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Question Number 192226 Answers: 0 Comments: 0

Question Number 192220 Answers: 1 Comments: 2

prove that if n∈N, n>1 and n is odd then 1^n +...+(n−1)^n is divisible by n (dont use ≡(modn))

$$ \\ $$$$\mathrm{prove}\:\mathrm{that}\:\mathrm{if}\:\mathrm{n}\in\mathbb{N},\:\mathrm{n}>\mathrm{1}\:\mathrm{and}\:\mathrm{n}\:\mathrm{is}\:\mathrm{odd}\:\mathrm{then} \\ $$$$\:\mathrm{1}^{\mathrm{n}} +...+\left(\mathrm{n}−\mathrm{1}\right)^{\mathrm{n}} \:\mathrm{is}\:\mathrm{divisible}\:\mathrm{by}\:\mathrm{n} \\ $$$$\left(\mathrm{dont}\:\mathrm{use}\:\equiv\left(\mathrm{modn}\right)\right) \\ $$

Question Number 192212 Answers: 1 Comments: 0

∫−1^x dx

$$\int−\mathrm{1}^{{x}} {dx} \\ $$

Question Number 192208 Answers: 1 Comments: 0

Question Number 192202 Answers: 0 Comments: 0

Question Number 192204 Answers: 1 Comments: 0

Question Number 192203 Answers: 1 Comments: 0

Question Number 192187 Answers: 0 Comments: 0

{ ((z_1 + z_2 = a)),((z_2 + z_3 = b)),((z_3 + z_4 = c)),((z_1 + z_4 = d)) :} solve using gaussian elimination

$$\begin{cases}{{z}_{\mathrm{1}} \:+\:{z}_{\mathrm{2}} \:=\:{a}}\\{{z}_{\mathrm{2}} \:+\:{z}_{\mathrm{3}} \:=\:{b}}\\{{z}_{\mathrm{3}} \:+\:{z}_{\mathrm{4}} \:=\:{c}}\\{{z}_{\mathrm{1}} \:+\:{z}_{\mathrm{4}} \:=\:{d}}\end{cases} \\ $$$${solve}\:{using}\:{gaussian}\:{elimination} \\ $$

Question Number 192186 Answers: 3 Comments: 0

If α, β and γ are the roots of x^3 + px + q = 0, find Σα^4 .

$$\mathrm{If}\:\:\alpha,\:\beta\:\:\mathrm{and}\:\gamma\:\mathrm{are}\:\mathrm{the}\:\mathrm{roots}\:\mathrm{of}\:\:\:\mathrm{x}^{\mathrm{3}} \:\:+\:\:\mathrm{px}\:\:+\:\:\mathrm{q}\:\:=\:\:\mathrm{0},\:\:\:\:\mathrm{find}\:\:\:\Sigma\alpha^{\mathrm{4}} . \\ $$

Question Number 192177 Answers: 0 Comments: 0

Question Number 192181 Answers: 1 Comments: 0

Question Number 192179 Answers: 0 Comments: 3

Question Number 192171 Answers: 1 Comments: 2

2^x^x^x =2^(√2) x=?

$$\mathrm{2}^{{x}^{{x}^{{x}} } } =\mathrm{2}^{\sqrt{\mathrm{2}}} \\ $$$${x}=? \\ $$

Question Number 192173 Answers: 1 Comments: 0

prove that (x^2 +a^2 )^4 = (x^4 −6x^2 a^2 +a^4 )^2 +(4x^3 a−4xa^3 )^2

$${prove}\:{that} \\ $$$$\left({x}^{\mathrm{2}} +{a}^{\mathrm{2}} \right)^{\mathrm{4}} \:=\:\left({x}^{\mathrm{4}} −\mathrm{6}{x}^{\mathrm{2}} {a}^{\mathrm{2}} +{a}^{\mathrm{4}} \right)^{\mathrm{2}} +\left(\mathrm{4}{x}^{\mathrm{3}} {a}−\mathrm{4}{xa}^{\mathrm{3}} \right)^{\mathrm{2}} \\ $$

