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Question Number 175280    Answers: 2   Comments: 0

A biology class at central high school predicted that a local population of animals will double in size every 12 years. The population at the beginning of 2014 was estimated to be 50 animals. If P represents the population n years after 2014. Find the equations represents the class model of the population over time?

$$\mathrm{A}\:\mathrm{biology}\:\:\mathrm{class}\:\mathrm{at}\:\mathrm{central}\:\:\mathrm{high}\:\:\mathrm{school} \\ $$$$\mathrm{predicted}\:\:\mathrm{that}\:\:\mathrm{a}\:\:\mathrm{local}\:\:\mathrm{population}\:\:\mathrm{of}\:\:\:\mathrm{animals} \\ $$$$\mathrm{will}\:\:\mathrm{double}\:\:\mathrm{in}\:\:\mathrm{size}\:\:\mathrm{every}\:\:\mathrm{12}\:\:\mathrm{years}. \\ $$$$\mathrm{The}\:\:\mathrm{population}\:\:\mathrm{at}\:\:\mathrm{the}\:\:\mathrm{beginning}\:\:\mathrm{of}\:\:\mathrm{2014} \\ $$$$\mathrm{was}\:\:\mathrm{estimated}\:\:\mathrm{to}\:\mathrm{be}\:\:\mathrm{50}\:\:\mathrm{animals}. \\ $$$$\mathrm{If}\:\:{P}\:\:\mathrm{represents}\:\:\mathrm{the}\:\:\mathrm{population}\:\:{n}\:\:\mathrm{years}\:\:\mathrm{after}\:\:\mathrm{2014}. \\ $$$$\mathrm{Find}\:\:\mathrm{the}\:\:\mathrm{equations}\:\:\mathrm{represents}\:\:\mathrm{the}\:\:\mathrm{class}\:\:\mathrm{model} \\ $$$$\mathrm{of}\:\:\mathrm{the}\:\:\mathrm{population}\:\:\mathrm{over}\:\:\mathrm{time}? \\ $$

Question Number 175277    Answers: 0   Comments: 0

Question Number 175279    Answers: 3   Comments: 0

Find the angle between the lines r = ((1),(0),(4) ) + λ ((2),((−1)),((−1)) ) r = ((1),(0),(4) ) + λ ((3),(0),(1) )

$$\mathrm{Find}\:\:\mathrm{the}\:\:\mathrm{angle}\:\:\mathrm{between}\:\:\mathrm{the}\:\:\mathrm{lines} \\ $$$${r}\:=\:\begin{pmatrix}{\mathrm{1}}\\{\mathrm{0}}\\{\mathrm{4}}\end{pmatrix}\:\:+\:\lambda\begin{pmatrix}{\mathrm{2}}\\{−\mathrm{1}}\\{−\mathrm{1}}\end{pmatrix} \\ $$$${r}\:=\:\begin{pmatrix}{\mathrm{1}}\\{\mathrm{0}}\\{\mathrm{4}}\end{pmatrix}\:\:+\:\lambda\begin{pmatrix}{\mathrm{3}}\\{\mathrm{0}}\\{\mathrm{1}}\end{pmatrix} \\ $$$$ \\ $$

Question Number 175275    Answers: 1   Comments: 0

Question Number 181521    Answers: 2   Comments: 0

Find the range of x such that { ((sinx>0)),(((√3)sinx+cosx>0)),((0<x<2π)) :}

$$\mathrm{Find}\:\mathrm{the}\:\mathrm{range}\:\mathrm{of}\:{x}\:\:\mathrm{such}\:\mathrm{that} \\ $$$$\begin{cases}{\mathrm{sin}{x}>\mathrm{0}}\\{\sqrt{\mathrm{3}}\mathrm{sin}{x}+\mathrm{cos}{x}>\mathrm{0}}\\{\mathrm{0}<{x}<\mathrm{2}\pi}\end{cases} \\ $$

Question Number 175266    Answers: 0   Comments: 1

Find matrix ∣A∣A^(-1) given that matrix A= (((√2),(-1),1,( 0)),(4,3,2,(-1)),(0,2,3,( 1)),(1,(-1),0,( 1)) ) using row operations

$$\mathrm{Find}\:\mathrm{matrix}\:\mid\mathrm{A}\mid\mathrm{A}^{-\mathrm{1}} \: \\ $$$$\mathrm{given}\:\mathrm{that}\:\mathrm{matrix}\: \\ $$$$\mathrm{A}=\begin{pmatrix}{\sqrt{\mathrm{2}}}&{-\mathrm{1}}&{\mathrm{1}}&{\:\mathrm{0}}\\{\mathrm{4}}&{\mathrm{3}}&{\mathrm{2}}&{-\mathrm{1}}\\{\mathrm{0}}&{\mathrm{2}}&{\mathrm{3}}&{\:\mathrm{1}}\\{\mathrm{1}}&{-\mathrm{1}}&{\mathrm{0}}&{\:\mathrm{1}}\end{pmatrix} \\ $$$$\mathrm{using}\:\mathrm{row}\:\mathrm{operations} \\ $$

