CUET Formula Sheets - Complete Collection

One-stop resource for all important formulas across Mathematics, Physics, and Chemistry. Perfect for quick revision before exams, mock tests, and last-minute preparation.

🎯 How to Use Formula Sheets

Effective Revision Strategy:

✅ Daily Practice: Write each formula 3 times daily ✅ Application: Solve 2-3 problems using each formula ✅ Flashcards: Create cards for difficult formulas ✅ Mock Tests: Keep sheets handy during practice ✅ Last Week: Focus only on formula revision

Memory Techniques:

1. Understand, Don’t Memorize: Know why the formula works
2. Visual Association: Create mental images
3. Mnemonics: Use memory tricks for complex formulas
4. Teach Others: Explaining helps retention
5. Regular Review: Space out revision (Day 1, 3, 7, 15)

📐 Mathematics Formula Sheets

1. Algebra

Quadratic Equations

Standard Form: ax² + bx + c = 0

Quadratic Formula:

x = [-b ± √(b² - 4ac)] / 2a

Discriminant (D):

D = b² - 4ac

If D > 0  → Two distinct real roots
If D = 0  → Two equal real roots (repeated root)
If D < 0  → No real roots (complex roots)

Sum and Product of Roots:

Sum of roots (α + β) = -b/a
Product of roots (αβ) = c/a

Forming Equation from Roots:

x² - (sum of roots)x + (product of roots) = 0
x² - (α + β)x + αβ = 0

Progressions

Arithmetic Progression (AP):

nth term: an = a + (n-1)d
Sum of n terms: Sn = n/2 [2a + (n-1)d]
               or Sn = n/2 [a + l]  where l = last term

Geometric Progression (GP):

nth term: an = ar^(n-1)
Sum of n terms: Sn = a(r^n - 1)/(r - 1)  if r ≠ 1
               or Sn = a(1 - r^n)/(1 - r)
Sum to infinity: S∞ = a/(1-r)  if |r| < 1

Harmonic Progression (HP):

If a, b, c are in HP, then 1/a, 1/b, 1/c are in AP
nth term: 1/an = 1/a + (n-1)d

Matrices & Determinants

Matrix Operations:

(AB)^T = B^T A^T
(AB)^-1 = B^-1 A^-1
(A^T)^-1 = (A^-1)^T

2×2 Determinant:

|a  b|
|c  d| = ad - bc

Inverse of 2×2 Matrix:

If A = |a  b|
       |c  d|

A^-1 = (1/|A|) |d   -b|
                |-c   a|

Properties:

|A| = |A^T|
|AB| = |A| × |B|
|kA| = k^n |A|  (for n×n matrix)
|A^-1| = 1/|A|

2. Calculus

Differentiation Formulas

Basic Derivatives:

d/dx (x^n) = nx^(n-1)
d/dx (e^x) = e^x
d/dx (a^x) = a^x ln(a)
d/dx (ln x) = 1/x
d/dx (log_a x) = 1/(x ln a)

Trigonometric Derivatives:

d/dx (sin x) = cos x
d/dx (cos x) = -sin x
d/dx (tan x) = sec²x
d/dx (cot x) = -cosec²x
d/dx (sec x) = sec x tan x
d/dx (cosec x) = -cosec x cot x

Inverse Trigonometric Derivatives:

d/dx (sin^-1 x) = 1/√(1-x²)
d/dx (cos^-1 x) = -1/√(1-x²)
d/dx (tan^-1 x) = 1/(1+x²)
d/dx (cot^-1 x) = -1/(1+x²)
d/dx (sec^-1 x) = 1/(|x|√(x²-1))
d/dx (cosec^-1 x) = -1/(|x|√(x²-1))

Product & Quotient Rules:

Product Rule: d/dx (uv) = u(dv/dx) + v(du/dx)
Quotient Rule: d/dx (u/v) = [v(du/dx) - u(dv/dx)]/v²
Chain Rule: dy/dx = dy/du × du/dx

