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Question Number 23105 Answers: 0 Comments: 4

(∣m^2 −n^2 ∣,2mn,m^2 +n^2 ) is pythagorean triplet for all m,n∈N. This can be proved easily.Is the vice versa of the statement is also true? I-E If for a,b,c∈N ,a^2 +b^2 =c^2 then there exist m,n∈N such that m^2 +n^2 =c and {a,b}={∣m^2 −n^2 ∣,2mn}

$$\left(\mid\mathrm{m}^{\mathrm{2}} −\mathrm{n}^{\mathrm{2}} \mid,\mathrm{2mn},\mathrm{m}^{\mathrm{2}} +\mathrm{n}^{\mathrm{2}} \right)\:\mathrm{is}\:\mathrm{pythagorean} \\ $$$$\mathrm{triplet}\:\mathrm{for}\:\mathrm{all}\:\mathrm{m},\mathrm{n}\in\mathbb{N}.\:\mathrm{This}\:\mathrm{can}\:\mathrm{be} \\ $$$$\mathrm{proved}\:\mathrm{easily}.\mathrm{Is}\:\mathrm{the}\:\mathrm{vice}\:\mathrm{versa}\:\mathrm{of} \\ $$$$\mathrm{the}\:\mathrm{statement}\:\mathrm{is}\:\mathrm{also}\:\mathrm{true}? \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\mathrm{I}-\mathrm{E} \\ $$$$\mathrm{If}\:\:\mathrm{for}\:\mathrm{a},\mathrm{b},\mathrm{c}\in\mathbb{N}\:,\mathrm{a}^{\mathrm{2}} +\mathrm{b}^{\mathrm{2}} =\mathrm{c}^{\mathrm{2}} \:\mathrm{then}\:\mathrm{there}\: \\ $$$$\mathrm{exist}\:\mathrm{m},\mathrm{n}\in\mathbb{N}\:\mathrm{such}\:\mathrm{that}\:\mathrm{m}^{\mathrm{2}} +\mathrm{n}^{\mathrm{2}} =\mathrm{c}\:\mathrm{and} \\ $$$$\left\{\mathrm{a},\mathrm{b}\right\}=\left\{\mid\mathrm{m}^{\mathrm{2}} −\mathrm{n}^{\mathrm{2}} \mid,\mathrm{2mn}\right\} \\ $$

Question Number 23104 Answers: 1 Comments: 0

show tht the curve with parametric equcations x=t^2 ,y=t^3 −9t intersect at the point (9,0).

$${show}\:{tht}\:{the}\:{curve}\:{with}\:{parametric}\:{equcations} \\ $$$${x}={t}^{\mathrm{2}} \:,{y}={t}^{\mathrm{3}} \:−\mathrm{9}{t}\:{intersect}\:{at}\:{the}\:{point}\:\left(\mathrm{9},\mathrm{0}\right). \\ $$

Question Number 23103 Answers: 0 Comments: 1

Can anyone please solve question number 23057

$${Can}\:{anyone}\:{please}\:{solve}\:{question} \\ $$$${number}\:\mathrm{23057} \\ $$

Question Number 23096 Answers: 0 Comments: 1

Solve: 3^(x ) = ((27)/x) + 18

$$\mathrm{Solve}:\:\:\mathrm{3}^{\mathrm{x}\:} =\:\frac{\mathrm{27}}{\mathrm{x}}\:+\:\mathrm{18} \\ $$

Question Number 23073 Answers: 1 Comments: 1

Question Number 23069 Answers: 0 Comments: 0

Why ionic radii of^(35) Cl <^(37) Cl^− ?

$$\mathrm{Why}\:\mathrm{ionic}\:\mathrm{radii}\:\mathrm{of}\:^{\mathrm{35}} \mathrm{Cl}\:<\:^{\mathrm{37}} \mathrm{Cl}^{−} ? \\ $$

Question Number 23066 Answers: 1 Comments: 7

A rocket accelerates straight up by ejecting gas downwards. In a small time interval Δt, it ejects a gas of mass Δm at a relative speed u. Calculate KE of the entire system at t + Δt and t and show that the device that ejects gas does work = ((1/2))Δmu^2 in this time interval (neglect gravity).

