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AllQuestion and Answers: Page 1107

Question Number 106319 Answers: 1 Comments: 2

Question Number 106318 Answers: 0 Comments: 0

∫_0 ^(log(2)) ((e^x −4)/(x +2))dx = ..... {(a)log(2) (b)1−log(2) (c) 1−2log(2) (d)−log(2)}

$$\int_{\mathrm{0}} ^{\mathrm{log}\left(\mathrm{2}\right)} \frac{{e}^{{x}} \:−\mathrm{4}}{{x}\:+\mathrm{2}}{dx}\:\:\:=\:..... \\ $$$$\left\{\left({a}\right){log}\left(\mathrm{2}\right)\:\:\:\:\:\:\:\left({b}\right)\mathrm{1}−{log}\left(\mathrm{2}\right)\:\:\:\:\:\:\:\left({c}\right)\right. \\ $$$$\left.\mathrm{1}−\mathrm{2}{log}\left(\mathrm{2}\right)\:\:\:\:\:\:\:\:\left({d}\right)−{log}\left(\mathrm{2}\right)\right\} \\ $$

Question Number 106315 Answers: 1 Comments: 0

show that ⊛∀ (x_1 ,x_2 ,.....,x_(n ) )∈R^n (Σ_(k=1) ^n x_k )^2 ≤nΣ_(k=1) ^n x_k ^2 ⊛a,b>0 p(x)=x^n +ax+b=0 could not have more than 3 reals solutions

$$\boldsymbol{{show}}\:\boldsymbol{{that}} \\ $$$$\circledast\forall\:\left(\boldsymbol{{x}}_{\mathrm{1}} ,\boldsymbol{{x}}_{\mathrm{2}} ,.....,\boldsymbol{{x}}_{\boldsymbol{{n}}\:} \right)\in\mathbb{R}^{\boldsymbol{{n}}} \\ $$$$\left(\underset{\boldsymbol{{k}}=\mathrm{1}} {\overset{\boldsymbol{{n}}} {\sum}}\boldsymbol{{x}}_{\boldsymbol{{k}}} \right)^{\mathrm{2}} \leqslant\boldsymbol{{n}}\underset{\boldsymbol{{k}}=\mathrm{1}} {\overset{\boldsymbol{{n}}} {\sum}}\boldsymbol{{x}}_{\boldsymbol{{k}}} ^{\mathrm{2}} \\ $$$$\circledast\boldsymbol{{a}},\boldsymbol{{b}}>\mathrm{0} \\ $$$$\boldsymbol{{p}}\left(\boldsymbol{{x}}\right)=\boldsymbol{{x}}^{\boldsymbol{{n}}} +\boldsymbol{{ax}}+\boldsymbol{{b}}=\mathrm{0} \\ $$$$\boldsymbol{{could}}\:\boldsymbol{{not}}\:\boldsymbol{{have}}\:\boldsymbol{{more}}\:\boldsymbol{{than}}\:\mathrm{3} \\ $$$$\boldsymbol{{reals}}\:\boldsymbol{{solutions}} \\ $$

Question Number 106314 Answers: 3 Comments: 0

Question Number 106313 Answers: 1 Comments: 0

lim_(x→1) ((x∣x−1∣)/(x^2 −1))

$$\underset{{x}\rightarrow\mathrm{1}} {\mathrm{lim}}\:\frac{\mathrm{x}\mid\mathrm{x}−\mathrm{1}\mid}{\mathrm{x}^{\mathrm{2}} −\mathrm{1}} \\ $$

Question Number 106309 Answers: 1 Comments: 0

If g(x)= x+(√x) and lim_(x→2) ((f(x)−f(2))/(x^2 +ax+b))=(4/3) find the value of (f○g)′(1).

$$\mathrm{If}\:\mathrm{g}\left(\mathrm{x}\right)=\:\mathrm{x}+\sqrt{\mathrm{x}}\:\mathrm{and}\:\underset{{x}\rightarrow\mathrm{2}} {\mathrm{lim}}\:\frac{\mathrm{f}\left(\mathrm{x}\right)−\mathrm{f}\left(\mathrm{2}\right)}{\mathrm{x}^{\mathrm{2}} +\mathrm{ax}+\mathrm{b}}=\frac{\mathrm{4}}{\mathrm{3}} \\ $$$$\mathrm{find}\:\mathrm{the}\:\mathrm{value}\:\mathrm{of}\:\left(\mathrm{f}\circ\mathrm{g}\right)'\left(\mathrm{1}\right). \\ $$

Question Number 106305 Answers: 1 Comments: 0

what is probability of 5 coming up at least one if a die is rolled 3 times

$$\mathrm{what}\:\mathrm{is}\:\mathrm{probability}\:\mathrm{of}\:\mathrm{5}\:\mathrm{coming}\:\mathrm{up}\:\mathrm{at}\:\mathrm{least}\:\mathrm{one}\:\mathrm{if}\:\mathrm{a}\:\mathrm{die}\: \\ $$$$\mathrm{is}\:\mathrm{rolled}\:\mathrm{3}\:\mathrm{times}\: \\ $$

