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AllQuestion and Answers: Page 278

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Question Number 192289 Answers: 1 Comments: 0

∫_(−e) ^e ((sin x)/(sec^2 x+1)) dx =?

$$\:\:\:\:\:\:\:\:\underset{−\mathrm{e}} {\overset{\mathrm{e}} {\int}}\:\frac{\mathrm{sin}\:\mathrm{x}}{\mathrm{sec}\:^{\mathrm{2}} \mathrm{x}+\mathrm{1}}\:\mathrm{dx}\:=? \\ $$

Question Number 192286 Answers: 1 Comments: 0

Determine whether f(x)=(1/x)(2x^2 +1)is: 1.A function 2. injective 3. surjective 4. bijective

$$\mathrm{Determine}\:\mathrm{whether}\:\mathrm{f}\left(\mathrm{x}\right)=\frac{\mathrm{1}}{\mathrm{x}}\left(\mathrm{2x}^{\mathrm{2}} +\mathrm{1}\right)\mathrm{is}: \\ $$$$\mathrm{1}.\mathrm{A}\:\mathrm{function} \\ $$$$\mathrm{2}.\:\mathrm{injective} \\ $$$$\mathrm{3}.\:\mathrm{surjective} \\ $$$$\mathrm{4}.\:\mathrm{bijective} \\ $$

Question Number 192282 Answers: 1 Comments: 0

find the value of tan (π/9) + 4sin (π/9) = ?

$$\:\:\:{find}\:{the}\:{value}\:{of} \\ $$$$\:\:\:\:\mathrm{tan}\:\frac{\pi}{\mathrm{9}}\:+\:\mathrm{4sin}\:\frac{\pi}{\mathrm{9}}\:=\:? \\ $$

Question Number 192281 Answers: 0 Comments: 0

help me proving this f:R→R dan c∈R lim_(x→c) f(x)=L ⇔ lim_(c→0) f(x+c)=L

$$ \\ $$$$ \\ $$$$\:{help}\:{me}\:{proving}\:{this} \\ $$$$\:{f}:\mathbb{R}\rightarrow\mathbb{R}\:{dan}\:{c}\in\mathbb{R} \\ $$$$\:\underset{{x}\rightarrow{c}} {\mathrm{lim}}\:{f}\left({x}\right)={L}\:\Leftrightarrow\:\underset{{c}\rightarrow\mathrm{0}} {\mathrm{lim}}\:{f}\left({x}+{c}\right)={L} \\ $$$$\: \\ $$$$ \\ $$

Question Number 192280 Answers: 2 Comments: 0

Question Number 192278 Answers: 1 Comments: 2

S=arctan((2/1^2 ))+artan((2/2^2 ))+........

$${S}={arctan}\left(\frac{\mathrm{2}}{\mathrm{1}^{\mathrm{2}} }\right)+{artan}\left(\frac{\mathrm{2}}{\mathrm{2}^{\mathrm{2}} }\right)+........ \\ $$

Question Number 192277 Answers: 1 Comments: 0

Let a_1 , a_2 , a_3 ,..., a_n be real numbers such that: (√a_1 ) + (√(a_2 −1 )) +(√(a_3 −2 )) +...+(√(a_n −(n−1) ))=(1/2)(a_1 +a_2 +a_3 +...+a_n )−((n(n−3))/4) Then find the value of [ Σ_(i=1) ^(100) (a_i )].

$$ \\ $$$${Let}\:{a}_{\mathrm{1}} ,\:{a}_{\mathrm{2}} ,\:{a}_{\mathrm{3}} ,...,\:{a}_{{n}} \:{be}\:{real}\:{numbers}\:{such}\:{that}: \\ $$$$\sqrt{{a}_{\mathrm{1}} }\:+\:\sqrt{{a}_{\mathrm{2}} −\mathrm{1}\:\:}\:+\sqrt{{a}_{\mathrm{3}} −\mathrm{2}\:}\:+...+\sqrt{{a}_{{n}} −\left({n}−\mathrm{1}\right)\:}=\frac{\mathrm{1}}{\mathrm{2}}\left({a}_{\mathrm{1}} +{a}_{\mathrm{2}} +{a}_{\mathrm{3}} +...+{a}_{{n}} \right)−\frac{{n}\left({n}−\mathrm{3}\right)}{\mathrm{4}} \\ $$$$\boldsymbol{{Then}}\:\boldsymbol{{find}}\:\boldsymbol{{the}}\:\boldsymbol{{value}}\:\boldsymbol{{of}}\:\left[\:\underset{\boldsymbol{{i}}=\mathrm{1}} {\overset{\mathrm{100}} {\sum}}\left(\boldsymbol{{a}}_{\boldsymbol{{i}}} \right)\right]. \\ $$

