this as your anchor equation.

TRICK: When two equations are given and the question asks for (x+y) or (x−y) or (2x+3y), don't solve for x and y individually — add or subtract the equations directly to get the desired expression.

TRAP: 'Infinite solutions' does NOT mean the equation is wrong. It means the two equations are the same line. 'No solution' means parallel lines — inconsistent system.

TRICK: For mixture/ratio linear equations — use alligation to avoid algebra entirely. Alligation gives you the ratio of mixing directly.

10 questions · 1 min 30 sec each · Instant feedback

Averages ▾

Arithmetic Mean

Sum / Count

Average = total ÷ number of items.

New Average (add element)

Old avg + (New value − Old avg)/(n+1)

Faster than recalculating from scratch.

New Average (remove element)

Old avg − (Removed value − Old avg)/(n−1)

Missing Value

Total − Sum of known

If average and all but one values are known.

Average Speed (equal dist)

2S₁S₂/(S₁+S₂)

Harmonic mean of two speeds for equal distances.

Weighted Average

Σ(wᵢxᵢ)/Σwᵢ

Each value multiplied by its weight.

Average of AP

(First + Last)/2

Average of any arithmetic progression.

Average of 1 to n

(n+1)/2

e.g. Average of 1 to 99 = 50.

Average of first n odds

n

1,3,5,…,(2n−1). Average = n.

Average of first n evens

n+1

2,4,6,…,2n. Average = n+1.

TRICK: If all values increase by k, average increases by k. If all values multiply by k, average multiplies by k. Use for instant recalculation when a constant is added/multiplied.

TRICK: Batting average problems — new average = old average + (new score − old average)/(new innings). If new score > old average, average rises. If less, it falls.

TRICK: Class average of A and B: (Avg_A × nA + Avg_B × nB)/(nA + nB). Draw the alligation cross to find ratio nA:nB when combined average is given.

TRICK: Average of n consecutive integers starting from a = a + (n−1)/2. Middle term IS the average for any AP.

TRAP: Average speed is NOT (S₁+S₂)/2 unless time is equal. For equal distance, use harmonic mean: 2S₁S₂/(S₁+S₂). This is one of the most frequently wrong answers in CAT.

TRICK: 5 observations average=M. One obs corrected from x to y. New average = M + (y−x)/5. General: New avg = Old avg + (correction)/n.

10 questions · 1 min 30 sec each · Instant feedback

Quadratic Equations ▾

Standard Form

ax² + bx + c = 0

a ≠ 0

Quadratic Formula

x = [−b ± √(b²−4ac)] / 2a

Universal. Works for all quadratics.

Discriminant

Δ = b² − 4ac

Δ>0: 2 real roots. Δ=0: 1 root. Δ<0: no real roots.

Sum of Roots

α + β = −b/a

Vieta's formula.

Product of Roots

αβ = c/a

Vieta's formula.

Nature (Δ = perfect sq)

Roots are rational

If a,b,c are integers.

Form from Roots

x² − (α+β)x + αβ = 0

Build equation if roots are known.

α² + β²

= (α+β)² − 2αβ

Express in terms of sum and product.

α³ + β³

= (α+β)³ − 3αβ(α+β)

Cubic identity using Vieta's.

|α − β|

= √[(α+β)² − 4αβ] = √Δ/|a|

Difference of roots.

TRICK: Don't always solve the quadratic. If asked for α²+β², α³+β³, αβ(α+β), use Vieta's sum and product directly. No need to find α and β individually.

TRICK: If one root is reciprocal of other: αβ = 1 → c/a = 1 → c = a.

TRICK: If one root is negative of other: α+β = 0 → −b/a = 0 → b = 0.

TRICK: If roots are equal: Δ = 0 → b² = 4ac. Check this instead of solving.

TRICK: For quadratic in disguise — if equation has x⁴−5x²+4=0, substitute t=x² to get t²−5t+4=0. Solve for t, then x.

TRAP: x² = 9 gives x = ±3, NOT just x = 3. Always consider both roots unless the domain restricts to positives.

TRICK: Max/min of ax²+bx+c — minimum at x=−b/2a (if a>0), value = c − b²/4a. This is the vertex of the parabola.

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Mixtures and Alligations ▾

Alligation Cross

(Higher − Mean) : (Mean − Lower)

Ratio to mix two ingredients for a desired mean.

Mean Concentration

(C₁n₁ + C₂n₂)/(n₁+n₂)

Weighted average of mixture.

Replacement Formula

Final conc = C × (1 − r/V)ⁿ

n replacements of quantity r from vessel of volume V.

Two Solutions

x/y = (C₂−C)/(C−C₁)

Alligation: amounts inversely proportional to gaps.

Alligation for Price

Cheap : Costly = (Mean−Cheap's price) inverted

Draw the cross: difference from mean gives opposite ratio.

TRICK: Alligation works for ANY quantity — price, concentration, speed, age, marks, salary. Whenever you need to find ratio of mixing for a desired average, draw the cross.

TRICK: Replacement formula (1−r/V)ⁿ — if 1/4 of mixture replaced n times, final original fraction = (3/4)ⁿ. No need to track each step.

TRICK: When two vessels have different concentrations and are mixed in ratio a:b, final concentration = (a×C₁ + b×C₂)/(a+b). Same as weighted average.

TRAP: In replacement problems, the removed liquid is always in the CURRENT ratio of the mixture, not the original. This changes the calculation after each step.

TRICK: Equal quantities mixed — if two mixtures in ratio a:b and p:q are mixed equally, find each component in each mixture, add them, simplify ratio.

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SI and CI ▾

Simple Interest

SI = PRT/100

P=Principal, R=Rate%, T=Time(years).

Amount (SI)

A = P(1 + RT/100)

Total = Principal + SI.