Integration Formulas

Basic Integrals:

∫ x^n dx = x^(n+1)/(n+1) + C  (n ≠ -1)
∫ 1/x dx = ln|x| + C
∫ e^x dx = e^x + C
∫ a^x dx = a^x/ln(a) + C
∫ 1/(x²+a²) dx = (1/a) tan^-1(x/a) + C
∫ 1/√(a²-x²) dx = sin^-1(x/a) + C

Trigonometric Integrals:

∫ sin x dx = -cos x + C
∫ cos x dx = sin x + C
∫ tan x dx = ln|sec x| + C = -ln|cos x| + C
∫ cot x dx = ln|sin x| + C
∫ sec x dx = ln|sec x + tan x| + C
∫ cosec x dx = ln|cosec x - cot x| + C
∫ sec²x dx = tan x + C
∫ cosec²x dx = -cot x + C

Definite Integration Properties:

∫[a to b] f(x)dx = ∫[a to b] f(t)dt
∫[a to b] f(x)dx = -∫[b to a] f(x)dx
∫[a to b] f(x)dx = ∫[a to c] f(x)dx + ∫[c to b] f(x)dx
∫[0 to a] f(x)dx = ∫[0 to a] f(a-x)dx
∫[-a to a] f(x)dx = 2∫[0 to a] f(x)dx  if f(x) is even
                  = 0                 if f(x) is odd

3. Trigonometry

Basic Identities:

sin²θ + cos²θ = 1
1 + tan²θ = sec²θ
1 + cot²θ = cosec²θ

Compound Angle Formulas:

sin(A ± B) = sin A cos B ± cos A sin B
cos(A ± B) = cos A cos B ∓ sin A sin B
tan(A ± B) = (tan A ± tan B)/(1 ∓ tan A tan B)

Double Angle Formulas:

sin 2A = 2 sin A cos A
cos 2A = cos²A - sin²A = 2cos²A - 1 = 1 - 2sin²A
tan 2A = 2tan A/(1 - tan²A)

Triple Angle Formulas:

sin 3A = 3sin A - 4sin³A
cos 3A = 4cos³A - 3cos A
tan 3A = (3tan A - tan³A)/(1 - 3tan²A)

Sum to Product Formulas:

sin A + sin B = 2 sin[(A+B)/2] cos[(A-B)/2]
sin A - sin B = 2 cos[(A+B)/2] sin[(A-B)/2]
cos A + cos B = 2 cos[(A+B)/2] cos[(A-B)/2]
cos A - cos B = -2 sin[(A+B)/2] sin[(A-B)/2]

4. Coordinate Geometry

Distance Formula:

Distance between (x₁, y₁) and (x₂, y₂):
d = √[(x₂-x₁)² + (y₂-y₁)²]

Section Formula:

Internal division (m:n):
x = (mx₂ + nx₁)/(m+n)
y = (my₂ + ny₁)/(m+n)

Midpoint (m=n=1):
x = (x₁+x₂)/2
y = (y₁+y₂)/2

Straight Line:

Slope-intercept form: y = mx + c
Point-slope form: y - y₁ = m(x - x₁)
Two-point form: (y - y₁)/(y₂ - y₁) = (x - x₁)/(x₂ - x₁)
Intercept form: x/a + y/b = 1
General form: Ax + By + C = 0

Slope from general form: m = -A/B
Distance from point (x₁, y₁) to line Ax+By+C=0:
d = |Ax₁ + By₁ + C|/√(A² + B²)

Circle:

Standard form: (x-h)² + (y-k)² = r²
  Center: (h, k), Radius: r

General form: x² + y² + 2gx + 2fy + c = 0
  Center: (-g, -f)
  Radius: √(g² + f² - c)

5. Statistics & Probability

Mean, Median, Mode:

Mean: x̄ = Σx/n = Σfx/Σf (for grouped data)
Median position: (n+1)/2
Mode: Value with highest frequency

