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📐 KCET 2026 — Important Maths Formulas

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1. ALGEBRA

Sets & Relations

• n(A ∪ B) = n(A) + n(B) − n(A ∩ B)
• n(A ∪ B ∪ C) = n(A) + n(B) + n(C) − n(A∩B) − n(B∩C) − n(A∩C) + n(A∩B∩C)

Quadratic Equations

• Roots: x = (−b ± √(b²−4ac)) / 2a
• Sum of roots: α + β = −b/a
• Product of roots: αβ = c/a
• Nature of roots: D = b² − 4ac

Progressions

• AP: aₙ = a + (n−1)d | Sₙ = n/2 [2a + (n−1)d]
• GP: aₙ = arⁿ⁻¹ | Sₙ = a(rⁿ−1)/(r−1) | S∞ = a/(1−r), |r| < 1
• Sum of natural numbers: Σn = n(n+1)/2
• Sum of squares: Σn² = n(n+1)(2n+1)/6
• Sum of cubes: Σn³ = [n(n+1)/2]²

Binomial Theorem

• (a + b)ⁿ = Σ ⁿCᵣ · aⁿ⁻ʳ · bʳ
• General term: Tᵣ₊₁ = ⁿCᵣ · aⁿ⁻ʳ · bʳ
• Middle term (n even): T_(n/2+1)

Permutations & Combinations

• nPr = n! / (n−r)!
• nCr = n! / [r!(n−r)!]
• nCr = nC(n−r)
• nC0 = nCn = 1

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2. TRIGONOMETRY

Basic Identities

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

Compound Angles

• sin(A±B) = sinA cosB ± cosA sinB
• cos(A±B) = cosA cosB ∓ sinA sinB
• tan(A±B) = (tanA ± tanB) / (1 ∓ tanA tanB)

Multiple Angles

• sin2A = 2sinA cosA = 2tanA/(1+tan²A)
• cos2A = cos²A − sin²A = 1−2sin²A = 2cos²A−1 = (1−tan²A)/(1+tan²A)
• tan2A = 2tanA/(1−tan²A)
• sin3A = 3sinA − 4sin³A
• cos3A = 4cos³A − 3cosA

Product to Sum / Sum to Product

• 2sinA cosB = sin(A+B) + sin(A−B)
• 2cosA cosB = cos(A−B) + cos(A+B)
• 2sinA sinB = cos(A−B) − cos(A+B)
• sinC + sinD = 2sin((C+D)/2)cos((C−D)/2)
• cosC + cosD = 2cos((C+D)/2)cos((C−D)/2)

Inverse Trig

• sin⁻¹x + cos⁻¹x = π/2
• tan⁻¹x + cot⁻¹x = π/2
• tan⁻¹x + tan⁻¹y = tan⁻¹[(x+y)/(1−xy)], xy < 1
• 2tan⁻¹x = sin⁻¹(2x/(1+x²)) = cos⁻¹((1−x²)/(1+x²)) = tan⁻¹(2x/(1−x²))

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3. COORDINATE GEOMETRY

Straight Lines

• Slope: m = (y₂−y₁)/(x₂−x₁)
• Point-slope form: y−y₁ = m(x−x₁)
• Slope-intercept: y = mx + c
• Distance between two points: d = √[(x₂−x₁)² + (y₂−y₁)²]
• Distance from point to line: d = |ax₁+by₁+c| / √(a²+b²)
• Area of triangle: ½|x₁(y₂−y₃) + x₂(y₃−y₁) + x₃(y₁−y₂)|

Circles

• Standard: x² + y² = r²
• General: x² + y² + 2gx + 2fy + c = 0; centre (−g, −f), radius = √(g²+f²−c)
• Length of tangent from external point: √(x₁²+y₁²+2gx₁+2fy₁+c)

Parabola (y² = 4ax)

• Focus: (a, 0) | Directrix: x = −a
• Length of latus rectum: 4a
• Parametric: (at², 2at)

Ellipse (x²/a² + y²/b² = 1, a > b)

• b² = a²(1−e²)
• Foci: (±ae, 0) | e = c/a where c² = a²−b²

Hyperbola (x²/a² − y²/b² = 1)

