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Question Number 1354    Answers: 1   Comments: 1

Slightly modified form of Q 1343. 3^(log(3x+4)) =4^(log(4x+3)) ,solve for x.

$${Slightly}\:{modified}\:{form}\:{of}\:{Q}\:\mathrm{1343}. \\ $$$$\mathrm{3}^{{log}\left(\mathrm{3}{x}+\mathrm{4}\right)} =\mathrm{4}^{{log}\left(\mathrm{4}{x}+\mathrm{3}\right)} ,{solve}\:{for}\:{x}. \\ $$

Question Number 1352    Answers: 1   Comments: 0

Solve the following DE : y(dy/dx)+6x+5y=0 (x≠0,y≠0)

$${Solve}\:{the}\:{following}\:{DE}\:: \\ $$$${y}\frac{{dy}}{{dx}}+\mathrm{6}{x}+\mathrm{5}{y}=\mathrm{0}\:\:\:\:\:\left({x}\neq\mathrm{0},{y}\neq\mathrm{0}\right) \\ $$

Question Number 1347    Answers: 0   Comments: 0

Solve the following DE (dy/dx)+(c_1 /(yx^2 ))=c_2

$$\mathrm{Solve}\:\mathrm{the}\:\mathrm{following}\:\mathrm{DE} \\ $$$$\frac{{dy}}{{dx}}+\frac{{c}_{\mathrm{1}} }{{yx}^{\mathrm{2}} }={c}_{\mathrm{2}} \\ $$

Question Number 1339    Answers: 0   Comments: 3

f:C→C,z_0 ∈C such that f(z)−f(z_0 )=(z−z_0 )f(z−z_0 ) does lim_(z→z_0 ) f(z)=f(z_0 )?

$${f}:\mathbb{C}\rightarrow\mathbb{C},{z}_{\mathrm{0}} \in\mathbb{C}\:\mathrm{such}\:\mathrm{that} \\ $$$${f}\left({z}\right)−{f}\left({z}_{\mathrm{0}} \right)=\left({z}−{z}_{\mathrm{0}} \right){f}\left({z}−{z}_{\mathrm{0}} \right) \\ $$$$\mathrm{does}\:\underset{{z}\rightarrow{z}_{\mathrm{0}} } {\mathrm{lim}}{f}\left({z}\right)={f}\left({z}_{\mathrm{0}} \right)? \\ $$

Question Number 1335    Answers: 1   Comments: 0

if a≤x≤b and c≤y≤d then did (a/d)≤(x/y)≤(b/c)? a,b,c,d,x,y∈R 0<c 0<a

$$\mathrm{if}\:{a}\leqslant{x}\leqslant{b}\:\mathrm{and}\:{c}\leqslant{y}\leqslant{d}\:\mathrm{then}\:\mathrm{did} \\ $$$$\frac{{a}}{{d}}\leqslant\frac{{x}}{{y}}\leqslant\frac{{b}}{{c}}? \\ $$$${a},{b},{c},{d},{x},{y}\in\mathbb{R} \\ $$$$\mathrm{0}<{c} \\ $$$$\mathrm{0}<{a} \\ $$

Question Number 1331    Answers: 1   Comments: 2

If α and β are different complex numbers with ∣β∣=1 then ∣((β−α)/(1−αβ))∣=?

$$\mathrm{If}\:\alpha\:\mathrm{and}\:\beta\:\mathrm{are}\:\mathrm{different}\:\mathrm{complex}\:\mathrm{numbers} \\ $$$$\mathrm{with}\:\mid\beta\mid=\mathrm{1}\:\mathrm{then}\:\mid\frac{\beta−\alpha}{\mathrm{1}−\alpha\beta}\mid=? \\ $$

Question Number 1322    Answers: 0   Comments: 2

a_0 =1 a_1 =2 a_n −a_0 =a_(n−1) −a_(n−2) ,n≥2 a_(10) =? lim_(n→+∞) a_n =? lim_(n→+∞) a_(6n+r) =? r∈Z_6

$${a}_{\mathrm{0}} =\mathrm{1} \\ $$$${a}_{\mathrm{1}} =\mathrm{2} \\ $$$${a}_{{n}} −{a}_{\mathrm{0}} ={a}_{{n}−\mathrm{1}} −{a}_{{n}−\mathrm{2}} ,{n}\geqslant\mathrm{2} \\ $$$${a}_{\mathrm{10}} =? \\ $$$$\underset{{n}\rightarrow+\infty} {\mathrm{lim}}\:{a}_{{n}} =? \\ $$$$\underset{{n}\rightarrow+\infty} {\mathrm{lim}}\:{a}_{\mathrm{6}{n}+{r}} =?\:\:\:\:\:\:\:\:{r}\in\mathbb{Z}_{\mathrm{6}} \\ $$

