Question Number 120636 by TANMAY PANACEA last updated on 01/Nov/20
$${selective}\:{Binomial} \\ $$
Commented by TANMAY PANACEA last updated on 01/Nov/20
Commented by TANMAY PANACEA last updated on 01/Nov/20
Commented by TANMAY PANACEA last updated on 01/Nov/20
Commented by TANMAY PANACEA last updated on 01/Nov/20
Commented by TANMAY PANACEA last updated on 01/Nov/20
Commented by TANMAY PANACEA last updated on 01/Nov/20
Commented by TANMAY PANACEA last updated on 01/Nov/20
Commented by Dwaipayan Shikari last updated on 01/Nov/20
$${S}=\frac{\mathrm{1}}{\mathrm{1}!\left({n}−\mathrm{1}\right)!}+\frac{\mathrm{1}}{\mathrm{3}!\left({n}−\mathrm{3}\right)!}+… \\ $$$${S}=\frac{{n}}{{n}!}+\frac{{n}\left({n}−\mathrm{1}\right)\left({n}−\mathrm{2}\right)}{\mathrm{3}!{n}!}+\frac{{n}\left({n}−\mathrm{1}\right)\left({n}−\mathrm{2}\right)\left({n}−\mathrm{3}\right)\left({n}−\mathrm{4}\right)}{\mathrm{5}!{n}!}+… \\ $$$$=\frac{\mathrm{1}}{{n}!}\left({n}+\frac{{n}\left({n}−\mathrm{1}\right)\left({n}−\mathrm{2}\right)}{\mathrm{3}!}+\frac{{n}\left({n}−\mathrm{1}\right)\left({n}−\mathrm{2}\right)\left({n}−\mathrm{3}\right)\left({n}−\mathrm{4}\right)}{\mathrm{5}!}+…\right) \\ $$$$=\frac{\mathrm{2}^{{n}−\mathrm{1}} }{{n}!} \\ $$$$\left(\mathrm{1}+{x}\right)^{{n}} =\mathrm{1}+{nx}+\frac{{n}\left({n}−\mathrm{1}\right)}{\mathrm{2}!}{x}^{\mathrm{2}} +… \\ $$$$\left(\mathrm{1}−{x}\right)^{{n}} =\mathrm{1}−{nx}+\frac{{n}\left({n}−\mathrm{1}\right)}{\mathrm{2}!}{x}^{\mathrm{2}} −.. \\ $$$$\frac{\left(\mathrm{1}+{x}\right)^{{n}} −\left(\mathrm{1}−{x}\right)^{{n}} }{\mathrm{2}}={nx}+\frac{{n}\left({n}−\mathrm{1}\right)\left({n}−\mathrm{2}\right)}{\mathrm{3}!}{x}^{\mathrm{3}} +… \\ $$$$\mathrm{2}^{{n}−\mathrm{1}} =\left({n}+\frac{{n}\left({n}−\mathrm{1}\right)\left({n}−\mathrm{2}\right)}{\mathrm{3}!}+…\right)\:\:\:\left({x}=\mathrm{1}\right) \\ $$
Commented by Dwaipayan Shikari last updated on 01/Nov/20
$$\left(\mathrm{1}+{x}\right)^{{n}} ={C}_{\mathrm{0}} +{C}_{\mathrm{1}} {x}+… \\ $$$$\int_{\mathrm{0}} ^{\mathrm{1}} \left(\mathrm{1}+{x}\right)^{{n}} =\int_{\mathrm{0}} ^{\mathrm{1}} {C}_{\mathrm{0}} +{C}_{\mathrm{1}} {x}+.. \\ $$$$\frac{\mathrm{2}^{{n}+\mathrm{1}} }{{n}+\mathrm{1}}={C}_{\mathrm{0}} +\frac{\mathrm{1}}{\mathrm{2}}{C}_{\mathrm{1}} +\frac{\mathrm{1}}{\mathrm{3}}{C}_{\mathrm{2}} +…\frac{\mathrm{1}}{{n}+\mathrm{1}}{C}_{{n}} \\ $$
Commented by Dwaipayan Shikari last updated on 01/Nov/20