Compound Interest

A = P(1 + R/100)ⁿ

Compounded annually.

CI Half-yearly

A = P(1 + R/200)²ⁿ

Rate halved, time doubled.

CI Quarterly

A = P(1 + R/400)⁴ⁿ

Rate quartered, time×4.

CI − SI (2 years)

= P(R/100)²

Difference for exactly 2 years.

CI − SI (3 years)

= P(R/100)²(3 + R/100)

Difference for exactly 3 years.

Rule of 72

Years to double ≈ 72/R

Quick estimate. e.g. 8% → doubles in 9 years.

Effective Annual Rate

= (1 + R/n)ⁿ − 1

n = compounding frequency per year.

TRICK: CI − SI for 2 years = P(r/100)². This is because the extra interest in CI is the 'interest on interest' from year 1. Instant formula for 2-year difference questions.

TRICK: If CI for 2nd year − CI for 1st year = x, then rate R = x/CI₁ × 100. The ratio of successive CI years gives the rate directly.

TRICK: If a sum doubles in n years at SI, it triples in 2n years and becomes (k+1) times in kn years. Linear growth.

TRICK: If a sum doubles in n years at CI, it becomes 4× in 2n years and 2ᵏ× in kn years. Exponential growth — powers of 2.

TRAP: SI and CI give same interest only in year 1. From year 2 onwards, CI > SI always (for same P, R, T). The gap grows each year.

TRICK: Rule of 72 — for quick approximation. At 6% CI, money doubles in 72/6 = 12 years. At 9%, in 8 years. Very useful for elimination in MCQs.

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Time, Speed and Distance ▾

Speed

Distance / Time

S = D/T

Distance

Speed × Time

D = S × T

Average Speed (equal dist)

2S₁S₂/(S₁+S₂)

Harmonic mean — NOT arithmetic mean.

Relative Speed (opposite)

S₁ + S₂

Trains/people approaching each other.

Relative Speed (same dir)

S₁ − S₂

Overtaking. The faster one gains.

Train crosses pole

L / S

L = length of train.

Train crosses platform

(L_train + L_platform) / S

Sum of both lengths.

Train crosses train (opposite)

(L₁+L₂)/(S₁+S₂)

Train crosses train (same dir)

(L₁+L₂)/(S₁−S₂)

Boat upstream

V_boat − V_current

Against current.

Boat downstream

V_boat + V_current

With current.

Still water speed

(Downstream + Upstream)/2

Current speed

(Downstream − Upstream)/2

Meeting point (circular)

Track / Relative Speed

Time to first meeting on circular track.

TRICK: Average speed for equal distances = 2ab/(a+b) ALWAYS. Never add and divide by 2. CAT sets traps on this every year.

TRICK: If A and B start from ends of a straight path and meet for the first time at point P, they have together covered the full distance D. Second meeting: 3D. Third: 5D. Pattern: odd multiples of D.

TRICK: Escalator problems — treat escalator as current. If escalator moves at e steps/sec and person at p steps/sec, effective speed up = p+e, down = p−e. Exactly like boat-stream.

TRICK: A leaves at time t₁ and B at time t₂ travelling same route. B overtakes A when: Head start of A = Relative speed × time B travels. Set equal and solve.

TRAP: km/h to m/s multiply by 5/18. m/s to km/h multiply by 18/5. Getting this backwards is extremely common.

TRICK: If a person walking at speed S reaches destination t₁ min late and at speed S₂ reaches t₂ min early, distance = S₁×S₂×(t₁+t₂)/(S₂−S₁). Memorise this shortcut.

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Linear and Circular Motion ▾

Linear Meeting (opposite)

Time = D/(S₁+S₂)

Two objects moving towards each other.

Linear Overtaking

Time = Head Start / (S₁−S₂)

Faster one closes the gap.

Circular Same Direction

First meet = Track/(S₁−S₂)

Faster laps the slower.

Circular Opposite Direction

First meet = Track/(S₁+S₂)

They approach each other.

All Meet at Start

LCM of individual lap times

When all participants return to start simultaneously.

3 people on circle

A & B meet at T₁, A & C at T₂. All 3 meet at LCM(T₁,T₂)

Combine pairwise meeting times.

Race: A beats B by d m in L m race

B covers L−d when A covers L

Translate 'beats by d metres' to this.

Race: A beats B by t sec

B takes t sec more to finish

Translate to time difference.

TRICK: For circular track — number of meetings = |relative speed/track| × time. In time T, faster meets slower exactly T×(S₁−S₂)/Track times (same direction).

TRICK: A and B run on circular track. A's time per lap = a, B's = b. They start together. They meet for the kth time at time = k × ab/|a−b| (same dir) or k × ab/(a+b) (opposite).

TRICK: Race by distance vs race by time. 'A gives B a start of 20m' = B starts 20m ahead. 'A gives B a start of 5 sec' = B starts 5 sec earlier. Convert both to distance at start before solving.

TRICK: If A, B, C are on a circular track and A meets B every t₁ minutes and B meets C every t₂ minutes (all same direction), all three meet at LCM(t₁, t₂).

TRAP: 'Meets for the nth time' vs 'returns to start for the nth time' — completely different calculations. First is about relative lapping, second is about absolute laps.

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Time and Work ▾

Work Rate

1 unit of work / n days = rate of 1/n per day

If a task is done in n days.

Combined Rate

1/A + 1/B

A and B together per day.

Together Time

AB/(A+B)

Equivalent to harmonic mean/2.

Man-Days

Work = Men × Days × Hours/day

All three factors multiply.

Work done in d days

d × (1/n)

Fraction of work completed.

Pipe Fill + Empty

1/F − 1/E

Fill pipe adds, empty pipe subtracts.

Efficiency Ratio

Inversely proportional to time

More efficient → less time → higher daily rate.

