Maths

Gain in surface energy, $\Delta U=T\Delta A$  

from volume centenary, $\frac{4}{3}\pi R^3=64*\frac{4}{3}\pi r^3$  

$\Rightarrow r=\frac{R}{4}$                                            

Initial surface area, Ai = 4pR²

final surface area, $A_f=64*4\pi r^2$  

$\therefore \Delta U=12\pi R^2.T=0.075*12*3.14*10^{-4}=2.8*10^{-4}J$          

$f\left(x\right)={\int }_1^3?\frac{\sqrt[]{x}dx}{(1+x{)}^2}={\int }_1^{\sqrt[]{3}}?\frac{t.2tdt}{{\left(1+t^2\right)}^2}\mathrm{}$ (put $\sqrt[]{x}=t$ )

$={\left(-\frac{\mathrm{t}}{1+{\mathrm{t}}^2}\right)}_1^{\sqrt[]{3}}+{\left({\mathrm{t}\mathrm{a}\mathrm{n}}^{-1}?\mathrm{t}\right)}_1^{\sqrt[]{3}}\mathrm{}$ [Applying by parts]

$\begin{array}{rr}& =-\left(\frac{\sqrt[]{3}}{4}-\frac{1}{2}\right)+\frac{\pi }{3}-\frac{\pi }{4}\\ & =\frac{1}{2}-\frac{\sqrt[]{3}}{4}+\frac{\pi }{12}\end{array}$

$[x{]}^2+2[x+2]-7=0$

$\begin{array}{rr}& \Rightarrow [x{]}^2+2[x]+4-7=0\\ & \Rightarrow \left[x\right]=1,-3\\ & \Rightarrow x\in \left[\mathrm{1,2}\right)\cup [-3,-2)\end{array}$

This is a Fill in the blanks Type Questions as classified in NCERT Exemplar

$\begin{array}{l}Lettheverticesofthe\Delta ABCbe\\ A\left(x_1,y_1,z_1\right),B\left(x_2,y_2,z_2\right)andC\left(x_3,y_3,z_3\right)\\ Disthemid-pointofAB\\ \therefore 1=\frac{x_1+x_2}{2}\Rightarrow x_1+x_2=2\dots \left(i\right)\\ 2=\frac{y_1+y_2}{2}\Rightarrow y_1+y_2=4\dots \left(ii\right)\\ and-3=\frac{z_1+z_2}{2}\Rightarrow z_1+z_2=-6\dots \left(iii\right)\\ Eisthemid-pointofBC\\ \therefore 3=\frac{x_2+x_3}{2}\Rightarrow x_2+x_3=6\dots \left(iv\right)\\ 0=\frac{y_2+y_3}{2}\Rightarrow y_2+y_3=0\dots \left(v\right)\\ and1=\frac{z_2+z_3}{2}\Rightarrow z_2+z_3=2\dots \left(vi\right)\\ Fisthemid-pointofAC\\ \therefore -1=\frac{x_1+x_3}{2}\Rightarrow x_1+x_3=-2\dots \left(vii\right)\\ 1=\frac{y_1+y_3}{2}\Rightarrow y_1+y_3=2\dots \left(viii\right)\\ and-4=\frac{z_1+z_3}{2}\Rightarrow z_1+z_3=-8\dots \left(ix\right)\\ Addingeq.\left(i\right),\left(iv\right)and\left(vii\right)weget\\ 2\left(x_1+x_2+x_3\right)=2+6-2\\ \therefore x_1+x_2+x_3=3\dots \left(x\right)\\ Addingeq.\left(ii\right),\left(v\right)and\left(viii\right)weget\\ 2\left(y_1+y_2+y_3\right)=4+0+2\\ \therefore y_1+y_2+y_3=3\dots \left(xi\right)\\ Addingeq.\left(iii\right),\left(vi\right)and\left(ix\right)weget\\ 2\left(z_1+z_2+z_3\right)=-6+2-8\\ \therefore z_1+z_2+z_3=-6\dots \left(xii\right)\\ Subtracting\left(i\right)fromeq.\left(x\right)weget\\ x_3=3-2=1\Rightarrow x_3=1\\ Subtracting\left(iv\right)fromeq.\left(x\right)weget\\ x_1=3-6=-3\Rightarrow x_1=-3\\ Subtracting\left(vii\right)fromeq.\left(x\right)weget\\ x_2=3-\left(-2\right)=5\Rightarrow x_2=5\end{array}$

