Maths

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

(2) $a^2-d^2={\mathrm{c}\mathrm{o}\mathrm{s}}^2?5\theta -{\mathrm{c}\mathrm{o}\mathrm{s}}^2?5\theta =0$

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

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

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

$\begin{array}{l}yandzcoordinates.\\ Hence,thevalueofthefillerisyandzcoordinates.\end{array}$

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

$\begin{array}{l}z-coordinates.\\ Hence,thevalueofthefillerisz-coordinates.\end{array}$

$\int {\left(\frac{x}{x\mathrm{s}\mathrm{i}\mathrm{n}?x+\mathrm{c}\mathrm{o}\mathrm{s}?x}\right)}^{2}dx=\int \left(\frac{x}{\mathrm{c}\mathrm{o}\mathrm{s}?x}\right)\cdot \frac{x\mathrm{c}\mathrm{o}\mathrm{s}?xdx}{(x\mathrm{s}\mathrm{i}\mathrm{n}?x+\mathrm{c}\mathrm{o}\mathrm{s}?x{)}^2}$

$\begin{array}{rr}& =\frac{x}{\mathrm{c}\mathrm{o}\mathrm{s}?x}\left(-\frac{1}{x\mathrm{s}\mathrm{i}\mathrm{n}?x+\mathrm{c}\mathrm{o}\mathrm{s}?x}\right)+\int \left(\frac{\mathrm{c}\mathrm{o}\mathrm{s}?x+x\mathrm{s}\mathrm{i}\mathrm{n}?x}{{\mathrm{c}\mathrm{o}\mathrm{s}}^2?x}\right)\left(\frac{1}{x\mathrm{s}\mathrm{i}\mathrm{n}?x+\mathrm{c}\mathrm{o}\mathrm{s}?x}\right)dx\\ & =-\frac{x\mathrm{s}\mathrm{e}\mathrm{c}?x}{x\mathrm{s}\mathrm{i}\mathrm{n}?x+\mathrm{c}\mathrm{o}\mathrm{s}?x}+\int {\mathrm{s}\mathrm{e}\mathrm{c}}^2?xdx\\ & =-\frac{x\mathrm{s}\mathrm{e}\mathrm{c}?x}{x\mathrm{s}\mathrm{i}\mathrm{n}?x+\mathrm{c}\mathrm{o}\mathrm{s}?x}+\mathrm{t}\mathrm{a}\mathrm{n}?x+C\end{array}$