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AllQuestion and Answers: Page 170

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?

Question Number 204879 Answers: 2 Comments: 2

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

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

Question Number 204853 Answers: 2 Comments: 0

f(x)=(√(1−log_((2x+5)) ((x+1)^2 ))) Find the domain of this function

$${f}\left({x}\right)=\sqrt{\mathrm{1}−\mathrm{log}_{\left(\mathrm{2}{x}+\mathrm{5}\right)} \left(\left({x}+\mathrm{1}\right)^{\mathrm{2}} \right)} \\ $$$$\mathrm{Find}\:\mathrm{the}\:\mathrm{domain}\:\mathrm{of}\:\mathrm{this}\:\mathrm{function} \\ $$

Question Number 204851 Answers: 1 Comments: 0

Question Number 204845 Answers: 1 Comments: 0

Question Number 204844 Answers: 2 Comments: 1

find ⌊Σ_(n=1) ^(80) (1/( (√n)))⌋=?

$${find}\:\lfloor\underset{{n}=\mathrm{1}} {\overset{\mathrm{80}} {\sum}}\frac{\mathrm{1}}{\:\sqrt{{n}}}\rfloor=? \\ $$

Question Number 204841 Answers: 1 Comments: 1

Racionalizar el denominador ((1 − x)/( (x)^(1/3) − (√(√x))))

$${Racionalizar}\:{el}\:{denominador} \\ $$$$\frac{\mathrm{1}\:−\:{x}}{\:\sqrt[{\mathrm{3}}]{{x}}\:−\:\sqrt{\sqrt{{x}}}} \\ $$

Question Number 204826 Answers: 1 Comments: 11

A rectangular enclosure is to be made against a straight wall using three lengths of fencing. The total length of the fencing available is 50m. Show that the area of the enclosure is 50x − 2x^2 , where x is the length of the sides perpendicular to the wall. Hence find the maximum area of the enclosure.

$${A}\:{rectangular}\:{enclosure}\:{is}\:{to}\:{be}\:{made} \\ $$$${against}\:{a}\:{straight}\:{wall}\:{using}\:{three} \\ $$$${lengths}\:{of}\:{fencing}.\:{The}\:{total}\:{length}\:{of} \\ $$$${the}\:{fencing}\:{available}\:{is}\:\mathrm{50}{m}.\:{Show} \\ $$$${that}\:{the}\:{area}\:{of}\:{the}\:{enclosure}\:{is} \\ $$$$\mathrm{50}{x}\:−\:\mathrm{2}{x}^{\mathrm{2}} ,\:{where}\:{x}\:{is}\:{the}\:{length}\:{of}\:{the} \\ $$$${sides}\:{perpendicular}\:{to}\:{the}\:{wall}.\:{Hence} \\ $$$${find}\:{the}\:{maximum}\:{area}\:{of}\:{the} \\ $$$${enclosure}. \\ $$

Question Number 204815 Answers: 2 Comments: 0

Given (3p^2 −p+q^3 )^(12) , find the coefficient of p^(10) q^6

$$\:\:\:\:\:\mathrm{Given}\:\left(\mathrm{3p}^{\mathrm{2}} −\mathrm{p}+\mathrm{q}^{\mathrm{3}} \right)^{\mathrm{12}} \:,\:\mathrm{find}\:\mathrm{the}\: \\ $$$$\:\:\:\:\mathrm{coefficient}\:\mathrm{of}\:\mathrm{p}^{\mathrm{10}} \mathrm{q}^{\mathrm{6}} \\ $$

Question Number 204805 Answers: 0 Comments: 5

Why lim_(n→∞) (x^n u_n )=0 ,When u_n is bounded

$$\mathrm{Why}\:\underset{{n}\rightarrow\infty} {\mathrm{lim}}\left({x}^{{n}} {u}_{{n}} \right)=\mathrm{0}\:,\mathrm{When}\:{u}_{{n}} \:\mathrm{is}\:\mathrm{bounded}\: \\ $$

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