pab2 long topics

  1. PixVerse AI LEAKED: Sora & Runway DESTROYED?! Next-Gen Video Generator Just Changed YouTube!

  2. OpenAI’s GPT-4.1 Is HERE – The Most Powerful ChatGPT Ever Released (1M Tokens!)

  3. Elon Musk’s GROK Just Invaded Microsoft Azure | OpenAI’s Exclusive Deal is OVER!

  4. Meta’s SHOCKING AGI Announcement: Superintelligence Labs to Beat ChatGPT, Gemini, & DeepMind?!

  5. Copilot’s Zero-Click AI Hack EXPOSED — Microsoft Didn’t Want You to Know

  6. Top 5 Side Hustles You Must Try in 2025

  7. AI in Space Exploration: Will Machines Be Our Next Astronauts?

  8. Earn Passive Income Like a Pro with Affiliate Marketing

  9. Will AI Ever Achieve Consciousness? A Look into the Future of Machine Intelligence

  10. Turn Digital Products into a Steady Income — Beginner’s Guide

  11. The Future of AI: How Artificial Intelligence Will Shape the Next Decade

  12. Monetize Your Social Media Platforms in 2025 — Here’s How

  13. Will AI Replace Human Jobs? The Future of Employment in an Automated World

  14. Freelancing in 2025: Your Complete Roadmap to Success

  15. The Future of AI in Entertainment: Virtual Reality, Gaming, and Beyond

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Chapter no 1

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chapter 2

  1: Compute the Fourier coefficients for the function f(t) = t (0 ≤ t ≤ 1) Solution: To compute the Fourier series of the function f ( t ) = t f(t) = t f ( t ) = t on the interval 0 ≤ t ≤ 1 0 \leq t \leq 1 0 ≤ t ≤ 1 , we first need to express it as a periodic function of period 1. The Fourier series of a function f ( t ) f(t) f ( t ) on the interval [ 0 , 1 ] [0,1] [ 0 , 1 ] is given by: f ( t ) = a 0 + ∑ n = 1 ∞ ( a n cos ⁡ ( 2 π n t ) + b n sin ⁡ ( 2 π n t ) ) f(t) = a_0 + \sum_{n=1}^{\infty} \left( a_n \cos(2\pi nt) + b_n \sin(2\pi nt) \right) f ( t ) = a 0 ​ + n = 1 ∑ ∞ ​ ( a n ​ cos ( 2 πn t ) + b n ​ sin ( 2 πn t ) ) Where: a 0 = ∫ 0 1 f ( t )   d t = ∫ 0 1 t   d t = 1 2 a_0 = \int_0^1 f(t) \, dt = \int_0^1 t \, dt = \frac{1}{2} a 0 ​ = ∫ 0 1 ​ f ( t ) d t = ∫ 0 1 ​ t d t = 2 1 ​ a n = 2 ∫ 0 1 t cos ⁡ ( 2 π n t )   d t = 0 ( since this is an odd function about  t = 0.5 ) a_n = 2 \int_0^1 t \cos(2\pi nt) \, dt = 0 \quad (\text{since...
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chapter no 3

  1. The message is split into 10 frames, each with an 80% chance of arriving undamaged. What is the expected number of times the message must be sent to get the entire message through? Answer: Let the probability of a frame being transmitted successfully be 0.8, and the probability of a frame being damaged (and requiring retransmission) be 0.2. For one frame to arrive undamaged, the expected number of transmissions is the reciprocal of the probability of success: E [ transmissions per frame ] = 1 0.8 = 1.25 E[\text{transmissions per frame}] = \frac{1}{0.8} = 1.25 E [ transmissions per frame ] = 0.8 1 ​ = 1.25 Since there are 10 frames, the total expected number of transmissions is: E [ total transmissions ] = 10 × 1.25 = 12.5 E[\text{total transmissions}] = 10 \times 1.25 = 12.5 E [ total transmissions ] = 10 × 1.25 = 12.5 Thus, on average, the message will need to be sent  12.5 times  to get the entire message through. 2. Character Encoding...
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