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Question Number 205734    Answers: 1   Comments: 0

Question Number 205733    Answers: 0   Comments: 0

If a,b,c∈R^+ and a+b+c=6 Prove that: ((a^2 −4)/(4a^2 −9a + 6)) + ((b^2 −4)/(4b^2 −9b + 6)) + ((c^2 −4)/(4c^2 −9c + 6)) ≤ 0

$$\mathrm{If}\:\:\mathrm{a},\mathrm{b},\mathrm{c}\in\mathbb{R}^{+} \:\:\mathrm{and}\:\:\mathrm{a}+\mathrm{b}+\mathrm{c}=\mathrm{6} \\ $$$$\mathrm{Prove}\:\mathrm{that}: \\ $$$$\frac{\mathrm{a}^{\mathrm{2}} −\mathrm{4}}{\mathrm{4a}^{\mathrm{2}} −\mathrm{9a}\:+\:\mathrm{6}}\:+\:\frac{\mathrm{b}^{\mathrm{2}} −\mathrm{4}}{\mathrm{4b}^{\mathrm{2}} −\mathrm{9b}\:+\:\mathrm{6}}\:+\:\frac{\mathrm{c}^{\mathrm{2}} −\mathrm{4}}{\mathrm{4c}^{\mathrm{2}} −\mathrm{9c}\:+\:\mathrm{6}}\:\leqslant\:\mathrm{0} \\ $$

Question Number 205727    Answers: 1   Comments: 3

calculate lim_(x→0) ((e^x −cosx)/x^2 )

$${calculate}\:{lim}_{{x}\rightarrow\mathrm{0}} \frac{{e}^{{x}} −{cosx}}{{x}^{\mathrm{2}} } \\ $$

Question Number 205716    Answers: 2   Comments: 0

lim_(n→∞) (((2n+1)(2n+3)...(4n+1))/((2n)(2n+2)...(4n))) = ?

$$\underset{{n}\rightarrow\infty} {\mathrm{lim}}\:\frac{\left(\mathrm{2}{n}+\mathrm{1}\right)\left(\mathrm{2}{n}+\mathrm{3}\right)...\left(\mathrm{4}{n}+\mathrm{1}\right)}{\left(\mathrm{2}{n}\right)\left(\mathrm{2}{n}+\mathrm{2}\right)...\left(\mathrm{4}{n}\right)}\:\:=\:\:? \\ $$

Question Number 205726    Answers: 0   Comments: 0

101 is chosen arbitrarily from the numbers 1,2,3,...,199,200. Prove that two of these selected numbers can be found such that one is divisible by the other.

$$ \\ $$101 is chosen arbitrarily from the numbers 1,2,3,...,199,200. Prove that two of these selected numbers can be found such that one is divisible by the other.

Question Number 205685    Answers: 2   Comments: 0

Question Number 205684    Answers: 0   Comments: 0

Figure Shows that Object A is connected to Object B by thread along the Slope it shows a costant acceleration motion the mass of A of B are 3m , 2m respectively and when A communicates from point P to Q B′s the decrease in potential energy is 10 times the decrease in Kinetic energy of B find accerate of A (3 point)

$$\mathrm{Figure}\:\mathrm{Shows}\:\mathrm{that}\:\mathrm{Object}\:\boldsymbol{\mathrm{A}}\:\mathrm{is}\:\mathrm{connected}\:\mathrm{to} \\ $$$$\mathrm{Object}\:\boldsymbol{\mathrm{B}}\:\mathrm{by}\:\mathrm{thread}\:\mathrm{along}\:\mathrm{the}\:\mathrm{Slope} \\ $$$$\mathrm{it}\:\mathrm{shows}\:\mathrm{a}\:\mathrm{costant}\:\mathrm{acceleration}\:\mathrm{motion} \\ $$$$\mathrm{the}\:\mathrm{mass}\:\mathrm{of}\:\boldsymbol{\mathrm{A}}\:\mathrm{of}\:\boldsymbol{\mathrm{B}}\:\mathrm{are}\:\mathrm{3m}\:,\:\mathrm{2m} \\ $$$$\mathrm{respectively}\:\mathrm{and}\:\mathrm{when}\:\boldsymbol{\mathrm{A}}\:\mathrm{communicates}\:\mathrm{from}\:\mathrm{point}\:\boldsymbol{\mathrm{P}}\:\mathrm{to}\:\boldsymbol{\mathrm{Q}}\: \\ $$$$\boldsymbol{\mathrm{B}}'{s}\:\mathrm{the}\:\mathrm{decrease}\:\mathrm{in}\:\mathrm{potential}\:\mathrm{energy}\:\mathrm{is}\:\mathrm{10}\:\:\mathrm{times} \\ $$$$\mathrm{the}\:\mathrm{decrease}\:\mathrm{in}\:\mathrm{Kinetic}\:\mathrm{energy}\:\mathrm{of}\:\boldsymbol{\mathrm{B}}\: \\ $$$$\mathrm{find}\:\mathrm{accerate}\:\mathrm{of}\:\boldsymbol{\mathrm{A}}\:\left(\mathrm{3}\:\mathrm{point}\right) \\ $$

