Pr 6 5a: Fill & Download for Free

*Please check it regularly as I will update it time to time, with reference later** Before Marking it “Needing Improvement”, please comment or message me directly what is the issue. And if possible I will make necessary edit. But if you mark it “Needing Improvement”, the answer becomes invisible and many students will not able to see it.***Also I Suggest you to take some problems which you have done earlier and apply these methods, check and compare the results also the working time and let me know in the comments if you face any problems.Thanks A2AI can recall some of themTaylor Series[math]f (x) = f (a) + f^{ \prime} (a) (x - a) + \dfrac {f ^{\prime \prime} (a)}{2!} (x - a)^2 + \dfrac {f^{(3)} (a)}{3!} (x - a)^3 + \cdots + \dfrac {f^{(n)} (a)}{n!} (x - a)^n + \cdots[/math]Now if we take [math]a=0[/math][math]f (x) = f (0) + f^{ \prime} (0) x + \dfrac {f^{ \prime \prime} (0)}{2!} x ^2 + \dfrac {f^{(3)}(0)}{3!} x^3 + \cdots + \dfrac {f^{(n)} (0)}{n!} x^n + \cdots[/math]Here [math]a[/math] is the center of expansion you don’t need to know thisbut I want to say, you know[math]e^x = 1+ x + \dfrac{x^2}{2!} + \dfrac{x^3}{3!}+ \cdots [/math]We can write this also in[math]e^x = e^2 + e^2 (x - 2) + \dfrac {e^2}{2!} (x - 2)^2 + \dfrac {e^2}{3!} (x - 2)^3 + \cdots + \dfrac {e^2}{n!} (x - 2)^n + \cdots[/math]The only difference is the center of expansion is [math]a=2[/math] { i.e [math](x-2)[/math] term is there } whereas previously the center is [math]a=0[/math] { i.e [math]x[/math] }Eigenvalues and Eigen-vectors (Matrices):It is a very big topic and you probably need only a particular part of it. For me which is characteristic equationWhy you need this?In exam suppose there may be question “let [math]A[/math] is a [math](3\times 3)[/math] matrix where [math]A^3 +24A +56I = \hat {0}[/math], Where [math]I[/math] is [math](3\times 3)[/math] identity matrix and [math]\hat {0}[/math] is [math](3\times 3)[/math] zero matrix, Find [math]A^{99}[/math]"Now most of you can do it. But if the equation [math]A^3 +24A +56I = \hat {0}[/math], not given. Only the matrix [math]A[/math] is given, then how will you do it?For this you should know how the equation [math]A^3 + 24A + 56I = \hat {0}[/math] comes? This equation is called characteristic equation of the matrix [math]A[/math]So for this you should know the definition of Eigenvalues and Eigen-vectorsSo let [math]A[/math] is a [math](n\times n)[/math] square matrix and [math]x[/math] is a non zero [math](n\times 1)[/math] column vector. Now if this happens[math]\qquad \qquad \qquad \boxed {Ax = \lambda x}[/math]Where [math]\lambda[/math] is a constant or scalar, then [math]\lambda[/math] is called Eigenvalue of the matrix [math]A[/math] and [math]x[/math] is the corresponding Eigen-vector of Eigenvalue [math]\lambda[/math]Now we can write[math]Ax = \lambda x[/math][math]\Rightarrow ( A - \lambda I)x = \hat {\hat {0}}[/math]Where [math]I[/math] is [math](n\times n)[/math] identity matrix and [math]\hat {\hat {0}}[/math] is [math](n\times 1)[/math] zero vectorAnd now to find the characteristic equation you have to do first write[math]det(A - \lambda I) =0[/math]and replace [math]\lambda[/math] with [math]A[/math]Example:Let[math]A = \begin {pmatrix} 1 & -3 & -3 \\\\ 3 & -5 & 3 \\\\ [/math][math]6[/math][math] & -6 & 4 \end {pmatrix}[/math]Now we have to find [math]A - \lambda I[/math][math]= \begin {pmatrix} 1 & -3 & -3 \\\\ 3 & -5 & 3 \\\\ [/math][math]6[/math][math] & -6 & 4 \end {pmatrix} - \begin {pmatrix} \lambda & 0 & 0 \\\\ 0 & \lambda & 0 \\\\ 0 & 0 & \lambda \end {pmatrix}[/math][math]= \begin {pmatrix} 1- \lambda & -3 & -3 \\\\ 3 & -5- \lambda & 3 \\\\ [/math][math]6[/math][math] & -6 & 4- \lambda \end {pmatrix}[/math]Now