2nd/2 ways: Show 1+3+5+ ... +(2n-1) + (2n+1) + (2n-1)+...+5+3+1= n^2 + (n+1)^2

3 ways: Find 1+2+3+ ... + n^2.

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MPPET 1996 Hindustan: trig integral ∫ sin^2(x) -cos^2(x)/(sin^2(x) cos^2(x)) dx

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AI 2008 Hindustan: Evaluate integral ∫ 2/3 cos(x)/sin^2(x) dx

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1st way: exp ∫(e^x)/x (1+x log(x)) dx

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1st/3parts: Checkpoint 1.1 STPM Math T Sem2

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Swinburne U 2nd way: e^(-e^x) e^x dx

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2nd way: integral ∫ log √x(x^n) dx

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MP 2000 Hindustan: Evaluate ∫(tan(x)+cot(x))^2 dx

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Review: Linear form, Quadratic form & Bilinear form

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A box contains 8 blue & 6 red socks. What is the number of ways that 2 socks can be drawn ...

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logic: tautology (¬p→q)∧(¬p→¬q) →p

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2nd way: derivative (x^x)^x

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Given events J & K:

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Find the expected value & the variance of a geometric r.v. X

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A random variable X has the following parameters: E(X)=a & Var(X)=2a^2-1 a∈R

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alternate way: ∫ √(x^2+a^2 )dx via "expressing in 2 ways"

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Screen Recording 2023-12-06 at 3.07.30 AM

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prob stat In a sample of n= 6 scores, 5 of the scores are each above the mean by one point.

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prob stat: A population of N=10 scores has a mean of 30.

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A bag contains 20 identical red marbles, 20 identical green marbles, & 20 identical blue marbles.

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