Question and Answers Forum

A1) Events A and B are such that P(A) = (9/(30)), P(B)= (2/5) and P(A ∪ B) = (4/5) find a) P(A ∩ B) b) P(A∣B) c) P(B^ ) 2) A bag contains 3 black balls and 5 white balls. Two balls selected are random without replacement.Find the probability a) of selecting a black and white ball in order b) that the balls are of different colours c) the second ball is white

$$\left.\:{A}\mathrm{1}\right)\:\mathrm{Events}\:\mathrm{A}\:\mathrm{and}\:\mathrm{B}\:\mathrm{are}\:\mathrm{such}\:\mathrm{that} \\ $$$$\:\:{P}\left({A}\right)\:=\:\frac{\mathrm{9}}{\mathrm{30}},\:{P}\left(\boldsymbol{{B}}\right)=\:\frac{\mathrm{2}}{\mathrm{5}}\:{and}\: \\ $$$${P}\left({A}\:\cup\:{B}\right)\:=\:\frac{\mathrm{4}}{\mathrm{5}} \\ $$$$\left.{find}\:{a}\right)\:{P}\left({A}\:\cap\:{B}\right) \\ $$$$\left.\:\:\:\:\:\:\:\:\:\:\:{b}\right)\:{P}\left({A}\mid{B}\right) \\ $$$$\left.\:\:\:\:\:\:\:\:\:\:{c}\right)\:{P}\left(\bar {{B}}\right) \\ $$$$\left.\mathrm{2}\right)\:\mathrm{A}\:\mathrm{bag}\:\mathrm{contains}\:\mathrm{3}\:\mathrm{black}\:\mathrm{balls}\:\mathrm{and} \\ $$$$\mathrm{5}\:\mathrm{white}\:\mathrm{balls}.\:\mathrm{Two}\:\mathrm{balls}\:\mathrm{selected}\:\mathrm{are} \\ $$$$\mathrm{random}\:\mathrm{without}\:\mathrm{replacement}.\mathrm{Find} \\ $$$$\mathrm{the}\:\mathrm{probability} \\ $$$$\left.\:\:\:\:\:\:\:\:\:\:{a}\right)\:{of}\:{selecting}\:{a}\:{black}\:{and}\:{white} \\ $$$${ball}\:{in}\:{order} \\ $$$$\left.\:\:\:\:\:\:\:\:\:\:{b}\right)\:{that}\:{the}\:{balls}\:{are}\:{of}\:{different}\:{colours} \\ $$$$\left.\:\:\:\:\:\:\:\:{c}\right)\:{the}\:{second}\:{ball}\:{is}\:{white}\: \\ $$

Question Number 34439 Answers: 1 Comments: 0

lim_(x→∞) (−1)^(x−1) sin (π(√(x^2 +0.5x+1))), where x∈N.

$$\underset{{x}\rightarrow\infty} {\mathrm{lim}}\:\left(−\mathrm{1}\right)^{{x}−\mathrm{1}} \mathrm{sin}\:\left(\pi\sqrt{{x}^{\mathrm{2}} +\mathrm{0}.\mathrm{5}{x}+\mathrm{1}}\right), \\ $$$${where}\:{x}\in\mathbb{N}. \\ $$

Question Number 34438 Answers: 0 Comments: 0

Question Number 34985 Answers: 1 Comments: 0

Question Number 34433 Answers: 1 Comments: 1

find lim_(n→+∞) (1/n)Σ_(k=1) ^(n−1) (√((n+k)/(n−k)))

$${find}\:{lim}_{{n}\rightarrow+\infty} \frac{\mathrm{1}}{{n}}\sum_{{k}=\mathrm{1}} ^{{n}−\mathrm{1}} \:\sqrt{\frac{{n}+{k}}{{n}−{k}}} \\ $$

Question Number 34429 Answers: 2 Comments: 0

Prove that 3^m +3^n +1 is not a perfect square. where m and n are positive integers.

$${Prove}\:{that} \\ $$$$\mathrm{3}^{{m}} +\mathrm{3}^{{n}} +\mathrm{1}\:{is}\:{not}\:{a}\:{perfect}\:{square}. \\ $$$${where}\:{m}\:{and}\:{n}\:{are}\:{positive}\:{integers}. \\ $$