you write [math]det(A - \lambda I) =0[/math][math]\Rightarrow \begin {vmatrix} 1- \lambda & -3 & -3 \\\\ 3 & -5- \lambda & 3 \\\\ [/math][math]6[/math][math] & -6 & 4- \lambda \end {vmatrix} = 0[/math]By simplifying you can get[math]\Rightarrow \begin {vmatrix} 1- \lambda & -3 & -3 \\\\ 3 & -5- \lambda & 3 \\\\ [/math][math]6[/math][math] & -6 & 4- \lambda \end {vmatrix} = {-\lambda}^3 - 24\lambda - 56 = 0[/math][math]\Rightarrow {\lambda}^3 + 24\lambda + 56 =0[/math]Now you replace [math]\lambda[/math] with [math]A[/math], multiply constant by [math]I[/math] and replace [math]0[/math] by [math]\hat {0}[/math] you will get[math]\qquad \qquad \boxed {A^3 + 24 A +56I = \hat {0}} [/math]This is the characteristic equation of the matrix [math]A[/math]. With this method you can go for further calculation. You don’t need to know anything more than this in Eigenvalues and Eigen-VectorsThis characteristic equation has some propertiesNo of non zero values of [math]\lambda[/math] in the characteristic equation is rank of the matrix [math]A[/math] Here [math]{\lambda}^3 +24 \lambda +56 = 0[/math] has a nonzero constant term [math]56[/math] and maximum power of [math]\lambda[/math] is [math]3[/math]. So all [math]\lambda[/math]'s are non zero. So rank [math]3[/math]Determinant of matrix [math]A[/math] is product of the eigenvalues of [math]A[/math] Here [math]{\lambda}^3 +24 \lambda +56 = 0,[/math] product of [math]\lambda[/math]'s i.e [math]\lambda_1.\lambda_2.\lambda_3 = -56[/math]. So determinant of [math]A[/math] is [math]-56[/math]But please note you have to make the highest power of the characteristic equation equal to the dimension of the matrix to find the determinant of matrix [math]A[/math].ExampleIf characteristic equation of a matrix [math]A[/math] is[math]A^2 + 5A + 6I = \hat {0}[/math]but [math]A[/math] is a [math](3\times 3)[/math] matrix, you have to multiply the equation with [math]A[/math] again to make the highest power [math]3[/math] (Dimension [math]3[/math])So [math]A(A^2 + 5A + 6I) = \hat {0}[/math][math]\Rightarrow A^3 + 5A^2 + 6A = \hat {0}[/math]Now product of [math]\lambda[/math]'s i.e [math]\lambda_1.\lambda_2.\lambda_3 = 0[/math].So determinant of [math]A[/math] is [math]0[/math]So by ease from characteristic equation of a matrix, you can tell inverse of Matrix possible or not but don’t do this[math]A^2 + 5A + 6I = \hat {0}[/math] and [math]A[/math] is a [math](3\times 3)[/math]So this is wrong[math]A^{-1}(A^2 + 5A + 6I) = \hat {0}[/math][math]\Rightarrow A + 5I + 6A^{-1} = \hat {0}[/math][math]\Rightarrow A^{-1} = -\dfrac {A + 5I}{6}[/math]As from previous point Det [math]A[/math] is zero, so this will not possibleCaley Hamilton Theorem:It is also a good method to reduce a matrix polynomial. But it is helpful only if you can able to find the roots of the characteristic equation of matrix [math]A[/math]By this you can find [math]e^A , \sin {A}, \tan{A}[/math] etc. Where [math]A[/math] is a matrixSee this answer “Rajdeep Biswas's answer to How do you solve the question in the description below?” For details.Beta Function :The main form is[math]\qquad \qquad \boxed {\beta (m,n) = \displaystyle \int_0^{1} t^m (1 - t)^n \,dt}[/math]You need to know the alternative form of this which is[math]\qquad \qquad \boxed {\beta (m,n) = \displaystyle 2 \int_0^{\frac {\pi}{2} } \sin^{2m-1}{x} \cos^{2n-1}{x} \,dx}[/math]Where [math]m>0[/math] and [math]n>0[/math]You can convert the above from from the main form very easilyNow why you need this Beta function?This is because you can convert Beta to Gamma function easilyNowGamma Function :[math]\qquad \qquad \boxed { \Gamma (n) = \displaystyle \int_0^{\infty } t^{n-1} e^{-t} \,dt}[/math]Where [math]n>0[/math]And there are some