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AllQuestion and Answers: Page 610

Question Number 155630    Answers: 1   Comments: 0

Question Number 155628    Answers: 1   Comments: 0

Question Number 155625    Answers: 2   Comments: 0

lim_(x→0) (((tanx)/x))^(1/x^2 ) = ?

$$\underset{{x}\rightarrow\mathrm{0}} {\mathrm{lim}}\:\left(\frac{{tanx}}{{x}}\right)^{\frac{\mathrm{1}}{{x}^{\mathrm{2}} }} =\:? \\ $$

Question Number 155621    Answers: 1   Comments: 0

Question Number 155620    Answers: 1   Comments: 0

Find: 𝛀 = lim_(xβ†’1) ((∫_(x-1) ^(e^x -e) cos(t^5 )dt)/(3^x - 3)) = ?

$$\mathrm{Find}:\:\:\:\boldsymbol{\Omega}\:=\:\underset{\boldsymbol{\mathrm{x}}\rightarrow\mathrm{1}} {\mathrm{lim}}\frac{\underset{\boldsymbol{\mathrm{x}}-\mathrm{1}} {\overset{\boldsymbol{\mathrm{e}}^{\boldsymbol{\mathrm{x}}} -\boldsymbol{\mathrm{e}}} {\int}}\mathrm{cos}\left(\mathrm{t}^{\mathrm{5}} \right)\mathrm{dt}}{\mathrm{3}^{\boldsymbol{\mathrm{x}}} \:-\:\mathrm{3}}\:=\:? \\ $$

Question Number 155619    Answers: 1   Comments: 0

𝛀 =∫_( 0) ^( 1) (x^(49) /(1 + x + x^2 + x^3 ... x^(100) )) dx = ?

$$\boldsymbol{\Omega}\:=\underset{\:\mathrm{0}} {\overset{\:\mathrm{1}} {\int}}\:\frac{\mathrm{x}^{\mathrm{49}} }{\mathrm{1}\:+\:\mathrm{x}\:+\:\mathrm{x}^{\mathrm{2}} \:+\:\mathrm{x}^{\mathrm{3}} \:...\:\mathrm{x}^{\mathrm{100}} }\:\mathrm{dx}\:=\:? \\ $$

Question Number 155618    Answers: 0   Comments: 0

if x;y∈R and x^3 +y^3 =16 prove that: x^4 + y^4 + 2x^2 + y^2 β‰₯ 4x + 36

$$\mathrm{if}\:\:\mathrm{x};\mathrm{y}\in\mathbb{R}\:\:\mathrm{and}\:\:\mathrm{x}^{\mathrm{3}} +\mathrm{y}^{\mathrm{3}} =\mathrm{16}\:\:\mathrm{prove}\:\mathrm{that}: \\ $$$$\mathrm{x}^{\mathrm{4}} \:+\:\mathrm{y}^{\mathrm{4}} \:+\:\mathrm{2x}^{\mathrm{2}} \:+\:\mathrm{y}^{\mathrm{2}} \:\geqslant\:\mathrm{4x}\:+\:\mathrm{36} \\ $$

Question Number 155604    Answers: 1   Comments: 0

Question Number 155594    Answers: 1   Comments: 0

Question Number 155586    Answers: 1   Comments: 0

How to proof f:Xβ†’Y f is 1 to 1 ⇐⇒ f(E)\f(F)=f(E\F)

$${How}\:{to}\:{proof}\:\:\:\:\:{f}:{X}\rightarrow{Y} \\ $$$${f}\:{is}\:\mathrm{1}\:{to}\:\mathrm{1}\:\Leftarrow\Rightarrow\:{f}\left({E}\right)\backslash{f}\left({F}\right)={f}\left({E}\backslash{F}\right) \\ $$

Question Number 155585    Answers: 0   Comments: 0

Evaluate the limit and prove by the Ξ΅βˆ’Ξ΄ definition that as nβ†’βˆž for zβ‰₯1 (2(z)^(1/n) βˆ’ 1)^n β†’ z^2

$$\mathrm{Evaluate}\:\mathrm{the}\:\mathrm{limit}\:\mathrm{and}\:\mathrm{prove}\:\mathrm{by}\:\mathrm{the} \\ $$$$\varepsilonβˆ’\delta\:\mathrm{definition}\:\mathrm{that}\:\mathrm{as}\:\mathrm{n}\rightarrow\infty\:\mathrm{for}\:\mathrm{z}\geqslant\mathrm{1} \\ $$$$\left(\mathrm{2}\sqrt[{\boldsymbol{\mathrm{n}}}]{\mathrm{z}}\:βˆ’\:\mathrm{1}\right)^{\boldsymbol{\mathrm{n}}} \:\rightarrow\:\mathrm{z}^{\mathrm{2}} \\ $$

