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AllQuestion and Answers: Page 655

Question Number 146702 Answers: 1 Comments: 0

Question Number 146701 Answers: 3 Comments: 2

Question Number 146697 Answers: 0 Comments: 0

∀t≥−1,F(t)=(2/π)∫_0 ^(π/2) (√(1+tcos^2 ϕ))dϕ 1) Show that ∀t≤−1 F(t)=(√(1+t))F(−(1/(1+t))) 2) show that if 0≤t_1 , 0≤F(t_2 )−F(t_1 )≤((t_2 −t_1 )/4)

$$\forall{t}\geqslant−\mathrm{1},{F}\left({t}\right)=\frac{\mathrm{2}}{\pi}\int_{\mathrm{0}} ^{\frac{\pi}{\mathrm{2}}} \sqrt{\mathrm{1}+{tcos}^{\mathrm{2}} \varphi}{d}\varphi \\ $$$$\left.\mathrm{1}\right)\:{Show}\:{that}\:\forall{t}\leqslant−\mathrm{1}\:{F}\left({t}\right)=\sqrt{\mathrm{1}+{t}}{F}\left(−\frac{\mathrm{1}}{\mathrm{1}+{t}}\right) \\ $$$$\left.\mathrm{2}\right)\:{show}\:{that}\:{if}\:\mathrm{0}\leqslant{t}_{\mathrm{1}} \:, \\ $$$$\mathrm{0}\leqslant{F}\left({t}_{\mathrm{2}} \right)−{F}\left({t}_{\mathrm{1}} \right)\leqslant\frac{{t}_{\mathrm{2}} −{t}_{\mathrm{1}} }{\mathrm{4}} \\ $$

Question Number 146689 Answers: 1 Comments: 0

∫ ((−3x+5)/( (((3x^2 −10x)^2 ))^(1/3) )) dx = ... ? Solve it without substitution method.

$$\int\:\:\frac{−\mathrm{3}{x}+\mathrm{5}}{\:\sqrt[{\mathrm{3}}]{\left(\mathrm{3}{x}^{\mathrm{2}} −\mathrm{10}{x}\right)^{\mathrm{2}} }}\:\:{dx}\:\:=\:\:\:...\:\:? \\ $$$${Solve}\:\:{it}\:\:{without}\:\:{substitution}\:\:{method}. \\ $$

Question Number 146687 Answers: 0 Comments: 0

E = MC^(2 ) ∫requency = E/M

$${E}\:=\:{MC}^{\mathrm{2}\:} \:\:\:\:\:\:\:\:\:\int{requency}\:=\:{E}/{M} \\ $$

Question Number 146680 Answers: 2 Comments: 0

Compare: 100^(101) and 101^(100)

$${Compare}:\:\:\mathrm{100}^{\mathrm{101}} \:\:{and}\:\:\:\mathrm{101}^{\mathrm{100}} \\ $$

Question Number 146678 Answers: 2 Comments: 0

∫ (√(2 + x^2 )) dx = ?

$$\int\:\sqrt{\mathrm{2}\:+\:{x}^{\mathrm{2}} }\:{dx}\:=\:? \\ $$

Question Number 146677 Answers: 1 Comments: 0

if f(x) = 3^(x+1) find ((f(2x + 1))/(f(x + 1))) = ?

$${if}\:\:\:{f}\left({x}\right)\:=\:\mathrm{3}^{\boldsymbol{{x}}+\mathrm{1}} \:\:\:{find}\:\:\:\frac{{f}\left(\mathrm{2}{x}\:+\:\mathrm{1}\right)}{{f}\left({x}\:+\:\mathrm{1}\right)}\:=\:? \\ $$

Question Number 146672 Answers: 1 Comments: 0

Question Number 146669 Answers: 1 Comments: 0

∫_0 ^π (a−e^(−ix) )^n (a−e^(ix) )^n cos(nx)dx

$$\int_{\mathrm{0}} ^{\pi} \left(\mathrm{a}−\mathrm{e}^{−\mathrm{ix}} \right)^{\mathrm{n}} \left(\mathrm{a}−\mathrm{e}^{\mathrm{ix}} \right)^{\mathrm{n}} \mathrm{cos}\left(\mathrm{nx}\right)\mathrm{dx} \\ $$

