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Question Number 63457 Answers: 1 Comments: 1

((3/4))^x ((2/3))^y =((32)/(27)) find the value of x and yy y.

$$\left(\frac{\mathrm{3}}{\mathrm{4}}\right)^{{x}} \left(\frac{\mathrm{2}}{\mathrm{3}}\right)^{{y}} =\frac{\mathrm{32}}{\mathrm{27}}\:{find}\:{the}\:{value}\:{of}\:{x}\:{and}\:{yy} \\ $$$${y}. \\ $$

Question Number 63449 Answers: 0 Comments: 2

Do you remember that π≈3.1415926535897932⋱ 38462643383279502884⋱ 197...

$${Do}\:{you}\:{remember}\:{that} \\ $$$$\pi\approx\mathrm{3}.\mathrm{1415926535897932}\ddots \\ $$$$\mathrm{38462643383279502884}\ddots \\ $$$$\mathrm{197}... \\ $$

Question Number 63447 Answers: 1 Comments: 2

How to calculate ⌈(n) using gamma function ∀n∈R

$${How}\:{to}\:{calculate}\:\lceil\left({n}\right)\: \\ $$$${using}\:{gamma}\:{function} \\ $$$$\forall{n}\in{R} \\ $$

Question Number 63438 Answers: 1 Comments: 2

Question Number 63433 Answers: 1 Comments: 1

∫_1 ^x x^2 −3x(√x)dx =((−716)/(15)) then calculate ∫_x ^(x+1) (1/(x+3))dx

$$\underset{\mathrm{1}} {\overset{{x}} {\int}}{x}^{\mathrm{2}} −\mathrm{3}{x}\sqrt{{x}}{dx}\:=\frac{−\mathrm{716}}{\mathrm{15}} \\ $$$${then}\:{calculate}\:\underset{{x}} {\overset{{x}+\mathrm{1}} {\int}}\frac{\mathrm{1}}{{x}+\mathrm{3}}{dx} \\ $$

Question Number 63430 Answers: 1 Comments: 0

Question Number 63428 Answers: 0 Comments: 2

It is given that S_n =Σ_(r=1) ^n (3r^(2 ) −3r−1). Use the the formulae of Σ_(r=1) ^n r^(2 ) and Σ_(r=1) ^n r to show that S_n =n^3 . sir Forkum Michael

$${It}\:{is}\:{given}\:{that}\:{S}_{{n}} =\underset{{r}=\mathrm{1}} {\overset{{n}} {\sum}}\left(\mathrm{3}{r}^{\mathrm{2}\:} −\mathrm{3}{r}−\mathrm{1}\right).\:{Use}\:{the}\:{the}\:{formulae} \\ $$$${of}\:\underset{{r}=\mathrm{1}} {\overset{{n}} {\sum}}{r}^{\mathrm{2}\:\:} {and}\:\underset{{r}=\mathrm{1}} {\overset{{n}} {\sum}}{r}\:\:{to}\:{show}\:{that}\:{S}_{{n}} ={n}^{\mathrm{3}} . \\ $$$${sir}\:{Forkum}\:{Michael} \\ $$

Question Number 63427 Answers: 0 Comments: 4

(1) A plane contains the lines ((x+1)/2)=((4−y)/2)=((z−2)/3) and r= (2i+2j + 12k)+t(−i+2j +4k). find (a) the angle between these lines. (b) A cartesian equation of the plane. (2) Given the lines l_1 :((x−10)/3)=((y−1)/1)=((z−9)/4) l_2 :r=(−9j+13k)+μ(i+2j−3k) where μ is a parameter; l_3 :((x+10)/4)=((y+5)/3)=((z+4)/1). a) show that the point (4,−1,1) is common to l_1 and l_2 . Find b) the point of intersection of l_2 and l_3 . c) A vector parametric equation of the plane containing the lines l_2 and l_3 . sir Forkum Michael