Question Number 192276 Answers: 1 Comments: 0

lim_(n→∞) (Σ_(k=0) ^n [((k(n−k)!+(k+1))/((k+1)!(n−k)!))])

$$\underset{{n}\rightarrow\infty} {\mathrm{lim}}\left(\underset{{k}=\mathrm{0}} {\overset{{n}} {\sum}}\left[\frac{{k}\left({n}−{k}\right)!+\left({k}+\mathrm{1}\right)}{\left({k}+\mathrm{1}\right)!\left({n}−{k}\right)!}\right]\right) \\ $$

Question Number 192275 Answers: 2 Comments: 0

2009^3^(2016n+2013) +2010^2^(2016n+2013) ≡x mod(11) where n is any integer ≥0

$$\mathrm{2009}^{\mathrm{3}^{\mathrm{2016}{n}+\mathrm{2013}} } +\mathrm{2010}^{\mathrm{2}^{\mathrm{2016}{n}+\mathrm{2013}} } \equiv{x}\:{mod}\left(\mathrm{11}\right)\:{where}\:{n}\:{is}\:{any}\:{integer}\:\geq\mathrm{0} \\ $$$$ \\ $$

Question Number 192274 Answers: 1 Comments: 0

prove that lim_(n→∞) (Σ_(k=1) ^n (n^2 /( (√(n^6 +k)))))=1

$${prove}\:{that}\:\underset{{n}\rightarrow\infty} {\mathrm{lim}}\left(\underset{{k}=\mathrm{1}} {\overset{{n}} {\sum}}\frac{{n}^{\mathrm{2}} }{\:\sqrt{{n}^{\mathrm{6}} +{k}}}\right)=\mathrm{1} \\ $$$$ \\ $$$$ \\ $$

Question Number 192270 Answers: 2 Comments: 0

prove that sin 50° + sin 10° = cos 20°

$$\:{prove}\:{that} \\ $$$$\:\:\mathrm{sin}\:\mathrm{50}°\:+\:\mathrm{sin}\:\mathrm{10}°\:=\:\mathrm{cos}\:\mathrm{20}° \\ $$

Question Number 192266 Answers: 1 Comments: 0

Question Number 192265 Answers: 1 Comments: 0

Question Number 192261 Answers: 0 Comments: 0

f(x)=⌊ (( 1)/(1+(√x))) ⌋ is derivable on ( 0 , k ). find the value of k_( max) =?

$$ \\ $$$$\:\:\:{f}\left({x}\right)=\lfloor\:\frac{\:\mathrm{1}}{\mathrm{1}+\sqrt{{x}}}\:\rfloor\:{is}\:{derivable}\: \\ $$$$\:\:{on}\:\:\left(\:\mathrm{0}\:,\:\:{k}\:\right).\:{find}\:{the}\:{value}\:{of} \\ $$$$\:\:\:\:\: \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:{k}_{\:{max}} =?\: \\ $$$$\:\:\: \\ $$

Question Number 192257 Answers: 0 Comments: 3

A bullet with a velocity of 30 ms^(−1) after pentrating a 6 cm whole tree the velocity is reduced by one−third and then the bullet travels for 1s more. Will the bullet penetratee th tree? Analyze mathematically.

$$ \\ $$$$\mathrm{A}\:\mathrm{bullet}\:\mathrm{with}\:\mathrm{a}\:\mathrm{velocity}\:\mathrm{of}\:\mathrm{30}\:\mathrm{ms}^{−\mathrm{1}} \:\mathrm{after} \\ $$$$\mathrm{pentrating}\:\mathrm{a}\:\mathrm{6}\:{cm}\:\mathrm{whole}\:\mathrm{tree}\:\mathrm{the}\:\mathrm{velocity}\:\mathrm{is}\: \\ $$$$\mathrm{reduced}\:\mathrm{by}\:\mathrm{one}−\mathrm{third}\:\mathrm{and}\:\mathrm{then}\:\mathrm{the}\:\mathrm{bullet} \\ $$$$\mathrm{travel}{s}\:\mathrm{for}\:\mathrm{1s}\:\mathrm{more}.\: \\ $$$$ \\ $$$$ \\ $$$${Will}\:\mathrm{the}\:\mathrm{bullet}\:\mathrm{penetratee} \\ $$$$\mathrm{th}\:\mathrm{tree}?\:\mathrm{Analyze}\:\mathrm{mathematically}. \\ $$

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