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AllQuestion and Answers: Page 181

Question Number 201516 Answers: 1 Comments: 0

Question Number 201515 Answers: 1 Comments: 0

Question Number 201555 Answers: 1 Comments: 0

Question Number 201557 Answers: 2 Comments: 0

5 ∙ 555...5_( 50) find the sum of the digits of the product.

$$\mathrm{5}\:\centerdot\:\underset{\:\mathrm{50}} {\underbrace{\mathrm{555}...\mathrm{5}}} \\ $$$$\mathrm{find}\:\mathrm{the}\:\mathrm{sum}\:\mathrm{of}\:\mathrm{the}\:\mathrm{digits}\:\mathrm{of}\:\mathrm{the} \\ $$$$\mathrm{product}. \\ $$

Question Number 201510 Answers: 1 Comments: 0

Question Number 201509 Answers: 1 Comments: 0

Question Number 201502 Answers: 0 Comments: 3

A generation is about one-third of a lifetime.Approximately about how many generations have passed since the year 0AD?

$${A}\:{generation}\:{is}\:{about}\:{one}-{third}\:{of}\:{a} \\ $$$${lifetime}.{Approximately}\:{about}\:{how} \\ $$$${many}\:{generations}\:{have}\:{passed}\:{since} \\ $$$${the}\:{year}\:\mathrm{0}{AD}? \\ $$

Question Number 201514 Answers: 0 Comments: 0

Question Number 201495 Answers: 1 Comments: 1

(Un)_(n≥1 ;) (1/(nC_(2n) ^n )) study convergence

$$ \\ $$$$ \\ $$$$ \\ $$$$ \\ $$$$\:\:\left({Un}\right)_{{n}\geqslant\mathrm{1}\:;} \:\:\:\:\frac{\mathrm{1}}{{nC}_{\mathrm{2}{n}} ^{{n}} \:} \\ $$$$ \\ $$$$\:\:\:\:{study}\:\:{convergence} \\ $$$$ \\ $$$$ \\ $$$$ \\ $$

Question Number 201486 Answers: 0 Comments: 0

Question Number 201477 Answers: 1 Comments: 0

how to prove that (3d_3 +4d_2 +3d_1 )^2 ≤5(d_1 ^2 +d_2 ^2 +d_3 ^2 +(d_2 +d_1 )^2 +(d_3 +d_2 )^2 +(d_1 +d_2 +d_3 )^2 )

Question Number 201475 Answers: 0 Comments: 5

Question Number 201473 Answers: 0 Comments: 0

∫_0 ^1 ((Li_3 (−x^2 ))/(1+x))dx

$$\int_{\mathrm{0}} ^{\mathrm{1}} \frac{\boldsymbol{\mathrm{Li}}_{\mathrm{3}} \left(−\boldsymbol{\mathrm{x}}^{\mathrm{2}} \right)}{\mathrm{1}+\boldsymbol{\mathrm{x}}}\boldsymbol{\mathrm{dx}} \\ $$

Question Number 201464 Answers: 2 Comments: 0

if f(2) = 3 and f(4) = 5 find ∫_2 ^( 4) f(x) ∙ f^′ (x) dx = ?

$$\mathrm{if}\:\:\:\mathrm{f}\left(\mathrm{2}\right)\:=\:\mathrm{3}\:\:\:\mathrm{and}\:\:\:\mathrm{f}\left(\mathrm{4}\right)\:=\:\mathrm{5} \\ $$$$\mathrm{find}\:\:\:\int_{\mathrm{2}} ^{\:\mathrm{4}} \:\mathrm{f}\left(\mathrm{x}\right)\:\centerdot\:\mathrm{f}\:^{'} \left(\mathrm{x}\right)\:\mathrm{dx}\:=\:? \\ $$

Question Number 201452 Answers: 2 Comments: 0

Question Number 201447 Answers: 0 Comments: 0

Question Number 201445 Answers: 0 Comments: 0

Question Number 201443 Answers: 1 Comments: 5

Question Number 201446 Answers: 0 Comments: 0

Question Number 201441 Answers: 0 Comments: 0

Let f(x) and g(x) be given by f(x)= (1/x) +(1/(x−2)) +(1/(x−4)) + ... +(1/(x−2018)) and g(x)=(1/(x−1)) +(1/(x−3)) +(1/(x−5)) +...+ (1/(x−2017)). Prove that ∣ f(x)−g(x)∣ >2 for any non−integer real number x satisfying 0<x<2018.

