2nd Year Math Formulas
All 2nd Year Math formulas, chapter by chapter, with search and a printable layout.
Chapters are placed in 1st or 2nd Year following the usual Punjab and Federal board outline. Your board may order chapters differently, so check your own syllabus.
Chapters: Coordinate Geometry · Calculus · Integration · Vectors · Conic Sections · Limits · Differentiation · 3D Geometry · Inequalities
Coordinate Geometry (16)
| Formula | Expression |
|---|
| Distance Formula | d = √((x₂-x₁)² + (y₂-y₁)²) |
| Midpoint | M = ((x₁+x₂)/2, (y₁+y₂)/2) |
| Slope | m = (y₂-y₁)/(x₂-x₁) |
| Line Equation | y = mx + c |
| Circle Equation | (x-h)² + (y-k)² = r² |
| Section Formula | P = ((mx₂+nx₁)/(m+n), (my₂+ny₁)/(m+n)) |
| Area of Triangle (coords) | A = ½|x₁(y₂−y₃)+x₂(y₃−y₁)+x₃(y₁−y₂)| |
| Perpendicular Distance (point-line) | d = |Ax₁+By₁+C|/√(A²+B²) |
| Point-Slope Form | y − y₁ = m(x − x₁) |
| Ellipse Equation | x²/a² + y²/b² = 1 |
| Hyperbola Equation | x²/a² − y²/b² = 1 |
| Parabola Equation | y² = 4ax |
| Eccentricity of Ellipse | e = √(1 − b²/a²) |
| Focus of Parabola | F = (a, 0), for y² = 4ax |
| Angle Between Two Lines | tanθ = |(m₁−m₂)/(1+m₁m₂)| |
| Centroid of Triangle | G = ((x₁+x₂+x₃)/3, (y₁+y₂+y₃)/3) |
Calculus (33)
| Formula | Expression |
|---|
| Power Rule | d/dx(xⁿ) = nxⁿ⁻¹ |
| Product Rule | d/dx(uv) = u'v + uv' |
| Quotient Rule | d/dx(u/v) = (u'v - uv')/v² |
| Chain Rule | dy/dx = dy/du × du/dx |
| d/dx(sin x) | cos x |
| d/dx(cos x) | -sin x |
| d/dx(tan x) | sec²x |
| d/dx(eˣ) | eˣ |
| d/dx(ln x) | 1/x |
| d/dx(secx) | secx tanx |
| d/dx(cscx) | -cscx cotx |
| d/dx(cotx) | -csc²x |
| Logarithmic Differentiation (concept) | take ln of both sides before differentiating |
| d/dx(cot x) | −cosec²x |
| d/dx(sec x) | sec x tan x |
| d/dx(cosec x) | −cosec x cot x |
| d/dx(logₐ x) | 1/(x ln a) |
| d/dx(aˣ) | aˣ ln a |
| d/dx(sin⁻¹x) | 1/√(1−x²) |
| d/dx(cos⁻¹x) | −1/√(1−x²) |
| d/dx(tan⁻¹x) | 1/(1+x²) |
| Second Derivative Test (Maxima) | f'(x)=0, f''(x)<0 |
| Second Derivative Test (Minima) | f'(x)=0, f''(x)>0 |
| Slope of Tangent | m = f'(x₀) |
| Equation of Tangent Line | y − y₀ = f'(x₀)(x − x₀) |
| Equation of Normal Line | y − y₀ = −1/f'(x₀) (x − x₀) |
| Rate of Change | dy/dt = dy/dx × dx/dt |
| L'Hopital's Rule | lim f/g = lim f'/g' (0/0 or infinity/infinity form) |
| Definition of Derivative | f'(x) = lim(h→0) [f(x+h)−f(x)]/h |
| Implicit Differentiation (concept) | differentiate both sides w.r.t x treating y as f(x) |
| Concavity Test | f''(x) > 0 concave up, f''(x) < 0 concave down |
| Point of Inflection | f''(x) = 0 and changes sign |
| Related Rates Setup (concept) | differentiate a relating equation w.r.t time |
Integration (24)
| Formula | Expression |
|---|
| Power Rule | ∫xⁿ dx = xⁿ⁺¹/(n+1) + C |
| ∫1/x dx | ln|x| + C |
| ∫eˣ dx | eˣ + C |
| ∫sin x dx | -cos x + C |
| ∫cos x dx | sin x + C |