Question Number 155583    Answers: 2   Comments: 0

Question Number 155576    Answers: 7   Comments: 3

the polynomial 4x^3 +ax^2 +bx+9, where a and b consant, is denoted by f(x). when f(x) is divide by (xβˆ’2) the remainder is r and when divided by (xβˆ’3) the remaider is 6r. its further given that (x+3) is a factor of f(x). Show that bβˆ’a=14 and hence find a and b

$${the}\:{polynomial}\:\mathrm{4}{x}^{\mathrm{3}} +{ax}^{\mathrm{2}} +{bx}+\mathrm{9}, \\ $$$${where}\:\boldsymbol{{a}}\:\:{and}\:\:\boldsymbol{{b}}\:{consant},\:{is}\:{denoted}\:{by} \\ $$$$\:{f}\left({x}\right).\:{when}\:{f}\left({x}\right)\:{is}\:{divide}\:{by}\:\left({x}βˆ’\mathrm{2}\right)\:{the} \\ $$$${remainder}\:{is}\:\boldsymbol{{r}}\:{and}\:{when}\:{divided}\:{by} \\ $$$$\:\left({x}βˆ’\mathrm{3}\right)\:{the}\:{remaider}\:{is}\:\mathrm{6}\boldsymbol{{r}}.\:{its}\:{further} \\ $$$$\:{given}\:{that}\:\left({x}+\mathrm{3}\right)\:{is}\:{a}\:{factor}\:{of}\:{f}\left({x}\right). \\ $$$$\:{Show}\:{that}\:\boldsymbol{{b}}βˆ’\boldsymbol{{a}}=\mathrm{14} \\ $$$$\:{and}\:{hence}\:{find}\:\boldsymbol{{a}}\:{and}\:\boldsymbol{{b}} \\ $$

Question Number 155574    Answers: 1   Comments: 0

The mean and standard deviation of 20 observation are found to be 10 and 2 respectively .On rechecking it was found that an observation 8 was incorrect.Calculate the incorrect mean and standard deviation (a)If the wrong iterm was ommited (b) If it is replaced by 12

$$\mathrm{The}\:\mathrm{mean}\:\mathrm{and}\:\mathrm{standard}\:\mathrm{deviation} \\ $$$$\mathrm{of}\:\mathrm{20}\:\mathrm{observation}\:\:\mathrm{are}\:\mathrm{found}\:\mathrm{to}\:\mathrm{be}\: \\ $$$$\mathrm{10}\:\mathrm{and}\:\mathrm{2}\:\mathrm{respectively}\:.\mathrm{On}\:\mathrm{rechecking} \\ $$$$\mathrm{it}\:\mathrm{was}\:\mathrm{found}\:\mathrm{that}\:\:\mathrm{an}\:\mathrm{observation} \\ $$$$\mathrm{8}\:\mathrm{was}\:\mathrm{incorrect}.\mathrm{Calculate}\:\mathrm{the}\:\mathrm{incorrect} \\ $$$$\mathrm{mean}\:\mathrm{and}\:\mathrm{standard}\:\mathrm{deviation} \\ $$$$\left(\boldsymbol{\mathrm{a}}\right)\boldsymbol{\mathrm{I}}\mathrm{f}\:\boldsymbol{\mathrm{the}}\:\boldsymbol{\mathrm{wrong}}\:\boldsymbol{\mathrm{iterm}}\:\boldsymbol{\mathrm{was}}\:\:\boldsymbol{\mathrm{ommited}} \\ $$$$\left(\boldsymbol{\mathrm{b}}\right)\:\boldsymbol{\mathrm{If}}\:\boldsymbol{\mathrm{it}}\:\boldsymbol{\mathrm{is}}\:\boldsymbol{\mathrm{replaced}}\:\boldsymbol{\mathrm{by}}\:\mathrm{12} \\ $$

Question Number 155573    Answers: 0   Comments: 1

Question Number 155572    Answers: 2   Comments: 0

Find Z^4 =1. Hence show that 1+w+w^2 +w^3 =0

$$\mathrm{Find}\:\mathrm{Z}^{\mathrm{4}} =\mathrm{1}. \\ $$$$\mathrm{Hence}\:\mathrm{show}\:\mathrm{that}\:\mathrm{1}+\mathrm{w}+\mathrm{w}^{\mathrm{2}} +\mathrm{w}^{\mathrm{3}} =\mathrm{0} \\ $$

