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Question Number 187373 Answers: 0 Comments: 0
$$\mathrm{Solve}\:\mathrm{for}\:\mathrm{natural}\:\mathrm{numbers}: \\ $$$$\mathrm{31}\:+\:\underset{\boldsymbol{\mathrm{k}}=\mathrm{1}} {\overset{\boldsymbol{\mathrm{n}}} {\sum}}\:\left(\begin{pmatrix}{\mathrm{n}}\\{\mathrm{k}}\end{pmatrix}\:\:\centerdot\:\underset{\boldsymbol{\mathrm{m}}=\mathrm{1}} {\overset{\boldsymbol{\mathrm{n}}} {\sum}}\:\mathrm{m}^{\boldsymbol{\mathrm{n}}−\boldsymbol{\mathrm{k}}} \right)\:=\:\mathrm{31}^{\mathrm{30}} \\ $$
Question Number 187364 Answers: 2 Comments: 3
Question Number 187362 Answers: 2 Comments: 1
Question Number 187359 Answers: 1 Comments: 0
$$ \\ $$$${what}\:{are}\:{the}\:{two}\:{complex}\:{solution}\:{to} \\ $$$${X}^{−{x}} +\left(−{X}\right)^{{x}} =\mathrm{0}\:{in}\:{addition}\:{to}\:\pm\mathrm{1}\:? \\ $$
Question Number 187357 Answers: 1 Comments: 0
Question Number 187355 Answers: 1 Comments: 2
Question Number 187340 Answers: 0 Comments: 0
Question Number 187336 Answers: 1 Comments: 0
$${solve}\:{for}\:{x}\:{if} \\ $$$${X}^{{x}} \bullet\mathrm{5}^{{x}} −\mathrm{5}^{\mathrm{2}+{x}} =\mathrm{0} \\ $$$$ \\ $$
Question Number 187335 Answers: 1 Comments: 0
$${Apply}\:{the}\:{rotation}\:{of}\:{coordinates}\:{given} \\ $$$${by}\:{the}\:{following}\:{matrix}\:{to}\:{the}\:{equation}\: \\ $$$${xy}=\mathrm{1};\:{what}\:{is}\:{the}\:{equation}\:{in}\:{th}\:{uv}\:{coordinate}\: \\ $$$${system}? \\ $$$$\begin{bmatrix}{{u}}\\{{v}}\end{bmatrix}=\begin{bmatrix}{{cos}\mathrm{45}\:\:\:\:\:\:\:\:\:\:\:\:\:\:{sin}\mathrm{45}}\\{−{sin}\mathrm{45}\:\:\:\:\:\:\:\:\:\:{cos}\mathrm{45}}\end{bmatrix}\begin{bmatrix}{{x}}\\{{y}}\end{bmatrix} \\ $$
Question Number 187330 Answers: 1 Comments: 0
$$ \\ $$$${show}\:{that}\:\frac{\mathrm{1}−{cos}\theta}{\mathrm{1}+{cos}\theta}={tan}^{\mathrm{2}} \left(\frac{\mathrm{1}}{\mathrm{2}}\theta\right)\:\:\: \\ $$
Question Number 187324 Answers: 1 Comments: 0
$${calculate}\:\:{the}\:{unit}\:{of}\:{the}\:\:{chord}\:{which}\:{is} \\ $$$$\mathrm{6}{cm}\:{from}\:{the}\:{center}\:{of}\:\:{the}\:\:{circle}\:{of}\:{radius}\:\mathrm{10}{cm} \\ $$$$ \\ $$$$ \\ $$
Question Number 187317 Answers: 2 Comments: 0
$$\int\frac{{dx}}{{x}\sqrt{\mathrm{1}−\mathrm{2}{x}}} \\ $$
Question Number 187312 Answers: 0 Comments: 1
Question Number 187311 Answers: 0 Comments: 2
$$\mathrm{If}\:{f}_{{k}} \left({x}\right)=\frac{\mathrm{1}}{{k}}\left(\mathrm{sin}\:^{{k}} {x}+\mathrm{cos}\:^{{k}} {x}\right)\:\mathrm{find}\:{f}_{\mathrm{4}} \left({x}\right)−{f}_{\mathrm{6}} \left({x}\right) \\ $$$${f}_{\mathrm{4}} \left({x}\right)−{f}_{\mathrm{6}} \left({x}\right)=\frac{\mathrm{1}}{\mathrm{4}}\left(\mathrm{sin}\:^{\mathrm{4}} {x}\:+\mathrm{cos}\:^{\mathrm{4}} {x}\right)−\frac{\mathrm{1}}{\mathrm{6}}\left(\mathrm{sin}\:^{\mathrm{6}} {x}+\mathrm{cos}\:^{\mathrm{6}} {x}\right) \\ $$$$\:\:\:=\frac{\mathrm{1}}{\mathrm{2}}\mathrm{sin}\:^{\mathrm{4}} {x}\left(\frac{\mathrm{1}}{\mathrm{2}}−\frac{\mathrm{1}}{\mathrm{3}}\mathrm{sin}\:^{\mathrm{2}} {x}\right)+\frac{\mathrm{1}}{\mathrm{2}}\mathrm{cos}\:^{\mathrm{4}} {x}\left(\frac{\mathrm{1}}{\mathrm{2}}−\frac{\mathrm{1}}{\mathrm{3}}\mathrm{cos}\:^{\mathrm{2}} {x}\right) \\ $$
