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Question Number 157184    Answers: 1   Comments: 0

p(x)=x^3 −3ax^2 +(3a^2 +1)x−(a^3 +a) p(2)<0 prove that 2<a<3

$${p}\left({x}\right)={x}^{\mathrm{3}} −\mathrm{3}{ax}^{\mathrm{2}} +\left(\mathrm{3}{a}^{\mathrm{2}} +\mathrm{1}\right){x}−\left({a}^{\mathrm{3}} +{a}\right) \\ $$$${p}\left(\mathrm{2}\right)<\mathrm{0} \\ $$$${prove}\:{that}\:\mathrm{2}<{a}<\mathrm{3} \\ $$

Question Number 157220    Answers: 1   Comments: 0

x = (1/2) find (√(x + 1 + (√(x^2 + 1)))) = ?

$$\mathrm{x}\:=\:\frac{\mathrm{1}}{\mathrm{2}} \\ $$$$\mathrm{find}\:\:\sqrt{\mathrm{x}\:+\:\mathrm{1}\:+\:\sqrt{\mathrm{x}^{\mathrm{2}} \:+\:\mathrm{1}}}\:=\:? \\ $$

Question Number 157177    Answers: 1   Comments: 0

Question Number 157217    Answers: 0   Comments: 2

show that ((cos2x+cos3x+cos8x)/(sin2x+sin3x+sin8x))=tanx

$${show}\:{that} \\ $$$$\:\frac{{cos}\mathrm{2}{x}+{cos}\mathrm{3}{x}+{cos}\mathrm{8}{x}}{{sin}\mathrm{2}{x}+{sin}\mathrm{3}{x}+{sin}\mathrm{8}{x}}={tanx} \\ $$

Question Number 157216    Answers: 1   Comments: 1

lim_(x→∞) (((n!)/n^n ))^(1/n) =? lim_(x→0) ((√(tan 4x))/(tan(√(4x))))=?

$$\underset{{x}\rightarrow\infty} {\mathrm{lim}}\:\left(\frac{{n}!}{{n}^{{n}} }\right)^{\frac{\mathrm{1}}{{n}}} =? \\ $$$$\underset{{x}\rightarrow\mathrm{0}} {\mathrm{lim}}\:\frac{\sqrt{\mathrm{tan}\:\mathrm{4}{x}}}{\mathrm{tan}\sqrt{\mathrm{4}{x}}}=? \\ $$

Question Number 157222    Answers: 2   Comments: 0

Question Number 157223    Answers: 2   Comments: 0

Question Number 157175    Answers: 2   Comments: 1

Question Number 157171    Answers: 1   Comments: 0

calculate lim_(n→+∞) (((n+ln(n)+1)/((5+(√n))^2 )))

$${calculate}\:\underset{{n}\rightarrow+\infty} {{lim}}\left(\frac{{n}+{ln}\left({n}\right)+\mathrm{1}}{\left(\mathrm{5}+\sqrt{{n}}\right)^{\mathrm{2}} }\right) \\ $$

Question Number 157169    Answers: 1   Comments: 0

Question Number 157168    Answers: 1   Comments: 0

x , y and z are numbers. Show that max(x, y)=((x+y+∣x−y∣)/2) and min(x,y)=((x+y−∣x−y∣)/2) then find a formula for max(x,y,z).

$${x}\:,\:{y}\:{and}\:{z}\:{are}\:{numbers}. \\ $$$${Show}\:{that}\:{max}\left({x},\:{y}\right)=\frac{{x}+{y}+\mid{x}−{y}\mid}{\mathrm{2}}\:{and}\:{min}\left({x},{y}\right)=\frac{{x}+{y}−\mid{x}−{y}\mid}{\mathrm{2}} \\ $$$${then}\:{find}\:{a}\:{formula}\:{for}\: \\ $$$${max}\left({x},{y},{z}\right). \\ $$

Question Number 157156    Answers: 1   Comments: 0

{ ((a_1 =((√3)/2))),((a_(n+1) =4a_n ^3 −3a_n ; ∀n≥1)) :} a_n =?

$$ \\ $$$$\:\begin{cases}{{a}_{\mathrm{1}} =\frac{\sqrt{\mathrm{3}}}{\mathrm{2}}}\\{{a}_{{n}+\mathrm{1}} =\mathrm{4}{a}_{{n}} ^{\mathrm{3}} −\mathrm{3}{a}_{{n}} \:;\:\forall{n}\geqslant\mathrm{1}}\end{cases} \\ $$$$\:{a}_{{n}} =? \\ $$

Question Number 157163    Answers: 1   Comments: 1

Find the positive integer solution of the equation: x^3 + y^3 = 7∙(14xy + 3)

$$\mathrm{Find}\:\mathrm{the}\:\mathrm{positive}\:\mathrm{integer}\:\mathrm{solution} \\ $$$$\mathrm{of}\:\mathrm{the}\:\mathrm{equation}: \\ $$$$\mathrm{x}^{\mathrm{3}} \:+\:\mathrm{y}^{\mathrm{3}} \:=\:\mathrm{7}\centerdot\left(\mathrm{14xy}\:+\:\mathrm{3}\right) \\ $$

Question Number 157150    Answers: 0   Comments: 0

Question Number 157209    Answers: 2   Comments: 2

Question Number 157207    Answers: 0   Comments: 1

sin lnxdx

$$\mathrm{sin}\:\mathrm{ln}{xdx} \\ $$

Question Number 157141    Answers: 1   Comments: 1

{ (((d^2 y/dx^2 )−2 (dy/dx)+y =3e^(4x) )),((y(0)=−(2/3) ; (dy/dx)∣_(x=0) = ((13)/3))) :}

$$\:\begin{cases}{\frac{{d}^{\mathrm{2}} {y}}{{dx}^{\mathrm{2}} }−\mathrm{2}\:\frac{{dy}}{{dx}}+{y}\:=\mathrm{3}{e}^{\mathrm{4}{x}} }\\{{y}\left(\mathrm{0}\right)=−\frac{\mathrm{2}}{\mathrm{3}}\:;\:\frac{{dy}}{{dx}}\mid_{{x}=\mathrm{0}} =\:\frac{\mathrm{13}}{\mathrm{3}}}\end{cases}\: \\ $$