Question Number 205683    Answers: 1   Comments: 0

$$\:\:\:\:\: \\ $$

Question Number 205682    Answers: 1   Comments: 2

Question Number 205681    Answers: 2   Comments: 0

Question Number 205680    Answers: 1   Comments: 0

solve ⌊x ⌋ + ⌊ x^2 ⌋ = ⌊ x^3 ⌋

$$ \\ $$$$\:\:\:\:\:\:\:\:\:{solve}\: \\ $$$$ \\ $$$$\:\:\:\:\:\:\:\lfloor{x}\:\rfloor\:+\:\lfloor\:{x}^{\mathrm{2}} \rfloor\:=\:\lfloor\:{x}^{\mathrm{3}} \:\rfloor \\ $$$$ \\ $$

Question Number 205690    Answers: 0   Comments: 3

Question Number 205673    Answers: 0   Comments: 0

Question Number 205672    Answers: 1   Comments: 0

Question Number 205671    Answers: 1   Comments: 0

Question Number 205670    Answers: 1   Comments: 0

Question Number 205669    Answers: 1   Comments: 0

Question Number 205660    Answers: 1   Comments: 0

Question Number 205657    Answers: 1   Comments: 0

Question Number 205656    Answers: 0   Comments: 0

Question Number 205725    Answers: 0   Comments: 0

Question Number 205645    Answers: 1   Comments: 0

Find: Ω = ∫_0 ^( (𝛑/2)) ((sin^2 x)/(2 cosx + 3 sinx)) dx = ?

$$\mathrm{Find}:\:\:\Omega\:=\:\int_{\mathrm{0}} ^{\:\frac{\boldsymbol{\pi}}{\mathrm{2}}} \:\frac{\mathrm{sin}^{\mathrm{2}} \mathrm{x}}{\mathrm{2}\:\mathrm{cosx}\:+\:\mathrm{3}\:\mathrm{sinx}}\:\mathrm{dx}\:=\:? \\ $$

Question Number 205643    Answers: 0   Comments: 0

If a,b,c>0 and a^2 + b^2 + c^2 = abc Prove that: (a/(a^2 + bc)) + (b/(b^2 + ac)) + (c/(c^2 + ab)) ≤ (1/2)

$$\mathrm{If}\:\:\mathrm{a},\mathrm{b},\mathrm{c}>\mathrm{0}\:\:\mathrm{and}\:\:\mathrm{a}^{\mathrm{2}} \:+\:\mathrm{b}^{\mathrm{2}} \:+\:\mathrm{c}^{\mathrm{2}} \:=\:\mathrm{abc} \\ $$$$\mathrm{Prove}\:\mathrm{that}: \\ $$$$\frac{\mathrm{a}}{\mathrm{a}^{\mathrm{2}} \:+\:\mathrm{bc}}\:+\:\frac{\mathrm{b}}{\mathrm{b}^{\mathrm{2}} \:+\:\mathrm{ac}}\:+\:\frac{\mathrm{c}}{\mathrm{c}^{\mathrm{2}} \:+\:\mathrm{ab}}\:\leqslant\:\frac{\mathrm{1}}{\mathrm{2}} \\ $$

Question Number 205640    Answers: 0   Comments: 0

If a,b,c>0 and abc≥1 Prove that: a + b + c ≥ ((1+a)/(1+b)) + ((1+b)/(1+c)) + ((1+c)/(1+a))

$$\mathrm{If}\:\:\mathrm{a},\mathrm{b},\mathrm{c}>\mathrm{0}\:\:\mathrm{and}\:\:\mathrm{abc}\geqslant\mathrm{1} \\ $$$$\mathrm{Prove}\:\mathrm{that}: \\ $$$$\mathrm{a}\:+\:\mathrm{b}\:+\:\mathrm{c}\:\geqslant\:\frac{\mathrm{1}+\mathrm{a}}{\mathrm{1}+\mathrm{b}}\:+\:\frac{\mathrm{1}+\mathrm{b}}{\mathrm{1}+\mathrm{c}}\:+\:\frac{\mathrm{1}+\mathrm{c}}{\mathrm{1}+\mathrm{a}} \\ $$

Question Number 205639    Answers: 2   Comments: 0

Question Number 205631    Answers: 1   Comments: 0

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