Question Number 187302 Answers: 0 Comments: 0
$${f}\left(\mathrm{4}\right)=\mathrm{44},\:{f}\left({m}\right)=\mathrm{52},{f}\left({l}\right)=−\mathrm{33} \\ $$$${l},{m}\:\mathrm{are}\:\mathrm{positive}\:\mathrm{integers}\:\mathrm{such}\:\mathrm{that}\:\mathrm{4}<{m}<{l} \\ $$$$\mathrm{and}\:{f}\left({x}\right)\:\mathrm{is}\:\mathrm{a}\:\mathrm{polynomial}\:\mathrm{with}\:\mathrm{integer}\:\mathrm{coefficients}. \\ $$$$\mathrm{Find}\:{l}+{m}. \\ $$
Question Number 187301 Answers: 2 Comments: 0
Question Number 187300 Answers: 0 Comments: 0
$$\mathrm{Find}\:\mathrm{the}\:\mathrm{number}\:\mathrm{of}\:\mathrm{integral}\:\mathrm{solutions}\:\mathrm{of} \\ $$$$\left({p}+{q}\right)\left({q}+{r}\right)\left({r}+{p}\right)=\mathrm{8}{pqr}+\mathrm{2} \\ $$
Question Number 187298 Answers: 0 Comments: 3
$${if}\: \\ $$$$\mathrm{2}^{{y}} =\mathrm{25}\:{and}\:\mathrm{5}^{{x}} =\mathrm{16},\:{find}\:{x}+{y} \\ $$
Question Number 187296 Answers: 0 Comments: 1
$$\mathrm{Prove}\:\mathrm{that}\:\mathrm{for}\:\mathrm{n}\geqslant\mathrm{4},\:\mathrm{S}_{{n}} =\:\underset{{k}=\mathrm{1}} {\overset{{n}} {\sum}}{k}^{{k}!} \:\mathrm{is}\:\mathrm{never}\:\mathrm{a}\:\mathrm{perfect}\:\mathrm{cube}. \\ $$
Question Number 187295 Answers: 0 Comments: 2
$${solve}: \\ $$$$\left(\sqrt{\left.\mathrm{4}+\sqrt{\mathrm{4}}\right)\:^{{x}} }\:+\left[\sqrt{\mathrm{4}−\sqrt{\left.\mathrm{4}\right]^{{x}} }}\:=\mathrm{4}^{{x}} \right.\right. \\ $$
Question Number 187294 Answers: 0 Comments: 1
$${solve}: \\ $$$$−\mathrm{1}−\left[{x}^{{x}} −\mathrm{5}{x}+\mathrm{6}\right]^{{x}} =\mathrm{1} \\ $$
Question Number 187288 Answers: 1 Comments: 0
$${logx}+{x}!=\mathrm{2} \\ $$$${x}=? \\ $$
Question Number 187284 Answers: 1 Comments: 2
Question Number 187270 Answers: 3 Comments: 0
$$\underset{{x}\rightarrow\pi} {\mathrm{lim}}\frac{\mathrm{cos}\:{x}\:+\:\mathrm{sin}\left(\mathrm{2}{x}\right)\:+\mathrm{1}}{{x}^{\mathrm{2}} −\pi^{\mathrm{2}} }\:= \\ $$$$\mathrm{A}.\:\frac{\mathrm{1}}{\mathrm{2}\pi}\:\:\:\:\:\:\:\:\:{C}.\:\mathrm{1} \\ $$$${B}.\:\frac{\mathrm{1}}{\pi}\:\:\:\:\:\:\:\:\:{D}.\:\mathrm{Does}\:\mathrm{Not}\:\mathrm{exist} \\ $$
Question Number 187257 Answers: 0 Comments: 2
Question Number 187255 Answers: 1 Comments: 0
$${Find}\:{the}\:{directional}\:{derivatives}\:{of}\:{the} \\ $$$${function}\: \\ $$$${f}\left({x},\mathrm{y},\mathrm{z}\right)=\mathrm{2}{x}^{\mathrm{2}} +\mathrm{3}{y}^{\mathrm{2}} +{z}^{\mathrm{2}} \:{at}\:{the}\:{point}\:{p}\left(\mathrm{2},\mathrm{1},\mathrm{3}\right) \\ $$
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