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Question Number 175593    Answers: 0   Comments: 3

If 0<a≤b then prove that: determinant ((1,a,(log(a^a ))),(1,((√((a^2 +b^2 )/2)) (√((a^2 +b^2 )/8)) ∙ log(((a^2 +b^2 )/2))),),(1,b,(log(b^b ))))≥ 0

$$\mathrm{If}\:\:\:\mathrm{0}<\mathrm{a}\leqslant\mathrm{b}\:\:\:\mathrm{then}\:\mathrm{prove}\:\mathrm{that}: \\ $$$$\begin{vmatrix}{\mathrm{1}}&{\mathrm{a}}&{\mathrm{log}\left(\mathrm{a}^{\boldsymbol{\mathrm{a}}} \right)}\\{\mathrm{1}}&{\sqrt{\frac{\mathrm{a}^{\mathrm{2}} +\mathrm{b}^{\mathrm{2}} }{\mathrm{2}}}\:\sqrt{\frac{\mathrm{a}^{\mathrm{2}} +\mathrm{b}^{\mathrm{2}} }{\mathrm{8}}}\:\centerdot\:\mathrm{log}\left(\frac{\mathrm{a}^{\mathrm{2}} +\mathrm{b}^{\mathrm{2}} }{\mathrm{2}}\right)}&{}\\{\mathrm{1}}&{\mathrm{b}}&{\mathrm{log}\left(\mathrm{b}^{\boldsymbol{\mathrm{b}}} \right)}\end{vmatrix}\geqslant\:\mathrm{0} \\ $$

Question Number 175588    Answers: 2   Comments: 1

Question Number 175585    Answers: 1   Comments: 6

in AB^Δ C prove that: sin ((( A)/2) ) ≤ (( a)/( b + c)) ■

$$ \\ $$$$\:\:\:{in}\:{A}\overset{\Delta} {{B}C}\:\:{prove}\:\:{that}: \\ $$$$\:\:\: \\ $$$$\:\:\:\:\:\:\:\:{sin}\:\left(\frac{\:{A}}{\mathrm{2}}\:\right)\:\leqslant\:\frac{\:{a}}{\:{b}\:+\:{c}}\:\:\:\:\:\:\:\blacksquare \\ $$$$ \\ $$

Question Number 175583    Answers: 0   Comments: 1

∫_0 ^2 x^t dt=3 solve for x

$$\int_{\mathrm{0}} ^{\mathrm{2}} {x}^{{t}} {dt}=\mathrm{3} \\ $$$${solve}\:{for}\:{x} \\ $$

Question Number 175579    Answers: 0   Comments: 0

lim_(x→∞) ((x^(1−sin ((1/x))) .(x^(sin ((1/x))) −1))/(ln x))

$$\:\underset{{x}\rightarrow\infty} {\mathrm{lim}}\:\frac{\mathrm{x}^{\mathrm{1}−\mathrm{sin}\:\left(\frac{\mathrm{1}}{\mathrm{x}}\right)} .\left(\mathrm{x}^{\mathrm{sin}\:\left(\frac{\mathrm{1}}{\mathrm{x}}\right)} −\mathrm{1}\right)}{\mathrm{ln}\:\mathrm{x}} \\ $$

Question Number 175573    Answers: 0   Comments: 0

Question Number 175572    Answers: 0   Comments: 0

Question Number 175571    Answers: 2   Comments: 0

Question Number 175567    Answers: 0   Comments: 0

in how many ways can you put 40 identical balls into 20 identical boxes such that each box obtains at least one ball and at most 5 balls?

$${in}\:{how}\:{many}\:{ways}\:{can}\:{you}\:{put}\:\mathrm{40} \\ $$$${identical}\:{balls}\:{into}\:\mathrm{20}\:{identical}\:{boxes} \\ $$$${such}\:{that}\:{each}\:{box}\:{obtains}\:{at}\:{least}\:{one} \\ $$$${ball}\:{and}\:{at}\:{most}\:\mathrm{5}\:{balls}? \\ $$

