Question and Answers Forum

AllQuestion and Answers: Page 1775

Question Number 32074 Answers: 0 Comments: 1

A machine with a velocity ratio of 5 requires 150J of work to raise a 500N load through a vertical distance of 200cm.Calculate a)the efficiency of the machine b)the mechanical advantage of the machine.

$${A}\:{machine}\:{with}\:{a}\:{velocity}\:{ratio} \\ $$$${of}\:\mathrm{5}\:{requires}\:\mathrm{150}{J}\:{of}\:{work}\:{to}\:{raise} \\ $$$${a}\:\mathrm{500}{N}\:{load}\:{through}\:{a}\:{vertical} \\ $$$${distance}\:{of}\:\mathrm{200}{cm}.{Calculate} \\ $$$$\left.{a}\right){the}\:{efficiency}\:{of}\:{the}\:{machine} \\ $$$$\left.{b}\right){the}\:{mechanical}\:{advantage}\:{of} \\ $$$${the}\:{machine}. \\ $$

Question Number 32091 Answers: 0 Comments: 0

Question Number 32049 Answers: 2 Comments: 0

If ∣z−6−8i∣≤4 then Minimum value of ∣z∣ is A) 4 B) 5 C) 6 D) 8.

$$\boldsymbol{{I}}{f}\:\mid\boldsymbol{{z}}−\mathrm{6}−\mathrm{8}\boldsymbol{{i}}\mid\leqslant\mathrm{4}\:{then}\:\boldsymbol{{M}}{inimum}\:{value} \\ $$$${of}\:\mid\boldsymbol{{z}}\mid\:{is}\: \\ $$$$\left.{A}\right)\:\mathrm{4}\: \\ $$$$\left.{B}\right)\:\mathrm{5} \\ $$$$\left.{C}\right)\:\mathrm{6} \\ $$$$\left.{D}\right)\:\mathrm{8}. \\ $$

Question Number 32047 Answers: 0 Comments: 1

Question Number 32045 Answers: 0 Comments: 0

find lim_(n→∞) ∫_0 ^∞ e^(−t) sin^n t dt .

$${find}\:{lim}_{{n}\rightarrow\infty} \:\int_{\mathrm{0}} ^{\infty} \:\:{e}^{−{t}} \:{sin}^{{n}} {t}\:{dt}\:\:. \\ $$

Question Number 32044 Answers: 0 Comments: 1

fimd lim_(x→0) (1/x^3 ) ∫_0 ^x t^2 ln(1+sint) dt .

$${fimd}\:\:{lim}_{{x}\rightarrow\mathrm{0}} \:\:\:\:\:\frac{\mathrm{1}}{{x}^{\mathrm{3}} }\:\int_{\mathrm{0}} ^{{x}} \:{t}^{\mathrm{2}} \:{ln}\left(\mathrm{1}+{sint}\right)\:{dt}\:. \\ $$

Question Number 32043 Answers: 0 Comments: 0

let f(x)= ∫_x ^x^2 (dt/(lnt)) with x>0 and x≠1 1) prove that ∀ x>1 ∫_x ^x^2 ((xdt)/(tlnt)) ≤f(x)≤ ∫_x ^x^2 ((x^2 dt)/(tlnt)) after find lim_(x→1) f(x) 2) calculate f^′ (x) .

$${let}\:{f}\left({x}\right)=\:\int_{{x}} ^{{x}^{\mathrm{2}} } \:\:\:\frac{{dt}}{{lnt}}\:\:{with}\:{x}>\mathrm{0}\:{and}\:{x}\neq\mathrm{1} \\ $$$$\left.\mathrm{1}\right)\:{prove}\:{that}\:\forall\:{x}>\mathrm{1}\:\int_{{x}} ^{{x}^{\mathrm{2}} } \:\:\frac{{xdt}}{{tlnt}}\:\leqslant{f}\left({x}\right)\leqslant\:\int_{{x}} ^{{x}^{\mathrm{2}} } \:\frac{{x}^{\mathrm{2}} {dt}}{{tlnt}}\:\:{after} \\ $$$${find}\:{lim}_{{x}\rightarrow\mathrm{1}} {f}\left({x}\right) \\ $$$$\left.\mathrm{2}\right)\:{calculate}\:{f}^{'} \left({x}\right)\:. \\ $$

Question Number 32042 Answers: 0 Comments: 1

let u_n =∫_0 ^1 x^n sin(πx)dx 1) prove that Σ u_n converges 2) prove that Σ u_n = ∫_0 ^π ((sint)/t)dt .

$${let}\:{u}_{{n}} \:\:=\int_{\mathrm{0}} ^{\mathrm{1}} \:\:{x}^{{n}} \:{sin}\left(\pi{x}\right){dx} \\ $$$$\left.\mathrm{1}\right)\:{prove}\:{that}\:\Sigma\:{u}_{{n}} \:{converges} \\ $$$$\left.\mathrm{2}\right)\:{prove}\:{that}\:\Sigma\:{u}_{{n}} =\:\int_{\mathrm{0}} ^{\pi} \:\:\frac{{sint}}{{t}}{dt}\:. \\ $$

