161 - Mathematics Questions Answers

$$\begin{gathered} Probability to get following differences \\ difference probability \\ 0 6/36 \\ 1 10/36 \\ 2 8/36 \\ 3 6/36 \\ 4 4/36 \\ 5 2/36 \\ So required probability =1-2\left[\frac{6}{36}\left(\frac{4}{36}+\frac{2}{36}\right)+\frac{10}{36}\left(\frac{2}{36}\right)\right] \\ now solve \\ problem is based on integral values so area is not very useful \end{gathered}$$

$$\begin{gathered} \frac{x}{1-x^2}=\frac{1}{1-x}-\frac{1}{1-x^2} \\ break all terms similarly and solve \end{gathered}$$

1+x²+x⁴ = (1+x+x²)(1-x+x²)

now solve

g(x) = 3-5sin(2x-4)

for getting reminder put x³= x so

x¹³⁵ + x¹²⁵ - x¹¹⁵ + x⁵ +1 will give

 x⁴⁵ + x⁴¹*x² - x³⁸*x +x*x² +1

x¹⁵ + x¹³*x²*x² - x¹²*x²*x +x*x² +1

 x¹⁷ + x³ +1

x⁵*x² + x + 1

x⁷ + x + 1

x²*x + x +1 

x³ + x +1 

x+x+1

2x+1

now put x = 10

two times sign change so 2 positive roots

if ax²+bx+c is positive for real x

then b²< 4ac similarly others

on adding all inequalities we get b² + c² + a²< 4(ac+ab+bc)

now get answer 

general term of the series = r(n+1-r)² = r(n+1)² + r³ - 2(n+1)r²

now put sigma then solve