Basic Mathematics (Matrices) and Their Use in NEET Exam

Introduction to the Topic

• This lecture is a crash course in Physics that can be significant for the NEET exam.
• Difficult problems can only be solved with knowledge of basic mathematical concepts.
• At the end of the lecture, PDF notes can be downloaded from the Physics Wala app.

Basic Trigonometry

• Values of sin theta, cos theta, and tan theta:
  • Remember the values of sin 0°, 30°, 45°, 60°, 90°, etc.
  • Also, remember the reciprocals of these values.
• Important triangles:
  • Understanding the 3-4-5 triangle: Recognition of triangles with 30°-60° angles.

Conversion between Radians and Degrees

• Theta = L/r (L = arc length, r = radius)
• π radians = 180°
• Formulas for converting values:
  • 360° = 2π radians
  • 180° = π radians
  • 90° = π/2 radians
  • 60° = π/3 radians
  • 45° = π/4 radians

Trigonometric Identities

1. sin²θ + cos²θ = 1
2. sec²θ - tan²θ = 1
3. cosec²θ - cot²θ = 1

Double Angle Formulas

• sin 2θ = 2 sin θ cos θ
• cos 2θ = cos² θ - sin² θ = 2 cos² θ - 1 = 1 - 2 sin² θ
• tan 2θ = 2 tan θ / (1 - tan² θ)
• cot 2θ = (cot² θ - 1) / 2 cot θ

Addition and Subtraction Formulas

• sin (A ± B)= sin A cos B ± cos A sin B
• cos (A ± B)= cos A cos B ∓ sin A sin B
• tan (A ± B)= (tan A ± tan B) / (1 ∓ tan A tan B)
• cot (A ± B)= (cot A cot B ∓ 1) / (cot B ± cot A)

Maximum and Minimum Values

• The slope at any point (dy/dx) = 0.
• The second derivative test can determine if the point is a maximum or minimum.
  • d²y/dx² < 0 => Maxima
  • d²y/dx² > 0 => Minima

Derivatives and Their Uses

• Power Rule: d/dx (xⁿ) = n*xⁿ⁻¹
• Common Derivatives:
  • d/dx (sin x) = cos x
  • d/dx (cos x) = -sin x
  • d/dx (tan x) = sec² x
  • d/dx (cot x) = -cosec² x
  • d/dx (sec x) = sec x tan x
  • d/dx (cosec x) = -cosec x cot x
• Logarithmic Derivatives:
  • d/dx (log x) = 1/x
  • d/dx (e^x) = e^x

Integration

• Basic formula of Integration:
  • ∫ xⁿ dx = (xⁿ⁺¹)/(n+1) + C, where n ≠ -1
  • ∫ 1/x dx = ln|x| + C
• Some important integrals:
  • ∫ dx = x + C
  • ∫ x dx = (x²)/2 + C
  • ∫ x² dx = (x³)/3 + C
  • ∫ e^x dx = e^x + C
  • ∫ sin x dx = -cos x + C
  • ∫ cos x dx = sin x + C
  • ∫ tan x dx = ln|sec x| + C

Advice and Conclusion

• Watch the lecture carefully and download the PDF notes afterward.
• Solve practice sheets after each lecture.
• Notes and practice sheets from the Ummed Batch can be downloaded from the Physics Wala app.
• Using derivatives and integrals will make solving difficult questions easier.

Note: Always keep revising new concepts to stay updated.