Partial Work

If A works for d days then leaves, B finishes

Set up: d/A + x/B = 1. Solve for x.

TRICK: LCM method — instead of fractions, take LCM of all days as the 'total work'. Express each person's rate as whole numbers. Much faster than fraction arithmetic.

TRICK: A is twice as fast as B means A takes half the time. If B takes 20 days, A takes 10. Together = 20×10/(20+10) = 200/30 = 6.67 days.

TRICK: Pipe problems — net rate = sum of fill rates − sum of drain rates. If result is positive, tank fills. If negative, it empties. Time = Total capacity / |Net rate|.

TRICK: If A and B together take T days, and A alone takes a days, then B alone takes = aT/(a−T) days. Rearrange the combined formula.

TRAP: If work is doubled, time does NOT halve if the number of workers also changes. Be careful when both work amount AND workforce change simultaneously.

TRICK: Alternating workers problem — find work done in one complete cycle (e.g. 2 days with A, then B), multiply cycles, handle the remainder separately.

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Diophantine Equations ▾

Integer Solution Exists

ax + by = c has solutions iff GCD(a,b) | c

Necessary and sufficient condition.

General Solution

x = x₀ + bt/d, y = y₀ − at/d

d = GCD(a,b). t ∈ ℤ. x₀,y₀ is one particular solution.

Positive Solutions

Count values of t for which x>0 and y>0

Finite if a,b have same sign in equation.

Chicken-Rabbit type

x + y = n and ax + by = S

Two equations, two unknowns. Solve simultaneously.

Coin Problems

Σ(value × count) = Total amount

Set up and solve for integer solutions.

TRICK: For ax + by = c, find one solution by inspection or extended Euclidean. Then generate all solutions using the general formula. The step size between x-values is b/GCD(a,b).

TRICK: Chicken-Rabbit: total heads=H, total legs=L. Chickens = (4H−L)/2. Rabbits = (L−2H)/2. Derive once, memorise the result.

TRICK: For positive solutions only — substitute extreme values. Let x=0 to get max y, y=0 to get max x. Count integers in the valid range.

TRAP: Always verify GCD(a,b) divides c before looking for solutions. If GCD=2 and c is odd, there are NO integer solutions. Many students skip this check.

TRICK: CAT Diophantine questions are usually disguised as: 'In how many ways can you pay Rs 50 using Rs 2 and Rs 5 coins?' — Set up 2x+5y=50 and count positive integer solutions.

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Polynomials ▾

Remainder Theorem

P(a) = remainder when P(x) ÷ (x−a)

Evaluate polynomial at x=a directly.

Factor Theorem

P(a) = 0 ⟺ (x−a) is a factor of P(x)

Special case of Remainder Theorem.

Sum of Roots (quadratic)

α+β = −b/a

ax²+bx+c.

Product of Roots (quadratic)

αβ = c/a

ax²+bx+c.

Sum of Roots (cubic)

α+β+γ = −b/a

ax³+bx²+cx+d.

Sum of products (cubic)

αβ+βγ+γα = c/a

ax³+bx²+cx+d.

Product of Roots (cubic)

αβγ = −d/a

ax³+bx²+cx+d.

Degree of Polynomial

Highest power of variable

deg(3x⁵+2x+1) = 5.

a³+b³

= (a+b)(a²−ab+b²)

Factoring sum of cubes.

a³−b³

= (a−b)(a²+ab+b²)

Factoring difference of cubes.

(a+b+c)²

= a²+b²+c²+2(ab+bc+ca)

Expansion of trinomial square.

TRICK: Remainder when P(x) is divided by (x−a) = P(a). No long division needed. e.g. Remainder of x³−2x+1 ÷ (x−2) = 2³−4+1 = 5.

TRICK: If P(1)=0, sum of all coefficients = 0. If P(−1)=0, alternating sum of coefficients = 0. Quick test for (x−1) and (x+1) as factors.

TRICK: a+b+c=0 ⟹ a³+b³+c³ = 3abc. This appears as a direct question or as a disguised form in options.

TRICK: (x+y+z)² = x²+y²+z² + 2(xy+yz+zx). So x²+y²+z² = (x+y+z)² − 2(xy+yz+zx). Use Vieta's roots to find this directly.

TRAP: Remainder theorem gives remainder for linear divisors only. For quadratic divisor (x²+ax+b), the remainder is of the form (rx+s), not a constant.

TRICK: Synthetic division — for quickly dividing by (x−k), write coefficients, bring down, multiply by k, add column by column. 10× faster than long division.

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Modulus ▾

|x|

x if x≥0 ; −x if x<0

Distance from zero on number line.

Triangle Inequality

|a+b| ≤ |a| + |b|

Equality when a,b have same sign.

Reverse Triangle

|a−b| ≥ ||a| − |b||

Lower bound on |a−b|.

|x| = a (a>0)

x = a or x = −a

Two solutions.

|x| < a

−a < x < a

Open interval.

|x| > a

x < −a or x > a

Two separate intervals.

|x − a| < r

a−r < x < a+r

All points within distance r of a.

|ab|

= |a| × |b|

Modulus distributes over multiplication.

|a/b|

= |a| / |b|

Modulus distributes over division.

TRICK: |x−a| = |x−b| means x is equidistant from a and b. Solution: x = (a+b)/2. The midpoint. No algebra needed.

TRICK: |x−3| + |x−7| = 4 means x lies between 3 and 7 (inclusive). For any x in [3,7], the sum equals exactly 4. Outside this range, sum > 4.

TRICK: Minimum of |x−a| + |x−b| = |a−b|. Achieved when x is between a and b. Minimum of |x−a| + |x−b| + |x−c| — achieved at the MEDIAN of a,b,c.

TRICK: For |f(x)| < g(x): solve −g(x) < f(x) < g(x). For |f(x)| > g(x): solve f(x) > g(x) OR f(x) < −g(x).

