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Question Number 159180    Answers: 1   Comments: 0

Question Number 159179    Answers: 1   Comments: 0

X^3 −4X^2 −6X−24 = 0

$$\:\:\mathcal{X}^{\mathrm{3}} −\mathrm{4}\mathcal{X}^{\mathrm{2}} −\mathrm{6}\mathcal{X}−\mathrm{24}\:=\:\mathrm{0} \\ $$

Question Number 159173    Answers: 1   Comments: 0

Question Number 159172    Answers: 1   Comments: 0

Question Number 159171    Answers: 1   Comments: 0

Prove by absurd that log 2 is the number irrational

$${Prove}\:{by}\:{absurd}\:{that}\:\mathrm{log}\:\mathrm{2}\:{is}\:{the} \\ $$$${number}\:{irrational} \\ $$

Question Number 159170    Answers: 1   Comments: 1

Question Number 159300    Answers: 1   Comments: 0

Question Number 159163    Answers: 2   Comments: 1

Question Number 159159    Answers: 1   Comments: 0

Three students are runing for school SRC president, kada′s probability of winning is (1/8), Atiga′s probability of winnig (1/3) and kada is half as likely to win as Apio. i. what is the probability that both Atiga and Apio draw ii. what is the probability that only one wins the presidency.

$$\:\mathrm{Three}\:\mathrm{students}\:\mathrm{are}\:\mathrm{runing}\:\mathrm{for}\:\mathrm{school} \\ $$$$\:\mathrm{SRC}\:\mathrm{president},\:\mathrm{kada}'\mathrm{s}\:\mathrm{probability}\:\mathrm{of} \\ $$$$\mathrm{winning}\:\mathrm{is}\:\frac{\mathrm{1}}{\mathrm{8}},\:\mathrm{Atiga}'\mathrm{s}\:\mathrm{probability}\: \\ $$$$\mathrm{of}\:\mathrm{winnig}\:\frac{\mathrm{1}}{\mathrm{3}}\:\mathrm{and}\:\mathrm{kada}\:\mathrm{is}\:\mathrm{half}\:\mathrm{as}\:\mathrm{likely} \\ $$$$\mathrm{to}\:\mathrm{win}\:\mathrm{as}\:\mathrm{Apio}.\: \\ $$$$\mathrm{i}.\:\mathrm{what}\:\mathrm{is}\:\mathrm{the}\:\mathrm{probability}\:\mathrm{that}\:\mathrm{both}\: \\ $$$$\:\mathrm{Atiga}\:\mathrm{and}\:\mathrm{Apio}\:\mathrm{draw} \\ $$$$\mathrm{ii}.\:\mathrm{what}\:\mathrm{is}\:\mathrm{the}\:\mathrm{probability}\:\mathrm{that}\:\mathrm{only} \\ $$$$\mathrm{one}\:\mathrm{wins}\:\mathrm{the}\:\mathrm{presidency}. \\ $$

Question Number 159292    Answers: 1   Comments: 0

Find: 𝛀 =∫_( 0) ^( ∞) ((x arctan(x))/((x + 1)(x^2 + 1))) dx

$$\mathrm{Find}:\:\:\boldsymbol{\Omega}\:=\underset{\:\mathrm{0}} {\overset{\:\infty} {\int}}\frac{\mathrm{x}\:\mathrm{arctan}\left(\mathrm{x}\right)}{\left(\mathrm{x}\:+\:\mathrm{1}\right)\left(\mathrm{x}^{\mathrm{2}} \:+\:\mathrm{1}\right)}\:\mathrm{dx} \\ $$$$ \\ $$

Question Number 159157    Answers: 1   Comments: 0

Question Number 159184    Answers: 0   Comments: 2

Question Number 159153    Answers: 0   Comments: 0

Question Number 159152    Answers: 1   Comments: 0

Question Number 159146    Answers: 0   Comments: 0

Find the absolute extrema of f(x)= 2 csc x + cot x on the interval ((π/2), ((3π)/2) ]

$${Find}\:{the}\:{absolute}\:{extrema}\:{of} \\ $$$${f}\left({x}\right)=\:\mathrm{2}\:\mathrm{csc}\:{x}\:+\:\mathrm{cot}\:{x}\:{on}\:{the}\: \\ $$$${interval}\:\left(\frac{\pi}{\mathrm{2}},\:\frac{\mathrm{3}\pi}{\mathrm{2}}\:\right] \\ $$

Question Number 159143    Answers: 1   Comments: 0

Question Number 159142    Answers: 2   Comments: 1

lim_(x→−(π/4)) (((π/( (√8))) −(√2) x. tan x)/(sin x+cos x)) =?

$$\:\underset{{x}\rightarrow−\frac{\pi}{\mathrm{4}}} {\mathrm{lim}}\:\frac{\frac{\pi}{\:\sqrt{\mathrm{8}}}\:−\sqrt{\mathrm{2}}\:{x}.\:\mathrm{tan}\:{x}}{\mathrm{sin}\:{x}+\mathrm{cos}\:{x}}\:=? \\ $$

