Mathematics IB — Formula Sheet
All key formulas for TS Inter 1st Year Maths IB — Straight Lines, Pair of Lines, 3D Geometry, Limits, Differentiation, and Applications. Based on official TGBIE Annual Plan 2025-26.
Ch. 3 The Straight Line — Key Formulas
- 1Slope of line through (x₁,y₁) and (x₂,y₂): m = (y₂−y₁)/(x₂−x₁)
- 2Slope-intercept form: y = mx + c
- 3Point-slope form: y − y₁ = m(x − x₁)
- 4Two-point form: (y−y₁)/(y₂−y₁) = (x−x₁)/(x₂−x₁)
- 5Normal form: x·cosα + y·sinα = p
- 6Symmetric form: (x−x₁)/cosθ = (y−y₁)/sinθ = r
- 7Intercept form: x/a + y/b = 1
- 8General form: ax + by + c = 0; slope = −a/b
- 9Angle between two lines: tanθ = |(m₁−m₂)/(1+m₁m₂)|
- 10Perpendicular: m₁·m₂ = −1; Parallel: m₁ = m₂
- 11Distance from point (x₁,y₁) to ax+by+c=0: d = |ax₁+by₁+c|/√(a²+b²)
- 12Distance between parallel lines ax+by+c₁=0 and ax+by+c₂=0: d = |c₁−c₂|/√(a²+b²)
Ch. 4 Pair of Straight Lines — Key Formulas
- 1Combined equation of pair through origin: ax²+2hxy+by²=0
- 2Slope product: m₁·m₂ = a/b; slope sum: m₁+m₂ = −2h/b
- 3Angle between pair: tanθ = 2√(h²−ab)/(a+b)
- 4Perpendicular pair: a + b = 0
- 5Coincident pair: h² − ab = 0
- 6Angle bisectors of ax²+2hxy+by²=0: (x²−y²)/(a−b) = xy/h
- 7Second degree equation ax²+2hxy+by²+2gx+2fy+c=0 represents a pair of lines if Δ = abc+2fgh−af²−bg²−ch² = 0
Ch. 5 & 6 Three Dimensional Geometry — Key Formulas
- 1Distance between (x₁,y₁,z₁) and (x₂,y₂,z₂): √((x₂−x₁)²+(y₂−y₁)²+(z₂−z₁)²)
- 2Section formula (internal): ((mx₂+nx₁)/(m+n), (my₂+ny₁)/(m+n), (mz₂+nz₁)/(m+n))
- 3Centroid of triangle: ((x₁+x₂+x₃)/3, (y₁+y₂+y₃)/3, (z₁+z₂+z₃)/3)
- 4Direction cosines: l²+m²+n²=1
- 5Angle between lines: cosθ = l₁l₂+m₁m₂+n₁n₂
- 6Perpendicular condition: l₁l₂+m₁m₂+n₁n₂ = 0
Ch. 8 Limits — Standard Limits
- 1lim(x→a) [xⁿ−aⁿ]/(x−a) = naⁿ⁻¹
- 2lim(x→0) sinx/x = 1 (x in radians)
- 3lim(x→0) tanx/x = 1
- 4lim(x→0) (1−cosx)/x² = 1/2
- 5lim(x→0) (aˣ−1)/x = logₑa; lim(x→0) (eˣ−1)/x = 1
- 6lim(x→0) (1+x)^(1/x) = e; lim(n→∞) (1+1/n)ⁿ = e
- 7lim(x→0) log(1+x)/x = 1
Ch. 9 Differentiation — Key Formulas
- 1d/dx(xⁿ) = nxⁿ⁻¹; d/dx(constant) = 0
- 2d/dx(sinx) = cosx; d/dx(cosx) = −sinx
- 3d/dx(tanx) = sec²x; d/dx(cotx) = −cosec²x
- 4d/dx(secx) = secx·tanx; d/dx(cosecx) = −cosecx·cotx
- 5d/dx(eˣ) = eˣ; d/dx(aˣ) = aˣ·logₑa
- 6d/dx(logx) = 1/x; d/dx(logₐx) = 1/(x·logₑa)
- 7d/dx(sin⁻¹x) = 1/√(1−x²); d/dx(cos⁻¹x) = −1/√(1−x²)
- 8d/dx(tan⁻¹x) = 1/(1+x²); d/dx(cot⁻¹x) = −1/(1+x²)
- 9Product rule: d/dx(uv) = u·v' + v·u'
- 10Quotient rule: d/dx(u/v) = (v·u' − u·v')/v²
- 11Chain rule: dy/dx = (dy/du)·(du/dx)
Ch. 10 Applications of Derivatives — Key Formulas
- 1Slope of tangent at (x₁,y₁): m = (dy/dx) at (x₁,y₁)
- 2Equation of tangent: y − y₁ = m(x − x₁)
- 3Equation of normal: y − y₁ = (−1/m)(x − x₁)
- 4Length of tangent: y₁√(1 + (dx/dy)²)
- 5Length of normal: y₁√(1 + (dy/dx)²)
- 6Rolle's theorem: f(a)=f(b) and f continuous on [a,b] → ∃c∈(a,b): f'(c)=0
- 7Lagrange's MVT: ∃c∈(a,b): f'(c) = (f(b)−f(a))/(b−a)
- 8f increasing on I if f'(x) > 0 ∀ x ∈ I; decreasing if f'(x) < 0
- 9Local max: f'(c) = 0 and f''(c) < 0; Local min: f'(c) = 0 and f''(c) > 0