Maths IIB — Formula Sheet
Key formulas for all 8 chapters of TS Intermediate 2nd Year Mathematics IIB as per the TGBIE Annual Plan 2025-26. Paper: 75 marks · 3 hours.
1.
Circle
- General form: x² + y² + 2gx + 2fy + c = 0; centre (−g, −f), radius r = √(g²+f²−c)
- Standard form: (x−h)² + (y−k)² = r²; centre (h,k), radius r
- Tangent at (x₁,y₁) on x²+y²+2gx+2fy+c=0: xx₁+yy₁+g(x+x₁)+f(y+y₁)+c=0
- Length of tangent from (x₁,y₁): √(x₁²+y₁²+2gx₁+2fy₁+c)
- Condition for two circles to be orthogonal: 2g₁g₂ + 2f₁f₂ = c₁ + c₂
2.
System of Circles
- Radical axis of S₁=0 and S₂=0: S₁ − S₂ = 0
- Radical centre: point where all three radical axes of three circles meet
- Angle θ between two circles: cos θ = (r₁²+r₂²−d²)/(2r₁r₂)
- Orthogonal circles: d² = r₁² + r₂²
3.
Parabola
- Standard form: y² = 4ax; focus (a,0), directrix x = −a, vertex (0,0)
- Latus rectum length: 4a; endpoints: (a, ±2a)
- Parametric point: (at², 2at)
- Tangent at (x₁,y₁): yy₁ = 2a(x + x₁)
- Tangent in parametric form: ty = x + at²
- Normal at (at², 2at): y + tx = 2at + at³
4.
Ellipse
- Standard form: x²/a² + y²/b² = 1 (a > b); c² = a² − b²
- Eccentricity: e = c/a < 1; foci: (±ae, 0); directrices: x = ±a/e
- Latus rectum length: 2b²/a; semi-latus rectum: b²/a
- Parametric point: (a cos θ, b sin θ)
- Tangent at (x₁,y₁): xx₁/a² + yy₁/b² = 1
- Tangent in slope form: y = mx ± √(a²m²+b²)
5.
Hyperbola
- Standard form: x²/a² − y²/b² = 1; c² = a² + b²; e = c/a > 1
- Asymptotes: y = ±(b/a)x; combined equation: x²/a² − y²/b² = 0
- Rectangular hyperbola: xy = c²; parametric point: (ct, c/t)
- Tangent at (x₁,y₁): xx₁/a² − yy₁/b² = 1
6.
Integration
- ∫xⁿ dx = xⁿ⁺¹/(n+1) + C (n ≠ −1); ∫(1/x)dx = ln|x| + C
- ∫eˣ dx = eˣ + C; ∫aˣ dx = aˣ/ln a + C
- ∫sin x dx = −cos x + C; ∫cos x dx = sin x + C
- ∫tan x dx = ln|sec x| + C; ∫sec²x dx = tan x + C
- Integration by parts: ∫u dv = uv − ∫v du (ILATE order)
- ∫1/(a²−x²) dx = (1/2a)ln|(a+x)/(a−x)| + C
- ∫1/√(a²−x²) dx = sin⁻¹(x/a) + C
- ∫1/(a²+x²) dx = (1/a)tan⁻¹(x/a) + C
7.
Definite Integrals
- ∫ₐᵇ f(x)dx = F(b) − F(a) where F'(x) = f(x)
- ∫ₐᵇ f(x)dx = ∫ₐᵇ f(a+b−x)dx
- ∫₀ᵃ f(x)dx = ∫₀ᵃ f(a−x)dx
- ∫₀²ᵃ f(x)dx = 2∫₀ᵃ f(x)dx if f(2a−x)=f(x); = 0 if f(2a−x)=−f(x)
- Reduction formula: ∫₀^(π/2) sinⁿx dx = [(n−1)/n]·[(n−3)/(n−2)]·… ×(π/2 or 1)
8.
Differential Equations
- Variables separable: f(x)dx = g(y)dy → ∫f(x)dx = ∫g(y)dy
- Homogeneous DE: dy/dx = f(y/x); substitute y = vx
- Non-homogeneous: dy/dx = (ax+by+c)/(px+qy+r); shift origin to intersection point
- Linear DE: dy/dx + Py = Q; Integrating Factor (IF) = e^∫P dx
- Solution: y(IF) = ∫Q(IF)dx + C