Question Number 175265    Answers: 1   Comments: 0

Given that matrix B= { (( (√3))),((-(√5))) :} {: (( (√2))),(( (√7))) } find B^(-1) using row operation

$$\mathrm{Given}\:\mathrm{that}\:\mathrm{matrix}\: \\ $$$$\mathrm{B}=\begin{cases}{\:\:\sqrt{\mathrm{3}}}\\{-\sqrt{\mathrm{5}}}\end{cases}\left.\begin{matrix}{\:\:\:\:\sqrt{\mathrm{2}}}\\{\:\:\:\:\sqrt{\mathrm{7}}}\end{matrix}\right\}\:\mathrm{find}\: \\ $$$$\mathrm{B}^{-\mathrm{1}} \:\mathrm{using}\:\mathrm{row}\:\mathrm{operation} \\ $$

Question Number 175264    Answers: 1   Comments: 0

Question Number 175251    Answers: 1   Comments: 0

Given that 4(x−3)^2 +9(y+2)^2 =27 graph the ellipse

$$\boldsymbol{\mathrm{Given}}\:\boldsymbol{\mathrm{that}}\: \\ $$$$\:\mathrm{4}\left(\boldsymbol{\mathrm{x}}−\mathrm{3}\right)^{\mathrm{2}} +\mathrm{9}\left(\boldsymbol{\mathrm{y}}+\mathrm{2}\right)^{\mathrm{2}} =\mathrm{27} \\ $$$$\:\boldsymbol{\mathrm{graph}}\:\boldsymbol{\mathrm{the}}\:\boldsymbol{\mathrm{ellipse}} \\ $$

Question Number 175250    Answers: 0   Comments: 0

If in △ABC and A = ((2π)/3) then: ((√(1/(sinB))))^3 + ((√(1/(sinC))))^3 ≥ 4 (√2)

$$\mathrm{If}\:\mathrm{in}\:\:\bigtriangleup\mathrm{ABC}\:\:\mathrm{and}\:\:\mathrm{A}\:=\:\frac{\mathrm{2}\pi}{\mathrm{3}}\:\:\mathrm{then}: \\ $$$$\left(\sqrt{\frac{\mathrm{1}}{\mathrm{sinB}}}\right)^{\mathrm{3}} +\:\left(\sqrt{\frac{\mathrm{1}}{\mathrm{sinC}}}\right)^{\mathrm{3}} \geqslant\:\mathrm{4}\:\sqrt{\mathrm{2}} \\ $$

Question Number 175247    Answers: 2   Comments: 0

Question Number 175244    Answers: 1   Comments: 0

prove that ∫_0 ^a ∫_0 ^(√(a^2 −x^2 )) ((dx dy)/((1+e^y )(√(a^2 −x^2 −y^2 )))) = (π/2)log ((2e^a )/(1+e^a ))

$$\:{prove}\:{that} \\ $$$$\:\int_{\mathrm{0}} ^{{a}} \int_{\mathrm{0}} ^{\sqrt{{a}^{\mathrm{2}} −{x}^{\mathrm{2}} }} \frac{{dx}\:{dy}}{\left(\mathrm{1}+{e}^{{y}} \right)\sqrt{{a}^{\mathrm{2}} −{x}^{\mathrm{2}} −{y}^{\mathrm{2}} }}\:=\:\frac{\pi}{\mathrm{2}}\mathrm{log}\:\frac{\mathrm{2}{e}^{{a}} }{\mathrm{1}+{e}^{{a}} } \\ $$

Question Number 175237    Answers: 0   Comments: 0

The equations of the sides AC, BC and AB of a right−angled triangled with lengths a, b and c are y = −7, x=11 and 4x−3y−5=0 respectively. Find the equation of the inscribed circle of the triangle, if its radius r, is given by r = ((a+b−c)/2).