Question Number 159137    Answers: 1   Comments: 4

Yesterday,Mr John earned $50 mowing lawns, today Mr. John earned 60percent of what he earned yesterday mowing lawns. how much money did Mr Jacob earned mowing lawns today??

$$\mathrm{Yesterday},\mathrm{Mr}\:\mathrm{John}\:\mathrm{earned}\:\$\mathrm{50} \\ $$$$\mathrm{mowing}\:\mathrm{lawns},\:\mathrm{today}\:\mathrm{Mr}.\:\mathrm{John} \\ $$$$\mathrm{earned}\:\mathrm{60percent}\:\mathrm{of}\:\mathrm{what}\:\mathrm{he}\:\mathrm{earned} \\ $$$$\mathrm{yesterday}\:\mathrm{mowing}\:\mathrm{lawns}.\:\mathrm{how} \\ $$$$\mathrm{much}\:\mathrm{money}\:\mathrm{did}\:\mathrm{Mr}\:\mathrm{Jacob}\:\mathrm{earned} \\ $$$$\mathrm{mowing}\:\mathrm{lawns}\:\mathrm{today}?? \\ $$

Question Number 159136    Answers: 0   Comments: 0

Question Number 159135    Answers: 1   Comments: 4

Question Number 159131    Answers: 1   Comments: 0

Question Number 159130    Answers: 0   Comments: 0

f ∈ C^0 (R,R) , ((f(x))/x)→_(x→+∞) 0 Does lim_(x→+∞) f(x+1) − f(x) exist?

$${f}\:\in\:\mathcal{C}^{\mathrm{0}} \left(\mathbb{R},\mathbb{R}\right)\:,\:\frac{{f}\left({x}\right)}{{x}}\underset{{x}\rightarrow+\infty} {\rightarrow}\mathrm{0} \\ $$$${Does}\:\underset{{x}\rightarrow+\infty} {\mathrm{lim}}\:{f}\left({x}+\mathrm{1}\right)\:−\:{f}\left({x}\right)\:{exist}? \\ $$

Question Number 159123    Answers: 2   Comments: 0

Question Number 159122    Answers: 0   Comments: 0

In order to monitor buses in a travel agency, the manager decides to monitor the number of break downs of the buses using the sequence {x_n } defined by x_(n+1) = 1.05 x_n + 4. Given that x_0 = 40. is the number of break downs by the buses from the 1^(st) of january 2000, and that for every n∈N, we denote x_n the number of breakdowns of the buses as from 1^(st) of january of the year (2000 + n) (a) Calculate x_1 , x_2 , x_3 (b) Consider the sequence {y_n } defined by y_n = x_n + 80 for all n ∈ N (i) express y_(n+1) in terms of y_n and deduce the nature of the sequence {y_n }. (ii) Express y_n in terms of n. deduce x_n in terms of n (iv) find the number of break downs that will be registered by 1^(st) january 2021.