Question Number 175568    Answers: 1   Comments: 1

Question Number 175554    Answers: 2   Comments: 0

x^(√x) =(√x^x ) find x

$${x}^{\sqrt{{x}}} =\sqrt{{x}^{{x}} } \\ $$$${find}\:{x} \\ $$

Question Number 175553    Answers: 1   Comments: 0

Question Number 175548    Answers: 1   Comments: 4

determinant ((( determinant (((2+424+44244+4442444+∙∙∙n terms=?_ ^ _() ^() )))_ ^ ^(∣•∣_(−) ^(−) ) )))

$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\: \\ $$$$\begin{array}{|c|}{\overset{\underset{−} {\overline {\mid\bullet\mid}}} {\:\begin{array}{|c|}{\underset{} {\overset{} {\mathrm{2}+\mathrm{424}+\mathrm{44244}+\mathrm{4442444}+\centerdot\centerdot\centerdot{n}\:{terms}=?_{} ^{} }}}\\\hline\end{array}_{} ^{} }}\\\hline\end{array} \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\: \\ $$

Question Number 175547    Answers: 2   Comments: 0

Question Number 175544    Answers: 1   Comments: 1

tan^6 (10°)+tan^6 (50°)+tan^6 (70°)=?

$$\:\mathrm{tan}\:^{\mathrm{6}} \left(\mathrm{10}°\right)+\mathrm{tan}\:^{\mathrm{6}} \left(\mathrm{50}°\right)+\mathrm{tan}\:^{\mathrm{6}} \left(\mathrm{70}°\right)=? \\ $$

Question Number 175531    Answers: 2   Comments: 0

∫ (dt/(5cos t+6sin t)) =?

$$\:\int\:\frac{{dt}}{\mathrm{5cos}\:{t}+\mathrm{6sin}\:{t}}\:=? \\ $$

Question Number 175511    Answers: 0   Comments: 1

Question Number 175516    Answers: 1   Comments: 0

Question Number 175505    Answers: 1   Comments: 2

N=64990691606209 is a semi-prime number. That is, N=pq where p and q are prime numbers. Find p and q:

$${N}=\mathrm{64990691606209}\:\mathrm{is}\:\mathrm{a}\:\mathrm{semi}-\mathrm{prime}\:\mathrm{number}. \\ $$$$\mathrm{That}\:\mathrm{is},\:{N}={pq}\:\mathrm{where}\:{p}\:\mathrm{and}\:{q}\:\mathrm{are}\:\mathrm{prime}\:\mathrm{numbers}. \\ $$$$\mathrm{Find}\:{p}\:\mathrm{and}\:{q}: \\ $$

Question Number 175493    Answers: 1   Comments: 1

tan^(−1) (asin θ)=sin^(−1) b−θ find θ.

$$\mathrm{tan}^{−\mathrm{1}} \left({a}\mathrm{sin}\:\theta\right)=\mathrm{sin}^{−\mathrm{1}} {b}−\theta \\ $$$${find}\:\theta. \\ $$

Question Number 175490    Answers: 1   Comments: 0

solve f(x)f(y)= f(x+y)+xy f:R⇒R

$${solve} \\ $$$${f}\left({x}\right){f}\left({y}\right)=\:{f}\left({x}+{y}\right)+{xy} \\ $$$${f}:\mathbb{R}\Rightarrow\mathbb{R} \\ $$

Question Number 175487    Answers: 0   Comments: 0

Question Number 175483    Answers: 2   Comments: 0

Solve it by horner′s method and get the quotient. 2x^3 y+3xy−5x^2 y^2 +12÷(2x−4)=?

$${Solve}\:{it}\:{by}\:{horner}'{s}\:{method}\:{and}\:{get} \\ $$$${the}\:{quotient}. \\ $$$$\mathrm{2}{x}^{\mathrm{3}} {y}+\mathrm{3}{xy}−\mathrm{5}{x}^{\mathrm{2}} {y}^{\mathrm{2}} +\mathrm{12}\boldsymbol{\div}\left(\mathrm{2}{x}−\mathrm{4}\right)=? \\ $$

Question Number 175476    Answers: 1   Comments: 1

Question Number 175471    Answers: 1   Comments: 0

Question Number 175470    Answers: 1   Comments: 0

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