Question Number 1318    Answers: 0   Comments: 1

η:C^2 →R_+ η(x,y)=∣∣x∣−∣y∣∣ η(x,y)=0⇔^? x=y η(x,y)=^? η(y,x) η(x,z)≤^? η(x,y)+η(y,z) x∈C y∈C z∈C

$$\eta:\mathbb{C}^{\mathrm{2}} \rightarrow\mathbb{R}_{+} \\ $$$$\eta\left({x},{y}\right)=\mid\mid{x}\mid−\mid{y}\mid\mid \\ $$$$\eta\left({x},{y}\right)=\mathrm{0}\overset{?} {\Leftrightarrow}{x}={y} \\ $$$$\eta\left({x},{y}\right)\overset{?} {=}\eta\left({y},{x}\right) \\ $$$$\eta\left({x},{z}\right)\overset{?} {\leqslant}\eta\left({x},{y}\right)+\eta\left({y},{z}\right) \\ $$$${x}\in\mathbb{C} \\ $$$${y}\in\mathbb{C} \\ $$$${z}\in\mathbb{C} \\ $$

Question Number 1304    Answers: 1   Comments: 0

If 3^(log 3x) =4^(log 4x) , find x.

$${If}\:\mathrm{3}^{{log}\:\mathrm{3}{x}} =\mathrm{4}^{{log}\:\mathrm{4}{x}} ,\:{find}\:{x}. \\ $$$$ \\ $$

Question Number 1302    Answers: 1   Comments: 0

Generalization of Q No 1296 f(x+k)=[ f(x) ]^n f(x)=?

$${Generalization}\:{of}\:{Q}\:{No}\:\mathrm{1296} \\ $$$${f}\left({x}+{k}\right)=\left[\:{f}\left({x}\right)\:\right]^{{n}} \\ $$$${f}\left({x}\right)=? \\ $$

Question Number 1351    Answers: 0   Comments: 2

W{f(x)}(t)=∫_0 ^(1/t) f(x)ln(xt)dx,t>0 W{f(x)+g(x)}(t)=^? W{f(x)}(t)+W{g(x)}(t) W{cf(x)}(t)=^? cW{f(x)}(t) W{1}(t)=? W{x}(t)=? W{x^n }(t)=?,n∈N W{f′(x)}(t)=?

$$\mathcal{W}\left\{{f}\left({x}\right)\right\}\left({t}\right)=\underset{\mathrm{0}} {\overset{\mathrm{1}/{t}} {\int}}{f}\left({x}\right)\mathrm{ln}\left({xt}\right){dx},{t}>\mathrm{0} \\ $$$$\mathcal{W}\left\{{f}\left({x}\right)+{g}\left({x}\right)\right\}\left({t}\right)\overset{?} {=}\mathcal{W}\left\{{f}\left({x}\right)\right\}\left({t}\right)+\mathcal{W}\left\{{g}\left({x}\right)\right\}\left({t}\right) \\ $$$$\mathcal{W}\left\{{cf}\left({x}\right)\right\}\left({t}\right)\overset{?} {=}{c}\mathcal{W}\left\{{f}\left({x}\right)\right\}\left({t}\right) \\ $$$$\mathcal{W}\left\{\mathrm{1}\right\}\left({t}\right)=? \\ $$$$\mathcal{W}\left\{{x}\right\}\left({t}\right)=? \\ $$$$\mathcal{W}\left\{{x}^{{n}} \right\}\left({t}\right)=?,{n}\in\mathbb{N} \\ $$$$\mathcal{W}\left\{{f}'\left({x}\right)\right\}\left({t}\right)=? \\ $$

Question Number 1350    Answers: 0   Comments: 3

Evaluate the following integral: I=∫_(π/4) ^( π/2) (cos2x+sin2x)ln(cosx+sinx) dx

$${Evaluate}\:{the}\:{following}\:{integral}: \\ $$$${I}=\int_{\pi/\mathrm{4}} ^{\:\pi/\mathrm{2}} \left({cos}\mathrm{2}{x}+{sin}\mathrm{2}{x}\right){ln}\left({cosx}+{sinx}\right)\:{dx} \\ $$