TRAP: |x²| = x² always (since x² ≥ 0). But |x|² = x² too. These are equal. However |x+y|² ≠ |x|²+|y|². Don't expand modulus like a square.

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Logarithms ▾

Definition

log_a(b) = x ⟺ aˣ = b

a>0, a≠1, b>0.

Product Rule

log(mn) = log m + log n

Log of product = sum of logs.

Quotient Rule

log(m/n) = log m − log n

Log of quotient = difference of logs.

Power Rule

log(mⁿ) = n·log m

Exponent becomes multiplier.

Change of Base

log_a(b) = log b / log a

Convert to any convenient base.

Special Values

log_a(1)=0 | log_a(a)=1

Universal for any valid base a.

Reciprocal

log_a(b) = 1/log_b(a)

Flip base and argument.

log_a(aˣ) = x

a^(log_a x) = x

Log and exponent are inverses.

Chain Rule

log_a(b) × log_b(c) = log_a(c)

Telescoping of bases.

Inequalities

If a>1: log_a is increasing. If 0

Direction of inequality changes for base < 1.

Inequalities ▾

Linear Inequality

ax+b > 0 → x > −b/a (if a>0)

Flip inequality sign when multiplying/dividing by negative.

Quadratic Inequality

ax²+bx+c > 0

Use sign chart. Roots divide number line into intervals.

AM ≥ GM

(a+b)/2 ≥ √(ab)

Equality iff a = b.

AM ≥ GM ≥ HM

For positive numbers

HM = 2ab/(a+b). Always AM ≥ GM ≥ HM.

AM-GM (n numbers)

(a₁+a₂+…+aₙ)/n ≥ (a₁a₂…aₙ)^(1/n)

Equality iff all are equal.

Cauchy-Schwarz

(a₁b₁+a₂b₂)² ≤ (a₁²+a₂²)(b₁²+b₂²)

Powerful bound — rarely needed in CAT but useful.

Min of x + 1/x

≥ 2 (for x>0)

By AM-GM: x+1/x ≥ 2√(x·1/x) = 2. Equality at x=1.

Modulus Inequality

|x−a| < b ↔ a−b < x < a+b

Interval around a.

TRICK: AM-GM gold standard — to minimise a+b given ab = constant, set a=b. To maximise ab given a+b = constant, set a=b. Equality condition = equal values.

TRICK: For x+y = S (fixed), maximum of xy = S²/4 (at x=y=S/2). For xy = P (fixed), minimum of x+y = 2√P (at x=y=√P).

TRICK: x²+y²+z² ≥ xy+yz+zx always. Rearranges to: (x−y)²+(y−z)²+(z−x)² ≥ 0. Equality when x=y=z.

TRAP: When multiplying inequality by a variable (like x), you must consider two cases: x>0 and x<0. Skipping this is the #1 error.

TRICK: For questions asking 'minimum value of expression' — try AM-GM. For 'maximum value' — try AM-GM on the reciprocal. For range questions — complete the square.

TRICK: If a+b+c = constant, then max(a²+b²+c²) is when values are as unequal as possible. Min(a²+b²+c²) is when a=b=c. Opposite of AM-GM direction.

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Remainders ▾

Basic Remainder

N = DQ + R, 0 ≤ R < D

N÷D gives quotient Q and remainder R.

Addition

(A+B) mod D = [(A mod D) + (B mod D)] mod D

Reduce each term first.

Multiplication

(A×B) mod D = [(A mod D)(B mod D)] mod D

Reduce each factor first.

Fermat's Little

aᵖ⁻¹ ≡ 1 (mod p), gcd(a,p)=1

p prime. Key for large power remainders.

Euler's Theorem

aᵠ⁽ᴺ⁾ ≡ 1 (mod N), gcd(a,N)=1

Generalises Fermat. φ(N) = Euler's totient.

Chinese Remainder Theorem

Unique solution mod LCM(m₁,m₂,…)

For system x≡a₁(mod m₁), x≡a₂(mod m₂), … when mᵢ are co-prime.

Wilson's Theorem

(p−1)! ≡ −1 (mod p)

For prime p. Useful for factorial remainders.

Cyclicity for Unit Digit

2,3,7,8→cycle4 | 4,9→cycle2 | 0,1,5,6→cycle1

Find unit digit of large power.

Last 2 digits

Find pattern of aⁿ mod 100

Cycle usually found within 20 terms.

TRICK: For 2ⁿ mod 7 — cycle is 2,4,1 (period 3). For 3ⁿ mod 7 — cycle is 3,2,6,4,5,1 (period 6). For 2ⁿ mod 9 — cycle is 2,4,8,7,5,1 (period 6). Memorise these.

TRICK: (x−1)ⁿ mod x = (−1)ⁿ mod x. So 99ⁿ mod 100 = (−1)ⁿ. Instant: 99¹⁰⁰ mod 100 = 1. 99⁹⁹ mod 100 = 99.

TRICK: (x+1)ⁿ mod x = 1 always. So 101ⁿ mod 100 = 1 always. 8ⁿ mod 7 = 1 always.

TRICK: N! mod p = 0 for any prime p ≤ N. So 7! mod 5 = 0, 10! mod 7 = 0. Instant zero.

TRICK: Remainder of sum 1!+2!+3!+…+n! divided by k — from k! onwards, all terms are divisible by k! and hence by k (for k ≤ n). Only compute first (k−1) terms.

TRICK: For remainder of aᵇ ÷ m — first find aᵇ mod φ(m) to reduce the exponent (Euler), then find the remainder. Two-step approach for very large exponents.

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Functions ▾

Domain

Set of all valid inputs x

Function is defined only for these x values.

Range

Set of all possible outputs f(x)

Subset of codomain.