 ${\sum }_{r=0}^{20}?{}^{50-r}{\mathrm{C}}_6={}^{50}{\mathrm{C}}_6+{}^{49}{\mathrm{C}}_6+{}^{48}{\mathrm{C}}_6+\dots .+{}^{30}{\mathrm{C}}_6$

$\begin{array}{rr}& ={}^{50}{\mathrm{C}}_6+{}^{49}{\mathrm{C}}_6+{}^{48}{\mathrm{C}}_6+\dots .+\left({}^{30}{\mathrm{C}}_6+{}^{30}{\mathrm{C}}_7\right)-{}^{30}{\mathrm{C}}_7\\ & ={}^{50}{\mathrm{C}}_6+{}^{49}{\mathrm{C}}_6+{}^{48}{\mathrm{C}}_6+\dots .+\left({}^{31}{\mathrm{C}}_6+{}^{31}{\mathrm{C}}_7\right)-{}^{30}{\mathrm{C}}_7\\ & ={}^{50}{\mathrm{C}}_6+{}^{50}{\mathrm{C}}_7-{}^{30}{\mathrm{C}}_7\\ & ={}^{51}{\mathrm{C}}_7-{}^{30}{\mathrm{C}}_7\end{array}$

This is a Fill in the blanks Type Questions as classified in NCERT Exemplar

$\begin{array}{l}Thegiven\text{\hspace{0.17em}points}are\left(a,2,1\right)and\left(1,-1,1\right)\\ \therefore \text{\hspace{0.17em}Distance}=\sqrt[]{{\left(a-1\right)}^2+{\left(2+1\right)}^2+{\left(1-1\right)}^2}\\ 5=\sqrt[]{a^2+1-2a+9}\\ Squaringbothsides,wehave\\ 25=a^2+1-2a+9\\ \Rightarrow a^2-2a-15=0\\ \Rightarrow a^2-5a+3a-15=0\\ \Rightarrow a\left(a-5\right)+3\left(a-5\right)=0\\ \Rightarrow \left(a+3\right)\left(a-5\right)=0\\ \therefore a=-3or5\\ Hence,thevalueofthefilleris5or-3.\end{array}$

This is a Fill in the blanks Type Questions as classified in NCERT Exemplar

$Thegiven\text{\hspace{0.17em}dimensions}$
$\begin{array}{l}are10,13and8\\ Leta=10,b=13andc=8\\ \therefore \text{\hspace{0.17em}Required}length=\sqrt[]{a^2+b^2+c^2}\\ =\sqrt[]{{\left(10\right)}^2+{\left(13\right)}^2+{\left(8\right)}^2}=\sqrt[]{100+169+64}=\sqrt[]{333}\\ Hence,thevalueofthefilleris\sqrt[]{333}.\end{array}$

This is a Fill in the blanks Type Questions as classified in NCERT Exemplar

$\begin{array}{l}Theplaneparalleltoyz-planeisperpendiculartox-axis.\\ Hence,thevalueofthefillerisx-axis.\end{array}$

This is a Fill in the blanks Type Questions as classified in NCERT Exemplar

$x=arepresentsaplaneparalleltoyz-plane.$

$A^2=\left(\begin{array}{ll}\mathrm{c}\mathrm{o}\mathrm{s}?2\theta & i\mathrm{s}\mathrm{i}\mathrm{n}?2\theta \\ i\mathrm{s}\mathrm{i}\mathrm{n}?2\theta & \mathrm{c}\mathrm{o}\mathrm{s}?2\theta \end{array}\right)$

Similarly, $A^5=\left(\begin{array}{cc}\mathrm{c}\mathrm{o}\mathrm{s}?5\theta & i\mathrm{s}\mathrm{i}\mathrm{n}?5\theta \\ i\mathrm{s}\mathrm{i}\mathrm{n}?5\theta & \mathrm{c}\mathrm{o}\mathrm{s}?5\theta \end{array}\right)=\left(\begin{array}{ll}a& b\\ c& d\end{array}\right)$

(1) $a^2+b^2={\mathrm{c}\mathrm{o}\mathrm{s}}^2?5\theta -{\mathrm{s}\mathrm{i}\mathrm{n}}^2?5\theta =\mathrm{c}\mathrm{o}\mathrm{s}?10\theta =\mathrm{c}\mathrm{o}\mathrm{s}?{75}^{?}$