Question Number 1295    Answers: 1   Comments: 2

Solve for f(x) f(x+1)=[f(x)]^3

$$\mathrm{Solve}\:\mathrm{for}\:{f}\left({x}\right) \\ $$$${f}\left({x}+\mathrm{1}\right)=\left[{f}\left({x}\right)\right]^{\mathrm{3}} \\ $$

Question Number 1343    Answers: 1   Comments: 0

3^(log 3x+4) =4^(log 4x+3)

$$\mathrm{3}^{{log}\:\mathrm{3}{x}+\mathrm{4}} =\mathrm{4}^{{log}\:\mathrm{4}{x}+\mathrm{3}} \\ $$

Question Number 1289    Answers: 1   Comments: 0

Prove that ′′ f( f(x) ) is polynomial ′′ ⇒ ′′ f(x) is rational expression ′′ Or give a counter example .

$${Prove}\:{that}\: \\ $$$$\:\:''\:{f}\left(\:{f}\left({x}\right)\:\right)\:{is}\:{polynomial}\:''\:\Rightarrow\:''\:{f}\left({x}\right)\:{is}\:{rational}\:{expression}\:'' \\ $$$$\:{Or}\:{give}\:{a}\:{counter}\:{example}\:. \\ $$$$ \\ $$

Question Number 1280    Answers: 0   Comments: 2

f ( (1/(f(x))))=f(x) f(x)=?

$$\:{f}\:\left(\:\frac{\mathrm{1}}{{f}\left({x}\right)}\right)={f}\left({x}\right)\: \\ $$$${f}\left({x}\right)=? \\ $$

Question Number 1278    Answers: 1   Comments: 2

Determine f(x) when f( f(x))=x .

$${Determine}\:{f}\left({x}\right)\:{when}\:{f}\left(\:{f}\left({x}\right)\right)={x}\:\:. \\ $$

Question Number 1268    Answers: 1   Comments: 0

Prove that AM > HM.

$$\boldsymbol{\mathrm{Prove}}\:\boldsymbol{\mathrm{that}}\:\boldsymbol{\mathrm{AM}}\:>\:\boldsymbol{\mathrm{HM}}. \\ $$

Question Number 1260    Answers: 0   Comments: 4

What is degree of (x/y) ? (x and y are both variables) Is (x/y) and a constant of same degree?

$$\mathrm{What}\:\mathrm{is}\:\mathrm{degree}\:\mathrm{of}\:\frac{\mathrm{x}}{\mathrm{y}}\:?\:\left(\mathrm{x}\:\mathrm{and}\:\mathrm{y}\:\mathrm{are}\:\mathrm{both}\:\mathrm{variables}\right) \\ $$$$\mathrm{Is}\:\frac{\mathrm{x}}{\mathrm{y}}\:\:\mathrm{and}\:\mathrm{a}\:\mathrm{constant}\:\mathrm{of}\:\mathrm{same}\:\mathrm{degree}? \\ $$

Question Number 1255    Answers: 1   Comments: 1

(X,F(X)), X∈C, F(X)∈C, can be plotted in R^2 (Im on Y, Real on X axes) as a directed line segment, ^ (X)−−≫(F(X))^() . This has the advantage of showing vector ffields, fixed points and bifurcations.

$$\left(\mathrm{X},\mathrm{F}\left(\mathrm{X}\right)\right),\:\mathrm{X}\in{C},\:\mathrm{F}\left(\mathrm{X}\right)\in{C},\:\:\mathrm{can}\:\mathrm{be}\:\: \\ $$$$\mathrm{plotted}\:\mathrm{in}\:\mathrm{R}^{\mathrm{2}} \:\left(\mathrm{Im}\:\mathrm{on}\:\mathrm{Y},\:\mathrm{Real}\:\mathrm{on}\:\mathrm{X}\right. \\ $$$$\left.\mathrm{axes}\right)\:\mathrm{as}\:\mathrm{a}\:\mathrm{directed}\:\mathrm{line}\:\mathrm{segment}, \\ $$$$\overset{} {\:}\overset{} {\left(\mathrm{X}\right)−−\gg\left(\mathrm{F}\left(\mathrm{X}\right)\right)}.\:\mathrm{This}\:\mathrm{has} \\ $$$$\mathrm{the}\:\mathrm{advantage}\:\mathrm{of}\:\mathrm{showing}\:\mathrm{vector}\: \\ $$$$\mathrm{ffields},\:\mathrm{fixed}\:\mathrm{points}\:\mathrm{and}\:\mathrm{bifurcations}. \\ $$$$ \\ $$