Composite

(f∘g)(x) = f(g(x))

Apply g first, then f.

Inverse

f⁻¹(y) = x iff f(x) = y

Exists only if f is one-one and onto.

Even Function

f(−x) = f(x)

Symmetric about y-axis. e.g. x², cos x.

Odd Function

f(−x) = −f(x)

Symmetric about origin. e.g. x³, sin x.

Periodic

f(x+T) = f(x)

T = period.

Floor Function

⌊x⌋ = greatest integer ≤ x

⌊3.7⌋=3, ⌊−1.2⌋=−2.

Ceiling Function

⌈x⌉ = smallest integer ≥ x

⌈3.2⌉=4, ⌈−1.2⌉=−1.

Fractional Part

{x} = x − ⌊x⌋

Always 0 ≤ {x} < 1.

TRICK: For f(f(x)) type questions — apply f once, simplify, apply again. Don't substitute the full expression at once. Step by step.

TRICK: f(x)+f(1−x) = 1 type identity — add f(x) and f(1−x) and the complicated parts cancel. Very common CAT pattern. Sum from x=1/n to x=(n−1)/n often pairs up.

TRICK: Number of integers in interval [a,b] = ⌊b⌋ − ⌈a⌉ + 1 (for inclusive interval). For (a,b): ⌊b⌋ − ⌊a⌋ (excluding endpoints if they're integers).

TRICK: ⌊x⌋ + ⌊x+1/2⌋ = ⌊2x⌋. This is Hermite's identity. Useful for simplifying floor expressions.

TRAP: Domain of log(f(x)) requires f(x) > 0. Domain of √(f(x)) requires f(x) ≥ 0. Domain of 1/f(x) requires f(x) ≠ 0. Check all three types.

TRICK: If f is increasing and f(a) > f(b), then a > b. If f is decreasing, a < b. Use monotonicity to compare values without computing.

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Graphs ▾

Straight Line

y = mx + c

m = slope, c = y-intercept.

Slope

m = (y₂−y₁)/(x₂−x₁) = tan θ

θ = angle with positive x-axis.

Parabola

y = ax²+bx+c

Opens up if a>0, down if a<0. Vertex at x = −b/2a.

Circle

(x−h)²+(y−k)²=r²

Centre (h,k), radius r.

Modulus graph

y = |x|

V-shape with vertex at origin.

Intersection of lines

Solve equations simultaneously

Number of solutions = number of intersection points.

Parallel lines

Equal slopes, different intercepts

m₁ = m₂, c₁ ≠ c₂.

Perpendicular lines

m₁ × m₂ = −1

Product of slopes = −1.

Distance from point to line

|ax₀+by₀+c| / √(a²+b²)

Point (x₀,y₀) to line ax+by+c=0.

TRICK: Number of solutions to f(x) = g(x) = number of intersections of their graphs. Draw both curves quickly and count — often faster than solving algebraically.

TRICK: |x| + |y| = k forms a diamond (rhombus) with vertices at (±k,0) and (0,±k). Area = 2k².

TRICK: y = |f(x)| — flip all negative portions of y=f(x) above the x-axis. y = f(|x|) — keep right half, mirror it to the left.

TRICK: For y = f(x+a) — shift f(x) LEFT by a units. For y = f(x−a) — shift RIGHT. For y = f(x)+a — shift UP. Confusing left/right shifts is the top error.

TRICK: Maximum/minimum of ax+by subject to constraints is always at a corner of the feasible region (Linear Programming). Check all vertices.

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Lines and Angles ▾

Supplementary

∠A + ∠B = 180°

Linear pair on a straight line.

Complementary

∠A + ∠B = 90°

Sum is a right angle.

Vertically Opposite

Equal angles

Formed by two intersecting lines.

Corresponding Angles

Equal (parallel lines + transversal)

Same position at each intersection.

Alternate Interior

Equal (parallel lines + transversal)

On opposite sides of transversal, inside.

Co-interior (same side)

Supplementary: sum = 180°

Also called consecutive interior angles.

Angle Bisector

Divides angle into two equal parts

Locus of points equidistant from two lines.

Exterior Angle

= Sum of two non-adjacent interior angles

Triangle exterior angle theorem.

TRICK: When a transversal cuts parallel lines, mark one angle. All 8 angles formed are either equal to it or supplementary to it. Just track these two values.

TRICK: Exterior angle of a triangle = sum of the two opposite interior angles. Faster than finding all three angles.

TRICK: Two parallel lines cut by a transversal — co-interior angles (same side) are supplementary. Remember: CO-interior → C-orrect → Co-supplement.

TRAP: 'Lines are parallel' must be PROVEN, not assumed, unless explicitly stated. Many CAT geometry questions hide the assumption.

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Triangles ▾

Angle Sum

∠A + ∠B + ∠C = 180°

Any triangle.

Area (base-height)

½ × base × height

Height must be perpendicular to base.

Area (two sides + angle)

½ab sin C

a,b are two sides, C is included angle.

Heron's Formula

√[s(s−a)(s−b)(s−c)]

s = (a+b+c)/2 is semi-perimeter.

Pythagoras Theorem

a² + b² = c²

Right triangle. c = hypotenuse.

Sine Rule

a/sinA = b/sinB = c/sinC = 2R

R = circumradius.

Cosine Rule

a² = b²+c²−2bc cosA

Any triangle. Generalises Pythagoras.

Median to hypotenuse

= hypotenuse/2

In a right triangle, median to hypotenuse = half hypotenuse.

Centroid

Divides median 2:1 from vertex

G = (x₁+x₂+x₃)/3, (y₁+y₂+y₃)/3.

Circumradius R

abc / 4K

K = area of triangle.

Inradius r

K / s

K = area, s = semi-perimeter.