Standard Deviation & Variance:

Variance: σ² = Σ(x - x̄)²/n
Standard Deviation: σ = √[Σ(x - x̄)²/n]
                       = √[Σx²/n - (Σx/n)²]

Probability:

P(A) = n(A)/n(S)  where n(S) = total outcomes

P(A ∪ B) = P(A) + P(B) - P(A ∩ B)
P(A ∩ B) = P(A) × P(B)  if independent
P(A') = 1 - P(A)  (complement)

Permutations & Combinations:

nPr = n!/(n-r)!
nCr = n!/[r!(n-r)!]

nC0 = 1
nCn = 1
nCr = nC(n-r)

⚛️ Physics Formula Sheets

1. Mechanics

Motion Equations (Constant Acceleration):

v = u + at
s = ut + (1/2)at²
v² = u² + 2as
s = (u+v)t/2

Newton’s Laws:

F = ma
Momentum: p = mv
Impulse: J = FΔt = Δp

Work, Energy, Power:

Work: W = F × s × cos θ
Kinetic Energy: KE = (1/2)mv²
Potential Energy: PE = mgh
Power: P = W/t = F × v

Circular Motion:

Angular velocity: ω = v/r = 2π/T
Centripetal acceleration: a = v²/r = ω²r
Centripetal force: F = mv²/r = mω²r

Gravitation:

F = Gm₁m₂/r²  (G = 6.67×10⁻¹¹ Nm²/kg²)
g = GM/R²
Orbital velocity: v₀ = √(GM/r)
Escape velocity: vₑ = √(2GM/R)

2. Electricity & Magnetism

Ohm’s Law & Resistance:

V = IR
R = ρL/A
Series: Rₜ = R₁ + R₂ + R₃ + ...
Parallel: 1/Rₜ = 1/R₁ + 1/R₂ + 1/R₃ + ...

Electric Power:

P = VI = I²R = V²/R
Energy: E = Pt = VIt

Capacitance:

C = Q/V
Parallel plate: C = ε₀εᵣA/d
Energy stored: U = (1/2)CV² = (1/2)QV = Q²/2C

Series: 1/Cₜ = 1/C₁ + 1/C₂ + ...
Parallel: Cₜ = C₁ + C₂ + ...

Magnetic Force:

F = qvB sin θ  (on moving charge)
F = BIL sin θ  (on current-carrying conductor)

Electromagnetic Induction:

Faraday's Law: ε = -dΦ/dt = -N(dΦ/dt)
Induced EMF: ε = BLv  (for moving conductor)

3. Optics

Mirror Formula:

1/f = 1/v + 1/u
Magnification: m = -v/u = h'/h

For concave: f is negative
For convex: f is positive

Lens Formula:

1/f = 1/v - 1/u
Power: P = 1/f  (in meters)
Magnification: m = v/u = h'/h

Lens Maker’s Formula:

1/f = (μ-1)(1/R₁ - 1/R₂)

Refraction:

Snell's Law: μ₁ sin i = μ₂ sin r
Refractive index: μ = c/v = sin i/sin r

Critical angle: sin C = μ₂/μ₁  (μ₁ > μ₂)
Total Internal Reflection occurs when i > C

4. Modern Physics

Photoelectric Effect:

Energy of photon: E = hf = hc/λ
  h = 6.63 × 10⁻³⁴ J·s (Planck's constant)
  c = 3 × 10⁸ m/s

Einstein's equation: KEₘₐₓ = hf - φ
  φ = work function

De Broglie Wavelength:

λ = h/p = h/mv

Radioactivity:

N = N₀ e⁻ᵏᵗ
Half-life: T₁/₂ = 0.693/λ

5. Thermodynamics

Heat & Temperature:

Q = mcΔT
Latent heat: Q = mL

Gas Laws:

Boyle's Law: PV = constant  (T constant)
Charles' Law: V/T = constant  (P constant)
Ideal Gas: PV = nRT
  R = 8.314 J/(mol·K)