Question Number 155571    Answers: 2   Comments: 0

Find the cube root of one .Hence show that the sum of the root is equal to zero

$$\mathrm{Find}\:\mathrm{the}\:\mathrm{cube}\:\mathrm{root}\:\mathrm{of}\:\mathrm{one}\:.\mathrm{Hence} \\ $$$$\mathrm{show}\:\mathrm{that}\:\mathrm{the}\:\mathrm{sum}\:\mathrm{of}\:\mathrm{the}\:\mathrm{root}\:\mathrm{is}\: \\ $$$$\mathrm{equal}\:\mathrm{to}\:\mathrm{zero} \\ $$

Question Number 155568    Answers: 2   Comments: 0

for a,b,c,d,e ∈R and a+b+c+d+e=5 find the minimum value of a^2 +2b^2 +3c^2 +4d^2 +5e^2 =?

$${for}\:{a},{b},{c},{d},{e}\:\in{R}\:{and}\:{a}+{b}+{c}+{d}+{e}=\mathrm{5} \\ $$$${find}\:{the}\:{minimum}\:{value}\:{of}\: \\ $$$${a}^{\mathrm{2}} +\mathrm{2}{b}^{\mathrm{2}} +\mathrm{3}{c}^{\mathrm{2}} +\mathrm{4}{d}^{\mathrm{2}} +\mathrm{5}{e}^{\mathrm{2}} =? \\ $$

Question Number 155562    Answers: 1   Comments: 2

Question Number 155561    Answers: 1   Comments: 0

Question Number 155560    Answers: 0   Comments: 0

Question Number 155553    Answers: 0   Comments: 0

∫_(βˆ’βˆž) ^( ∞) ((sin(x^2 )cos(x^3 ))/((ln((sin(x)cos(x))^2 ))^2 +1)) dx

$$\: \\ $$$$\int_{βˆ’\infty} ^{\:\infty} \:\frac{\mathrm{sin}\left({x}^{\mathrm{2}} \right)\mathrm{cos}\left({x}^{\mathrm{3}} \right)}{\left(\mathrm{ln}\left(\left(\mathrm{sin}\left({x}\right)\mathrm{cos}\left({x}\right)\right)^{\mathrm{2}} \right)\right)^{\mathrm{2}} +\mathrm{1}}\:\:{dx}\:\: \\ $$$$\: \\ $$

Question Number 155547    Answers: 1   Comments: 0

let a;b;c>0 and a+b+c=3 find min value of the expression: S = abc + (a-1)^2 + (b-1)^2 + (c-1)^2

$$\mathrm{let}\:\:\mathrm{a};\mathrm{b};\mathrm{c}>\mathrm{0}\:\:\mathrm{and}\:\mathrm{a}+\mathrm{b}+\mathrm{c}=\mathrm{3} \\ $$$$\mathrm{find}\:\mathrm{min}\:\mathrm{value}\:\mathrm{of}\:\mathrm{the}\:\mathrm{expression}: \\ $$$$\mathrm{S}\:=\:\mathrm{abc}\:+\:\left(\mathrm{a}-\mathrm{1}\right)^{\mathrm{2}} \:+\:\left(\mathrm{b}-\mathrm{1}\right)^{\mathrm{2}} \:+\:\left(\mathrm{c}-\mathrm{1}\right)^{\mathrm{2}} \\ $$

Question Number 155545    Answers: 1   Comments: 6

Solve in R ((5x)/( (√(5x^2 + 4)) + 7(√x))) + ((x + 2)/( (√(x^2 - 3x - 18)) + 2(√x)))

$$\mathrm{Solve}\:\mathrm{in}\:\mathbb{R} \\ $$$$\frac{\mathrm{5x}}{\:\sqrt{\mathrm{5x}^{\mathrm{2}} \:+\:\mathrm{4}}\:+\:\mathrm{7}\sqrt{\mathrm{x}}}\:+\:\frac{\mathrm{x}\:+\:\mathrm{2}}{\:\sqrt{\mathrm{x}^{\mathrm{2}} \:-\:\mathrm{3x}\:-\:\mathrm{18}}\:+\:\mathrm{2}\sqrt{\mathrm{x}}} \\ $$

Question Number 155542    Answers: 1   Comments: 4

find the tylor series expantion of ((z^2 βˆ’1)/((z+1)(z+3)))

$${find}\:{the}\:{tylor}\:{series}\:{expantion}\:{of}\:\frac{{z}^{\mathrm{2}} βˆ’\mathrm{1}}{\left({z}+\mathrm{1}\right)\left({z}+\mathrm{3}\right)} \\ $$

Question Number 155541    Answers: 0   Comments: 0

form a partial equation from (x^2 /2)+(y^2 /2)=z^2

$${form}\:{a}\:{partial}\:{equation}\:{from}\:\frac{{x}^{\mathrm{2}} }{\mathrm{2}}+\frac{{y}^{\mathrm{2}} }{\mathrm{2}}={z}^{\mathrm{2}} \\ $$

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