Question Number 34422 Answers: 2 Comments: 0

Show that ((1+x)/(1+(√(1+x)) )) +((1−x)/(1−(√(1−x )))) =1 when x=((√(3 ))/2)

$${Show}\:{that} \\ $$$$\frac{\mathrm{1}+{x}}{\mathrm{1}+\sqrt{\mathrm{1}+{x}}\:}\:+\frac{\mathrm{1}−{x}}{\mathrm{1}−\sqrt{\mathrm{1}−{x}\:}}\:=\mathrm{1}\:{when}\:{x}=\frac{\sqrt{\mathrm{3}\:}}{\mathrm{2}} \\ $$

Question Number 34420 Answers: 1 Comments: 1

let A = arctanx −arctany give another form of A if xy≠−1 .

$${let}\:{A}\:\:=\:{arctanx}\:−{arctany} \\ $$$${give}\:{another}\:{form}\:{of}\:{A}\:{if}\:\:{xy}\neq−\mathrm{1}\:. \\ $$

Question Number 34421 Answers: 0 Comments: 1

let A = ∫_(−∞) ^(+∞) (dx/(x^2 −j)) with j=e^(i((2π)/3)) extract ReA and Im(A) and calculste its values.

$${let}\:{A}\:\:=\:\int_{−\infty} ^{+\infty} \:\:\:\:\frac{{dx}}{{x}^{\mathrm{2}} \:−{j}}\:\:\:\:{with}\:{j}={e}^{{i}\frac{\mathrm{2}\pi}{\mathrm{3}}} \\ $$$${extract}\:\:{ReA}\:{and}\:{Im}\left({A}\right)\:{and}\:{calculste}\:{its}\:{values}. \\ $$

Question Number 34411 Answers: 0 Comments: 5

Question Number 34410 Answers: 0 Comments: 0

Prove that (((n^2 )!)/((n!)^(n+1) )) is always an integer for n∈N.

$${Prove}\:{that}\:\frac{\left({n}^{\mathrm{2}} \right)!}{\left({n}!\right)^{{n}+\mathrm{1}} }\:{is}\:{always}\:{an}\:{integer} \\ $$$${for}\:{n}\in{N}. \\ $$

Question Number 34408 Answers: 0 Comments: 1

what is the value of (√(5(√(5(√(5(√(5(√(5(√5))))))))))) ignoring 64.

$${what}\:{is}\:{the}\:{value}\:{of}\: \\ $$$$\:\sqrt{\mathrm{5}\sqrt{\mathrm{5}\sqrt{\mathrm{5}\sqrt{\mathrm{5}\sqrt{\mathrm{5}\sqrt{\mathrm{5}}}}}}} \\ $$$${ignoring}\:\mathrm{64}. \\ $$

Question Number 34385 Answers: 0 Comments: 0

p,q∈P m,n∈{0,1,2,...} How many pairs are there,whose LCM is p^m q^(n ) ,when: (i)(a,b) & (b,a) are considered same. (ii)(a,b) & (b,a) are considered different. (Generalization of Q# 34358)

$$\mathrm{p},\mathrm{q}\in\mathbb{P} \\ $$$$\mathrm{m},\mathrm{n}\in\left\{\mathrm{0},\mathrm{1},\mathrm{2},...\right\} \\ $$$$\mathrm{How}\:\mathrm{many}\:\mathrm{pairs}\:\mathrm{are}\:\mathrm{there},\mathrm{whose} \\ $$$$\mathrm{LCM}\:\mathrm{is}\:\mathrm{p}^{\mathrm{m}} \mathrm{q}^{\mathrm{n}\:} ,\mathrm{when}: \\ $$$$\left(\mathrm{i}\right)\left(\mathrm{a},\mathrm{b}\right)\:\&\:\left(\mathrm{b},\mathrm{a}\right)\:\mathrm{are}\:\mathrm{considered}\:\mathrm{same}. \\ $$$$\left(\mathrm{ii}\right)\left(\mathrm{a},\mathrm{b}\right)\:\&\:\left(\mathrm{b},\mathrm{a}\right)\:\mathrm{are}\:\mathrm{considered}\:\mathrm{different}. \\ $$$$\:\:\:\:\left(\mathrm{Generalization}\:\mathrm{of}\:\mathrm{Q}#\:\mathrm{34358}\right) \\ $$

Question Number 34382 Answers: 1 Comments: 0

how do i prove ∫_o ^t cos 2(ωt+α)dt=0

$$\mathrm{how}\:\mathrm{do}\:\mathrm{i}\:\mathrm{prove} \\ $$$$\int_{\mathrm{o}} ^{\mathrm{t}} \mathrm{cos}\:\mathrm{2}\left(\omega\mathrm{t}+\alpha\right)\mathrm{dt}=\mathrm{0} \\ $$

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