• b² = a²(e²−1)
• Asymptotes: y = ±(b/a)x

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4. CALCULUS

Limits

• lim(x→0) sinx/x = 1
• lim(x→0) tanx/x = 1
• lim(x→0) (eˣ−1)/x = 1
• lim(x→0) (aˣ−1)/x = logₑa
• lim(x→0) (1+x)^(1/x) = e
• lim(x→∞) (1+1/x)ˣ = e

Differentiation

• d/dx(xⁿ) = nxⁿ⁻¹
• d/dx(eˣ) = eˣ
• d/dx(aˣ) = aˣ logₑa
• d/dx(ln x) = 1/x
• 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(sin⁻¹x) = 1/√(1−x²)
• d/dx(cos⁻¹x) = −1/√(1−x²)
• d/dx(tan⁻¹x) = 1/(1+x²)
• Product rule: (uv)' = u'v + uv'
• Quotient rule: (u/v)' = (u'v − uv')/v²
• Chain rule: dy/dx = (dy/du)·(du/dx)

Integration

• ∫xⁿ dx = xⁿ⁺¹/(n+1) + C
• ∫eˣ dx = eˣ + C
• ∫1/x dx = ln|x| + C
• ∫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
• ∫1/√(1−x²) dx = sin⁻¹x + C
• ∫1/(1+x²) dx = tan⁻¹x + C
• ∫1/√(x²−a²) dx = ln|x+√(x²−a²)| + C
• By parts: ∫u dv = uv − ∫v du (ILATE rule)
• Definite integral property: ∫₀ᵃ f(x) dx = ∫₀ᵃ f(a−x) dx

Application of Derivatives

• Maxima/Minima: f'(x) = 0 and check f''(x)
• Rate of change: dy/dt = (dy/dx)·(dx/dt)
• Equation of tangent at (x₁,y₁): y−y₁ = f'(x₁)(x−x₁)
• Equation of normal: y−y₁ = −1/f'(x₁) · (x−x₁)

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5. MATRICES & DETERMINANTS

• det of 2×2: |a b; c d| = ad − bc
• Inverse: A⁻¹ = adjA / |A|
• |AB| = |A||B|
• (AB)⁻¹ = B⁻¹A⁻¹
• (Aᵀ)ᵀ = A
• Cramer's rule: x = D₁/D, y = D₂/D, z = D₃/D

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6. VECTORS & 3D GEOMETRY

Vectors

• |a⃗| = √(a₁²+a₂²+a₃²)
• a⃗ · b⃗ = |a||b|cosθ → cosθ = (a⃗·b⃗)/(|a||b|)
• a⃗ × b⃗ = |a||b|sinθ n̂
• |a⃗ × b⃗|² + (a⃗·b⃗)² = |a|²|b|²
• Scalar triple product: [a⃗ b⃗ c⃗] = a⃗·(b⃗×c⃗)

3D Geometry

• Distance formula: d = √[(x₂−x₁)²+(y₂−y₁)²+(z₂−z₁)²]
• Direction cosines: l²+m²+n² = 1
• Angle between lines: cosθ = |l₁l₂+m₁m₂+n₁n₂|
• Distance from point to plane: d = |ax₁+by₁+cz₁+d| / √(a²+b²+c²)

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7. PROBABILITY & STATISTICS

• P(A∪B) = P(A) + P(B) − P(A∩B)
• P(A|B) = P(A∩B)/P(B)
• Bayes' theorem: P(Aᵢ|B) = P(Aᵢ)·P(B|Aᵢ) / ΣP(Aⱼ)·P(B|Aⱼ)
• Mean of Binomial: μ = np
• Variance of Binomial: σ² = npq
• P(X=r) = ⁿCᵣ · pʳ · qⁿ⁻ʳ

Statistics

• Mean: x̄ = Σfx / Σf
• Variance: σ² = Σf(x−x̄)² / Σf
• Standard deviation: σ = √variance

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8. LINEAR PROGRAMMING

• Optimal value occurs at a corner point of the feasible region.
• Always evaluate the objective function Z = ax + by at all corner points.

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✅ KCET-Specific Tips

• Calculus (Limits, Derivatives, Integration) carries the highest weightage (~25–30% of maths questions)
• Coordinate Geometry and Algebra are the next most important
• Focus on standard integrals, application of derivatives, and conic sections — these are frequently tested
• Practice 1-mark MCQs — speed and accuracy matter more than lengthy solutions