| Integration by Parts | ∫u dv = uv - ∫v du |
| Definite Integral (FTC) | ∫ₐᵇ f(x)dx = F(b) − F(a) |
| Area Between Curves | A = ∫ₐᵇ [f(x) − g(x)] dx |
| Volume of Revolution | V = π∫ₐᵇ [f(x)]² dx |
| ∫sec²x dx | tanx + C |
| ∫secx tanx dx | secx + C |
| ∫cosec²x dx | −cot x + C |
| ∫sec x tan x dx | sec x + C |
| ∫cosec x cot x dx | −cosec x + C |
| ∫tan x dx | ln|sec x| + C |
| ∫aˣ dx | aˣ/ln a + C |
| ∫1/(1+x²) dx | tan⁻¹x + C |
| ∫1/√(1−x²) dx | sin⁻¹x + C |
| Definite Integral - Area Under Curve | A = ∫[a,b] f(x) dx |
| Fundamental Theorem of Calculus | ∫[a,b] f(x)dx = F(b) − F(a) |
| Volume by Disk Method | V = π∫[a,b] [f(x)]² dx |
| Area Between Two Curves | A = ∫[a,b] [f(x) − g(x)] dx |
| Integration by Substitution (concept) | let u = g(x), du = g'(x)dx |
| Average Value of Function | f_avg = 1/(b−a) ∫[a,b] f(x)dx |
Vectors (3)
| Formula | Expression |
|---|
| Vector Magnitude | |A| = √(x²+y²+z²) |
| Unit Vector | â = A/|A| |
| Scalar Triple Product | A·(B×C) |
Conic Sections (4)
| Formula | Expression |
|---|
| Parabola | y² = 4ax |
| Ellipse | x²/a² + y²/b² = 1 |
| Hyperbola | x²/a² − y²/b² = 1 |
| Eccentricity (ellipse) | e = √(1 − b²/a²) |
Limits (2)
| Formula | Expression |
|---|
| Standard Limit sinx/x | lim(x→0) sinx/x = 1 |
| L'Hôpital's Rule | lim f/g = lim f'/g' (0/0 or ∞/∞) |
Differentiation (2)
| Formula | Expression |
|---|
| Implicit Differentiation | d/dx[F(x,y)=0] → solve for dy/dx |
| Second Derivative Test | f''(x) > 0 ⇒ local min |
3D Geometry (8)
| Formula | Expression |
|---|
| Direction Cosines Relation | l² + m² + n² = 1 |
| Distance Between Two Points in 3D | d = √((x2-x1)² + (y2-y1)² + (z2-z1)²) |
| Equation of Plane (Normal Form) | ax + by + cz = d |
| Angle Between Two Planes | cos(theta) = |a1a2+b1b2+c1c2| / (|n1||n2|) |
| Equation of Line in Space | (x-x1)/l = (y-y1)/m = (z-z1)/n |
| Distance Between Point and Plane | d = |ax1+by1+cz1-d| / sqrt(a²+b²+c²) |
| Direction Ratios from Two Points | (x2-x1, y2-y1, z2-z1) |
| Angle Between Two Lines (3D) | cos(theta) = (l1l2+m1m2+n1n2) |
Inequalities (4)
| Formula | Expression |
|---|
| Triangle Inequality | |a+b| <= |a|+|b| |
| AM-GM Inequality (n terms) | (a1+a2+...+an)/n >= (a1a2...an)^(1/n) |
| Cauchy-Schwarz Inequality | (Sum aibi)² <= (Sum ai²)(Sum bi²) |
| Absolute Value Inequality | |x-a| < b means a-b < x < a+b |
More class-wise formulas
Formulas are for revision. Always follow the notation and rounding used in your own textbook and board papers.
FAQ
How many 2nd Year Math formulas are on this page?
96 formulas in 9 chapters.
Can I print the 2nd Year Math formulas?
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Do the formulas follow my board syllabus?
They follow the usual school and intermediate syllabus, but chapter lists differ between boards. Check your own syllabus for what is included in your exam.
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