Question Number 146667 Answers: 1 Comments: 0

Question Number 146730 Answers: 1 Comments: 2

Question Number 146653 Answers: 1 Comments: 0

Question Number 146722 Answers: 1 Comments: 0

D=lim_(n→+∝) (n^2 /(n−1))[((sin((88)/n))/(1+2)) + ((sin((88)/n))/(1+2+3)) + ...+ ((sin((88)/n))/(1+2+3+...+n))]

$$\mathrm{D}=\underset{\mathrm{n}\rightarrow+\propto} {\mathrm{lim}}\frac{\mathrm{n}^{\mathrm{2}} }{\mathrm{n}−\mathrm{1}}\left[\frac{\mathrm{sin}\frac{\mathrm{88}}{\mathrm{n}}}{\mathrm{1}+\mathrm{2}}\:+\:\frac{\mathrm{sin}\frac{\mathrm{88}}{\mathrm{n}}}{\mathrm{1}+\mathrm{2}+\mathrm{3}}\:+\:...+\:\frac{\mathrm{sin}\frac{\mathrm{88}}{\mathrm{n}}}{\mathrm{1}+\mathrm{2}+\mathrm{3}+...+\mathrm{n}}\right] \\ $$

Question Number 146636 Answers: 1 Comments: 2

4 + ((12)/(8 + (7/(3 + (4/(x + 1)))))) = 8 ⇒ x=?

$$\mathrm{4}\:+\:\frac{\mathrm{12}}{\mathrm{8}\:+\:\frac{\mathrm{7}}{\mathrm{3}\:+\:\frac{\mathrm{4}}{\boldsymbol{{x}}\:+\:\mathrm{1}}}}\:=\:\mathrm{8}\:\:\Rightarrow\:\:\boldsymbol{{x}}=? \\ $$

Question Number 146635 Answers: 1 Comments: 0

lim_(x→∞) (((2x + 1)/(x + 1)))^(1/x) = ?

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

Question Number 146634 Answers: 2 Comments: 0

log_(xyz) x = 2 and log_(xyz) y = 3 find log_(xyz) z = ?

$$\boldsymbol{{log}}_{\boldsymbol{{xyz}}} \:\boldsymbol{{x}}\:=\:\mathrm{2}\:\:\:{and}\:\:\:\boldsymbol{{log}}_{\boldsymbol{{xyz}}} \:\boldsymbol{{y}}\:=\:\mathrm{3} \\ $$$${find}\:\:\:\boldsymbol{{log}}_{\boldsymbol{{xyz}}} \:\boldsymbol{{z}}\:=\:? \\ $$

Question Number 146632 Answers: 0 Comments: 0

Question Number 146622 Answers: 1 Comments: 1

(1/( (6)^(1/8) + 1)) ∙ (1/( (6)^(1/4) + 1)) ∙ (1/( (√6) + 1)) + 1 = ?

$$\frac{\mathrm{1}}{\:\sqrt[{\mathrm{8}}]{\mathrm{6}}\:+\:\mathrm{1}}\:\centerdot\:\frac{\mathrm{1}}{\:\sqrt[{\mathrm{4}}]{\mathrm{6}}\:+\:\mathrm{1}}\:\centerdot\:\frac{\mathrm{1}}{\:\sqrt{\mathrm{6}}\:+\:\mathrm{1}}\:+\:\mathrm{1}\:=\:? \\ $$

Question Number 146621 Answers: 2 Comments: 0

∫_0 ^( ∞) ((xsin(4x))/(9 + 4x^2 ))dx =

$$\int_{\mathrm{0}} ^{\:\infty} \:\frac{{xsin}\left(\mathrm{4}{x}\right)}{\mathrm{9}\:+\:\mathrm{4}{x}^{\mathrm{2}} }{dx}\:=\: \\ $$

Question Number 146619 Answers: 1 Comments: 0

solve :: ( x ∈ R ) [ x ] = [ x^( 2) − x −6 ] note:: [x ] := max { q ∈ Z ∣ q ≤ x }

$$ \\ $$$$\:\:\:\:{solve}\:::\:\:\:\left(\:{x}\:\in\:\mathbb{R}\:\right) \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\left[\:{x}\:\right]\:=\:\left[\:{x}^{\:\mathrm{2}} −\:{x}\:−\mathrm{6}\:\right] \\ $$$$ \\ $$$$\:\:\:\:\:\:\:\:{note}::\:\:\:\left[{x}\:\right]\::=\:{max}\:\left\{\:{q}\:\in\:\mathbb{Z}\:\mid\:{q}\:\leqslant\:{x}\:\right\} \\ $$