Question Number 192160 Answers: 1 Comments: 3

if x,y,z are three distinct complex numbers such that (x/(y−z))+(y/(z−x))+(z/(x−y)) = 0 then find the value of Σ (x^2 /((y−z)^2 ))

$$\mathrm{if}\:\mathrm{x},\mathrm{y},\mathrm{z}\:\mathrm{are}\:\mathrm{three}\:\mathrm{distinct}\:\mathrm{complex}\:\mathrm{numbers} \\ $$$$\mathrm{such}\:\mathrm{that}\:\frac{\mathrm{x}}{\mathrm{y}−{z}}+\frac{\mathrm{y}}{\mathrm{z}−\mathrm{x}}+\frac{\mathrm{z}}{\mathrm{x}−\mathrm{y}}\:=\:\mathrm{0}\:\mathrm{then}\: \\ $$$$\mathrm{find}\:\mathrm{the}\:\mathrm{value}\:\mathrm{of}\:\:\Sigma\:\frac{\mathrm{x}^{\mathrm{2}} }{\left(\mathrm{y}−\mathrm{z}\right)^{\mathrm{2}} } \\ $$

Question Number 192149 Answers: 5 Comments: 0

Find: (1/2) + (3/2^3 ) + (5/2^5 ) + (7/2^7 ) + ...

$$\mathrm{Find}: \\ $$$$\frac{\mathrm{1}}{\mathrm{2}}\:+\:\frac{\mathrm{3}}{\mathrm{2}^{\mathrm{3}} }\:+\:\frac{\mathrm{5}}{\mathrm{2}^{\mathrm{5}} }\:+\:\frac{\mathrm{7}}{\mathrm{2}^{\mathrm{7}} }\:+\:... \\ $$

Question Number 192143 Answers: 1 Comments: 2

Find: (7/2) + ((77)/(22)) + ((777)/(222)) + ((7777)/(2222)) +...+ ((77777777)/(22222222))

$$\mathrm{Find}: \\ $$$$\frac{\mathrm{7}}{\mathrm{2}}\:+\:\frac{\mathrm{77}}{\mathrm{22}}\:+\:\frac{\mathrm{777}}{\mathrm{222}}\:+\:\frac{\mathrm{7777}}{\mathrm{2222}}\:+...+\:\frac{\mathrm{77777777}}{\mathrm{22222222}} \\ $$

Question Number 192142 Answers: 1 Comments: 0

Question let x=<a_n a_(n−1) ...a_1 a_0 > ∈N ; a_0 ≠0 & y=<a_n a_(n−1) ...a_1 > ∈N be two natural numbers such that (x/y)∈N find the number “ x ” ?

$${Question} \\ $$$${let}\:\:\:{x}=<{a}_{{n}} {a}_{{n}−\mathrm{1}} ...{a}_{\mathrm{1}} {a}_{\mathrm{0}} >\:\in\mathbb{N}\:;\:{a}_{\mathrm{0}} \neq\mathrm{0}\:\:\&\: \\ $$$$\:{y}=<{a}_{{n}} {a}_{{n}−\mathrm{1}} ...{a}_{\mathrm{1}} >\:\in\mathbb{N}\:\:{be}\: \\ $$$${two}\:{natural}\:{numbers}\: \\ $$$${such}\:{that}\:\:\frac{{x}}{{y}}\in\mathbb{N}\: \\ $$$${find}\:{the}\:{number}\:``\:{x}\:''\:? \\ $$$$ \\ $$

Question Number 192138 Answers: 1 Comments: 0

show that f(x,y) = {_(0 (x,y)=(0,0)) ^(((x^2 y)/(x^6 + 2y^2 )) (x,y)≠ (0,0)) has a directional derivative in the direction of an arbitrary unit vector φ at (0,0), but f is not continous at (0,0)

$$\mathrm{show}\:\mathrm{that}\: \\ $$$$\mathrm{f}\left(\mathrm{x},\mathrm{y}\right)\:=\:\left\{_{\mathrm{0}\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\left(\mathrm{x},\mathrm{y}\right)=\left(\mathrm{0},\mathrm{0}\right)} ^{\frac{\mathrm{x}^{\mathrm{2}} \mathrm{y}}{\mathrm{x}^{\mathrm{6}} \:+\:\mathrm{2y}^{\mathrm{2}} }\:\:\:\:\:\:\:\:\:\:\:\:\left(\mathrm{x},\mathrm{y}\right)\neq\:\left(\mathrm{0},\mathrm{0}\right)} \right. \\ $$$$\mathrm{has}\:\mathrm{a}\:\mathrm{directional}\:\mathrm{derivative}\:\mathrm{in}\:\mathrm{the} \\ $$$$\mathrm{direction}\:\mathrm{of}\:\mathrm{an}\:\mathrm{arbitrary}\:\mathrm{unit}\:\mathrm{vector} \\ $$$$\phi\:\mathrm{at}\:\left(\mathrm{0},\mathrm{0}\right),\:\mathrm{but}\:\mathrm{f}\:\:\mathrm{is}\:\mathrm{not}\:\mathrm{continous}\:\mathrm{at}\:\left(\mathrm{0},\mathrm{0}\right)\: \\ $$