easy results[math]\Gamma (n+1) = n\Gamma (n)[/math][math]\Gamma (1) = 1[/math][math]\Gamma {\huge (} \dfrac {1}{2}{\huge )} = \sqrt{\pi}[/math][math]\Gamma (p) \Gamma (1-p) = \dfrac{\pi}{\sin (\pi p)} \qquad {\forall p \epsilon (0,1)}[/math]Beta to Gamma Conversion:The popular Conversion[math]\qquad \qquad \boxed { \beta (m,n) = \dfrac { \Gamma (m). \Gamma (n)}{ \Gamma (m + n)}}[/math]By using this you can solve very complicated problems. For this you can see my solutions of these [math]2[/math] problems, you will get an idea on thisRajdeep Biswas' answer to What is the Integration [math]\displaystyle \int_0^{2\pi } \cos^{4}{x} \,dx[/math]?Rajdeep Biswas' answer to What will be the answer for [math]\displaystyle \int_{-\infty} ^{\infty} \dfrac{x^2 dx}{x^4+a^4} [/math]?Star Delta Conversion (Electrical Physics)This is also a very easy and powerful conversion for electrical networks mainly in Unbalanced Wheatstone Bridgeswe can change this two circuits one from the other, which isWhere[math]\qquad \qquad \boxed {\begin {matrix} P = \dfrac {AB}{A+ B +C} \\\\ Q = \dfrac {AC}{A+ B +C} \\\\ R = \dfrac {BC}{A+ B +C} \end {matrix}}[/math]And[math]\qquad \qquad \boxed {\begin {matrix} A = P + Q + \dfrac{PQ}{R} \\\\ B = P + R + \dfrac{PR}{Q} \\\\ C = R + Q + \dfrac{RQ}{P} \end {matrix}}[/math]Note that [math]A,B,C,P,Q,R[/math] are complex impedance. So these will be [math]R[/math], [math]j\omega L[/math] and [math]\dfrac {1}{ j\omega C}[/math] only or their combinationsExample :A Known Circuit (All Resistance Values are in Ohm)You have to find Equivalent resistance Between [math]A[/math] and [math]B[/math]. You know the answer already as it is a Balanced Wheatstone Bridge. But how to solve the problem using Star- Delta method . Please Follow the StepsStep 1:Here we will convert A delta to a equivalent star. Identify the delta and stars. There are 2 deltas and one star in this circuit….The two deltas are [math]ACD[/math] and [math]BCD[/math] and star is [math]ABCD[/math]Step 2:Chose appropriate Delta and convert it to equivalent Star or Chose appropriate Star and convert it to equivalent Delta, So that the circuit can be simplified…Here I choose Delta [math]BCD[/math]Step 3 :Find the values of equivalent Star parametersHere you have to find value of [math]R6[/math], [math]R7[/math] and [math]R8[/math]Step 4 :Calculate the parameters from the Star Delta Conversion Formula[math]R6 = \dfrac {R2.R4} {R2 + R4 + R5} = \dfrac {40.20} {40+20+30} = \dfrac {80}{9}[/math][math]R7 = \dfrac {R2.R5} {R2 + R4 + R5} = \dfrac {40.30} {40+20+30} = \dfrac {40}{3}[/math][math]R8 = \dfrac {R4.R5} {R2 + R4 + R5} = \dfrac {20.30} {40+20+30} = \dfrac {20}{3}[/math]SoStep 5 :Draw the final circuitSo equivalent resistance between [math]A[/math] and [math]B[/math] is[math]\{ (20 + \dfrac {40}{3} ) || (10 + \dfrac {20}{3} ) \} + \dfrac {80}{9}[/math][math]= \{ \dfrac {100}{3} || \dfrac {50}{3} \} +\dfrac {80}{9}[/math][math]= \dfrac {100}{9} + \dfrac {80}{9} = 20[/math]So equivalent resistance between [math]A[/math] and [math]B[/math] is [math]20[/math] ohmAlso You can try the star to delta conversion in the same circuit. Please let me know in comments will you able to do it or not. I will upload it later.You can check the answer with the method you used. Check it….………………………………………………………………………………………………………………..All the best for Jee Advanced 2017. I am looking forward you.

Let's consider the y-coordinate be a.then x-coordinate will be 2a.So,point P is (2a, a).According to question,we havePQ = PRThen by using distance formula between two points,PQ² = PR²=>(2a–2)²+ (a+5)²= (2a+3)² +(a–6)²=>4a²–8a+4+a²+10a+25= 4a²+12a+9+a²–12a+36=>5a² +2a +29 = 5a² +45=>2a+29= 45=>2a= 16a= 8Hence,point P= (2a,a) = (16, 8).I hope it helps you…!!!