$${C}_{\mathrm{0}} +\left({C}_{\mathrm{0}} +{C}_{\mathrm{1}} \right)+\left({C}_{\mathrm{0}} +{C}_{\mathrm{1}} +{C}_{\mathrm{2}} \right)+… \\ $$$${n}\left({C}_{\mathrm{0}} +{C}_{\mathrm{1}} +{C}_{\mathrm{2}} +…\right)−\left({C}_{\mathrm{1}} +\mathrm{2}{C}_{\mathrm{2}} +\mathrm{3}{C}_{\mathrm{3}} +..\right) \\ $$$${n}.\mathrm{2}^{{n}} −{n}.\mathrm{2}^{{n}−\mathrm{1}} ={n}\mathrm{2}^{{n}−\mathrm{1}} \left(\mathrm{2}−\mathrm{1}\right)={n}.\mathrm{2}^{{n}−\mathrm{1}} \\ $$$$\left[\left(\mathrm{1}+{x}\right)^{{n}} ={C}_{\mathrm{0}} +{C}_{\mathrm{1}} {x}+{C}_{\mathrm{3}} {x}^{\mathrm{2}} +..\right. \\ $$$${n}\left(\mathrm{1}+{x}\right)^{{n}−\mathrm{1}} ={C}_{\mathrm{1}} +\mathrm{2}{C}_{\mathrm{3}} {x}+.. \\ $$$$\left.{n}.\mathrm{2}^{{n}−\mathrm{1}} ={C}_{\mathrm{1}} +\mathrm{2}{C}_{\mathrm{3}} +…\right] \\ $$
Commented by Dwaipayan Shikari last updated on 01/Nov/20
$$\left(\mathrm{1}+{x}+\mathrm{2}{x}^{\mathrm{2}} \right)^{\mathrm{20}} ={a}_{\mathrm{0}} +{a}_{\mathrm{1}} {x}+… \\ $$$$\mathrm{1}^{\mathrm{20}} ={a}_{\mathrm{0}} \:\:\:\left({x}=\mathrm{0}\right) \\ $$$$\left(\mathrm{2}\right)^{\mathrm{40}} =\mathrm{1}+{a}_{\mathrm{1}} +{a}_{\mathrm{2}} +….\:\:\:\left({x}=\mathrm{1}\right) \\ $$$$\mathrm{2}^{\mathrm{20}} =\mathrm{1}−{a}_{\mathrm{1}} +{a}_{\mathrm{2}} −{a}_{\mathrm{3}} +….\left({x}=−\mathrm{1}\right) \\ $$$$\mathrm{2}^{\mathrm{40}} +\mathrm{2}^{\mathrm{20}} =\mathrm{2}\left(\mathrm{1}+{a}_{\mathrm{2}} +{a}_{\mathrm{4}} +…+{a}_{\mathrm{38}} \right) \\ $$$$\mathrm{2}^{\mathrm{39}} +\mathrm{2}^{\mathrm{19}} ={a}_{\mathrm{0}} +{a}_{\mathrm{2}} +…. \\ $$$$\mathrm{2}^{\mathrm{19}} \left(\mathrm{2}^{\mathrm{20}} +\mathrm{1}\right)={a}_{\mathrm{0}} +{a}_{\mathrm{2}} +…+{a}_{\mathrm{38}} \\ $$
Commented by TANMAY PANACEA last updated on 01/Nov/20
$${thank}\:{you} \\ $$
Commented by TANMAY PANACEA last updated on 01/Nov/20
$${i}\:{have}\:{posted}\:{questions}\:{collected}\:{from}\left[{books}\:{of}\:{IIT}\right. \\ $$
Commented by Dwaipayan Shikari last updated on 01/Nov/20
$$\left.{I}\:{am}\:{in}\:{class}\:\mathrm{11}\:\left({high}\:{school}\right)\::\right) \\ $$
Commented by prakash jain last updated on 02/Nov/20
$$\mathrm{You}\:\mathrm{are}\:\mathrm{doing}\:\mathrm{very}\:\mathrm{well}\:\mathrm{for}\:\mathrm{class}\:\mathrm{XI}. \\ $$$$\mathrm{Are}\:\mathrm{you}\:\mathrm{aiming}\:\mathrm{for}\:\mathrm{IIT}? \\ $$
Commented by Dwaipayan Shikari last updated on 02/Nov/20
$${I}\:{want}\:{to}\:{study}\:{physics}\:{in}\:{later}. \\ $$$${IIT}'{s}\:{are}\:{good}\:{but}\:{i}\:{want}\:{to}\:{study}\:{freely} \\ $$
Commented by Dwaipayan Shikari last updated on 02/Nov/20
$$\left.{But}\:{in}\mathrm{2022}\:{i}\:{have}\:\:{to}\:{take}\:{this}\:{exam}\::\right) \\ $$
Commented by prakash jain last updated on 02/Nov/20