Question Number 106302 Answers: 4 Comments: 0

Question Number 106303 Answers: 1 Comments: 0

Question Number 106295 Answers: 2 Comments: 0

(1/(cos x)) + ((√3)/(sin x)) = 4

$$\frac{\mathrm{1}}{\mathrm{cos}\:\mathrm{x}}\:+\:\frac{\sqrt{\mathrm{3}}}{\mathrm{sin}\:\mathrm{x}}\:=\:\mathrm{4}\: \\ $$

Question Number 106292 Answers: 1 Comments: 0

a box contains 4 blue, 3 green and 2 red identicall balls. if two balls are selected at random without replacement , what is the probability that two balls be of the same colours?

$$\mathrm{a}\:\mathrm{box}\:\mathrm{contains}\:\mathrm{4}\:\mathrm{blue},\:\mathrm{3}\:\mathrm{green}\:\mathrm{and}\:\mathrm{2}\:\mathrm{red}\:\mathrm{identicall}\:\mathrm{balls}.\: \\ $$$$\mathrm{if}\:\mathrm{two}\:\mathrm{balls}\:\mathrm{are}\:\mathrm{selected}\:\mathrm{at}\:\mathrm{random}\:\mathrm{without}\: \\ $$$$\mathrm{replacement}\:,\:\mathrm{what}\:\mathrm{is}\:\mathrm{the}\:\mathrm{probability} \\ $$$$\mathrm{that}\:\mathrm{two}\:\mathrm{balls}\:\mathrm{be}\:\mathrm{of}\:\mathrm{the}\:\mathrm{same}\:\mathrm{colours}? \\ $$

Question Number 106288 Answers: 1 Comments: 0

If 0<x<1<y<100<z, and satisfy these equations: { ((log _2 (xyz)=103)),(((1/(log_2 x))+(1/(log_2 y))+(1/(log_2 z))=(1/(103)))) :} Find xyz(x+y+z)−xy−yz−zx

$$\mathrm{If}\:\mathrm{0}<\mathrm{x}<\mathrm{1}<\mathrm{y}<\mathrm{100}<\mathrm{z},\:\mathrm{and}\:\mathrm{satisfy} \\ $$$$\mathrm{these}\:\mathrm{equations}: \\ $$$$\begin{cases}{\mathrm{log}\:_{\mathrm{2}} \left(\mathrm{xyz}\right)=\mathrm{103}}\\{\frac{\mathrm{1}}{\mathrm{log}_{\mathrm{2}} \mathrm{x}}+\frac{\mathrm{1}}{\mathrm{log}_{\mathrm{2}} \mathrm{y}}+\frac{\mathrm{1}}{\mathrm{log}_{\mathrm{2}} \mathrm{z}}=\frac{\mathrm{1}}{\mathrm{103}}}\end{cases} \\ $$$$\mathrm{Find}\:\mathrm{xyz}\left(\mathrm{x}+\mathrm{y}+\mathrm{z}\right)−\mathrm{xy}−\mathrm{yz}−\mathrm{zx} \\ $$

Question Number 106286 Answers: 2 Comments: 0

arc tan (((x+1)/(x−1)))+arc tan (((x−1)/x))=arc tan (−7) for x real number

$$\mathrm{arc}\:\mathrm{tan}\:\left(\frac{\mathrm{x}+\mathrm{1}}{\mathrm{x}−\mathrm{1}}\right)+\mathrm{arc}\:\mathrm{tan}\:\left(\frac{\mathrm{x}−\mathrm{1}}{\mathrm{x}}\right)=\mathrm{arc}\:\mathrm{tan}\:\left(−\mathrm{7}\right) \\ $$$$\mathrm{for}\:\mathrm{x}\:\mathrm{real}\:\mathrm{number} \\ $$

Question Number 106285 Answers: 2 Comments: 0

∫_0 ^∞ e^(−x^2 ) dx ?

$$\underset{\mathrm{0}} {\overset{\infty} {\int}}\mathrm{e}^{−\mathrm{x}^{\mathrm{2}} } \:\mathrm{dx}\:? \\ $$

Question Number 106281 Answers: 2 Comments: 0

(√(x+50))+(√(y+100))+(√(z+150))=((x+y+z)/4)+78 find x+y−z

$$\sqrt{\mathrm{x}+\mathrm{50}}+\sqrt{\mathrm{y}+\mathrm{100}}+\sqrt{\mathrm{z}+\mathrm{150}}=\frac{\mathrm{x}+\mathrm{y}+\mathrm{z}}{\mathrm{4}}+\mathrm{78} \\ $$$$\mathrm{find}\:\mathrm{x}+\mathrm{y}−\mathrm{z}\: \\ $$

Question Number 106272 Answers: 1 Comments: 1

(√(1 + (√(5 +(√(11 + (√(19 + (√(29 + (√…))))))))))) = ?