If we turn to the size of megacities, then the area of the city of Moscow is 2511 square kilometers. The area of the city of New York in square kilometers is 789, Washington, D.C in square kilometers is 179. Moscow, the date of its foundation - 1147, and New York - 1624, and Washington, D.C - 1790Moscow is a very big city and there is a very different architecture in this vast territory. I tried to make a small review of what might be interesting in order to see a small part of modern Moscow architecture.I will begin my story with some lovely buildings in Moscow ... If you are a Jew, remember the addressThe synagogue on Bolshaya Bronnaya was built at the end of the 19th century but reconstructed in modern times. This building looks like a big ship.“Synagogue on Big Bronnaya” - Inside, everything is very decent. There are kosher and book shop, there is an excellent restaurant upstairs. There is also a very nice veranda upstairs.Moscow Jewish Community Center (MEOC) is a Jewish religious public organization, the largest Jewish community center in Europe. Address:5a, 2nd Vysheslavtzev Pereulok Moscow. On the second floor of the building is a synagogue. The center was founded in 2000. President of the organization - Rabbi Mordechai WeisbergIn Moscow, built a lot of high-rise buildings - skyscrapers. They are houses for citizens and there are for некоторые из небоскребов as office or business centers.Skyscraper "Tower 2000" Architect Boris Thor, Moscow, Taras Shevchenko Embankment, 23aThe Federation Tower - a complex of two skyscrapers Architect Sergey Choban and Peter Schweger Moscow, Presnenskaya Embankment, 12Neva Towers - skyscraper Architect Sergey Choban, Alexey Ilyin Moscow, 1st Krasnogvardeysky pr-d, 22Evolution Tower Skyscraper - Architect Alexander Burkov, Moscow Presnenskaya Embankment, 2Embankment Tower (Block B and C) skyscraper - Project Architect: Architectural Bureau ENKA, Moscow Presnenskaya naberznaya, 10“Mercury City Tower” skyscraper - Architect: Frank Williams, Mikhail Posokhin, Moscow, 1st Krasnogvardeisky passage, 15OKO (Eye) Tower - skyscrapers - complex - North Tower, South Tower Moscow, Presnenskaya Embankment, 8, bldg. 1The Capital City is a complex - two tower Moscow and Tower Petersburg skyscrapers. Address: Moscow, 1st Krasnogvardeisky Ave, 21 p. 2"North Tower" skyscraper. Address: Moscow, Testovskaya ul., 10Tower Eurasia skyscraper. Architect Swanke Hayden Address: Moscow, Presnenskaya nab., 2Empire Tower skyscraper. Architect NBBJ Address: Moscow, Presnenskaya nab., 6, building 2Building "Gazprom", Architect Vladimir Iosifovich Havin Address ul. Nametkina, house 16, MoscowResidential complex "Triumph Palace" Architect Andrei Trofimov, Elena Treshchilina, Victor Shteller, Olga Markova Address Chapaevsky Pereulok, 3 MoscowResidential complex "House on Mosfilmovskaya" Address Street. Mosfilmovskaya, 8 building 1Residential complex "Vorob'evy Gory" Address: Mosfilmovskaya St., 70, MoscowResidential complex "Tricolor", Address pr-t Mira, 188B, MoscowResidential complex "Continental", Address Marshal Zhukov, 72-74, MoscowResidential complex "Scarlet Sails", Address Aviation 77-79, MoscowResidential complex "Edelweiss" Address Street. Davydkovskaya, 3, Moscow«Nordstar Tower», Address Begovaya, 3 pp. 1, MoscowResidential complex "Breath" Address: Moscow, Dmitrovskoye shosse, 13"White Square" on "Belarus" - Address Butyrsky Val ul., 10, MoscowResidential Complex "Nevsky" Address: Vyborg ul., 7, Moscow"Stanislavsky Factory" business center - Address ul. Stanislavsky, 21, building 2, MoscowBusiness Center "Kitezh" Address Kievskaya St., ow. 3-7, 17 MoscowResidential Complex "Golden Keys" Address: Minskaya ul., 1G building 2 building 1, MoscowAs you can see architecture and buildings are very different. For example, there are such buildings ...Residential Complex "Silhouette" Architects: MVRDV - Winy Maas, Jacob van Rijs and Nathalie de Vries Address Akademika Sakharova ave. 1, MoscowResidential building "Parus" - Address Grizodubova street, 2. Moscow. Included in the residential complex "Grand Park". The name is associated with the unusual shape of the building, formed by two arcs of different radius, and resembling a sail filled with wind. Architects: Boris Uborevich-Borovsky, Andrey BokovSpartak Stadium of Spartak Moscow Football Club Architect: Burkhard Grimm Address Volokolamskoye Shosse, 69, MoscowCopper House Residential Building, Architect Sergey Skuratov, Moscow, Butikovsky per., 5Residential complex “Embassy House”, Architect Alexander Skokan, Moscow, Borisoglebsky per., 13с3House-Egg, Moscow, st. Mashkova, 1/11 Architect Sergey TkachenkoMoscow in its territory as two cities of New York and therefore modern buildings are not so noticeable. The territory is large.In Moscow there are hundreds of modern residential complexes and interesting business centers.I hope you were interested to see some modern buildings in Moscow about which you probably did not knowIf you like what I wrote, mark as liked and I will be pleased.