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Question Number 32151 Answers: 1 Comments: 0

Question Number 32126 Answers: 1 Comments: 1

Question Number 32110 Answers: 1 Comments: 0

If y=1+x^2 +x^3 and x=1+α, where α is small, show that y≈3+5α. Hence, find the increase in y when x is increased from 1 to 1.02

$$\mathrm{If}\:\mathrm{y}=\mathrm{1}+\mathrm{x}^{\mathrm{2}} +\mathrm{x}^{\mathrm{3}} \:\mathrm{and}\:\mathrm{x}=\mathrm{1}+\alpha,\:\mathrm{where}\:\alpha\:\mathrm{is}\:\mathrm{small},\:\mathrm{show} \\ $$$$\mathrm{that}\:\mathrm{y}\approx\mathrm{3}+\mathrm{5}\alpha.\:\mathrm{Hence},\:\mathrm{find}\:\mathrm{the}\:\mathrm{increase}\:\mathrm{in}\:\mathrm{y}\:\mathrm{when} \\ $$$$\mathrm{x}\:\mathrm{is}\:\mathrm{increased}\:\mathrm{from}\:\mathrm{1}\:\mathrm{to}\:\mathrm{1}.\mathrm{02} \\ $$

Question Number 32109 Answers: 0 Comments: 0

Question Number 32100 Answers: 0 Comments: 0

Question Number 32099 Answers: 1 Comments: 0

Question Number 32098 Answers: 0 Comments: 14

Question Number 32094 Answers: 0 Comments: 0

Find the ordinary argument (arg z) and the principal argument (Arg z) of z=(i/(−2−2i))

$${Find}\:{the}\:{ordinary}\:{argument} \\ $$$$\left({arg}\:{z}\right)\:{and}\:{the}\:{principal}\:{argument} \\ $$$$\left({Arg}\:{z}\right)\:{of}\:{z}=\frac{{i}}{−\mathrm{2}−\mathrm{2}{i}} \\ $$

Question Number 32093 Answers: 1 Comments: 0

Find the number of ways of selecting 9 balls from 6 red balls, 5 white balls and 5 blue balls if each selection consists of 3 balls of each colour.

$${Find}\:{the}\:{number}\:{of}\:{ways}\:{of} \\ $$$${selecting}\:\mathrm{9}\:{balls}\:{from}\:\mathrm{6}\:{red}\:{balls}, \\ $$$$\mathrm{5}\:{white}\:{balls}\:{and}\:\mathrm{5}\:{blue}\:{balls}\:{if} \\ $$$${each}\:{selection}\:{consists}\:{of}\:\mathrm{3}\:{balls} \\ $$$${of}\:{each}\:{colour}. \\ $$

Question Number 32076 Answers: 1 Comments: 2

Question Number 32075 Answers: 0 Comments: 1

A rigid uniform bar of length 2.4m is pivoted horizontally at its mid-point. Weights are hung from two points of the bar such that a 200N is at the 0.4m mark and a 300N is at 1.6m mark.To maintain horizontal equilibrum a couple is applied to the bar. What is the torque and direction of the couple?

$${A}\:{rigid}\:{uniform}\:{bar}\:{of}\:{length}\:\mathrm{2}.\mathrm{4}{m} \\ $$$${is}\:{pivoted}\:{horizontally}\:{at}\:{its}\:{mid}-{point}. \\ $$$${Weights}\:{are}\:{hung}\:{from}\:{two}\:{points} \\ $$$${of}\:{the}\:{bar}\:{such}\:{that}\:{a}\:\mathrm{200}{N}\:{is}\:{at} \\ $$$${the}\:\mathrm{0}.\mathrm{4}{m}\:{mark}\:{and}\:{a}\:\mathrm{300}{N}\:{is}\:{at} \\ $$$$\mathrm{1}.\mathrm{6}{m}\:{mark}.{To}\:{maintain} \\ $$$${horizontal}\:{equilibrum}\:{a}\:{couple}\:{is} \\ $$$${applied}\:{to}\:{the}\:{bar}.\:{What}\:{is}\:{the} \\ $$$${torque}\:{and}\:{direction}\:{of}\:{the}\:{couple}? \\ $$

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 .

$${let}\:{consider}\:{the}\:{sequence}\:\:\left({u}_{{n}} \right)\:\:/{u}_{\mathrm{0}} \in\left[\mathrm{0},\mathrm{1}\right]\:{and} \\ $$$$\forall{n}\in{N}\:\:{u}_{{n}+\mathrm{1}} =\:{u}_{{n}} \:−{u}_{{n}} ^{\mathrm{2}} \\ $$$$\left.\mathrm{1}\right)\:{give}\:{a}\:{simple}\:{equivalent}\:{of}\:\:{u}_{{n}} \\ $$$$\left.\mathrm{2}\right)\:{find}\:{the}\:{nature}\:{of}\:\Sigma\:{u}_{{n}} . \\ $$

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