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Question Number 38235 Answers: 0 Comments: 4

A man 2m 50cm tall stands a distance of 3m in front of a large vertical plane mirror. i)what is the shortest length of the mirror that will enable the man see himself fully? ii)what is the answer of the above if the man were 5m away?

$${A}\:{man}\:\mathrm{2}{m}\:\mathrm{50}{cm}\:{tall}\:{stands}\:{a} \\ $$$${distance}\:{of}\:\mathrm{3}{m}\:{in}\:{front}\:{of}\:{a}\:{large} \\ $$$${vertical}\:{plane}\:{mirror}. \\ $$$$\left.{i}\right){what}\:{is}\:{the}\:{shortest}\:{length}\:{of}\:{the} \\ $$$${mirror}\:{that}\:{will}\:{enable}\:{the}\:{man}\:{see} \\ $$$${himself}\:{fully}? \\ $$$$\left.{ii}\right){what}\:{is}\:{the}\:{answer}\:{of}\:{the}\:{above} \\ $$$${if}\:{the}\:{man}\:{were}\:\mathrm{5}{m}\:{away}? \\ $$

Question Number 38261 Answers: 1 Comments: 0

Question Number 38232 Answers: 4 Comments: 1

Differentiate tan^(−1) ((((√(1+x^2 ))−1)/x)) without using any trigonometric substitution !

$$\mathrm{Differentiate}\: \\ $$$$\mathrm{tan}^{−\mathrm{1}} \left(\frac{\sqrt{\mathrm{1}+{x}^{\mathrm{2}} }−\mathrm{1}}{{x}}\right)\:\: \\ $$$${without}\:{using}\:{any}\:{trigonometric}\: \\ $$$${substitution}\:! \\ $$

Question Number 38222 Answers: 0 Comments: 1

If U={−5, −4, −3, −2, −1, 0, 1, 2, 3, 4, 5} A={x/x^2 =25, x ∈ Z} B={x/x^2 +5=9, x ∈ Z} and C={x/−2≤ x ≤ 2, x ∈ Z} then (A ∩ B ∩ C)^c ∩ (A△B)^c =?

$$\mathrm{If}\:\mathbb{U}=\left\{−\mathrm{5},\:−\mathrm{4},\:−\mathrm{3},\:−\mathrm{2},\:−\mathrm{1},\:\mathrm{0},\:\mathrm{1},\:\mathrm{2},\:\mathrm{3},\:\mathrm{4},\:\mathrm{5}\right\} \\ $$$$\mathrm{A}=\left\{{x}/{x}^{\mathrm{2}} =\mathrm{25},\:{x}\:\in\:\mathrm{Z}\right\} \\ $$$$\mathrm{B}=\left\{{x}/{x}^{\mathrm{2}} +\mathrm{5}=\mathrm{9},\:{x}\:\in\:\mathrm{Z}\right\}\:\mathrm{and} \\ $$$$\mathrm{C}=\left\{{x}/−\mathrm{2}\leqslant\:{x}\:\leqslant\:\mathrm{2},\:{x}\:\in\:\mathrm{Z}\right\}\:\mathrm{then} \\ $$$$\left(\mathrm{A}\:\cap\:\mathrm{B}\:\cap\:\mathrm{C}\right)^{\mathrm{c}} \:\cap\:\left(\mathrm{A}\bigtriangleup\mathrm{B}\right)^{\mathrm{c}} =? \\ $$

Question Number 38211 Answers: 0 Comments: 2

let x>0 and F(x)= ∫_0 ^(+∞) ((arctan(xt^2 ))/(1+t^2 ))dt 1) find a simple form of F(x) 2)find the value of ∫_0 ^∞ ((arctan(2t^2 ))/(1+t^2 ))dt 3)find the value of ∫_0 ^∞ ((arctan(3t^2 ))/(1+t^2 ))dt.

$${let}\:{x}>\mathrm{0}\:{and}\:{F}\left({x}\right)=\:\int_{\mathrm{0}} ^{+\infty} \:\frac{{arctan}\left({xt}^{\mathrm{2}} \right)}{\mathrm{1}+{t}^{\mathrm{2}} }{dt} \\ $$$$\left.\mathrm{1}\right)\:{find}\:{a}\:{simple}\:{form}\:{of}\:{F}\left({x}\right) \\ $$$$\left.\mathrm{2}\right){find}\:{the}\:{value}\:{of}\:\int_{\mathrm{0}} ^{\infty} \:\:\:\frac{{arctan}\left(\mathrm{2}{t}^{\mathrm{2}} \right)}{\mathrm{1}+{t}^{\mathrm{2}} }{dt} \\ $$$$\left.\mathrm{3}\right){find}\:{the}\:{value}\:{of}\:\int_{\mathrm{0}} ^{\infty} \frac{{arctan}\left(\mathrm{3}{t}^{\mathrm{2}} \right)}{\mathrm{1}+{t}^{\mathrm{2}} }{dt}. \\ $$