First Law of Thermodynamics:

ΔQ = ΔU + ΔW

🧪 Chemistry Formula Sheets

1. Physical Chemistry

Mole Concept:

Number of moles: n = m/M = V/22.4  (at STP)
  m = mass (g)
  M = molar mass (g/mol)
  V = volume (L) for gases at STP

Avogadro's number: Nₐ = 6.022 × 10²³
Number of particles: N = n × Nₐ

Percentage Composition:

% of element = (Mass of element in formula/Molecular mass) × 100

Empirical & Molecular Formula:

n = Molecular formula mass / Empirical formula mass
Molecular formula = n × Empirical formula

Gas Laws:

Combined Gas Law: (P₁V₁)/T₁ = (P₂V₂)/T₂
Ideal Gas Equation: PV = nRT
Dalton's Law: Pₜₒₜₐₗ = P₁ + P₂ + P₃ + ...
Graham's Law: r₁/r₂ = √(M₂/M₁)

Thermochemistry:

ΔH = Hₚᵣₒ𝒹ᵤ𝒸ₜₛ - Hᵣₑₐ𝒸ₜₐₙₜₛ
ΔH < 0 → Exothermic
ΔH > 0 → Endothermic

Hess's Law: ΔH = ΣΔHₚᵣₒ𝒹ᵤ𝒸ₜₛ - ΣΔHᵣₑₐ𝒸ₜₐₙₜₛ

Chemical Equilibrium:

Kc = [C]^c [D]^d / [A]^a [B]^b  for aA + bB ⇌ cC + dD

Kp = (Pc)^c (Pd)^d / (Pa)^a (Pb)^b

Relation: Kp = Kc(RT)^Δn
  Δn = (c+d) - (a+b)

Le Chatelier's Principle:
- ↑Concentration → Shift to consume
- ↑Pressure → Shift to fewer moles
- ↑Temperature → Shift to endothermic side

Acids & Bases:

pH = -log[H⁺]
pOH = -log[OH⁻]
pH + pOH = 14  (at 25°C)

[H⁺][OH⁻] = 10⁻¹⁴  (at 25°C)

For weak acids: pH = (1/2)[pKa - log C]
For weak bases: pOH = (1/2)[pKb - log C]

Electrochemistry:

E°cell = E°cathode - E°anode
ΔG° = -nFE°cell
  F = 96500 C/mol (Faraday constant)

Nernst Equation:
E = E° - (RT/nF) ln Q
  or E = E° - (0.059/n) log Q  at 25°C

2. Inorganic Chemistry

Periodic Table Trends:

Across a period (→):
- Atomic radius: Decreases
- Ionization energy: Increases
- Electronegativity: Increases
- Metallic character: Decreases

Down a group (↓):
- Atomic radius: Increases
- Ionization energy: Decreases
- Electronegativity: Decreases
- Metallic character: Increases

Oxidation States (Important):

Alkali metals: +1
Alkaline earth: +2
Aluminum: +3
Hydrogen: +1 (usually), -1 (with metals)
Oxygen: -2 (usually), -1 (peroxides)
Halogens: -1 (usually)

3. Organic Chemistry

Nomenclature (IUPAC):

Alkanes: CnH2n+2  (suffix: -ane)
Alkenes: CnH2n    (suffix: -ene)
Alkynes: CnH2n-2  (suffix: -yne)

Functional Groups (Priority order):
1. -COOH (Carboxylic acid)
2. -SO₃H (Sulfonic acid)
3. -COOR (Ester)
4. -COCl (Acid halide)
5. -CONH₂ (Amide)
6. -CN (Nitrile)
7. -CHO (Aldehyde)
8. -CO- (Ketone)
9. -OH (Alcohol)
10. -NH₂ (Amine)

Important Reactions:

