Mathematics Formula Sheet for CUET UG 2025

📋 Comprehensive Collection of Essential Mathematics Formulas

This formula sheet covers all essential topics for CUET UG Mathematics. Keep this handy for quick revision and last-minute preparation.

🔢 Algebra

Quadratic Equations

• Standard Form: ax² + bx + c = 0
• Root Formula: x = [-b ± √(b² - 4ac)] / 2a
• Discriminant: Δ = b² - 4ac
• Sum of Roots: α + β = -b/a
• Product of Roots: αβ = c/a

Progressions

Arithmetic Progression (AP)

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

Geometric Progression (GP)

• nth Term: aₙ = ar^(n-1)
• Sum of n terms: Sₙ = a(rⁿ - 1)/(r - 1), r ≠ 1
• Sum to infinity: S∞ = a/(1 - r), |r| < 1

Harmonic Progression (HP)

• Reciprocal of AP: If a, b, c are in AP, then 1/a, 1/b, 1/c are in HP
• nth Term: Hₙ = 1/Aₙ where Aₙ is nth term of corresponding AP

Binomial Theorem

• (a + b)ⁿ = Σ(nCr) a^(n-r) b^r, where r = 0 to n
• General Term: T_(r+1) = nCr a^(n-r) b^r
• Middle Term:
  • If n is even: (n/2 + 1)th term
  • If n is odd: (n+1)/2th and (n+3)/2th terms

Permutations and Combinations

• Permutation: nPr = n!/(n-r)!
• Combination: nCr = n!/[r!(n-r)!]
• Fundamental Principle:
  • Addition: m + n ways
  • Multiplication: m × n ways

Matrices and Determinants

• Matrix Addition: A + B = [a_ij + b_ij]
• Matrix Multiplication: AB = C where c_ij = Σ(a_ik × b_kj)
• 2×2 Determinant: |a b; c d| = ad - bc
• 3×3 Determinant: |a b c; d e f; g h i| = a(ei - fh) - b(di - fg) + c(dh - eg)

📐 Trigonometry

Basic Trigonometric Ratios

• sin θ = Opposite/Hypotenuse
• cos θ = Adjacent/Hypotenuse
• tan θ = Opposite/Adjacent = sin θ/cos θ
• cot θ = Adjacent/Opposite = cos θ/sin θ
• sec θ = Hypotenuse/Adjacent = 1/cos θ
• cosec θ = Hypotenuse/Opposite = 1/sin θ

Trigonometric Identities

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

Angle Addition 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 = 1 - 2 sin²A = 2 cos²A - 1
• tan 2A = 2 tan A/(1 - tan²A)

Triple Angle Formulas

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

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)

📏 Coordinate Geometry

Distance and Section Formula

• Distance between points: d = √[(x₂-x₁)² + (y₂-y₁)²]
• Section Formula (Internal): [(mx₂ + nx₁)/(m+n), (my₂ + ny₁)/(m+n)]
• Section Formula (External): [(mx₂ - nx₁)/(m-n), (my₂ - ny₁)/(m-n)]

Straight Lines

• Slope: m = (y₂-y₁)/(x₂-x₁)
• Equation forms:
  • Slope-intercept: y = mx + c
  • Point-slope: y - y₁ = m(x - x₁)
  • Two-point: (y-y₁)/(y₂-y₁) = (x-x₁)/(x₂-x₁)
  • General: ax + by + c = 0
• Angle between lines: tan θ = |(m₂-m₁)/(1+m₁m₂)|
• Parallel lines: m₁ = m₂
• Perpendicular lines: m₁m₂ = -1

Circles

• Standard Form: (x-a)² + (y-b)² = r²
• General Form: x² + y² + 2gx + 2fy + c = 0
• Center: (-g, -f)
• Radius: √(g² + f² - c)
• Circle through origin: x² + y² + 2gx + 2fy = 0

Parabola

• Standard Form: y² = 4ax (rightward), x² = 4ay (upward)
• Focus: (a, 0) for y² = 4ax, (0, a) for x² = 4ay
• Directrix: x = -a for y² = 4ax, y = -a for x² = 4ay
• Latus Rectum: 4a

Ellipse

• Standard Form: x²/a² + y²/b² = 1 (a > b)
• Major axis: 2a, Minor axis: 2b
• Foci: (±c, 0) where c² = a² - b²
• Eccentricity: e = c/a
• Directrices: x = ±a/e

Hyperbola

• Standard Form: x²/a² - y²/b² = 1
• Transverse axis: 2a, Conjugate axis: 2b
• Foci: (±c, 0) where c² = a² + b²
• Eccentricity: e = c/a
• Asymptotes: y = ±(b/a)x

📈 Calculus

Limits

• Basic Limits:
  • lim(x→0) sin x/x = 1
  • lim(x→0) (1 - cos x)/x = 0
  • lim(x→0) (a^x - 1)/x = ln a
  • lim(x→∞) (1 + 1/x)^x = e