Question Number 38210 Answers: 2 Comments: 4

let f(a)= ∫_0 ^π (dθ/(a +sin^2 θ)) (a from R) 1) find f(a) 2)calculate g(a)= ∫_0 ^π (dθ/((a+sin^2 θ)^2 )) 3)calculate ∫_0 ^π (dθ/(1+sin^2 θ)) and ∫_0 ^π (dθ/(2+sin^2 θ)) 4) calculate ∫_0 ^π (dθ/((3 +sin^2 θ)^2 )) .

$${let}\:{f}\left({a}\right)=\:\int_{\mathrm{0}} ^{\pi} \:\:\:\frac{{d}\theta}{{a}\:+{sin}^{\mathrm{2}} \theta}\:\:\:\left({a}\:{from}\:{R}\right) \\ $$$$\left.\mathrm{1}\right)\:{find}\:{f}\left({a}\right) \\ $$$$\left.\mathrm{2}\right){calculate}\:{g}\left({a}\right)=\:\int_{\mathrm{0}} ^{\pi} \:\:\:\frac{{d}\theta}{\left({a}+{sin}^{\mathrm{2}} \theta\right)^{\mathrm{2}} } \\ $$$$\left.\mathrm{3}\right){calculate}\:\int_{\mathrm{0}} ^{\pi} \:\:\:\:\frac{{d}\theta}{\mathrm{1}+{sin}^{\mathrm{2}} \theta}\:{and}\:\int_{\mathrm{0}} ^{\pi} \:\:\frac{{d}\theta}{\mathrm{2}+{sin}^{\mathrm{2}} \theta} \\ $$$$\left.\mathrm{4}\right)\:{calculate}\:\int_{\mathrm{0}} ^{\pi} \:\:\:\frac{{d}\theta}{\left(\mathrm{3}\:+{sin}^{\mathrm{2}} \theta\right)^{\mathrm{2}} }\:. \\ $$

Question Number 38209 Answers: 0 Comments: 2

let f(x)=e^(−x) cosx developp f at fourier serie 1) find the value of Σ_(n=−∞) ^(+∞) (((−1)^n )/(1+n^2 )) 2) calculate Σ_(n=0) ^∞ (1/(n^2 +1)) .

$${let}\:{f}\left({x}\right)={e}^{−{x}} {cosx} \\ $$$${developp}\:{f}\:{at}\:{fourier}\:{serie} \\ $$$$\left.\mathrm{1}\right)\:{find}\:{the}\:{value}\:{of}\:\sum_{{n}=−\infty} ^{+\infty} \:\frac{\left(−\mathrm{1}\right)^{{n}} }{\mathrm{1}+{n}^{\mathrm{2}} } \\ $$$$\left.\mathrm{2}\right)\:{calculate}\:\sum_{{n}=\mathrm{0}} ^{\infty} \:\:\frac{\mathrm{1}}{{n}^{\mathrm{2}} \:+\mathrm{1}}\:. \\ $$

Question Number 38208 Answers: 0 Comments: 3

let f(x)=ch(αx) developp f at fourier serie. (f 2π periodic even)

$${let}\:{f}\left({x}\right)={ch}\left(\alpha{x}\right)\: \\ $$$${developp}\:{f}\:{at}\:{fourier}\:{serie}. \\ $$$$\left({f}\:\mathrm{2}\pi\:{periodic}\:{even}\right) \\ $$

Question Number 38207 Answers: 0 Comments: 1

prove that coth(x)−(1/x) =Σ_(n=1) ^∞ ((2x)/(x^2 +n^2 π^2 )) (x≠0)

$${prove}\:{that}\:{coth}\left({x}\right)−\frac{\mathrm{1}}{{x}}\:=\sum_{{n}=\mathrm{1}} ^{\infty} \:\:\frac{\mathrm{2}{x}}{{x}^{\mathrm{2}} \:+{n}^{\mathrm{2}} \pi^{\mathrm{2}} } \\ $$$$\left({x}\neq\mathrm{0}\right) \\ $$

Question Number 38206 Answers: 1 Comments: 1

calculate lim_(x→0) ((x coth(x)−1)/x^2 )

$${calculate}\:{lim}_{{x}\rightarrow\mathrm{0}} \:\frac{{x}\:{coth}\left({x}\right)−\mathrm{1}}{{x}^{\mathrm{2}} } \\ $$

Question Number 38205 Answers: 0 Comments: 0

if (1/(sinx)) =Σ_(n=1) ^∞ a_n sin(nx) find the values of a_n .

