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Question Number 204978    Answers: 1   Comments: 0

Question Number 204970    Answers: 1   Comments: 1

Question Number 204961    Answers: 2   Comments: 0

∫ (1/(1+cot 3x)) dx =

$$\int\:\frac{\mathrm{1}}{\mathrm{1}+\mathrm{cot}\:\mathrm{3}{x}}\:{dx}\:=\:\: \\ $$

Question Number 204957    Answers: 2   Comments: 0

2×2 matrix A and B satisfy that AB+A=BA+B. Prove that (A−B)^2 =O.

$$\mathrm{2}×\mathrm{2}\:\mathrm{matrix}\:\boldsymbol{\mathrm{A}}\:\mathrm{and}\:\boldsymbol{\mathrm{B}}\:\mathrm{satisfy}\:\mathrm{that} \\ $$$$\boldsymbol{\mathrm{AB}}+\boldsymbol{\mathrm{A}}=\boldsymbol{\mathrm{BA}}+\boldsymbol{\mathrm{B}}. \\ $$$$\mathrm{Prove}\:\mathrm{that}\:\left(\boldsymbol{\mathrm{A}}−\boldsymbol{\mathrm{B}}\right)^{\mathrm{2}} =\boldsymbol{\mathrm{O}}. \\ $$

Question Number 204948    Answers: 0   Comments: 4

prove a^2 +b^2 +c^2 +((abc))^(1/3) ≥4 if ab+bc+ac=3

$${prove}\: \\ $$$${a}^{\mathrm{2}} +{b}^{\mathrm{2}} +{c}^{\mathrm{2}} +\sqrt[{\mathrm{3}}]{{abc}}\geqslant\mathrm{4} \\ $$$${if} \\ $$$${ab}+{bc}+{ac}=\mathrm{3} \\ $$

Question Number 204947    Answers: 1   Comments: 0

f′(x)+4x−6x.e^(x^2 −f(x)−1) =0 f(x)=¿

$${f}'\left({x}\right)+\mathrm{4}{x}−\mathrm{6}{x}.{e}^{{x}^{\mathrm{2}} −{f}\left({x}\right)−\mathrm{1}} =\mathrm{0} \\ $$$${f}\left({x}\right)=¿ \\ $$

Question Number 204944    Answers: 1   Comments: 1

Question Number 204941    Answers: 0   Comments: 0

Question Number 205204    Answers: 1   Comments: 0

Question Number 204929    Answers: 1   Comments: 0

lim_(n→∞) Π_(r=1) ^n ((n^2 −r)/(n^2 +r)) = ?

$$\:\:\:\:\:\:\:\:\:\underset{{n}\rightarrow\infty} {\mathrm{lim}}\:\underset{{r}=\mathrm{1}} {\overset{{n}} {\prod}}\:\frac{{n}^{\mathrm{2}} −{r}}{{n}^{\mathrm{2}} +{r}}\:\:=\:\:? \\ $$

Question Number 204926    Answers: 0   Comments: 2

Prove that in any △ABC (m_a + m_b + m_c )^2 ≥ 9(√3) F

$$\mathrm{Prove}\:\mathrm{that}\:\mathrm{in}\:\mathrm{any}\:\bigtriangleup\mathrm{ABC} \\ $$$$\left(\mathrm{m}_{\boldsymbol{\mathrm{a}}} \:+\:\mathrm{m}_{\boldsymbol{\mathrm{b}}} \:+\:\mathrm{m}_{\boldsymbol{\mathrm{c}}} \right)^{\mathrm{2}} \:\geqslant\:\mathrm{9}\sqrt{\mathrm{3}}\:\mathrm{F} \\ $$

Question Number 204921    Answers: 1   Comments: 0

Question Number 204920    Answers: 1   Comments: 0

16^(y+x^2 ) + 16^(y^2 +x) = 1 x+y =?

$$\:\:\:\:\:\mathrm{16}^{\mathrm{y}+\mathrm{x}^{\mathrm{2}} } \:+\:\mathrm{16}^{\mathrm{y}^{\mathrm{2}} +\mathrm{x}} \:=\:\mathrm{1}\: \\ $$$$\:\:\:\:\mathrm{x}+\mathrm{y}\:=? \\ $$

Question Number 204916    Answers: 0   Comments: 8

How many axes of symmetry does an open angle have?

$$ \\ $$How many axes of symmetry does an open angle have?

Question Number 204910    Answers: 1   Comments: 1

Question Number 204909    Answers: 1   Comments: 0

Ω= ∫_(1/e) ^( e) (( arctan(x))/x) dx=?

$$ \\ $$$$\:\:\:\:\:\:\:\Omega=\:\int_{\frac{\mathrm{1}}{{e}}} ^{\:{e}} \frac{\:{arctan}\left({x}\right)}{{x}}\:{dx}=? \\ $$

Question Number 204902    Answers: 2   Comments: 0

calculate ∫_0 ^1 (√(x(1−x)))dx

$$\boldsymbol{{calculate}}\:\int_{\mathrm{0}} ^{\mathrm{1}} \sqrt{\boldsymbol{{x}}\left(\mathrm{1}−\boldsymbol{{x}}\right)}\boldsymbol{{dx}} \\ $$

Question Number 204901    Answers: 0   Comments: 0

Question Number 204900    Answers: 0   Comments: 2

lim_(n→∞) n!(e−x_n ) = ? where x_(n ) = 1+(1/(1!))+(1/(2!))+...+(1/(n!))

$$\:\:\:\:\:\underset{{n}\rightarrow\infty} {\mathrm{lim}}\:\mathrm{n}!\left({e}−\mathrm{x}_{\mathrm{n}} \right)\:=\:? \\ $$$$\:\:\mathrm{where}\:\mathrm{x}_{\mathrm{n}\:} =\:\mathrm{1}+\frac{\mathrm{1}}{\mathrm{1}!}+\frac{\mathrm{1}}{\mathrm{2}!}+...+\frac{\mathrm{1}}{\mathrm{n}!} \\ $$

Question Number 204890    Answers: 2   Comments: 1

Question Number 204885    Answers: 1   Comments: 0

The density of a gas is 1.775kgm³ at 29°c and 10⁵N/m² pressure, its specific heat capacity at constant pressure is 856J/kg/K. Determine the ratio of its specific heat at constant pressure to that at constant volume?

