Maths

  $a_0{C}_0-a_1{C}_1+a_2{C}_2-a_3{C}_3+.....+a_{2012}{c}_{2012}$

$=a_0\left(C_0-C_1+C_2+....+C_{2012}\right)$            

  $-d\left(C_1-2C_2+3C_3-4C_4+....{-}^{2012}{C}_{2012}\right)$           

  $=a_0\left(0\right)-d\left(C_1-2C_2+3C_3{-}^4{C}_4\right)$           

+ …….-  

 =   $a_0\left(0\right)-d\left(0\right)$

= 0

$(x+y{)}^n={}^n{C}_r{x}^{n-r}{y}^r$

${\left(\frac{3}{2}{x}^2+\frac{2}{x}\right)}^{12}?{}^{12}{C}_r{\left(\frac{3}{2}{x}^2\right)}^{12-r}{\left(\frac{2}{x}\right)}^r$

For independent of X .

$x^{24-2r}{x}^{-r}=x^0;24=3r\Rightarrow r=8$

${}^{12}{C}_8{\left(\frac{3}{2}\right)}^{12-8}(2{)}^8;\frac{\lfloor 12}{\overline{12}44}*\frac{3^4}{2^4}*2^8\Rightarrow \lambda ;\frac{\lambda }{81}\Rightarrow 7920$

$Inx\in \left(-2,2\right)$  

 |x|- 1| is not differentiable at x = -1, 0, 1

|cospx| is not differentiable at x =  $\frac{3}{2},-\frac{1}{2},\frac{1}{2},\frac{3}{2}$ -  

$\int {\mathrm{c}\mathrm{o}\mathrm{t}}^{-1}?\left(\frac{1}{1+x}\right)dx$

$=\int {\mathrm{t}\mathrm{a}\mathrm{n}}^{-1}?(1+x)dx$

$=x{\mathrm{t}\mathrm{a}\mathrm{n}}^{-1}?(1+x)-\int \frac{x}{1+(1+x{)}^2}dx;=x{\mathrm{t}\mathrm{a}\mathrm{n}}^{-1}?(1+x)-\int \frac{x}{x^2+2x+2}dx$

$=x{\mathrm{t}\mathrm{a}\mathrm{n}}^{-1}?(1+x)-\int \frac{\frac{1}{2}(2x+2)-1}{x^2+2x+2}dx=x{\mathrm{t}\mathrm{a}\mathrm{n}}^{-1}?(1+x)-\frac{1}{2}\mathrm{l}\mathrm{o}\mathrm{g}?\left(x^2+2x+2\right)+\int \frac{dx}{(x+1{)}^2+1}$

$=x{\mathrm{t}\mathrm{a}\mathrm{n}}^{-1}?(1+x)-\frac{1}{2}\mathrm{l}\mathrm{o}\mathrm{g}?\left(x^2+2x+2\right)+{\mathrm{t}\mathrm{a}\mathrm{n}}^{-1}?(x+1)+c$

$A^2{\left|A\right|}^2-{\left|B\right|}^{3.2}B$

$=25A^2-\left|B\right|.B$

$\begin{array}{l}=25A^2+\left|B\right|A^2\\ =0\end{array}$

Put x³/² = t
√xdx = (2/3)dt
⇒ (2/3) ∫ dt/√ (1-t²) = (2/3)sin? ¹t + c
= (2/3)sin? ¹x³/² + c
⇒ g (x) = sin? ¹x
g (0) = 0

? P? = 6!/2! = 360

tanx + tan (x+π/3) + tan (x-π/3) = 3
tanx + (tanx+√3)/ (1-√3tanx) + (tanx-√3)/ (1+√3tanx) = 3
tanx + (8tanx)/ (1-3tan²x) = 3
(tanx (1-3tan²x) + 8tanx)/ (1-3tan²x) = 3
(3 (3tanx – tan³x)/ (1-3tan²x) = 3
⇒ 3tan3x = 3
tan3x = 1

$\sim \left(\sim q\wedge p\right)=\left(\sim \left(\sim q\right)\right)\vee \left(\sim p\right)$  

$=\sim p\vee q$            

$=p\to q$            

$\therefore contrapositiveis\sim q\to \sim p$            

 Hence, if $\sim$ q then $\sim$ p

A: runner succeeded exactly 3 times out of 5.
B: runner succeeds on the first trial.
P (B/A) = P (B ∩ A) / P (A) = (p (? C? p² (1−p)²) / (? C? p³ (1-p)²) = 3/5 = 0.6