Class 11th

= A + B (say)

(1 + x)¹⁵ =

differentiating 15 (1 + x)¹⁴ =   $\sum r{.}^{15}{C}_r{x}^{r-1}$

put  = x = -1

$\Rightarrow 0={-}^{15}{C}_1+2{.}^{15}{C}_2-.......=A$      

$B{=}^{14}{C}_{13}=2^{13}$    

$\therefore A+B=2^{13}-14$     

Total oscillation = $\frac{1}{2}+\frac{1}{8}=\frac{5}{8}$ $\frac{}{}\frac{}{}\frac{}{}$

$\Rightarrow timetaken=\frac{T}{2}+\frac{T}{12}=\frac{7T}{12}$

= 7

Let direction ratio of the normal to the required plane are l, m, n

$\begin{array}{l}\therefore 3l+m-2n=0\\ \therefore 2l-5m-n=0\\ \therefore \frac{l}{-11}=\frac{m}{-1}=\frac{n}{-17}\end{array}$             

Equation of required plane

11 (x – 1) + 1 (y – 2) + 17 (z + 3) = 0

Any point on line  $\frac{x-3}{1}=\frac{y-4}{2}=\frac{z-5}{2}=r$ is P (r + 3, 2r + 4, 2r + 5) lies on x + y + z = 17,

5r + 12 = 17

r = 1

$P\left(4,6,7\right)A\left(1,1,9\right)distAP=\sqrt[]{9+25+4}=\sqrt[]{38}$  

x² + y² = 36         ……(i)

y² = 9x      .(ii)

Solving (i) & (ii)

x² + 9x – 36 = 0

(x + 12) (x – 3) = 0

x = 3

Let $A={\displaystyle \underset{0}{\overset{3}{\int }}\left(\sqrt[]{36-x^2}-3\sqrt[]{x}\right)dx}$  

$={\left[\frac{x\sqrt[]{36-x^2}}{2}+18si{n}^{-1}\frac{x}{6}\right]}_0^3-3.{\frac{x^{3/2}}{3/2}]}_0^3$       

$=3\pi -\frac{3\sqrt[]{3}}{2}$        

Required area = $=\frac{1}{2}\pi {\left(6\right)}^2+2\left(3\pi -\frac{3\sqrt[]{3}}{2}\right)=\left(24\pi -3\sqrt[]{3}\right)sq.unit$

$v=\frac{\omega }{k}=\frac{30}{1}=30m/s$

$v=\sqrt[]{\frac{T}{\mu }}\Rightarrow T=\mu v^2=\frac{0.135*10^{-3}}{10^{-2}}*{\left(30\right)}^2=12.15N=1215*10^{-2}N$

mg – N = ma

N = m (g – a) = 60 * (10 – 1.8) = 60 * 8.2 = 492 N

$f\left(x\right)=2x-1g:R-\left\{1\right\}\to R$        

$g\left(x\right)=\frac{x-1/2}{x-1}$           

$f\left\{g\left(x\right)\right\}=2\left(\frac{x-1/2}{x-1}\right)-1=\frac{2x-1-x+1}{x-1}=\frac{x}{x-1},x\ne 1$           

$y=\frac{x}{x-1}\Rightarrow x=\frac{y}{y-1}\therefore y\ne 1$           

One – one but not onto

U = 3PV + 4

$\Rightarrow n{C}_{v}T=3PV+4$

$\Rightarrow n*\frac{fRT}{2}=3PV+4$

$\Rightarrow f=6+\frac{8}{PV}$

$\Rightarrow f>6\Rightarrow polyatomic$