Question Number 63426 Answers: 1 Comments: 5

show that (a) cos[2cos^(−1) (x) +sin^(−1) (x)]= −(√(1−x^2 )) (b) ((sinα + sinβ)/(cosα−cosβ))=cot(((β−α)/2)) (c) 2cos((π/3)+p)≊ 1−(√3) if p is small enough to neglect p^2 . (d) if θ =(1/2)sin^(−1) ((3/4)), show that sinθ−cosθ = ±(1/2) (e)write tan3A in terms of tanA (f) Factorise cosθ − cos3θ−cos5θ+cos7θ (g)i) verify that f(x)=((sin2θ+sin10θ)/(cos2θ+cos10θ))=((2tan3θ)/(1−tan^2 3θ)) ii) hence find in radians the general solution of f(x)=1 sir Forkum Michael

$${show}\:{that}\: \\ $$$$\left({a}\right)\:{cos}\left[\mathrm{2}{cos}^{−\mathrm{1}} \left({x}\right)\:+{sin}^{−\mathrm{1}} \left({x}\right)\right]=\:−\sqrt{\mathrm{1}−{x}^{\mathrm{2}} }\: \\ $$$$\left({b}\right)\:\frac{{sin}\alpha\:+\:{sin}\beta}{{cos}\alpha−{cos}\beta}={cot}\left(\frac{\beta−\alpha}{\mathrm{2}}\right) \\ $$$$\left({c}\right)\:\mathrm{2}{cos}\left(\frac{\pi}{\mathrm{3}}+{p}\right)\approxeq\:\mathrm{1}−\sqrt{\mathrm{3}}\:{if}\:{p}\:{is}\:{small}\:{enough}\:{to}\:{neglect}\:{p}^{\mathrm{2}} . \\ $$$$\left({d}\right)\:{if}\:\theta\:=\frac{\mathrm{1}}{\mathrm{2}}{sin}^{−\mathrm{1}} \left(\frac{\mathrm{3}}{\mathrm{4}}\right),\:{show}\:{that}\:{sin}\theta−{cos}\theta\:=\:\pm\frac{\mathrm{1}}{\mathrm{2}} \\ $$$$\left({e}\right){write}\:{tan}\mathrm{3}{A}\:{in}\:{terms}\:{of}\:{tanA} \\ $$$$\left({f}\right)\:{Factorise}\:{cos}\theta\:−\:{cos}\mathrm{3}\theta−{cos}\mathrm{5}\theta+{cos}\mathrm{7}\theta \\ $$$$\left.\left({g}\right){i}\right)\:{verify}\:{that}\:{f}\left({x}\right)=\frac{{sin}\mathrm{2}\theta+{sin}\mathrm{10}\theta}{{cos}\mathrm{2}\theta+{cos}\mathrm{10}\theta}=\frac{\mathrm{2}{tan}\mathrm{3}\theta}{\mathrm{1}−{tan}^{\mathrm{2}} \mathrm{3}\theta} \\ $$$$\left.\:\:{ii}\right)\:{hence}\:{find}\:{in}\:{radians}\:{the}\:{general}\:{solution}\:{of}\:\:{f}\left({x}\right)=\mathrm{1} \\ $$$${sir}\:{Forkum}\:{Michael} \\ $$

Question Number 63425 Answers: 0 Comments: 0

The probability that a vaccinated person(V) contracts a disease is (1/(20)). For a person vaccinated(V ′) , the probability of contracting a disease is (5/6). In a certain town 90%of thepopulation has been vaccinated against a disease. A person is selected at random from the town,find the probability that: (a) he has the disease, (b) he is vaccinated or he has the disease. sir Forkum Michael

$${The}\:{probability}\:{that}\:{a}\:{vaccinated}\:{person}\left({V}\right)\:{contracts}\:{a}\:{disease} \\ $$$${is}\:\frac{\mathrm{1}}{\mathrm{20}}.\:{For}\:{a}\:{person}\:{vaccinated}\left({V}\:'\right)\:,\:{the}\:{probability}\:{of}\:{contracting} \\ $$$${a}\:{disease}\:{is}\:\frac{\mathrm{5}}{\mathrm{6}}.\:{In}\:{a}\:{certain}\:{town}\:\mathrm{90\%}{of}\:{thepopulation}\:{has} \\ $$$${been}\:{vaccinated}\:{against}\:{a}\:{disease}.\:{A}\:{person}\:{is}\:{selected}\:{at} \\ $$$${random}\:{from}\:{the}\:{town},{find}\:{the}\:{probability}\:{that}: \\ $$$$\left({a}\right)\:{he}\:{has}\:{the}\:{disease}, \\ $$$$\left({b}\right)\:{he}\:{is}\:{vaccinated}\:{or}\:{he}\:{has}\:{the}\:{disease}. \\ $$$${sir}\:{Forkum}\:{Michael} \\ $$