Similar triangles

AA, SAS, SSS similarity

Ratios of corresponding sides are equal.

TRICK: Pythagorean triples to memorise: (3,4,5), (5,12,13), (7,24,25), (8,15,17), (9,40,41). Their multiples also work. Recognise these instantly.

TRICK: In a 30-60-90 triangle, sides are in ratio 1:√3:2. In a 45-45-90 triangle: 1:1:√2. Draw these once and commit to memory.

TRICK: Median divides triangle into two triangles of equal area. All 3 medians divide the triangle into 6 triangles of equal area. Useful for area-ratio problems.

TRICK: Area of triangle with vertices (x₁,y₁),(x₂,y₂),(x₃,y₃) = ½|x₁(y₂−y₃)+x₂(y₃−y₁)+x₃(y₁−y₂)|. Don't derive — use directly.

TRAP: In similar triangles, areas are in ratio of SQUARES of corresponding sides. Perimeters are in ratio of sides (linear ratio). Students often apply the linear ratio to areas.

TRICK: The line joining midpoints of two sides of a triangle is parallel to the third side and half its length. (Midpoint theorem). Appears in almost every CAT geometry set.

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Coordinate Geometry ▾

Distance

d = √[(x₂−x₁)² + (y₂−y₁)²]

Between two points.

Midpoint

((x₁+x₂)/2, (y₁+y₂)/2)

Midpoint of a segment.

Section Formula (internal)

((mx₂+nx₁)/(m+n), (my₂+ny₁)/(m+n))

Point dividing in ratio m:n internally.

Slope

m = (y₂−y₁)/(x₂−x₁) = tan θ

Line: slope-intercept

y = mx + c

m=slope, c=y-intercept.

Line: general

ax + by + c = 0

Slope = −a/b.

Two points form

(y−y₁)/(y₂−y₁) = (x−x₁)/(x₂−x₁)

Perpendicular distance

|ax₀+by₀+c| / √(a²+b²)

From point (x₀,y₀) to ax+by+c=0.

Area of triangle

½|x₁(y₂−y₃)+x₂(y₃−y₁)+x₃(y₁−y₂)|

Given 3 vertices.

Parallel: m₁=m₂

Same slope, c₁≠c₂

No intersection.

Perpendicular: m₁m₂=−1

Product of slopes = −1

Circle

(x−h)²+(y−k)²=r²

Centre (h,k), radius r.

TRICK: Centroid of a triangle = average of all 3 vertices: ((x₁+x₂+x₃)/3, (y₁+y₂+y₃)/3). The simplest formula in coordinate geometry.

TRICK: Reflection of point (a,b) about x-axis = (a,−b). About y-axis = (−a,b). About origin = (−a,−b). About y=x = (b,a). Memorise all four.

TRICK: If three points are collinear, area of triangle formed = 0. Use area formula to test collinearity: if |x₁(y₂−y₃)+…| = 0, they're collinear.

TRICK: Locus of point equidistant from two fixed points A and B = perpendicular bisector of AB. Locus equidistant from a point and a line = parabola.

TRAP: Slope is undefined for vertical lines (x=a). Horizontal lines have slope 0. Vertical is NOT slope 0 — this is a frequent confusion.

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Quadrilaterals ▾

Square

A=a², P=4a, d=a√2

All sides equal, all angles 90°.

Rectangle

A=lb, P=2(l+b), d=√(l²+b²)

Opposite sides equal, all angles 90°.

Parallelogram

A=base×height=ab sinθ

Diagonals bisect each other.

Rhombus

A=½d₁d₂, P=4a

Diagonals are perpendicular bisectors of each other.

Trapezium

A=½(a+b)×h

One pair of parallel sides (a,b). h=height.

Kite

A=½d₁d₂

One diagonal bisects the other perpendicularly.

Cyclic Quadrilateral

Opposite angles supplementary

Sum of opposite angles = 180°.

Ptolemy's Theorem

AC×BD = AB×CD + AD×BC

For cyclic quadrilateral.

TRICK: In a parallelogram, sum of squares of diagonals = 2(sum of squares of sides). d₁²+d₂² = 2(a²+b²). Useful for finding diagonals without trigonometry.

TRICK: Area of any quadrilateral with perpendicular diagonals = ½ d₁ d₂. This covers rhombus and kite as special cases.

TRICK: Cyclic quadrilateral — opposite angles sum to 180°. If you can show this, the quadrilateral is cyclic (and vice versa).

TRICK: The midpoints of any quadrilateral's sides form a parallelogram (Varignon's theorem). Area of this parallelogram = half the area of the quadrilateral.

TRAP: A rhombus has all sides equal but is NOT necessarily a square (angles need not be 90°). A square is a special rhombus.

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Polygons ▾

Sum of interior angles

(n−2) × 180°

n-sided polygon.

Each interior angle (regular)

(n−2) × 180° / n

Regular polygon: all sides and angles equal.

Each exterior angle (regular)

360° / n

Sum of all exterior angles = 360° always.

Number of diagonals

n(n−3)/2

For n-sided polygon.

Number of triangles (from one vertex)

n−2

Triangles formed by drawing diagonals from one vertex.

Area of regular n-gon

(n/4) × a² × cot(π/n)

a = side length.

TRICK: Each exterior angle of a regular polygon = 360°/n. Sum of exterior angles is ALWAYS 360° regardless of n. For interior: multiply exterior supplement by n.

TRICK: Regular hexagon with side a — area = (3√3/2)a². Diagonal = 2a (the long diagonal). Short diagonal = a√3. Hexagon = 6 equilateral triangles.

TRICK: Number of triangles formed by joining n points (no 3 collinear) = C(n,3). Number of diagonals = C(n,2) − n = n(n−1)/2 − n = n(n−3)/2.