Question Number 1239    Answers: 0   Comments: 4

Rasheed Ahmad (Rasheed Soomro) •For f(x) where x and f(x) both are real the (x,f(x)) can be plotted as a point easily. •Now consider F(X) where X and F(X) are complex numbers. How can (X,F(X)) be plotted? For a particular example: (3+2i,4−5i) how can be plotted?

$${Rasheed}\:{Ahmad}\:\left({Rasheed}\:{Soomro}\right) \\ $$$$\bullet\mathrm{For}\:\mathrm{f}\left(\mathrm{x}\right)\:\mathrm{where}\:\mathrm{x}\:\mathrm{and}\:\mathrm{f}\left(\mathrm{x}\right)\:\mathrm{both}\:{are} \\ $$$$\mathrm{real}\:\mathrm{the}\:\left(\mathrm{x},{f}\left({x}\right)\right)\:{can}\:{be}\:{plotted}\:{as} \\ $$$${a}\:{point}\:{easily}.\:\bullet{Now}\:{consider}\:{F}\left({X}\right) \\ $$$${where}\:{X}\:{and}\:{F}\left({X}\right)\:{are}\:{complex}\: \\ $$$${numbers}.\:{How}\:{can}\:\left({X},{F}\left({X}\right)\right)\:{be} \\ $$$${plotted}?\:{For}\:{a}\:{particular}\:{example}:\:\left(\mathrm{3}+\mathrm{2}{i},\mathrm{4}−\mathrm{5}{i}\right) \\ $$$${how}\:{can}\:{be}\:{plotted}? \\ $$$$ \\ $$$$ \\ $$

Question Number 1251    Answers: 0   Comments: 1

letsw f(x+y)=f(x)+f(y) f(1)=1 f(x)f((1/x))=1,x≠0 proof that f(x)=x∀x∈R

$$\mathrm{letsw} \\ $$$${f}\left({x}+{y}\right)={f}\left({x}\right)+{f}\left({y}\right) \\ $$$${f}\left(\mathrm{1}\right)=\mathrm{1} \\ $$$${f}\left({x}\right){f}\left(\frac{\mathrm{1}}{{x}}\right)=\mathrm{1},{x}\neq\mathrm{0} \\ $$$$\mathrm{proof}\:\mathrm{that}\:{f}\left({x}\right)={x}\forall{x}\in\mathbb{R} \\ $$

Question Number 1231    Answers: 2   Comments: 2

i^2 =i.i=(√(−1)).(√(−1))=(√(−1×−1))=(√1)=1??? i^2 =1⇒−1=1??? Resolve the contradiction.

$${i}^{\mathrm{2}} ={i}.{i}=\sqrt{−\mathrm{1}}.\sqrt{−\mathrm{1}}=\sqrt{−\mathrm{1}×−\mathrm{1}}=\sqrt{\mathrm{1}}=\mathrm{1}??? \\ $$$${i}^{\mathrm{2}} =\mathrm{1}\Rightarrow−\mathrm{1}=\mathrm{1}???\: \\ $$$${Resolve}\:{the}\:{contradiction}. \\ $$

Question Number 1229    Answers: 0   Comments: 6

f( f(x) )=x^2 −x+1 f(x)=? (Modification of Q 1147)

$${f}\left(\:{f}\left({x}\right)\:\right)={x}^{\mathrm{2}} −{x}+\mathrm{1} \\ $$$${f}\left({x}\right)=? \\ $$$$\left({Modification}\:{of}\:{Q}\:\mathrm{1147}\right) \\ $$

Question Number 1483    Answers: 1   Comments: 2

Prove that: a+(b+c)+d=(a+c)+(d+b)

$$\mathrm{Prove}\:\mathrm{that}: \\ $$$$\mathrm{a}+\left(\mathrm{b}+\mathrm{c}\right)+\mathrm{d}=\left(\mathrm{a}+\mathrm{c}\right)+\left(\mathrm{d}+\mathrm{b}\right) \\ $$

Question Number 1320    Answers: 0   Comments: 0

More generalized form of Q 1296 f(ax+b)=[f(x)]^n f(x)=?

$${More}\:{generalized}\:{form}\:\:{of}\:{Q}\:\mathrm{1296} \\ $$$${f}\left({ax}+{b}\right)=\left[{f}\left({x}\right)\right]^{{n}} \\ $$$${f}\left({x}\right)=? \\ $$

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