Question Number 146614 Answers: 0 Comments: 0

A university Department has 14 lectures. of these 7 teach pure mathematics, 6 tesch applied mathematics and 5 teach statistics. 3 teach pure mathematics and applied mathematics but no one teach pure mathematics and statistics. a) represent the infomation on a venn diagram. b) find the number of lectures who teach i. Applied mathematics and statistics ii. at most two subject

$$\mathrm{A}\:\mathrm{university}\:\mathrm{Department}\:\mathrm{has}\:\mathrm{14}\:\mathrm{lectures}. \\ $$$$\mathrm{of}\:\mathrm{these}\:\mathrm{7}\:\mathrm{teach}\:\mathrm{pure}\:\mathrm{mathematics},\: \\ $$$$\mathrm{6}\:\mathrm{tesch}\:\mathrm{applied}\:\mathrm{mathematics}\:\mathrm{and}\:\mathrm{5} \\ $$$$\mathrm{teach}\:\mathrm{statistics}.\: \\ $$$$\mathrm{3}\:\mathrm{teach}\:\mathrm{pure}\:\mathrm{mathematics}\:\mathrm{and}\: \\ $$$$\mathrm{applied}\:\mathrm{mathematics}\:\mathrm{but}\:\mathrm{no}\:\mathrm{one}\:\mathrm{teach} \\ $$$$\mathrm{pure}\:\mathrm{mathematics}\:\mathrm{and}\:\mathrm{statistics}. \\ $$$$\left.\mathrm{a}\right)\:\mathrm{represent}\:\mathrm{the}\:\mathrm{infomation}\:\mathrm{on}\:\mathrm{a}\:\mathrm{venn} \\ $$$$\mathrm{diagram}. \\ $$$$\left.\mathrm{b}\right)\:\mathrm{find}\:\mathrm{the}\:\mathrm{number}\:\mathrm{of}\:\mathrm{lectures}\:\mathrm{who}\: \\ $$$$\mathrm{teach} \\ $$$$\mathrm{i}.\:\mathrm{Applied}\:\mathrm{mathematics}\:\mathrm{and}\:\mathrm{statistics} \\ $$$$\mathrm{ii}.\:\mathrm{at}\:\mathrm{most}\:\mathrm{two}\:\mathrm{subject} \\ $$$$ \\ $$

Question Number 146613 Answers: 1 Comments: 0

Question Number 146611 Answers: 1 Comments: 0

y=x^2 +x , y=2x^2 +x+m and y=kx+b if one point intersects, find k∙b=?

$${y}={x}^{\mathrm{2}} +{x}\:,\:{y}=\mathrm{2}{x}^{\mathrm{2}} +{x}+{m}\:\:{and}\:\:{y}={kx}+{b} \\ $$$${if}\:{one}\:{point}\:{intersects},\:{find}\:\:\:{k}\centerdot{b}=? \\ $$

Question Number 146610 Answers: 1 Comments: 0

How many roots does the equation have? 5^(3x) + 27∙5^(−3x) + 9∙5^x + 27∙5^(−x) = 64

$${How}\:{many}\:{roots}\:{does}\:{the}\:{equation} \\ $$$${have}? \\ $$$$\mathrm{5}^{\mathrm{3}\boldsymbol{{x}}} \:+\:\mathrm{27}\centerdot\mathrm{5}^{−\mathrm{3}\boldsymbol{{x}}} \:+\:\mathrm{9}\centerdot\mathrm{5}^{\boldsymbol{{x}}} \:+\:\mathrm{27}\centerdot\mathrm{5}^{−\boldsymbol{{x}}} \:=\:\mathrm{64} \\ $$

Question Number 146608 Answers: 1 Comments: 0

((x (√x) - 1)/(x + (√x) + 1)) : (((√x) - 1)/( (√x) + 1)) = ?

$$\frac{{x}\:\sqrt{{x}}\:-\:\mathrm{1}}{{x}\:+\:\sqrt{{x}}\:+\:\mathrm{1}}\::\:\frac{\sqrt{{x}}\:-\:\mathrm{1}}{\:\sqrt{{x}}\:+\:\mathrm{1}}\:=\:? \\ $$

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