$${if}\:\:\frac{\mathrm{1}}{{sinx}}\:=\sum_{{n}=\mathrm{1}} ^{\infty} \:{a}_{{n}} {sin}\left({nx}\right)\:\:{find}\:{the}\:{values}\:{of} \\ $$$${a}_{{n}} . \\ $$

Question Number 38204 Answers: 0 Comments: 1

if (1/(cosx)) =(a_0 /2) +Σ_(n=1) ^∞ a_n cos(nx) calculate a_0 and a_n

$${if}\:\:\frac{\mathrm{1}}{{cosx}}\:=\frac{{a}_{\mathrm{0}} }{\mathrm{2}}\:+\sum_{{n}=\mathrm{1}} ^{\infty} \:{a}_{{n}} {cos}\left({nx}\right) \\ $$$${calculate}\:{a}_{\mathrm{0}} \:{and}\:{a}_{{n}} \\ $$

Question Number 38203 Answers: 0 Comments: 0

let x≠(π/2)+kπ,k∈Z.prove that (1/(2cosx)) =Σ_(n=0) ^∞ (−1)^n (cos(2n+1)x)

$${let}\:{x}\neq\frac{\pi}{\mathrm{2}}+{k}\pi,{k}\in{Z}.{prove}\:{that} \\ $$$$\frac{\mathrm{1}}{\mathrm{2}{cosx}}\:=\sum_{{n}=\mathrm{0}} ^{\infty} \left(−\mathrm{1}\right)^{{n}} \left({cos}\left(\mathrm{2}{n}+\mathrm{1}\right){x}\right) \\ $$

Question Number 38202 Answers: 0 Comments: 1

find the value of ∫_0 ^1 e^(−x) (√(1−e^(−2x) ))dx

$${find}\:{the}\:{value}\:{of}\:\:\int_{\mathrm{0}} ^{\mathrm{1}} \:{e}^{−{x}} \sqrt{\mathrm{1}−{e}^{−\mathrm{2}{x}} }{dx} \\ $$

Question Number 38201 Answers: 0 Comments: 0

calculate I = ∫_0 ^∞ xe^(−x^2 ) (√(1−e^(−2x^2 ) ))dx

$${calculate}\:{I}\:=\:\int_{\mathrm{0}} ^{\infty} \:\:\:{xe}^{−{x}^{\mathrm{2}} } \sqrt{\mathrm{1}−{e}^{−\mathrm{2}{x}^{\mathrm{2}} } }{dx} \\ $$

Question Number 38199 Answers: 1 Comments: 0

calculate ∫_0 ^1 ((√x)/(1+x^2 ))dx .

$${calculate}\:\:\int_{\mathrm{0}} ^{\mathrm{1}} \:\:\:\frac{\sqrt{{x}}}{\mathrm{1}+{x}^{\mathrm{2}} }{dx}\:. \\ $$

Question Number 38198 Answers: 0 Comments: 5

we give ∫_0 ^∞ e^(−x) ln(x)dx=−γ 1) calculate f(a)= ∫_0 ^∞ e^(−ax) ln(x)dx with a>0 2) let u_n = ∫_0 ^∞ e^(−nx) ln((x/n))dx find lim_(n→+∞) u_n

$${we}\:{give}\:\int_{\mathrm{0}} ^{\infty} \:\:{e}^{−{x}} {ln}\left({x}\right){dx}=−\gamma \\ $$$$\left.\mathrm{1}\right)\:{calculate}\:\:{f}\left({a}\right)=\:\int_{\mathrm{0}} ^{\infty} \:\:{e}^{−{ax}} {ln}\left({x}\right){dx}\:\:{with}\:{a}>\mathrm{0} \\ $$$$\left.\mathrm{2}\right)\:{let}\:{u}_{{n}} =\:\int_{\mathrm{0}} ^{\infty} \:\:{e}^{−{nx}} {ln}\left(\frac{{x}}{{n}}\right){dx}\:\:{find}\:{lim}_{{n}\rightarrow+\infty} \:{u}_{{n}} \\ $$

Question Number 38197 Answers: 0 Comments: 1

find a simple form of L(e^(−(√x)) ) L is laplace transform

$${find}\:{a}\:{simple}\:{form}\:{of}\:{L}\left({e}^{−\sqrt{{x}}} \right)\:\:{L}\:{is}\:{laplace}\:{transform} \\ $$

Question Number 38195 Answers: 0 Comments: 1

let x≥1 and δ(x)=Σ_(n=1) ^∞ (((−1)^n )/n^x ) 1) calculate δ(x) interms of ξ(x) if x>1 2)find δ(1) 3) find the value of Σ_(n=1) ^∞ (1/((2n+1)^2 )) 4) calculate δ(3) interms of ξ(3).

