Basic Mathematics for NEET Physics
Build unshakeable foundations — trigonometry, calculus, vectors & more, explained for NEET aspirants.
🔍 Detailed concept explanation
📐 1. Trigonometry fundamentals
Angles measured in radians (for calculus). For NEET, remember: sin(90°-θ)=cosθ, quadrants signs, and basic identities: sin²θ + cos²θ = 1, sec²θ - tan²θ = 1. Small angle approximations: sinθ ≈ θ, cosθ ≈ 1 - θ²/2, tanθ ≈ θ (θ in radians).
📈 2. Differentiation – rate of change
Derivative represents slope / velocity. If y = f(x), dy/dx = limₕ→₀ [f(x+h)-f(x)]/h. Key rules: d/dx (xⁿ) = n xⁿ⁻¹, d/dx (sin x) = cos x, d/dx (cos x) = – sin x, d/dx (tan x) = sec² x. Chain rule: dy/dx = (dy/du)·(du/dx). Product & quotient rules as needed.
📊 3. Integration – area & anti‑derivative
Integration is reverse differentiation. Indefinite: ∫xⁿ dx = xⁿ⁺¹/(n+1) + C (n≠-1). Trigonometric: ∫sin x dx = –cos x + C, ∫cos x dx = sin x + C, ∫sec² x dx = tan x + C. Definite integral ∫ₐᵇ f(x) dx gives area under curve from a to b.
🧭 4. Vectors – basic operations
Physical quantities with magnitude & direction. Unit vectors î, ĵ, k̂. Addition: triangle/parallelogram law. Dot product A·B = AB cosθ (scalar), cross product |A×B| = AB sinθ (vector perpendicular). Used in work, torque, etc.
📋 Complete formula sheet
1 + tan²θ = sec²θ
sin(π/2 ± θ) = cosθ, etc.
d/dx sin x = cos x
d/dx cos x = – sin x
∫sin x dx = –cos x + C
∫cos x dx = sin x + C
|A×B| = AB sinθ
direction ⊥ both (right‑hand rule)
| Differentiation (physics) | Integration (physics) |
|---|---|
| \( v = dx/dt \) (velocity) | \( x = \int v \, dt \) (displacement) |
| \( a = dv/dt \) (acceleration) | \( v = \int a \, dt \) (velocity) |
| max/min: set \( dy/dx = 0 \) | area under F-x graph = work |
✏️ Solved NEET-level examples
Example 1 (Differentiation)
The displacement of a particle is x = 5t² + 3t + 2 (metres). Find its velocity and acceleration at t = 2 s.
Answer: v = 23 m/s, a = 10 m/s².
Example 2 (Integration)
Acceleration of a particle a = 4t – 2 (m/s²). At t=0, velocity v₀ = 5 m/s. Find velocity as function of time.
Example 3 (Vectors)
Find the angle between A = 3î + 4ĵ and B = 2î – ĵ + 2k̂.
θ ≈ 82.3° (NEET often expects expression).
📈 Important graphs & key points
- y = sin θ : peaks at 1, -1; slope = cos θ
- y = cos θ : similar shape, phase shifted
- dy/dx as slope of x–t gives v–t
- ∫ v dt = area under v–t = displacement
⭐ In kinematics: slope of tangent on x–t → instantaneous velocity. Area under a–t → change in velocity.
⚡ Quick revision box
⚠️ Common mistakes to avoid
- Using degrees instead of radians in calculus
- Forgetting constant of integration (C) in indefinite integrals
- Sign errors: derivative of cos x = – sin x
- Dot product yields scalar; cross product yields vector
- Wrong quadrant signs for trigonometric values
- Setting dy/dx = 0 for maxima but forgetting second derivative check
🧠 Exam strategy tips
- Memorize basic derivatives/integrals – they are direct tools.
- In vector problems, resolve into components first.
- For motion questions, write given data and differentiate/integrate step by step.
- Use small-angle approximations only when explicitly allowed.
- Practice at least 5 mixed problems daily.
❓ Frequently asked questions
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