$${Let}\:{f}\left({x}\right)\:{and}\:{g}\left({x}\right)\:{be}\:{given}\:{by}\: \\ $$$$\:{f}\left({x}\right)=\:\frac{\mathrm{1}}{{x}}\:+\frac{\mathrm{1}}{{x}−\mathrm{2}}\:+\frac{\mathrm{1}}{{x}−\mathrm{4}}\:+\:...\:+\frac{\mathrm{1}}{{x}−\mathrm{2018}} \\ $$$$\:{and}\: \\ $$$$\:\:{g}\left({x}\right)=\frac{\mathrm{1}}{{x}−\mathrm{1}}\:+\frac{\mathrm{1}}{{x}−\mathrm{3}}\:+\frac{\mathrm{1}}{{x}−\mathrm{5}}\:+...+\:\frac{\mathrm{1}}{{x}−\mathrm{2017}}. \\ $$$$\:\:{Prove}\:{that}\:\:\mid\:{f}\left({x}\right)−{g}\left({x}\right)\mid\:>\mathrm{2} \\ $$$$\:\:{for}\:{any}\:{non}−{integer}\:{real}\:{number} \\ $$$$\:\:{x}\:{satisfying}\:\mathrm{0}<{x}<\mathrm{2018}.\: \\ $$

Question Number 201439 Answers: 0 Comments: 0

There is a field. Everyday kids throw some balls on the field. At night the farmer goes and place the bucket in a place where it will contain the most amount of balls. the field can be represented as a line of length 10. the bucket can be represented as a line of length 2. If the kids have thrown 3 balls into the field, what is the probability that the bucket will contain 2 balls how a sample looks like Where the blue line represents the field. and the red line represents the bucket. and the dots are the balls

$$ \\ $$$$ \\ $$There is a field. Everyday kids throw some balls on the field. At night the farmer goes and place the bucket in a place where it will contain the most amount of balls. the field can be represented as a line of length 10. the bucket can be represented as a line of length 2. If the kids have thrown 3 balls into the field, what is the probability that the bucket will contain 2 balls $$\mathrm{how}\:\mathrm{a}\:\mathrm{sample}\:\mathrm{looks}\:\mathrm{like} \\ $$$$ \\ $$$$ \\ $$$$ \\ $$$$ \\ $$Where the blue line represents the field. and the red line represents the bucket. and the dots are the balls $$ \\ $$

Question Number 201433 Answers: 3 Comments: 0

A truck, P, travelling at 54km/h passes a point at 10:30 am while another truck, Q travelling at 90km/h passes through this same point 30 minutes later. At what time will truck Q overtake P?

$${A}\:{truck},\:{P},\:{travelling}\:{at}\:\mathrm{54}{km}/{h}\:{passes} \\ $$$${a}\:{point}\:{at}\:\mathrm{10}:\mathrm{30}\:{am}\:{while}\:{another}\:{truck}, \\ $$$${Q}\:{travelling}\:{at}\:\mathrm{90}{km}/{h}\:{passes}\:{through} \\ $$$${this}\:{same}\:{point}\:\mathrm{30}\:{minutes}\:{later}.\:{At} \\ $$$${what}\:{time}\:{will}\:{truck}\:{Q}\:{overtake}\:{P}? \\ $$

Question Number 201430 Answers: 1 Comments: 0

Find: (2/(35)) + (2/(63)) + (2/(99)) + (2/(143)) = ?

$$\mathrm{Find}: \\ $$$$\frac{\mathrm{2}}{\mathrm{35}}\:+\:\frac{\mathrm{2}}{\mathrm{63}}\:+\:\frac{\mathrm{2}}{\mathrm{99}}\:+\:\frac{\mathrm{2}}{\mathrm{143}}\:=\:? \\ $$

Question Number 201427 Answers: 2 Comments: 0

{ ((sin(x+y)=cos(x−y))),((tanx−tany=1)) :} (x,y)=(?,?)

$$\begin{cases}{{sin}\left({x}+{y}\right)={cos}\left({x}−{y}\right)}\\{{tanx}−{tany}=\mathrm{1}}\end{cases} \\ $$$$\left({x},{y}\right)=\left(?,?\right) \\ $$

Question Number 201426 Answers: 1 Comments: 0

cosx−(√3)sinx=1 x=?

$${cosx}−\sqrt{\mathrm{3}}{sinx}=\mathrm{1} \\ $$$${x}=? \\ $$

Question Number 201425 Answers: 1 Comments: 0

prove that ((1−cosA+cosB−cos(A+B))/(1+cosA−cosB−cos(A+B)))=tan(A/2)∙cot(B/2)

$${prove}\:{that} \\ $$$$\frac{\mathrm{1}−{cosA}+{cosB}−{cos}\left({A}+{B}\right)}{\mathrm{1}+{cosA}−{cosB}−{cos}\left({A}+{B}\right)}={tan}\frac{{A}}{\mathrm{2}}\centerdot{cot}\frac{{B}}{\mathrm{2}} \\ $$

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