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DEFINATION-OF-QUADRATIC-FORM-A-Quadratic-form-is-a-homogeneous-polynomial-of-degree-two-in-multiple-variable-Q-X-T-AX-Here-Q-Quadratic-form-ax-2-by-2-cz

Question Number 212028 by siva12345 last updated on 27/Sep/24 $${DEFINATION}\:\:\:\:{OF}\:\:\:{QUADRATIC}\:\:{FORM}:\: \\ $$$$\:\:\:\:\:{A}\:\:{Quadratic}\:\:{form}\:\:{is}\:\:{a}\:{homogeneous}\:\:{polynomial}\:\:{of}\:\:{degree}\:{two}\:\:{in}\:\:{multiple}\:\:{variable}.\: \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:{Q}={X}^{{T}} {AX} \\ $$$${Here}\:\:{Q}={Quadratic}\:{form}. \\ $$$${ax}^{\mathrm{2}} +{by}^{\mathrm{2}} +{cz}^{\mathrm{2}} +\mathrm{2}{hxy}+\mathrm{2}{fyz}+\mathrm{2}{gzx}=\mathrm{0} \\ $$$${By}\:\:{using}\:\:{these}\:\:{Q}={X}^{{T}} {AX}\:\:\left[{we}\:\:{can}\:\:{write}\:{matrix}\:{A}\right]…

prove-lim-x-1-5-x-1-x-1-5-

Question Number 211812 by mokys last updated on 21/Sep/24 $${prove}\:\underset{{x}\rightarrow\infty} {{lim}}\:\left(\:\mathrm{1}\:+\:\frac{\mathrm{5}}{{x}}\:\right)^{\frac{\mathrm{1}}{{x}}} −\:\mathrm{1}\:=\:\mathrm{5}\: \\ $$ Commented by mr W last updated on 22/Sep/24 $${wrong}! \\ $$$${the}\:{result}\:{should}\:{be}\:\mathrm{0}.…

find-all-n-m-such-that-n-2-m-m-2-n-Z-

Question Number 211753 by alcohol last updated on 19/Sep/24 $${find}\:{all}\:\left({n},{m}\right)\:{such}\:{that}\:\frac{{n}^{\mathrm{2}} −{m}}{{m}^{\mathrm{2}} −{n}}\:\in\:\mathbb{Z} \\ $$ Answered by Frix last updated on 20/Sep/24 $$\mathrm{There}\:\mathrm{are}\:\mathrm{infinitely}\:\mathrm{many}\:\mathrm{solutions}. \\ $$$$\mathrm{3}\:\mathrm{ways}\:\mathrm{to}\:\mathrm{find}\:\mathrm{some}: \\…

Question-211632

Question Number 211632 by sonukgindia last updated on 15/Sep/24 Answered by A5T last updated on 15/Sep/24 $$\phi\left(\mathrm{1000}\right)=\mathrm{400} \\ $$$$\mathrm{2024}^{\mathrm{2024}} \equiv\mathrm{0}\left({mod}\:\mathrm{16}\right);\mathrm{2024}^{\mathrm{2024}} \equiv\mathrm{1}\left({mod}\:\mathrm{25}\right) \\ $$$$\Rightarrow\mathrm{2024}^{\mathrm{2024}} =\mathrm{25}{q}+\mathrm{1}\equiv\left(\mathrm{0}\:{mod}\:\mathrm{16}\right)\Rightarrow{q}\equiv\mathrm{7}\left({mod}\mathrm{16}\right) \\…

f-x-sin-x-e-x-1-Prove-that-f-x-has-only-2-zeros-in-pi-x-0-

Question Number 211637 by CrispyXYZ last updated on 15/Sep/24 $${f}\left({x}\right)\:=\:\mathrm{sin}\:{x}\:−\:\mathrm{e}^{{x}} \:+\:\mathrm{1}. \\ $$$$\mathrm{Prove}\:\mathrm{that}\:{f}\left({x}\right)\:\mathrm{has}\:\mathrm{only}\:\mathrm{2}\:\mathrm{zeros}\:\mathrm{in}\:−\pi\leqslant{x}\leqslant\mathrm{0}. \\ $$ Answered by mehdee1342 last updated on 15/Sep/24 $${f}\left(\mathrm{0}\right)=\mathrm{0} \\ $$$${f}\left(−\pi\right)=\mathrm{1}−{e}^{−\pi}…