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Question Number 170259 Answers: 0 Comments: 0

Question Number 170240 Answers: 0 Comments: 0

A deck of 2022 cards is numbered 1 to 2022. The first, 1, is placed at the bottom of the deck , the second,2, is discarded, the third ,3, placed at the bottom, the next,4, is discarded, and so on. the process continue till only one card is left. What is the number on the card? can you generalise? try some simpler example to help.

Question Number 170226 Answers: 1 Comments: 0

solve { ((x+y+z+w=0)),((x+y+z+2w=0)),((2x+2y+3z+4w=1)),((2x+3y+4z+5w=2)) :}

$${solve}\: \\ $$$$\begin{cases}{{x}+{y}+{z}+{w}=\mathrm{0}}\\{{x}+{y}+{z}+\mathrm{2}{w}=\mathrm{0}}\\{\mathrm{2}{x}+\mathrm{2}{y}+\mathrm{3}{z}+\mathrm{4}{w}=\mathrm{1}}\\{\mathrm{2}{x}+\mathrm{3}{y}+\mathrm{4}{z}+\mathrm{5}{w}=\mathrm{2}}\end{cases} \\ $$$$ \\ $$

Question Number 170221 Answers: 1 Comments: 1

lim_(x→0^+ ) (((((√x))^(√x) −x^x )/(((√x))^x −x^(√x) )))=? help me

$$\underset{{x}\rightarrow\mathrm{0}^{+} } {\mathrm{lim}}\:\left(\frac{\left(\sqrt{{x}}\right)^{\sqrt{{x}}} −{x}^{{x}} }{\left(\sqrt{{x}}\right)^{{x}} −{x}^{\sqrt{{x}}} }\right)=? \\ $$$${help}\:{me} \\ $$

Question Number 170212 Answers: 3 Comments: 0

which rectangle with integer length side have numerically the same area and perimeter? Find them all. Find a proof that convinces that you have found them all. what about right− angled triangle? how many solutions?

$$\:\mathrm{which}\:\mathrm{rectangle}\:\mathrm{with}\:\mathrm{integer}\:\mathrm{length}\: \\ $$$$\:\:\mathrm{side}\:\mathrm{have}\:\mathrm{numerically}\:\mathrm{the}\:\mathrm{same}\:\mathrm{area} \\ $$$$\:\:\mathrm{and}\:\mathrm{perimeter}?\:\mathrm{Find}\:\mathrm{them}\:\mathrm{all}.\:\mathrm{Find} \\ $$$$\:\mathrm{a}\:\mathrm{proof}\:\mathrm{that}\:\mathrm{convinces}\:\mathrm{that}\:\mathrm{you}\:\mathrm{have} \\ $$$$\:\mathrm{found}\:\mathrm{them}\:\mathrm{all}.\:\mathrm{what}\:\mathrm{about}\:\mathrm{right}− \\ $$$$\:\mathrm{angled}\:\mathrm{triangle}?\:\mathrm{how}\:\mathrm{many}\:\mathrm{solutions}? \\ $$

Question Number 170211 Answers: 1 Comments: 0

lim_(x→0) (((tan^(−1) (((x−x(√(1−x^2 )))/( (√(1−x^2 ))+x^2 ))))/x^3 ))=? pleas solve this

$$\underset{{x}\rightarrow\mathrm{0}} {\mathrm{lim}}\left(\frac{{tan}^{−\mathrm{1}} \left(\frac{{x}−{x}\sqrt{\mathrm{1}−{x}^{\mathrm{2}} }}{\:\sqrt{\mathrm{1}−{x}^{\mathrm{2}} }+{x}^{\mathrm{2}} }\right)}{{x}^{\mathrm{3}} }\right)=? \\ $$$${pleas}\:{solve}\:{this} \\ $$

Question Number 170206 Answers: 2 Comments: 0

1. { ((y^(log_x y) =x^4 )),((log_2 (y−x)=1)) :} 2. 25^(−x) −5^(−x+1) ≥50

$$\mathrm{1}.\:\:\:\:\begin{cases}{{y}^{{log}_{{x}} {y}} ={x}^{\mathrm{4}} }\\{{log}_{\mathrm{2}} \left({y}−{x}\right)=\mathrm{1}}\end{cases} \\ $$$$\mathrm{2}.\:\:\:\:\:\mathrm{25}^{−{x}} −\mathrm{5}^{−{x}+\mathrm{1}} \geqslant\mathrm{50} \\ $$

Question Number 170205 Answers: 1 Comments: 0

If x is nearer a than b in [a,b], is (√x) necessarily nearer (√a) than (√b) Give a proof of counterexample

$$\:\mathrm{If}\:\mathrm{x}\:\mathrm{is}\:\mathrm{nearer}\:\:\:\mathrm{a}\:\:\mathrm{than}\:\:\mathrm{b}\:\:\mathrm{in}\:\left[\mathrm{a},\mathrm{b}\right],\: \\ $$$$\mathrm{is}\:\sqrt{\mathrm{x}}\:\mathrm{necessarily}\:\mathrm{nearer}\:\sqrt{\mathrm{a}}\:\:\mathrm{than}\:\sqrt{\mathrm{b}} \\ $$$$ \\ $$$$\mathrm{Give}\:\mathrm{a}\:\mathrm{proof}\:\mathrm{of}\:\mathrm{counterexample} \\ $$

Question Number 170203 Answers: 1 Comments: 0

solve: 5^(lgx) =50−x^(lg5)