Question Number 157139    Answers: 1   Comments: 1

The figure given below is the section of the reinforced concrete short column. Calculate the stress in the concrete and the stress in the steel,if the axial load of 780KN is applied to the column. Young′s modulus of steel=210KN/mm^2 Young′s modulus of concrete=14KN/mm^2 diameter of the steel bars=20mm

$${The}\:{figure}\:{given}\:{below}\:{is}\:{the}\:{section} \\ $$$${of}\:{the}\:{reinforced}\:{concrete}\:{short}\:{column}. \\ $$$${Calculate}\:{the}\:{stress}\:{in}\:{the}\:{concrete} \\ $$$${and}\:{the}\:{stress}\:{in}\:{the}\:{steel},{if}\:{the} \\ $$$${axial}\:{load}\:{of}\:\mathrm{780}{KN}\:{is}\:{applied}\:{to} \\ $$$${the}\:{column}. \\ $$$$ \\ $$$${Young}'{s}\:{modulus}\:{of}\:{steel}=\mathrm{210}{KN}/{mm}^{\mathrm{2}} \\ $$$${Young}'{s}\:{modulus}\:{of}\:{concrete}=\mathrm{14}{KN}/{mm}^{\mathrm{2}} \\ $$$${diameter}\:{of}\:{the}\:{steel}\:{bars}=\mathrm{20}{mm} \\ $$

Question Number 157138    Answers: 0   Comments: 4

∫_0 ^1 x^x (1−x)^(1−x) sin(πx)dx=?

$$\int_{\mathrm{0}} ^{\mathrm{1}} \boldsymbol{\mathrm{x}}^{\boldsymbol{\mathrm{x}}} \left(\mathrm{1}−\boldsymbol{\mathrm{x}}\right)^{\mathrm{1}−\boldsymbol{\mathrm{x}}} \boldsymbol{\mathrm{sin}}\left(\pi\boldsymbol{\mathrm{x}}\right)\boldsymbol{\mathrm{dx}}=? \\ $$

Question Number 157126    Answers: 0   Comments: 9

how many ways can you exchange 1 dollar using (1,5,10,25,50) cents example 1 dollar(100 cents)=2×50cents =1×50+2×25=4×25 =1×50+1×25+1×10+1×5+10×1 and so on how many all the ways ?

$${how}\:{many}\:{ways}\:{can}\:{you} \\ $$$${exchange}\:\mathrm{1}\:{dollar}\:{using} \\ $$$$\left(\mathrm{1},\mathrm{5},\mathrm{10},\mathrm{25},\mathrm{50}\right)\:{cents} \\ $$$${example} \\ $$$$\mathrm{1}\:{dollar}\left(\mathrm{100}\:{cents}\right)=\mathrm{2}×\mathrm{50}{cents} \\ $$$$=\mathrm{1}×\mathrm{50}+\mathrm{2}×\mathrm{25}=\mathrm{4}×\mathrm{25} \\ $$$$=\mathrm{1}×\mathrm{50}+\mathrm{1}×\mathrm{25}+\mathrm{1}×\mathrm{10}+\mathrm{1}×\mathrm{5}+\mathrm{10}×\mathrm{1} \\ $$$${and}\:{so}\:{on} \\ $$$${how}\:{many}\:{all}\:{the}\:{ways}\:? \\ $$

Question Number 157123    Answers: 0   Comments: 1

lim_(x→0) ((log _(sin x) (cos x))/(log _(sin ((x/2))) (cos (x/2))))=?

$$\:\:\:\underset{{x}\rightarrow\mathrm{0}} {\mathrm{lim}}\:\frac{\mathrm{log}\:_{\mathrm{sin}\:{x}} \left(\mathrm{cos}\:{x}\right)}{\mathrm{log}\:_{\mathrm{sin}\:\left(\frac{{x}}{\mathrm{2}}\right)} \left(\mathrm{cos}\:\frac{{x}}{\mathrm{2}}\right)}=? \\ $$

Question Number 157120    Answers: 2   Comments: 0

Σ_(n=0) ^∞ ((1/((3n+1)(3n+2))))=?

$$\:\:\:\underset{{n}=\mathrm{0}} {\overset{\infty} {\sum}}\left(\frac{\mathrm{1}}{\left(\mathrm{3}{n}+\mathrm{1}\right)\left(\mathrm{3}{n}+\mathrm{2}\right)}\right)=? \\ $$

Question Number 157116    Answers: 1   Comments: 2

max ∧ min of f(x) =(√x) +4(√((1−x)/2))

$$\:{max}\:\wedge\:{min}\:{of}\:{f}\left({x}\right)\:=\sqrt{{x}}\:+\mathrm{4}\sqrt{\frac{\mathrm{1}−{x}}{\mathrm{2}}} \\ $$

Question Number 157114    Answers: 0   Comments: 2

Question Number 157198    Answers: 1   Comments: 0

Montrer que l′ensemble Q n′est ni ferme ni ouvert

$$\mathrm{M}{ontrer}\:{que}\:{l}'{ensemble}\:{Q}\:{n}'{est}\:{ni}\:{ferme}\:{ni}\:{ouvert} \\ $$

Question Number 157197    Answers: 1   Comments: 0

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