$$\mathrm{The}\:\mathrm{equations}\:\mathrm{of}\:\mathrm{the}\:\mathrm{sides}\:\:\mathrm{AC},\:\mathrm{BC} \\ $$$$\mathrm{and}\:\mathrm{AB}\:\mathrm{of}\:\:\mathrm{a}\:\mathrm{right}−\mathrm{angled}\:\mathrm{triangled} \\ $$$$\mathrm{with}\:\mathrm{lengths}\:{a},\:{b}\:\mathrm{and}\:{c}\:\mathrm{are}\:{y}\:=\:−\mathrm{7}, \\ $$$${x}=\mathrm{11}\:\mathrm{and}\:\mathrm{4}{x}−\mathrm{3}{y}−\mathrm{5}=\mathrm{0}\:\mathrm{respectively}. \\ $$$$\mathrm{Find}\:\mathrm{the}\:\mathrm{equation}\:\mathrm{of}\:\mathrm{the}\:\mathrm{inscribed} \\ $$$$\mathrm{circle}\:\mathrm{of}\:\mathrm{the}\:\mathrm{triangle},\:\mathrm{if}\:\mathrm{its}\:\mathrm{radius}\:{r}, \\ $$$$\mathrm{is}\:\mathrm{given}\:\mathrm{by}\:{r}\:=\:\frac{{a}+{b}−{c}}{\mathrm{2}}. \\ $$

Question Number 181485    Answers: 2   Comments: 0

two medians of a triange are 3 and 4 cm respectively. find the maximum area of the triangle.

$${two}\:{medians}\:{of}\:{a}\:{triange}\:{are}\:\mathrm{3}\:{and} \\ $$$$\mathrm{4}\:{cm}\:{respectively}.\:{find}\:{the}\:{maximum} \\ $$$${area}\:{of}\:{the}\:{triangle}. \\ $$

Question Number 181484    Answers: 0   Comments: 6

If a hen and a half lay an egg and a half in a day and a half how many eggs would one hen lay in one day?

$${If}\:{a}\:{hen}\:{and}\:{a}\:{half} \\ $$$${lay}\:{an}\:{egg}\:{and}\:{a}\:{half} \\ $$$${in}\:{a}\:{day}\:{and}\:{a}\:{half} \\ $$$${how}\:{many}\:{eggs}\:{would} \\ $$$${one}\:{hen}\:{lay}\:{in}\:{one} \\ $$$${day}? \\ $$

Question Number 175240    Answers: 1   Comments: 1

Question Number 175230    Answers: 1   Comments: 0

Question Number 175221    Answers: 2   Comments: 0

Question Number 175218    Answers: 1   Comments: 0

⌊ ( 1 + (√5) )^( 8) ⌋ = ?

$$ \\ $$$$\:\:\lfloor\:\:\left(\:\mathrm{1}\:+\:\sqrt{\mathrm{5}}\:\right)^{\:\mathrm{8}} \:\rfloor\:=\:? \\ $$$$\:\:\: \\ $$

Question Number 175217    Answers: 0   Comments: 1

lim_( x→0) { ⌊ ((cos(x))/x) ⌋ −⌊((1+cos(x))/(1−cos(x))) ⌋}

$$ \\ $$$$\:\:{lim}_{\:{x}\rightarrow\mathrm{0}} \left\{\:\lfloor\:\frac{{cos}\left({x}\right)}{{x}}\:\rfloor\:−\lfloor\frac{\mathrm{1}+{cos}\left({x}\right)}{\mathrm{1}−{cos}\left({x}\right)}\:\rfloor\right\} \\ $$

Question Number 175213    Answers: 2   Comments: 0

Solve by Method of variation parameter (d^2 y/dx^2 )−3(dy/dx)+2y=sinx M.m

$$\mathrm{Solve}\:\mathrm{by}\:\mathrm{Method}\:\mathrm{of}\:\mathrm{variation}\:\mathrm{parameter} \\ $$$$\frac{\mathrm{d}^{\mathrm{2}} \mathrm{y}}{\mathrm{dx}^{\mathrm{2}} }−\mathrm{3}\frac{\mathrm{dy}}{\mathrm{dx}}+\mathrm{2y}=\mathrm{sinx} \\ $$$$ \\ $$$$\mathrm{M}.\mathrm{m} \\ $$

Question Number 175211    Answers: 1   Comments: 0

Assume that the sequence terms tend to the constant value u, so that as n→∞, u_(n−1) →u and u_n →u. (i) show that u^2 +u−1=0 (ii) show that (1/(1+(1/(1+(1/(1+(1/(1+.....))))))))=((−1+(√5))/2)