Question Number 63424 Answers: 0 Comments: 0

A colony of bacteria if left undisturbed will grow at a rate proportional to the number of bacteria, P present at time,t. However,a toxic substance is being added slowly such that at time t, the bacteria also die at the rate μPt where μ is a positive constant. (a) Show that at time t the rate of growth of the bacteria in the colony is governed by the differential equation (dP/dt)= (k−μt)p where k is apositive constant. when t=0, (dP/dt)=2P and when t=1, (dP/dt)=((19)/(10))P (b) show that (dP/dt)= (1/(10))(20−t)P. Sir Forkum Michael.

$${A}\:{colony}\:{of}\:{bacteria}\:{if}\:{left}\:{undisturbed}\:{will}\:{grow}\:{at}\:{a}\:{rate} \\ $$$${proportional}\:{to}\:{the}\:{number}\:{of}\:{bacteria},\:{P}\:{present}\:{at}\:{time},{t}. \\ $$$${However},{a}\:{toxic}\:{substance}\:{is}\:{being}\:{added}\:{slowly}\:{such}\:{that} \\ $$$${at}\:{time}\:{t},\:{the}\:{bacteria}\:{also}\:{die}\:{at}\:{the}\:{rate}\:\mu{Pt}\:{where}\:\mu\:{is} \\ $$$${a}\:{positive}\:{constant}. \\ $$$$\left({a}\right)\:\:{Show}\:{that}\:{at}\:{time}\:{t}\:{the}\:{rate}\:{of}\:{growth}\:{of}\:{the}\:{bacteria}\:{in} \\ $$$${the}\:{colony}\:{is}\:{governed}\:{by}\:{the}\:{differential}\:{equation} \\ $$$$\:\frac{{dP}}{{dt}}=\:\left({k}−\mu{t}\right){p}\:{where}\:{k}\:{is}\:{apositive}\:{constant}. \\ $$$${when}\:{t}=\mathrm{0},\:\frac{{dP}}{{dt}}=\mathrm{2}{P}\:{and}\:{when}\:{t}=\mathrm{1},\:\frac{{dP}}{{dt}}=\frac{\mathrm{19}}{\mathrm{10}}{P} \\ $$$$\left({b}\right)\:{show}\:{that}\: \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\frac{{dP}}{{dt}}=\:\frac{\mathrm{1}}{\mathrm{10}}\left(\mathrm{20}−{t}\right){P}. \\ $$$$\:{Sir}\:{Forkum}\:{Michael}. \\ $$

Question Number 63423 Answers: 0 Comments: 1

Prove that ((sin^2 (36°))/(cos^2 (72°)))+sin^2 (72°)cos(36°)=((45+11(√5))/(16))

$${Prove}\:{that}\: \\ $$$$\frac{{sin}^{\mathrm{2}} \left(\mathrm{36}°\right)}{{cos}^{\mathrm{2}} \left(\mathrm{72}°\right)}+{sin}^{\mathrm{2}} \left(\mathrm{72}°\right){cos}\left(\mathrm{36}°\right)=\frac{\mathrm{45}+\mathrm{11}\sqrt{\mathrm{5}}}{\mathrm{16}} \\ $$

Question Number 63579 Answers: 0 Comments: 4

lim_(x→0) ((2^(sec(x)) − 2^(cos(x)) )/x^2 )

$$\:\underset{{x}\rightarrow\mathrm{0}} {\mathrm{lim}}\:\:\frac{\mathrm{2}^{\mathrm{sec}\left(\mathrm{x}\right)} \:−\:\mathrm{2}^{\mathrm{cos}\left(\mathrm{x}\right)} }{\mathrm{x}^{\mathrm{2}} } \\ $$

Question Number 63410 Answers: 2 Comments: 2