$$\mathrm{A}\:\mathrm{rocket}\:\mathrm{accelerates}\:\mathrm{straight}\:\mathrm{up}\:\mathrm{by} \\ $$$$\mathrm{ejecting}\:\mathrm{gas}\:\mathrm{downwards}.\:\mathrm{In}\:\mathrm{a}\:\mathrm{small} \\ $$$$\mathrm{time}\:\mathrm{interval}\:\Delta{t},\:\mathrm{it}\:\mathrm{ejects}\:\mathrm{a}\:\mathrm{gas}\:\mathrm{of}\:\mathrm{mass} \\ $$$$\Delta{m}\:\mathrm{at}\:\mathrm{a}\:\mathrm{relative}\:\mathrm{speed}\:{u}.\:\mathrm{Calculate}\:\mathrm{KE} \\ $$$$\mathrm{of}\:\mathrm{the}\:\mathrm{entire}\:\mathrm{system}\:\mathrm{at}\:{t}\:+\:\Delta{t}\:\mathrm{and}\:{t}\:\mathrm{and} \\ $$$$\mathrm{show}\:\mathrm{that}\:\mathrm{the}\:\mathrm{device}\:\mathrm{that}\:\mathrm{ejects}\:\mathrm{gas} \\ $$$$\mathrm{does}\:\mathrm{work}\:=\:\left(\frac{\mathrm{1}}{\mathrm{2}}\right)\Delta{mu}^{\mathrm{2}} \:\mathrm{in}\:\mathrm{this}\:\mathrm{time} \\ $$$$\mathrm{interval}\:\left(\mathrm{neglect}\:\mathrm{gravity}\right). \\ $$

Question Number 23061 Answers: 0 Comments: 0

∫_0 ^∞ j_4 (x)dx=?

$$\int_{\mathrm{0}} ^{\infty} \mathrm{j}_{\mathrm{4}} \left(\mathrm{x}\right)\mathrm{dx}=? \\ $$

Question Number 23059 Answers: 0 Comments: 3

Question Number 23057 Answers: 0 Comments: 1

Question Number 23050 Answers: 1 Comments: 0

how can demonstred that ∀a,b,c∈N a^2 +b^2 =c^2 ⇒ abc≡0[60]

$${how}\:{can}\:{demonstred}\:{that}\: \\ $$$$\:\:\:\forall{a},{b},{c}\in\mathbb{N}\: \\ $$$${a}^{\mathrm{2}} +{b}^{\mathrm{2}} ={c}^{\mathrm{2}} \:\:\Rightarrow \\ $$$${abc}\equiv\mathrm{0}\left[\mathrm{60}\right]\:\: \\ $$

Question Number 23049 Answers: 1 Comments: 0

A particle of mass m strikes on ground with angle of incidence 45°. If coefficient of restitution, e = (1/(√2)) , find the velocity after impact and angle of reflection.

$$\mathrm{A}\:\mathrm{particle}\:\mathrm{of}\:\mathrm{mass}\:{m}\:\mathrm{strikes}\:\mathrm{on}\:\mathrm{ground} \\ $$$$\mathrm{with}\:\mathrm{angle}\:\mathrm{of}\:\mathrm{incidence}\:\mathrm{45}°.\:\mathrm{If}\:\mathrm{coefficient} \\ $$$$\mathrm{of}\:\mathrm{restitution},\:{e}\:=\:\frac{\mathrm{1}}{\sqrt{\mathrm{2}}}\:,\:\mathrm{find}\:\mathrm{the}\:\mathrm{velocity} \\ $$$$\mathrm{after}\:\mathrm{impact}\:\mathrm{and}\:\mathrm{angle}\:\mathrm{of}\:\mathrm{reflection}. \\ $$

Question Number 23036 Answers: 1 Comments: 16

Question Number 23034 Answers: 0 Comments: 1

Question Number 23022 Answers: 1 Comments: 5

what are the conditions necessary for a body of mass mkg on an incline plane at angle θ to the horizontal to remain at rest?

$${what}\:{are}\:{the}\:{conditions}\:{necessary} \\ $$$${for}\:{a}\:{body}\:{of}\:{mass}\:{mkg}\:{on}\:{an}\:{incline}\:{plane} \\ $$$${at}\:{angle}\:\theta\:{to}\:{the}\:{horizontal}\:{to} \\ $$$${remain}\:{at}\:{rest}? \\ $$

Question Number 23020 Answers: 1 Comments: 3

∫_1 ^3 ((4x+1)/(2x^2 +x−2))dx=?

$$\underset{\mathrm{1}} {\overset{\mathrm{3}} {\int}}\frac{\mathrm{4}\boldsymbol{\mathrm{x}}+\mathrm{1}}{\mathrm{2}\boldsymbol{\mathrm{x}}^{\mathrm{2}} +\boldsymbol{\mathrm{x}}−\mathrm{2}}\boldsymbol{\mathrm{dx}}=? \\ $$

Question Number 23012 Answers: 2 Comments: 2

Question Number 23009 Answers: 0 Comments: 3

Question Number 23000 Answers: 0 Comments: 0

Let n be a positive integer and p_1 , p_2 , ..., p_n be n prime numbers all larger than 5 such that 6 divides p_1 ^2 + p_2 ^2 + ... + p_n ^2 . Prove that 6 divides n.