TRICK: Interior angle of regular polygon as n→∞ → 180°. As n increases, polygon approaches a circle. Interior angles approach 180° (straight line).

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Circles ▾

Circumference

2πr

r = radius.

Area

πr²

Arc Length

(θ/360°) × 2πr

θ = central angle in degrees.

Sector Area

(θ/360°) × πr²

Chord Length

2r sin(θ/2)

θ = central angle subtended by chord.

Tangent ⊥ Radius

90° at point of contact

Fundamental property.

Angle in Semicircle

= 90°

Angle subtended by diameter = right angle.

Inscribed Angle

= Half the central angle

Same arc.

Tangent-Tangent

PA = PB

Tangents from external point are equal.

Power of a Point

PA × PB = PC × PD

For two chords/secants through a point.

Tangent-Secant

PT² = PA × PB

PT = tangent length. PA, PB = secant segments.

TRICK: Angle in a semicircle = 90°. If you see a right angle inscribed in a circle, the hypotenuse IS the diameter. Converse also true — very useful.

TRICK: Angles subtended by the same arc on the same side are equal. Angles on opposite sides are supplementary. One property, two results.

TRICK: Two tangents from external point P to circle: PA=PB (equal lengths), and ∠APO = ∠BPO (angle bisected by line to centre). O,P and midpoint of AB are collinear.

TRICK: Common tangents — two circles with radii r₁, r₂ and centre distance d: External tangents = √(d²−(r₁−r₂)²). Internal tangents = √(d²−(r₁+r₂)²).

TRICK: Power of a point: for any point P and any line through P intersecting circle at A,B: PA×PB = constant (the power). Inside circle: power is negative. Outside: positive.

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Trigonometry ▾

sin θ

Opposite / Hypotenuse

SOH

cos θ

Adjacent / Hypotenuse

CAH

tan θ

Opposite / Adjacent = sin/cos

TOA

cosec θ

1/sin θ

sec θ

1/cos θ

cot θ

1/tan θ = cos/sin

sin²θ + cos²θ = 1

Fundamental identity

Most used.

1 + tan²θ = sec²θ

Second Pythagorean identity

sec²−tan²=1 also.

1 + cot²θ = cosec²θ

Third Pythagorean identity

cosec²−cot²=1 also.

sin(A±B)

sinA cosB ± cosA sinB

cos(A±B)

cosA cosB ∓ sinA sinB

tan(A+B)

(tanA+tanB)/(1−tanA tanB)

sin 2A

2 sinA cosA

cos 2A

cos²A − sin²A = 2cos²A−1 = 1−2sin²A

Three forms.

Special: 30°

sin=½, cos=√3/2, tan=1/√3

Special: 45°

sin=cos=1/√2, tan=1

Special: 60°

sin=√3/2, cos=½, tan=√3

TRICK: sec²θ − tan²θ = 1 always. If given sec θ + tan θ = k, then sec θ − tan θ = 1/k. Instant, no solving needed.

TRICK: sin θ + cosec θ ≥ 2 (by AM-GM). sin θ × cosec θ = 1. Similarly cos θ + sec θ ≥ 2. tan θ + cot θ ≥ 2.

TRICK: Height & Distance — angle of elevation from ground, object above. Angle of depression from height, object below. tan(angle) = Height/Distance always.

TRICK: For (sin θ + cos θ) = k, then sin θ × cos θ = (k²−1)/2. And sin⁴θ + cos⁴θ = 1 − 2sin²θcos²θ = 1 − (k²−1)²/2. Chain of identities.

TRAP: sin⁻¹(sin θ) = θ only when θ ∈ [−π/2, π/2]. Outside this range, it gives the principal value, not θ itself. Frequent trap in options.

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Mensuration ▾

Rectangle

A=lb, P=2(l+b)

Triangle

A=½bh or √[s(s−a)(s−b)(s−c)]

Heron's formula for all three sides.

Circle

A=πr², C=2πr

Cylinder

V=πr²h, CSA=2πrh, TSA=2πr(r+h)

Cone

V=⅓πr²h, l=√(r²+h²), CSA=πrl, TSA=πr(r+l)

l=slant height.

Sphere

V=⁴⁄₃πr³, SA=4πr²

Hemisphere

V=⅔πr³, CSA=2πr², TSA=3πr²

Cube

V=a³, SA=6a², d=a√3

d=space diagonal.

Cuboid

V=lbh, SA=2(lb+bh+lh), d=√(l²+b²+h²)

Prism

V=Base Area × Height

Any prism.

Pyramid

V=⅓ × Base Area × Height

Any pyramid.

TRICK: Sphere inscribed in cube of side a: r=a/2. Cube inscribed in sphere of radius r: side=2r/√3. Sphere inscribed in cylinder of radius r, height 2r: sphere radius=r (perfect fit).

TRICK: When a cone is melted and recast into spheres (or vice versa), VOLUME is conserved. Set volumes equal. No need to track dimensions of individual pieces.

TRICK: Diagonal of cuboid = √(l²+b²+h²). For a cube of side a = a√3. If any two sides are equal (square base), simplify before computing.

TRICK: Frustum of a cone (cone with top cut off): V = πh/3 × (r₁²+r₂²+r₁r₂). This appears regularly. Lateral surface = π(r₁+r₂)×l where l=slant height.

TRAP: TSA (Total Surface Area) includes the base(s). CSA (Curved Surface Area) does NOT. For a cylinder: TSA = 2πr(r+h). CSA = 2πrh. Getting this wrong changes your answer entirely.

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Permutations and Combinations ▾

Permutation

P(n,r) = n!/(n−r)!

Ordered arrangement of r items from n.

Combination

C(n,r) = n! / [r!(n−r)!]

Unordered selection of r from n.

Relation

P(n,r) = r! × C(n,r)

Permutation = Combination × arrangements of r items.