$$\mathrm{In}\:\mathrm{order}\:\mathrm{to}\:\mathrm{monitor}\:\mathrm{buses}\:\mathrm{in}\:\mathrm{a}\:\mathrm{travel} \\ $$$$\mathrm{agency},\:\mathrm{the}\:\mathrm{manager}\:\mathrm{decides}\:\mathrm{to}\:\mathrm{monitor} \\ $$$$\mathrm{the}\:\mathrm{number}\:\mathrm{of}\:\mathrm{break}\:\mathrm{downs}\:\mathrm{of}\:\mathrm{the}\:\mathrm{buses} \\ $$$$\mathrm{using}\:\mathrm{the}\:\mathrm{sequence}\:\left\{{x}_{{n}} \right\}\:\mathrm{defined}\:\mathrm{by} \\ $$$${x}_{{n}+\mathrm{1}} \:=\:\mathrm{1}.\mathrm{05}\:{x}_{{n}} \:+\:\mathrm{4}.\:\mathrm{Given}\:\mathrm{that}\:{x}_{\mathrm{0}} \:=\:\mathrm{40}. \\ $$$$\mathrm{is}\:\mathrm{the}\:\mathrm{number}\:\mathrm{of}\:\mathrm{break}\:\mathrm{downs}\:\mathrm{by}\:\mathrm{the}\:\mathrm{buses} \\ $$$$\mathrm{from}\:\mathrm{the}\:\mathrm{1}^{\mathrm{st}} \:\mathrm{of}\:\mathrm{january}\:\mathrm{2000},\:\mathrm{and}\:\mathrm{that} \\ $$$$\mathrm{for}\:\mathrm{every}\:{n}\in\mathbb{N},\:\mathrm{we}\:\mathrm{denote}\:{x}_{{n}} \:\mathrm{the}\:\mathrm{number} \\ $$$$\mathrm{of}\:\mathrm{breakdowns}\:\mathrm{of}\:\mathrm{the}\:\mathrm{buses}\:\mathrm{as}\:\mathrm{from}\:\mathrm{1}^{\mathrm{st}} \\ $$$$\mathrm{of}\:\mathrm{january}\:\mathrm{of}\:\mathrm{the}\:\mathrm{year}\:\left(\mathrm{2000}\:+\:{n}\right) \\ $$$$\left(\mathrm{a}\right)\:\mathrm{Calculate}\:{x}_{\mathrm{1}} ,\:{x}_{\mathrm{2}} \:,\:{x}_{\mathrm{3}} \\ $$$$\left(\mathrm{b}\right)\:\mathrm{Consider}\:\mathrm{the}\:\mathrm{sequence}\:\left\{{y}_{{n}} \right\}\:\mathrm{defined} \\ $$$$\mathrm{by}\:{y}_{{n}} \:=\:{x}_{{n}} \:+\:\mathrm{80}\:\mathrm{for}\:\mathrm{all}\:{n}\:\in\:\mathbb{N} \\ $$$$\left(\mathrm{i}\right)\:\mathrm{express}\:{y}_{{n}+\mathrm{1}} \:\mathrm{in}\:\mathrm{terms}\:\mathrm{of}\:{y}_{{n}} \:\mathrm{and} \\ $$$$\mathrm{deduce}\:\mathrm{the}\:\mathrm{nature}\:\mathrm{of}\:\mathrm{the}\:\mathrm{sequence}\:\left\{{y}_{{n}} \right\}. \\ $$$$\left(\mathrm{ii}\right)\:\mathrm{Express}\:{y}_{{n}} \:\mathrm{in}\:\mathrm{terms}\:\mathrm{of}\:{n}.\:\mathrm{deduce}\:{x}_{{n}} \\ $$$$\mathrm{in}\:\mathrm{terms}\:\mathrm{of}\:{n} \\ $$$$\left(\mathrm{iv}\right)\:\mathrm{find}\:\mathrm{the}\:\mathrm{number}\:\mathrm{of}\:\mathrm{break}\:\mathrm{downs} \\ $$$$\mathrm{that}\:\mathrm{will}\:\mathrm{be}\:\mathrm{registered}\:\mathrm{by}\:\mathrm{1}^{\mathrm{st}} \:\mathrm{january}\: \\ $$$$\mathrm{2021}. \\ $$

Question Number 159121    Answers: 0   Comments: 0

Consider f(x) = x^3 + 2x −1. Use the intermidiate value theorem and the Rolle theorem to establish that the equation f(x) = 0 has a unique solution denoted a_0 ∈] 0,1[.

$$\mathrm{Consider} \\ $$$${f}\left({x}\right)\:=\:{x}^{\mathrm{3}} \:+\:\mathrm{2}{x}\:−\mathrm{1}. \\ $$$$\mathrm{Use}\:\mathrm{the}\:\mathrm{intermidiate}\:\mathrm{value}\:\mathrm{theorem}\:\mathrm{and} \\ $$$$\mathrm{the}\:\mathrm{Rolle}\:\mathrm{theorem}\:\mathrm{to}\:\mathrm{establish}\:\mathrm{that}\:\mathrm{the} \\ $$$$\mathrm{equation}\:{f}\left({x}\right)\:=\:\mathrm{0}\:\mathrm{has}\:\mathrm{a}\:\mathrm{unique}\:\mathrm{solution} \\ $$$$\left.\mathrm{denoted}\:{a}_{\mathrm{0}} \in\right]\:\mathrm{0},\mathrm{1}\left[.\:\right. \\ $$

Question Number 159119    Answers: 0   Comments: 0

Assume x;y;z>0 and x^2 +y^2 +z^2 =12 Prove that: Σ_(cycl) (((x/y) + 1 + (y/x))/((1/x) + (1/y))) ≤ 9

$$\mathrm{Assume}\:\:\mathrm{x};\mathrm{y};\mathrm{z}>\mathrm{0}\:\:\mathrm{and}\:\:\mathrm{x}^{\mathrm{2}} +\mathrm{y}^{\mathrm{2}} +\mathrm{z}^{\mathrm{2}} =\mathrm{12} \\ $$$$\mathrm{Prove}\:\mathrm{that}:\:\:\underset{\boldsymbol{\mathrm{cycl}}} {\sum}\:\frac{\frac{\mathrm{x}}{\mathrm{y}}\:+\:\mathrm{1}\:+\:\frac{\mathrm{y}}{\mathrm{x}}}{\frac{\mathrm{1}}{\mathrm{x}}\:+\:\frac{\mathrm{1}}{\mathrm{y}}}\:\leqslant\:\mathrm{9} \\ $$

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