$$\boldsymbol{\mathrm{Assume}}\:\boldsymbol{\mathrm{that}}\:\boldsymbol{\mathrm{the}}\:\boldsymbol{\mathrm{sequence}}\:\boldsymbol{\mathrm{terms}}\:\boldsymbol{\mathrm{tend}} \\ $$$$\boldsymbol{\mathrm{to}}\:\boldsymbol{\mathrm{the}}\:\boldsymbol{\mathrm{constant}}\:\boldsymbol{\mathrm{value}}\:\boldsymbol{{u}},\:\boldsymbol{\mathrm{so}}\:\boldsymbol{\mathrm{that}}\:\boldsymbol{\mathrm{as}} \\ $$$$\boldsymbol{\mathrm{n}}\rightarrow\infty,\:\boldsymbol{{u}}_{\boldsymbol{{n}}−\mathrm{1}} \rightarrow\boldsymbol{{u}}\:\boldsymbol{\mathrm{and}}\:\boldsymbol{{u}}_{\boldsymbol{{n}}} \rightarrow\boldsymbol{{u}}. \\ $$$$\:\left(\boldsymbol{\mathrm{i}}\right)\:\boldsymbol{\mathrm{show}}\:\boldsymbol{\mathrm{that}}\:\:\boldsymbol{{u}}^{\mathrm{2}} +\boldsymbol{{u}}−\mathrm{1}=\mathrm{0} \\ $$$$\:\left(\boldsymbol{\mathrm{ii}}\right)\:\boldsymbol{\mathrm{show}}\:\boldsymbol{\mathrm{that}}\:\:\frac{\mathrm{1}}{\mathrm{1}+\frac{\mathrm{1}}{\mathrm{1}+\frac{\mathrm{1}}{\mathrm{1}+\frac{\mathrm{1}}{\mathrm{1}+.....}}}}=\frac{−\mathrm{1}+\sqrt{\mathrm{5}}}{\mathrm{2}} \\ $$

Question Number 175210    Answers: 0   Comments: 0

y′x + y = y^( 2) ln(x) u=y^( −1) ⇒ u′ =−y′y^( −2) −y′y^( −2) x −y^(−1) = −ln(x) u′x −u = −ln(x) u′−(1/x) u =((−ln(x))/x) u = e^( −∫−(1/x)dx) ( ∫−((ln(x))/x)e^( −∫(1/x)dx) dx +C) = x (−∫ ((ln(x))/x^( 2) )dx +C) ln(x)=t ∫te^( −t) dt= [ −e^( −t) .t +∫e^(−t) dt] = −(1/x)ln(x) −(1/x) u= −ln(x) −Cx −1 y = (1/(−ln(x)−Cx−1)) ✓

$$ \\ $$$$\:\:\:{y}'{x}\:+\:{y}\:=\:{y}^{\:\mathrm{2}} {ln}\left({x}\right) \\ $$$$\:\:\:\:{u}={y}^{\:−\mathrm{1}} \:\Rightarrow\:{u}'\:=−{y}'{y}^{\:−\mathrm{2}} \\ $$$$\:\:\:\:\:−{y}'{y}^{\:−\mathrm{2}} {x}\:−{y}^{−\mathrm{1}} =\:−{ln}\left({x}\right) \\ $$$$\:\:\:\:\:\:\:{u}'{x}\:−{u}\:=\:−{ln}\left({x}\right) \\ $$$$\:\:\:\:{u}'−\frac{\mathrm{1}}{{x}}\:{u}\:=\frac{−{ln}\left({x}\right)}{{x}}\: \\ $$$$\:\:\:\:\:{u}\:=\:{e}^{\:−\int−\frac{\mathrm{1}}{{x}}{dx}} \left(\:\int−\frac{{ln}\left({x}\right)}{{x}}{e}^{\:−\int\frac{\mathrm{1}}{{x}}{dx}} {dx}\:+{C}\right) \\ $$$$\:\:=\:\:{x}\:\left(−\int\:\frac{{ln}\left({x}\right)}{{x}^{\:\mathrm{2}} }{dx}\:+{C}\right) \\ $$$$\:\:\:\:{ln}\left({x}\right)={t} \\ $$$$\:\:\:\:\:\:\int{te}^{\:−{t}} {dt}=\:\left[\:−{e}^{\:−{t}} .{t}\:+\int{e}^{−{t}} {dt}\right] \\ $$$$\:\:\:\:\:\:=\:−\frac{\mathrm{1}}{{x}}{ln}\left({x}\right)\:−\frac{\mathrm{1}}{{x}} \\ $$$$\:\:\:\:\:\:{u}=\:−{ln}\left({x}\right)\:−{Cx}\:−\mathrm{1} \\ $$$$\:\:\:\:\:\:{y}\:=\:\frac{\mathrm{1}}{−{ln}\left({x}\right)−{Cx}−\mathrm{1}}\:\checkmark \\ $$

Question Number 175199    Answers: 1   Comments: 0

Question Number 175193    Answers: 2   Comments: 0

Question Number 175191    Answers: 2   Comments: 1

∫(√(1−(1/x)))dx=?

$$\int\sqrt{\mathrm{1}−\frac{\mathrm{1}}{{x}}}{dx}=? \\ $$

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