Alkenes + H₂ → Alkanes (Hydrogenation)
Alkenes + HX → Haloalkanes (Markovnikov's rule)
Alkenes + H₂O/H⁺ → Alcohols
Alcohols + Na → Sodium alkoxide + H₂
Alcohols + Oxidation → Aldehydes/Ketones/Carboxylic acids

📱 Quick Reference Tables

Mathematical Constants:

π ≈ 3.14159
e ≈ 2.71828
√2 ≈ 1.414
√3 ≈ 1.732
log₁₀ 2 ≈ 0.301
log₁₀ 3 ≈ 0.477
ln 2 ≈ 0.693
ln 10 ≈ 2.303

Physical Constants:

Speed of light (c) = 3 × 10⁸ m/s
Planck's constant (h) = 6.63 × 10⁻³⁴ J·s
Gravitational constant (G) = 6.67 × 10⁻¹¹ Nm²/kg²
Acceleration due to gravity (g) = 9.8 m/s²
Avogadro's number (Nₐ) = 6.022 × 10²³
Gas constant (R) = 8.314 J/(mol·K)
Faraday constant (F) = 96500 C/mol
Electron charge (e) = 1.6 × 10⁻¹⁹ C

Unit Conversions:

1 km = 1000 m
1 hour = 3600 s
1 eV = 1.6 × 10⁻¹⁹ J
1 atm = 1.01 × 10⁵ Pa
1 calorie = 4.18 J
0°C = 273 K

💡 Formula Memorization Tips

Week-by-Week Plan:

Week 1: Mathematics (Algebra + Calculus)

• Day 1-2: Write all formulas once
• Day 3-4: Solve 5 problems per formula
• Day 5-7: Mixed practice + revision

Week 2: Physics (Mechanics + Electricity)

• Same approach as Week 1
• Focus on unit consistency

Week 3: Chemistry + Remaining Topics

• Organic reactions need special attention
• Create reaction flowcharts

Week 4: Consolidation

• Daily revision of all formulas
• Focus on weak areas
• Flashcard practice

🎯 Last-Minute Revision (24 Hours Before Exam)

Priority Formula List:

Mathematics (Must Know):

1. Quadratic formula
2. Integration basics (10 formulas)
3. Differentiation basics (15 formulas)
4. Trigonometric identities (8 formulas)
5. Coordinate geometry (distance, section, line equations)

Physics (Must Know):

1. Motion equations (4 formulas)
2. Ohm’s law and power (P=VI, V=IR)
3. Mirror/Lens formula
4. Capacitor formulas
5. Photoelectric equation

Chemistry (Must Know):

1. Mole concept (n=m/M)
2. pH formulas
3. Gas laws (PV=nRT)
4. Nernst equation
5. Common organic reactions (5-6)

🔗 Related Resources

Practice & Application:

• Mock Test Series - Apply formulas
• Previous Year Analysis - See formula-based questions

Study Strategies:

• Last 30 Days Plan
• Last 15 Days Plan
• Last 7 Days Plan

Community Support:

• CUET Forum - Ask formula doubts
• Ask Expert - Live sessions
• Study Groups - Learn together

Subject Pages:

• Mathematics Preparation
• Physics Preparation
• Chemistry Preparation

📥 Downloadable Sheets

Coming Soon:

• PDF versions of all formula sheets
• Subject-wise flashcards
• Formula practice worksheets
• Quick revision posters

For now: Bookmark this page and access anytime!

🚀 Final Words

“Formulas are tools. Master them, and the exam becomes easy.”

Your Formula Mastery Plan:

1. Understand: Know the logic behind each formula
2. Practice: Use each formula in 10+ problems
3. Memorize: Write daily until automatic
4. Apply: Solve mixed problems
5. Revise: Regular spaced repetition

Remember:

• Understanding > Memorization
• Practice makes permanent
• Last-minute cramming doesn’t work
• Formula sheets are revision tools, not learning tools

Master these formulas and ace your CUET exam! 🎯📐✨

Need help with any formula? Ask on CUET Forum!