$${let}\:{x}\geqslant\mathrm{1}\:{and}\:\delta\left({x}\right)=\sum_{{n}=\mathrm{1}} ^{\infty} \:\:\frac{\left(−\mathrm{1}\right)^{{n}} }{{n}^{{x}} } \\ $$$$\left.\mathrm{1}\right)\:{calculate}\:\delta\left({x}\right)\:{interms}\:{of}\:\xi\left({x}\right)\:{if}\:{x}>\mathrm{1} \\ $$$$\left.\mathrm{2}\right){find}\:\:\delta\left(\mathrm{1}\right) \\ $$$$\left.\mathrm{3}\right)\:{find}\:{the}\:{value}\:{of}\:\:\sum_{{n}=\mathrm{1}} ^{\infty} \:\frac{\mathrm{1}}{\left(\mathrm{2}{n}+\mathrm{1}\right)^{\mathrm{2}} } \\ $$$$\left.\mathrm{4}\right)\:{calculate}\:\delta\left(\mathrm{3}\right)\:{interms}\:{of}\:\xi\left(\mathrm{3}\right). \\ $$

Question Number 38190 Answers: 1 Comments: 2

Question Number 38181 Answers: 2 Comments: 3

If y= x^((lnx)^(ln(lnx)) ) then (dy/dx) = ?

$$\mathrm{If}\:\mathrm{y}=\:\:{x}^{\left({lnx}\right)^{{ln}\left({lnx}\right)} } \:{then}\:\frac{{dy}}{{dx}}\:=\:? \\ $$

Question Number 38177 Answers: 0 Comments: 0

Question Number 38155 Answers: 0 Comments: 1

Propanol ,(C_3 H_7 OH) is an alcohol. a) State the functional group in propanol that makes it alcohol. b) Ethanol can be converted to a number of compounds as shown below. Ethene→ Ethane ↑ Ethylethanoate ⇆ Ethanol → Ethanoic acid ↓ soduimethanoate. when Soduim hydroxide solution reacts with Ethylethanoate ,Sodium Ethanoate and other products are obtained Give the name of the other products

Question Number 38154 Answers: 1 Comments: 0

Given that a,b,c are 3 consecutive term of a Geometric sequence f(n) , show that log a,logb,logc are the first 3 terms of an Arithmetic SequenceP(n).

$${Given}\:{that}\: \\ $$$$\:\:{a},{b},{c}\:{are}\:\mathrm{3}\:{consecutive}\:{term}\:{of}\: \\ $$$${a}\:{Geometric}\:{sequence}\:{f}\left({n}\right)\:,\:{show} \\ $$$${that}\:{log}\:{a},{logb},{logc}\:{are}\:{the}\:{first}\: \\ $$$$\mathrm{3}\:{terms}\:{of}\:{an}\:{Arithmetic}\:{SequenceP}\left({n}\right). \\ $$

Question Number 38151 Answers: 3 Comments: 8

a > b > 0 a^2 cos θ−b^2 sin θ=(a^2 −b^2 )sin θcos θ Find θ in terms of a, b. θ ∈ (0,(π/2)) .

$${a}\:>\:{b}\:>\:\mathrm{0} \\ $$$${a}^{\mathrm{2}} \mathrm{cos}\:\theta−{b}^{\mathrm{2}} \mathrm{sin}\:\theta=\left({a}^{\mathrm{2}} −{b}^{\mathrm{2}} \right)\mathrm{sin}\:\theta\mathrm{cos}\:\theta \\ $$$${Find}\:\theta\:{in}\:{terms}\:{of}\:{a},\:{b}. \\ $$$$\:\:\theta\:\in\:\left(\mathrm{0},\frac{\pi}{\mathrm{2}}\right)\:. \\ $$

Question Number 38130 Answers: 4 Comments: 0

1. ∫tan^3 (2x)sec^5 (2x) dx 2. ∫_0 ^(π/3) tan^5 (x)sec^6 (x) dx 3. ∫tan^6 (ay) dy

$$\mathrm{1}.\:\int\mathrm{tan}^{\mathrm{3}} \left(\mathrm{2}{x}\right)\mathrm{sec}^{\mathrm{5}} \left(\mathrm{2}{x}\right)\:{dx}\: \\ $$$$\mathrm{2}.\:\int_{\mathrm{0}} ^{\frac{\pi}{\mathrm{3}}} \mathrm{tan}^{\mathrm{5}} \left({x}\right)\mathrm{sec}^{\mathrm{6}} \left({x}\right)\:{dx}\: \\ $$$$\mathrm{3}.\:\int\mathrm{tan}^{\mathrm{6}} \left({ay}\right)\:{dy}\: \\ $$

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