The density of a gas is 1.775kgm³ at 29°c and 10⁵N/m² pressure, its specific heat capacity at constant pressure is 856J/kg/K. Determine the ratio of its specific heat at constant pressure to that at constant volume?

Question Number 204879    Answers: 2   Comments: 2

lim_(n→∞) n!(e−x_n ) = ? where x_(n ) = 1+(1/(1!))+(1/(2!))+...+(1/(n!))

$$\:\:\:\:\:\underset{{n}\rightarrow\infty} {\mathrm{lim}}\:\mathrm{n}!\left({e}−\mathrm{x}_{\mathrm{n}} \right)\:=\:? \\ $$$$\:\:\mathrm{where}\:\mathrm{x}_{\mathrm{n}\:} =\:\mathrm{1}+\frac{\mathrm{1}}{\mathrm{1}!}+\frac{\mathrm{1}}{\mathrm{2}!}+...+\frac{\mathrm{1}}{\mathrm{n}!} \\ $$

Question Number 204878    Answers: 2   Comments: 0

prove that: (e)^(1/4) < ∫_0 ^( 1) e^( t^2 ) dt< ((1 + e)/2)

$$ \\ $$$$\:\:{prove}\:{that}: \\ $$$$ \\ $$$$\:\:\:\:\sqrt[{\mathrm{4}}]{{e}}\:<\:\int_{\mathrm{0}} ^{\:\mathrm{1}} {e}^{\:{t}^{\mathrm{2}} } {dt}<\:\frac{\mathrm{1}\:+\:{e}}{\mathrm{2}} \\ $$

Question Number 204873    Answers: 1   Comments: 2

The figure below represents a design on the windows of a building. The curved part XY is an arc of a circle. The rise of the segmental arc is 10cm, its span is 100cm and XZ=ZY=120cm. calculate: (i) the radius of the circle (ii) the area of the segmental cap, correct to 2 significant figures. (iii) the total area of the design, correct to 3 significant figures.

$${The}\:{figure}\:{below}\:{represents}\:{a}\:{design} \\ $$$${on}\:{the}\:{windows}\:{of}\:{a}\:{building}.\:{The} \\ $$$${curved}\:{part}\:{XY}\:{is}\:{an}\:{arc}\:{of}\:{a}\:{circle}. \\ $$$${The}\:{rise}\:{of}\:{the}\:{segmental}\:{arc}\:{is}\:\mathrm{10}{cm}, \\ $$$${its}\:{span}\:{is}\:\mathrm{100}{cm}\:{and}\:{XZ}={ZY}=\mathrm{120}{cm}. \\ $$$${calculate}: \\ $$$$\left({i}\right)\:{the}\:{radius}\:{of}\:{the}\:{circle} \\ $$$$\left({ii}\right)\:{the}\:{area}\:{of}\:{the}\:{segmental}\:{cap}, \\ $$$${correct}\:{to}\:\mathrm{2}\:{significant}\:{figures}. \\ $$$$\left({iii}\right)\:{the}\:{total}\:{area}\:{of}\:{the}\:{design},\:{correct} \\ $$$${to}\:\mathrm{3}\:{significant}\:{figures}. \\ $$

Question Number 204869    Answers: 1   Comments: 0

How many distinct positive integer valued solution exist the equation (x^2 − 7x + 11)^((x^2 −13x + 42)) = 1 (a) 2 (b) 4 (c) 6 (d) 8

$$\mathrm{How}\:\mathrm{many}\:\mathrm{distinct}\:\mathrm{positive}\:\mathrm{integer}\:\mathrm{valued}\:\mathrm{solution}\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\: \\ $$$$\mathrm{exist}\:\mathrm{the}\:\mathrm{equation}\:\left({x}^{\mathrm{2}} \:−\:\mathrm{7}{x}\:+\:\mathrm{11}\right)^{\left({x}^{\mathrm{2}} \:−\mathrm{13}{x}\:+\:\mathrm{42}\right)} \:=\:\mathrm{1}\:\:\:\:\:\:\:\:\:\:\:\:\: \\ $$$$\left(\mathrm{a}\right)\:\mathrm{2}\:\:\:\:\:\:\:\:\:\:\:\:\:\:\left(\mathrm{b}\right)\:\mathrm{4}\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\left(\mathrm{c}\right)\:\mathrm{6}\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\left(\mathrm{d}\right)\:\mathrm{8} \\ $$

Question Number 204866    Answers: 1   Comments: 0

∫ ((x+3)/(x^2 (√(2x+3)))) dx=?

$$\int\:\frac{{x}+\mathrm{3}}{{x}^{\mathrm{2}} \sqrt{\mathrm{2}{x}+\mathrm{3}}}\:{dx}=? \\ $$

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