$$\mathrm{Are}\:\mathrm{u}\:\mathrm{interested}\:\mathrm{in}\:\mathrm{theoritical}\:\mathrm{physics}? \\ $$
Commented by TANMAY PANACEA last updated on 02/Nov/20
$$\mathrm{17}^{\mathrm{1983}} +\mathrm{11}^{\mathrm{1983}} −\mathrm{7}^{\mathrm{1983}} \\ $$$${answer}\:{is}\:=\mathrm{1} \\ $$$${explanation} \\ $$$$\mathrm{7}^{\mathrm{1}} ={last}\:{digit}\:\mathrm{7} \\ $$$$\mathrm{7}^{\mathrm{2}} ={l}.{d}\:\mathrm{9} \\ $$$$\mathrm{7}^{\mathrm{3}} ={l}.{d}\:\mathrm{3} \\ $$$$\mathrm{7}^{\mathrm{4}} =\mathrm{1} \\ $$$$\mathrm{7}^{\mathrm{5}} =\mathrm{7} \\ $$$${cycle}\:{of}\:\left(\mathrm{7}\:\mathrm{9}\:\mathrm{1}\:\mathrm{7}\right) \\ $$$$\frac{\mathrm{1983}}{\mathrm{7}}=\mathrm{283}+\frac{\mathrm{2}}{\mathrm{7}}\rightarrow{last}\:{digit}=\mathrm{9} \\ $$$${last}\:{digit}\:{of}\:\mathrm{7}^{\mathrm{1983}} \rightarrow=\mathrm{9} \\ $$$${last}\:{digit}\:{of}\:\mathrm{11}^{\mathrm{1983}} \:{is}\mathrm{1} \\ $$$${so}\:\left(\mathrm{9}+\mathrm{1}−\mathrm{9}\right)\rightarrow{last}\:{digit}=\mathrm{1} \\ $$$$\boldsymbol{{pls}}\:\boldsymbol{{chk}} \\ $$
Commented by TANMAY PANACEA last updated on 02/Nov/20
$$\mathrm{32}^{\mathrm{32}^{\mathrm{32}} } \:{devided}\:{by}\:\mathrm{7} \\ $$$$\left(\mathrm{4}×\mathrm{7}+\mathrm{5}\right)^{\mathrm{32}^{\mathrm{32}} } \\ $$$$={f}\left(\mathrm{7}\right)+\mathrm{5}^{\mathrm{32}^{\mathrm{32}} } \\ $$$${now}\:\mathrm{32}^{\mathrm{32}} ={even}\:{number} \\ $$$$\mathrm{32}^{\mathrm{32}} \\ $$$$\mathrm{2}^{\mathrm{1}} \rightarrow\mathrm{2} \\ $$$$\mathrm{2}^{\mathrm{2}} \rightarrow\mathrm{4} \\ $$$$\mathrm{2}^{\mathrm{3}} \rightarrow\mathrm{8} \\ $$$$\mathrm{2}^{\mathrm{4}} \rightarrow\mathrm{16} \\ $$$$\mathrm{2}^{\mathrm{5}} \rightarrow\mathrm{32} \\ $$$${cycle}\left({last}\:{digit}\:\:\:\mathrm{2}\:\mathrm{4}\:\mathrm{8}\:\mathrm{6}\:\right) \\ $$$$\mathrm{32}^{\mathrm{32}} =\mathrm{2}{k}\:{is}\:{a}\:{even}\:{number}\:{whose}\:{last}\:{digit}\:\mathrm{6} \\ $$$$\mathrm{5}^{\mathrm{2}{k}} ={last}\:{two}\:{digit}\:{is}\:\mathrm{25} \\ $$$${so}\:{when}\:{devided}\:{by}\:\mathrm{7}\rightarrow{last}\:{digit}\:{is}\:\mathrm{4} \\ $$$$\boldsymbol{{pls}}\:\boldsymbol{{chk}}\:\boldsymbol{{is}}\:\boldsymbol{{my}}\:\boldsymbol{{thinking}}\:\boldsymbol{{correct}} \\ $$$$ \\ $$$$ \\ $$$$ \\ $$
Commented by Dwaipayan Shikari last updated on 02/Nov/20
$${yes}\:,\:{Prakash}\:{sir}! \\ $$
Commented by Dwaipayan Shikari last updated on 03/Nov/20
$$\left(\mathrm{4}×\mathrm{7}+\mathrm{4}\right)=\mathrm{32}^{\mathrm{32}^{\mathrm{32}} } \\ $$$${typo} \\ $$
Commented by TANMAY PANACEA last updated on 03/Nov/20
$${yes}\:{yes}\: \\ $$