$$\mathrm{Let}\:{n}\:\mathrm{be}\:\mathrm{a}\:\mathrm{positive}\:\mathrm{integer}\:\mathrm{and}\:{p}_{\mathrm{1}} ,\:{p}_{\mathrm{2}} , \\ $$$$...,\:{p}_{{n}} \:\mathrm{be}\:{n}\:\mathrm{prime}\:\mathrm{numbers}\:\mathrm{all}\:\mathrm{larger} \\ $$$$\mathrm{than}\:\mathrm{5}\:\mathrm{such}\:\mathrm{that}\:\mathrm{6}\:\mathrm{divides}\:{p}_{\mathrm{1}} ^{\mathrm{2}} \:+\:{p}_{\mathrm{2}} ^{\mathrm{2}} \:+\:...\:+ \\ $$$${p}_{{n}} ^{\mathrm{2}} .\:\mathrm{Prove}\:\mathrm{that}\:\mathrm{6}\:\mathrm{divides}\:{n}. \\ $$

Question Number 22999 Answers: 0 Comments: 0

ΔH_f ^o of N_2 O is 82 kJ/mol and bond energy for N_2 , N = N, O = O and N = O bonds are 946, 418, 498 and 607 kJ/mol respectively, then resonance energy of N_2 O is

$$\Delta\mathrm{H}_{\mathrm{f}} ^{\mathrm{o}} \:\mathrm{of}\:\mathrm{N}_{\mathrm{2}} \mathrm{O}\:\mathrm{is}\:\mathrm{82}\:\mathrm{kJ}/\mathrm{mol}\:\mathrm{and}\:\mathrm{bond} \\ $$$$\mathrm{energy}\:\mathrm{for}\:\mathrm{N}_{\mathrm{2}} ,\:\mathrm{N}\:=\:\mathrm{N},\:\mathrm{O}\:=\:\mathrm{O}\:\mathrm{and}\:\mathrm{N}\:=\:\mathrm{O} \\ $$$$\mathrm{bonds}\:\mathrm{are}\:\mathrm{946},\:\mathrm{418},\:\mathrm{498}\:\mathrm{and}\:\mathrm{607}\:\mathrm{kJ}/\mathrm{mol} \\ $$$$\mathrm{respectively},\:\mathrm{then}\:\mathrm{resonance}\:\mathrm{energy}\:\mathrm{of} \\ $$$$\mathrm{N}_{\mathrm{2}} \mathrm{O}\:\mathrm{is} \\ $$

Question Number 22990 Answers: 1 Comments: 0

A train weighing 100 metric ton is running on a level track with a uniform speed of 72 km h^(−1) . If the frictional resistance amounts to 0.5 kg per metric ton, find the power of the engine.

$$\mathrm{A}\:\mathrm{train}\:\mathrm{weighing}\:\mathrm{100}\:\mathrm{metric}\:\mathrm{ton}\:\mathrm{is} \\ $$$$\mathrm{running}\:\mathrm{on}\:\mathrm{a}\:\mathrm{level}\:\mathrm{track}\:\mathrm{with}\:\mathrm{a}\:\mathrm{uniform} \\ $$$$\mathrm{speed}\:\mathrm{of}\:\mathrm{72}\:\mathrm{km}\:\mathrm{h}^{−\mathrm{1}} .\:\mathrm{If}\:\mathrm{the}\:\mathrm{frictional} \\ $$$$\mathrm{resistance}\:\mathrm{amounts}\:\mathrm{to}\:\mathrm{0}.\mathrm{5}\:\mathrm{kg}\:\mathrm{per}\:\mathrm{metric} \\ $$$$\mathrm{ton},\:\mathrm{find}\:\mathrm{the}\:\mathrm{power}\:\mathrm{of}\:\mathrm{the}\:\mathrm{engine}. \\ $$

Question Number 22975 Answers: 0 Comments: 5

Question Number 22972 Answers: 0 Comments: 0

Can anyone please solve question number 22956

$${Can}\:{anyone}\:{please}\:{solve}\:{question} \\ $$$${number}\:\mathrm{22956} \\ $$

Question Number 22993 Answers: 2 Comments: 0

For which of the following, the hydration energy of Mg^(2+) is larger? (1) Na^+ (2) Al^(3+) (3) Be^(2+) (4) Cr^(3+) .

$$\mathrm{For}\:\mathrm{which}\:\mathrm{of}\:\mathrm{the}\:\mathrm{following},\:\mathrm{the} \\ $$$$\mathrm{hydration}\:\mathrm{energy}\:\mathrm{of}\:\mathrm{Mg}^{\mathrm{2}+} \:\mathrm{is}\:\mathrm{larger}? \\ $$$$\left(\mathrm{1}\right)\:\mathrm{Na}^{+} \\ $$$$\left(\mathrm{2}\right)\:\mathrm{Al}^{\mathrm{3}+} \\ $$$$\left(\mathrm{3}\right)\:\mathrm{Be}^{\mathrm{2}+} \\ $$$$\left(\mathrm{4}\right)\:\mathrm{Cr}^{\mathrm{3}+} . \\ $$

Question Number 22968 Answers: 0 Comments: 2

Question Number 22959 Answers: 0 Comments: 0

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