Question Number 32041 Answers: 0 Comments: 1

find the nature of Σ u_n / u_n = (((√1) +(√2) +....+(√n))/n^3 ) .

$${find}\:{the}\:{nature}\:{of}\:\Sigma\:{u}_{{n}} \:\:/ \\ $$$${u}_{{n}} =\:\frac{\sqrt{\mathrm{1}}\:+\sqrt{\mathrm{2}}\:+....+\sqrt{{n}}}{{n}^{\mathrm{3}} }\:. \\ $$

Question Number 32040 Answers: 0 Comments: 2

let give f(x) =∫_0 ^(π/2) (dt/(1+x tant)) 1) find a simple form of f(x) 2) calculate ∫_0 ^(π/2) ((tant)/((1+xtant)^2 ))dt 3)give the value of ∫_0 ^(π/2) ((tant)/((1+(√3) tant)^2 )) dt .

$${let}\:{give}\:{f}\left({x}\right)\:=\int_{\mathrm{0}} ^{\frac{\pi}{\mathrm{2}}} \:\:\:\:\:\frac{{dt}}{\mathrm{1}+{x}\:{tant}}\:\: \\ $$$$\left.\mathrm{1}\right)\:{find}\:{a}\:{simple}\:{form}\:{of}\:{f}\left({x}\right) \\ $$$$\left.\mathrm{2}\right)\:{calculate}\:\:\int_{\mathrm{0}} ^{\frac{\pi}{\mathrm{2}}} \:\:\frac{{tant}}{\left(\mathrm{1}+{xtant}\right)^{\mathrm{2}} }{dt} \\ $$$$\left.\mathrm{3}\right){give}\:{the}\:{value}\:{of}\:\:\int_{\mathrm{0}} ^{\frac{\pi}{\mathrm{2}}} \:\:\:\:\frac{{tant}}{\left(\mathrm{1}+\sqrt{\mathrm{3}}\:{tant}\right)^{\mathrm{2}} }\:{dt}\:. \\ $$

Question Number 32039 Answers: 0 Comments: 3

a>−1 calculate ∫_0 ^(π/2) (dt/(1+a tan^2 t)) . 2) find ∫_0 ^(π/2) ((tan^2 t)/((1+atan^2 t)^2 )) dt 3) find the value of ∫_0 ^(π/2) ((tan^2 t)/((1+2tan^2 t)^2 ))dt.

$${a}>−\mathrm{1}\:\:{calculate}\:\:\int_{\mathrm{0}} ^{\frac{\pi}{\mathrm{2}}} \:\:\:\:\:\frac{{dt}}{\mathrm{1}+{a}\:{tan}^{\mathrm{2}} {t}}\:. \\ $$$$\left.\mathrm{2}\right)\:{find}\:\int_{\mathrm{0}} ^{\frac{\pi}{\mathrm{2}}} \:\:\:\frac{{tan}^{\mathrm{2}} {t}}{\left(\mathrm{1}+{atan}^{\mathrm{2}} {t}\right)^{\mathrm{2}} }\:{dt} \\ $$$$\left.\mathrm{3}\right)\:\:{find}\:{the}\:{value}\:{of}\:\:\int_{\mathrm{0}} ^{\frac{\pi}{\mathrm{2}}} \:\:\:\:\frac{{tan}^{\mathrm{2}} {t}}{\left(\mathrm{1}+\mathrm{2}{tan}^{\mathrm{2}} {t}\right)^{\mathrm{2}} }{dt}.\: \\ $$

Question Number 32037 Answers: 0 Comments: 1

let u_n =cos(π(√(n^2 +n+1))) find nature of Σ u_n .

$${let}\:\:{u}_{{n}} ={cos}\left(\pi\sqrt{{n}^{\mathrm{2}} \:+{n}+\mathrm{1}}\right)\:{find}\:{nature}\:{of}\:\Sigma\:{u}_{{n}} . \\ $$$$ \\ $$

Question Number 32036 Answers: 0 Comments: 0

nature of Σ u_n with u_n = (1/((ln(2))^2 +....+(ln(n))^2 )) .

$${nature}\:{of}\:\Sigma\:{u}_{{n}} \:\:{with}\:{u}_{{n}} =\:\:\:\frac{\mathrm{1}}{\left({ln}\left(\mathrm{2}\right)\right)^{\mathrm{2}} \:+....+\left({ln}\left({n}\right)\right)^{\mathrm{2}} }\:\:. \\ $$

Question Number 32034 Answers: 0 Comments: 0

let u_n = ∫_0 ^1 (dx/(1+x+...+x^n )) study the convergence of Σ u_n .

$${let}\:{u}_{{n}} =\:\int_{\mathrm{0}} ^{\mathrm{1}} \:\:\frac{{dx}}{\mathrm{1}+{x}+...+{x}^{{n}} }\:\:{study}\:{the}\:{convergence}\:{of} \\ $$$$\Sigma\:{u}_{{n}} \:\:. \\ $$

Question Number 32033 Answers: 0 Comments: 0

let consider the sequence (u_n ) /u_0 ∈[0,1] and ∀n∈N u_(n+1) = u_n −u_n ^2 1) give a simple equivalent of u_n 2) find the nature of Σ u_n .