Differentiation Formulas

• Power Rule: d/dx(x^n) = nx^(n-1)
• 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: d/dx[f(g(x))] = f’(g(x)) × g’(x)

Derivatives of Functions

• 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
• 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)

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

• Trigonometric Integrals:

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

Integration by Parts

• Formula: ∫u dv = uv - ∫v du

Definite Integrals

• Fundamental Theorem: ∫(a to b) f(x) dx = F(b) - F(a)
• Properties:
  • ∫(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
  • ∫(a to b) f(x) dx = ∫(a to b) f(a + b - x) dx

📊 Probability and Statistics

Probability

• Basic Probability: P(A) = n(A)/n(S)
• Addition Rule: P(A ∪ B) = P(A) + P(B) - P(A ∩ B)
• Multiplication Rule: P(A ∩ B) = P(A) × P(B|A)
• Conditional Probability: P(A|B) = P(A ∩ B)/P(B)

Permutations and Combinations

• nPn = n!
• nCr = nPr/r!
• nC0 = nCn = 1
• nC1 = nC(n-1) = n

Statistics

• Mean: x̄ = (Σx)/n

• Median: Middle value when arranged in order

• Mode: Most frequently occurring value

• Range: Maximum - Minimum

• Variance: σ² = Σ(x - x̄)²/n

• Standard Deviation: σ = √(Σ(x - x̄)²/n)

• Coefficient of Variation: CV = (σ/x̄) × 100%

Correlation

• Correlation Coefficient: r = Σ[(x - x̄)(y - ȳ)]/[√Σ(x - x̄)² × √Σ(y - ȳ)²]

🎯 Vectors

Vector Operations

• Magnitude: |r| = √(x² + y² + z²)
• Unit Vector: r̂ = r/|r|
• Dot Product: a · b = |a||b|cos θ
• Cross Product: a × b = |a||b|sin θ n̂

Vector in 3D

• Position Vector: r = xî + yĵ + zk̂
• Direction Cosines: l² + m² + n² = 1
• Section Formula: r = (mr₂ + nr₁)/(m + n)

🔵 Sets and Relations

Set Operations

• Union: A ∪ B = {x | x ∈ A or x ∈ B}
• Intersection: A ∩ B = {x | x ∈ A and x ∈ B}
• Complement: A’ = {x | x ∉ A}
• Difference: A - B = {x | x ∈ A and x ∉ B}

De Morgan’s Laws

• (A ∪ B)’ = A’ ∩ B’
• (A ∩ B)’ = A’ ∪ B’

📐 Geometry

Triangle Formulas

• Area: ½ × base × height
• Heron’s Formula: A = √[s(s-a)(s-b)(s-c)], where s = (a+b+c)/2
• Sine Rule: a/sin A = b/sin B = c/sin C = 2R
• Cosine Rule: a² = b² + c² - 2bc cos A

Circle Formulas

• Circumference: 2πr
• Area: πr²
• Arc Length: l = θr (θ in radians)
• Sector Area: A = ½r²θ

3D Geometry

• Cube: Surface Area = 6a², Volume = a³
• Cuboid: Surface Area = 2(lw + wh + hl), Volume = lwh
• Sphere: Surface Area = 4πr², Volume = (4/3)πr³
• Cylinder: Surface Area = 2πr(r + h), Volume = πr²h
• Cone: Surface Area = πr(r + l), Volume = (1/3)πr²h

🎲 Logarithms

Logarithmic Rules

• log₁ₐa = 1
• log₁ₐ1 = 0
• log₁ₐ(ab) = log₁ₐa + log₁ₐb
• log₁ₐ(a/b) = log₁ₐa - log₁ₐb
• log₁ₐa^n = n log₁ₐa
• Change of Base: log₁ₐb = log₁ₓb/log₁ₓa**

🔢 Complex Numbers

Complex Number Operations

• Basic Form: z = a + ib
• Modulus: |z| = √(a² + b²)
• Argument: θ = tan⁻¹(b/a)
• Polar Form: z = r(cos θ + i sin θ)
• Euler’s Form: z = re^(iθ)

De Moivre’s Theorem

• (cos θ + i sin θ)^n = cos nθ + i sin nθ

📚 Quick Reference Tables

Common Values

Squares and Cubes

Logarithm Values

🎯 Exam Tips

Quick Revision Strategy

1. Memorize basic formulas - These are frequently used
2. Practice application - Understand when to use each formula
3. Create formula cards - For quick reference
4. Solve previous year questions - To see patterns
5. Time management - Quick access saves time

Common Mistakes to Avoid

1. Sign errors in trigonometric functions
2. Domain restrictions in logarithmic functions
3. Absolute value in integration results
4. Plus C in indefinite integrals
5. Units in application problems

🔗 Additional Resources

Practice Materials

• Mathematics Practice Questions
• Previous Year Questions
• Mock Tests

Study Support

• Expert Guidance
• Study Groups

📌 Remember: Practice regularly with these formulas to build speed and accuracy in CUET UG Mathematics!

Last Updated: October 2024 | CUET UG 2025 Mathematics Formula Sheet