Question Number 192134 Answers: 1 Comments: 0

when ((√2)+1)^7 =(√(57125))+(√(57124)) then ((√2)−1)^7 =?

$$\mathrm{when}\:\left(\sqrt{\mathrm{2}}+\mathrm{1}\right)^{\mathrm{7}} =\sqrt{\mathrm{57125}}+\sqrt{\mathrm{57124}} \\ $$$$\mathrm{then}\:\:\:\left(\sqrt{\mathrm{2}}−\mathrm{1}\right)^{\mathrm{7}} =? \\ $$

Question Number 192132 Answers: 1 Comments: 0

What is the nearest point in f(x) to (5,2) where f(x)=−0.5x^2 +3

$$\mathrm{What}\:\mathrm{is}\:\mathrm{the}\:\mathrm{nearest}\:\mathrm{point}\:\mathrm{in}\:{f}\left({x}\right)\:\mathrm{to}\:\left(\mathrm{5},\mathrm{2}\right) \\ $$$$\mathrm{where}\:{f}\left({x}\right)=−\mathrm{0}.\mathrm{5}{x}^{\mathrm{2}} +\mathrm{3} \\ $$

Question Number 192129 Answers: 2 Comments: 0

prove that ∣a+(√(a^2 −b^2 ))∣ + ∣a − (√(a^2 −b^2 ))∣ = ∣a+b∣ +∣a−b∣ a,b ∈ C

$$\:{prove}\:{that} \\ $$$$\:\mid{a}+\sqrt{{a}^{\mathrm{2}} −{b}^{\mathrm{2}} }\mid\:+\:\mid{a}\:−\:\sqrt{{a}^{\mathrm{2}} −{b}^{\mathrm{2}} }\mid\:=\:\mid{a}+{b}\mid\:+\mid{a}−{b}\mid \\ $$$${a},{b}\:\in\:\mathbb{C} \\ $$

Question Number 192115 Answers: 1 Comments: 0

Ω = lim_( x→0) ( (( cot^( −1) ((1/x) ))/( x)) )^(1/x^( 2) ) = ?

$$ \\ $$$$\:\Omega\:=\:\mathrm{lim}_{\:{x}\rightarrow\mathrm{0}} \:\left(\:\:\frac{\:\mathrm{cot}^{\:−\mathrm{1}} \:\left(\frac{\mathrm{1}}{{x}}\:\right)}{\:{x}}\:\right)^{\frac{\mathrm{1}}{{x}^{\:\mathrm{2}} }} =\:?\:\:\:\:\:\: \\ $$$$\:\: \\ $$$$ \\ $$

Question Number 192112 Answers: 2 Comments: 2

prove that. 0!=1

$${prove}\:{that}.\:\:\:\:\:\:\:\mathrm{0}!=\mathrm{1} \\ $$

Question Number 192172 Answers: 1 Comments: 0

Q1 ∴ x=<1a_1 a_2 ...a_n >∈N & y=<a_1 a_2 ...a_n 1>∈N if y=3x then , find the smallest value of x Q2 ∴ with the above conditions ,what other values can be placed besides the number “ 1 ”

$${Q}\mathrm{1}\:\therefore\:\:{x}=<\mathrm{1}{a}_{\mathrm{1}} {a}_{\mathrm{2}} ...{a}_{{n}} >\in\mathbb{N}\:\:\&\:\:{y}=<{a}_{\mathrm{1}} {a}_{\mathrm{2}} ...{a}_{{n}} \mathrm{1}>\in\mathbb{N} \\ $$$${if}\:\:{y}=\mathrm{3}{x}\:\:{then}\:\:,\:{find}\:{the}\:{smallest}\: \\ $$$${value}\:{of}\:\:{x} \\ $$$${Q}\mathrm{2}\:\therefore\:{with}\:{the}\:{above}\:{conditions}\:,{what}\:{other}\:{values}\: \\ $$$${can}\:{be}\:{placed}\:\:{besides}\:{the}\:{number}\:``\:\mathrm{1}\:''\: \\ $$

Question Number 192105 Answers: 3 Comments: 0

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