Maths

f is increasing function

x < 5x < 7x

f (x) < f (5x) < f (7x)

  $\frac{f\left(x\right)}{f\left(x\right)}<\frac{f\left(5x\right)}{f\left(x\right)}<\frac{f\left(7x\right)}{f\left(x\right)}$           

$\underset{x\to \infty }{lim}\frac{f\left(x\right)}{f\left(x\right)}<\underset{x\to \infty }{lim}\frac{f\left(5x\right)}{f\left(x\right)}<\underset{x\to \infty }{lim}\frac{f\left(7x\right)}{f\left(x\right)}$             

-> $1<\underset{x\to \infty }{lim}\frac{f\left(5x\right)}{f\left(x\right)}<1$ $\underset{x\to \infty }{lim}\frac{f\left(5x\right)}{f\left(x\right)}=1$

$\underset{x\to \infty }{lim}\left(\frac{f\left(5x\right)}{f\left(x\right)}-1\right)=0$            

Area =   $\left|{\displaystyle \underset{1}{\overset{4}{\int }}\left[{\left(4-x\right)}^2-\frac{{\left(x-4\right)}^2}{3}\right]}\right|dx$

Area =   $=\left|\left(16-\frac{64}{3}\right)-\left(4-\frac{1}{3}+\frac{27}{9}\right)\right|$

$=\left|\left(16-\frac{64}{3}\right)-\left(4-\frac{1}{3}+\frac{27}{9}\right)\right|$

$=\left|16-\frac{64}{3}-4+\frac{1}{3}+3\right|$

$=\left|15-2\right|=6$

a + d, a + 7d and a + 43d are 1^st, 2^nd, 3^rd term of G.P.

$\frac{a+7d}{a+d}=\frac{a+43d}{a+7d}$              

⇒ (a + 7d)² = (a + d) (a + 43d)

⇒ a² + 49d² + 14d = a² + 44ad + 43d³

⇒ 6d² = 30ad

⇒ d² = 5d

⇒ d = 0, 5

a = 1, d = 5

  $S_{20}=\frac{20}{2}\left[2+\left(19\right)5\right]$          

= 10 [95 + 2]

= 970

  $\frac{a+b+68+44+40+60}{6}=55$

212 + a + b = 330

⇒ a + b = 118

$\frac{{\displaystyle \sum _{}^{}{x}_i^2}}{n}-{\left(\overline{x}\right)}^2=194$          

$\frac{a^2+b^2+{\left(68\right)}^2+{\left(44\right)}^2+{\left(40\right)}^2+{\left(60\right)}^2}{6}={\left(55\right)}^2=194$

= 3219

11760 + a² + b² = 19314

⇒ a² + b² = 19314 – 11760

= 7554

(a + b)² –2ab = 7554

From here b = 41.795

a + b = 118

⇒ a + b + 2b = 118 + 83.59

= 201.59

Let probability of tail is   $\frac{1}{3}$

⇒ Probability of getting head = $\frac{2}{3}$  

∴ Probability of getting 2 heads and 1 tail

$=\left(\frac{2}{3}*\frac{2}{3}*\frac{1}{3}\right)*3$

$=\frac{4}{27}*3$

$=\frac{4}{9}$                  

a = sin⁻¹ (sin5) = 5 − 2π

and b = cos⁻¹ (cos5) = 2π − 5

∴    a² + b² = (5 − 2π)² + (2π − 5)²

= 8π² − 40π + 50

The examples of the probability include - flipping a coin, rolling a die, drawing a card from a deck, and picking a ball from a bag.

There are various applications of the probability in the real life. It is used to take informed decisions and risks in the following areas - insurance, weather forecasting, finance, business, sports, computer Science and gaming.

There are two basic rules for finding the Probability of two events A and B - Multiplication Rule and Addition Rule.