$$\sqrt{\mathrm{1}\:+\:\sqrt{\mathrm{5}\:+\sqrt{\mathrm{11}\:+\:\sqrt{\mathrm{19}\:+\:\sqrt{\mathrm{29}\:+\:\sqrt{\ldots}}}}}}\:=\:? \\ $$

Question Number 106259 Answers: 2 Comments: 1

Question Number 106256 Answers: 0 Comments: 2

Question Number 106250 Answers: 0 Comments: 1

Question Number 106246 Answers: 3 Comments: 3

Question Number 106240 Answers: 1 Comments: 0

lim_(n→+∞) Σ_(k=1) ^n (((−1)^(k+1) )/k)=????

$$ \\ $$$$ \\ $$$$ \\ $$$$\boldsymbol{{li}}\underset{\boldsymbol{{n}}\rightarrow+\infty} {\boldsymbol{{m}}}\underset{{k}=\mathrm{1}} {\overset{{n}} {\sum}}\frac{\left(−\mathrm{1}\right)^{\boldsymbol{{k}}+\mathrm{1}} }{\boldsymbol{{k}}}=???? \\ $$

Question Number 106237 Answers: 1 Comments: 0

Question Number 106233 Answers: 2 Comments: 0

(x^2 + 1)(x − 1)^2 = 2017yz (y^2 + 1)(y − 1)^2 = 2017xz (z^2 + 1)(z − 1)^2 = 2017xy x ≥ 1 ; y ≥ 1 ; z ≥ 1 prove that x = y = z = ((1+ (√(2018)) + (√(2015+ 2(√(2018)))))/2)

$$\left({x}^{\mathrm{2}} \:+\:\mathrm{1}\right)\left({x}\:−\:\mathrm{1}\right)^{\mathrm{2}} \:=\:\mathrm{2017}{yz} \\ $$$$\left({y}^{\mathrm{2}} \:+\:\mathrm{1}\right)\left({y}\:−\:\mathrm{1}\right)^{\mathrm{2}} \:=\:\mathrm{2017}{xz} \\ $$$$\left({z}^{\mathrm{2}} \:+\:\mathrm{1}\right)\left({z}\:−\:\mathrm{1}\right)^{\mathrm{2}} \:=\:\mathrm{2017}{xy} \\ $$$${x}\:\geqslant\:\mathrm{1}\:;\:{y}\:\geqslant\:\mathrm{1}\:;\:{z}\:\geqslant\:\mathrm{1} \\ $$$${prove}\:{that}\:\:\:{x}\:=\:{y}\:=\:{z}\:= \\ $$$$\frac{\mathrm{1}+\:\sqrt{\mathrm{2018}}\:+\:\sqrt{\mathrm{2015}+\:\mathrm{2}\sqrt{\mathrm{2018}}}}{\mathrm{2}} \\ $$

Question Number 106232 Answers: 0 Comments: 0

find ∫ cos^n x ch(nx)dx /with n integr

$$\mathrm{find}\:\int\:\mathrm{cos}^{\mathrm{n}} \mathrm{x}\:\mathrm{ch}\left(\mathrm{nx}\right)\mathrm{dx}\:/\mathrm{with}\:\mathrm{n}\:\mathrm{integr} \\ $$

Question Number 106222 Answers: 1 Comments: 5

Determine x & y such that: lcm(x,y)−gcd(x,y)=x+y.

$$\mathcal{D}{etermine}\:{x}\:\&\:{y}\:{such}\:{that}: \\ $$$$\mathrm{lcm}\left({x},{y}\right)−\mathrm{gcd}\left({x},{y}\right)={x}+{y}. \\ $$

Question Number 106217 Answers: 1 Comments: 0

If f(x)= ((sin x−x cos x)/x^2 ) and f(0)=0 when x = 0 ,what will be the value of lim_(t→0) ((∫_0 ^t f(x) dx)/t^2 )

$$\mathrm{If}\:\mathrm{f}\left(\mathrm{x}\right)=\:\frac{\mathrm{sin}\:\mathrm{x}−\mathrm{x}\:\mathrm{cos}\:\mathrm{x}}{\mathrm{x}^{\mathrm{2}} }\:\mathrm{and}\:\mathrm{f}\left(\mathrm{0}\right)=\mathrm{0}\:\mathrm{when}\:\mathrm{x} \\ $$$$=\:\mathrm{0}\:,\mathrm{what}\:\mathrm{will}\:\mathrm{be}\:\mathrm{the}\:\mathrm{value}\:\mathrm{of}\:\underset{\mathrm{t}\rightarrow\mathrm{0}} {\mathrm{lim}}\frac{\underset{\mathrm{0}} {\overset{\mathrm{t}} {\int}}\:\mathrm{f}\left(\mathrm{x}\right)\:\mathrm{dx}}{\mathrm{t}^{\mathrm{2}} } \\ $$$$ \\ $$

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