Symmetry

C(n,r) = C(n,n−r)

Choosing r to include = choosing n−r to exclude.

Pascal's Identity

C(n,r) = C(n−1,r−1) + C(n−1,r)

Sum of all C(n,r)

2ⁿ

Total number of subsets of n items.

Circular Permutation

(n−1)!

Arrangements of n distinct items in a circle.

Necklace

(n−1)!/2

Circular, clockwise = anticlockwise.

Identical items

n! / (p! q! r! …)

Permutation with repeated items.

Distribution

Stars and bars: C(n+r−1, r−1)

r identical items into n distinct groups.

TRICK: Counting problems — always ask: Is order important? (Permutation) or not? (Combination). Then ask: Are items identical? Then: Are there restrictions? These three questions structure every problem.

TRICK: At least/at most problems — use complement. P(at least 1) = Total − P(none). P(at most 2) = P(0)+P(1)+P(2). Usually complement is faster.

TRICK: ABCDEF in a row, A and B always together: treat AB as a unit → 5! arrangements × 2! internal = 2×120=240.

TRICK: Derangements (no item in its original position): D(n) = n! × (1−1/1!+1/2!−1/3!+…+(−1)ⁿ/n!). D(1)=0, D(2)=1, D(3)=2, D(4)=9.

TRAP: Circular permutation — 'around a table' usually means circular (n−1)!. But 'in a ring/necklace' where flipping is same = (n−1)!/2. Distinguish carefully.

TRICK: Distributing n different items into r groups: if groups are distinct = rⁿ. If groups are identical (unordered) = Stirling numbers. For CAT, groups are usually distinct.

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Probability ▾

Basic

P(E) = n(E)/n(S)

Favourable outcomes / Total outcomes.

Complement

P(E') = 1 − P(E)

Probability event does NOT occur.

Addition Rule

P(A∪B) = P(A)+P(B)−P(A∩B)

Any two events.

Mutually Exclusive

P(A∪B) = P(A)+P(B)

P(A∩B) = 0.

Independent Events

P(A∩B) = P(A)×P(B)

Events don't affect each other.

Conditional Probability

P(A|B) = P(A∩B)/P(B)

Probability of A given B has occurred.

Bayes' Theorem

P(A|B) = P(B|A)×P(A)/P(B)

Reverse conditional probability.

Expected Value

E(X) = Σ xᵢ P(xᵢ)

Weighted average of outcomes.

Geometric Probability

P = Favourable length or area / Total length or area

Continuous sample space.

TRICK: 'At least one' problems — P(at least one) = 1 − P(none). Almost always faster. e.g. P(at least one head in 5 tosses) = 1 − (1/2)⁵ = 31/32.

TRICK: Independent vs Mutually Exclusive. Independent: P(A∩B)=P(A)×P(B). Mutually exclusive: P(A∩B)=0. Two events CANNOT be both independent AND mutually exclusive (unless one has probability 0).

TRICK: Conditional probability — P(A|B) = P(A∩B)/P(B). Always reduce the sample space to B first, then count favourable outcomes within B.

TRICK: Matching/Birthday problem — P(no match) = n!/nⁿ × nᵏ/... Use complement: P(at least one match) = 1 − P(all different). For birthday: 1 − 365×364×…/(365ⁿ).

TRAP: 'P(A and B)' means intersection. 'P(A or B)' means union. Students often compute P(A)×P(B) for 'or' — that's wrong unless A and B are independent AND exclusive.

TRICK: Expected value is linear: E(aX+b) = aE(X)+b. E(X+Y) = E(X)+E(Y) always. E(XY) = E(X)×E(Y) ONLY IF X and Y are independent.

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Sequences and Series ▾

AP nth term

aₙ = a + (n−1)d

a=first term, d=common difference.

AP Sum

Sₙ = n/2 × [2a + (n−1)d] = n/2 × (a+l)

l = last term.

GP nth term

aₙ = arⁿ⁻¹

r = common ratio.

GP Sum (finite)

Sₙ = a(rⁿ−1)/(r−1)

r ≠ 1.

GP Sum (infinite)

S∞ = a/(1−r)

Valid only when |r| < 1.

Sum of squares

n(n+1)(2n+1)/6

1²+2²+3²+…+n²

Sum of cubes

[n(n+1)/2]²

1³+2³+3³+…+n³

Arithmetic Mean

(a+b)/2

AM of two terms.

Geometric Mean

√(ab)

GM of two terms.

Harmonic Mean

2ab/(a+b)

HM of two terms.

AM ≥ GM ≥ HM

For positive numbers

Equality iff a=b.

AGP Sum

Combine AP × GP

Multiply AP term by GP term; use subtraction method.

TRICK: Sum of cubes = (Sum of first n naturals)². So 1³+2³+…+n³ = [n(n+1)/2]². This means sum of cubes is always a perfect square.

TRICK: In an AP, if you need sum of terms symmetric about the middle, use: Sₙ = n × (middle term) if n is odd, or Sₙ = n × (average of two middle terms) if n is even.

TRICK: For AGP (Arithmetic-Geometric Progression) like 1×1 + 2×2 + 3×4 + 4×8 + … — multiply the sum by the common ratio, subtract, and solve. Subtraction method.

TRICK: If three numbers are in AP: let them be (a−d), a, (a+d). Their sum = 3a. If in GP: let them be a/r, a, ar. Their product = a³. Choosing symmetric forms simplifies algebra dramatically.

TRAP: GP sum formula S = a(rⁿ−1)/(r−1) requires r≠1. For r=1, Sₙ=na. Don't blindly apply the formula when r=1.

TRICK: Infinite GP converges only if |r|<1. If |r|≥1, sum diverges (infinite). Always check this condition before applying S∞=a/(1−r).

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CAT Quantitative Aptitude Tool — All 34 Topics.