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Question Number 110798 Answers: 0 Comments: 0

Use Laplace transform to solve ∂u/∂x +∂u/∂t=x x>0,t>0 u(0,t)=0,u(x,0)=0,t>0,x>0

$${Use}\:{Laplace}\:{transform}\:{to}\:{solve} \\ $$$$\partial{u}/\partial{x}\:+\partial{u}/\partial{t}={x} \\ $$$${x}>\mathrm{0},{t}>\mathrm{0} \\ $$$${u}\left(\mathrm{0},{t}\right)=\mathrm{0},{u}\left({x},\mathrm{0}\right)=\mathrm{0},{t}>\mathrm{0},{x}>\mathrm{0} \\ $$

Question Number 110797 Answers: 0 Comments: 2

If p^→ = ((a),(b) ) and q^→ = ((c),(d) ), Prove that the area bounded by p^→ ,q^→ and p^→ −q^(→ ) is (((ad−bc))/2). Hints: Use cosine rule and sine rule

$$\mathrm{If}\:\overset{\rightarrow} {{p}}=\begin{pmatrix}{{a}}\\{{b}}\end{pmatrix}\:\mathrm{and}\:\overset{\rightarrow} {{q}}=\begin{pmatrix}{{c}}\\{{d}}\end{pmatrix}, \\ $$$$\mathrm{Prove}\:\mathrm{that}\:\mathrm{the}\:\mathrm{area}\:\mathrm{bounded}\:\mathrm{by}\: \\ $$$$\overset{\rightarrow} {{p}},\overset{\rightarrow} {{q}}\:\mathrm{and}\:\overset{\rightarrow} {{p}}−\overset{\rightarrow\:} {{q}}\mathrm{is}\:\frac{\left({ad}−{bc}\right)}{\mathrm{2}}. \\ $$$$ \\ $$$$\mathrm{Hints}:\:\mathrm{Use}\:\mathrm{cosine}\:\mathrm{rule}\:\mathrm{and}\:\mathrm{sine}\:\mathrm{rule} \\ $$

Question Number 110783 Answers: 0 Comments: 2

Find the number of rational numbers r, 0<r<1, such that when r is written as fraction in lowest term. The numerator and demominator have a sum of 1000.

$$\mathrm{Find}\:\mathrm{the}\:\mathrm{number}\:\mathrm{of}\:\mathrm{rational}\:\mathrm{numbers} \\ $$$$\mathrm{r},\:\mathrm{0}<\mathrm{r}<\mathrm{1},\:\mathrm{such}\:\mathrm{that}\:\mathrm{when}\:\mathrm{r}\:\mathrm{is}\:\mathrm{written} \\ $$$$\mathrm{as}\:\mathrm{fraction}\:\mathrm{in}\:\mathrm{lowest}\:\mathrm{term}.\:\mathrm{The} \\ $$$$\mathrm{numerator}\:\mathrm{and}\:\mathrm{demominator}\:\mathrm{have}\:\mathrm{a} \\ $$$$\mathrm{sum}\:\mathrm{of}\:\mathrm{1000}. \\ $$

Question Number 110782 Answers: 2 Comments: 0

A triangle has area 15 and circumradius 12. Find the product of its heights.

$$\mathrm{A}\:\mathrm{triangle}\:\mathrm{has}\:\mathrm{area}\:\mathrm{15}\:\mathrm{and} \\ $$$$\mathrm{circumradius}\:\mathrm{12}.\:\mathrm{Find}\:\mathrm{the}\:\mathrm{product}\:\mathrm{of} \\ $$$$\mathrm{its}\:\mathrm{heights}. \\ $$

Question Number 110779 Answers: 1 Comments: 0

If 0≤x≤(π/2), Prove that (2/π)x≤sin x≤x without graphical method.

$$\mathrm{If}\:\mathrm{0}\leqslant{x}\leqslant\frac{\pi}{\mathrm{2}}, \\ $$$$\mathrm{Prove}\:\mathrm{that}\:\frac{\mathrm{2}}{\pi}{x}\leqslant\mathrm{sin}\:{x}\leqslant{x} \\ $$$$\mathrm{without}\:\mathrm{graphical}\:\mathrm{method}. \\ $$

Question Number 110781 Answers: 1 Comments: 2

Between a square,a triangle and a circle of the same perimeter, which shape has the least area?

$$\mathrm{Between}\:\mathrm{a}\:\mathrm{square},\mathrm{a}\:\mathrm{triangle}\:\mathrm{and}\:\mathrm{a} \\ $$$$\mathrm{circle}\:\mathrm{of}\:\mathrm{the}\:\mathrm{same}\:\mathrm{perimeter},\:\mathrm{which} \\ $$$$\mathrm{shape}\:\mathrm{has}\:\mathrm{the}\:\mathrm{least}\:\mathrm{area}? \\ $$

Question Number 111284 Answers: 0 Comments: 11

Find the maximum area of a triangle whose vertices lie on a regular hexagon of unit area.

$$\mathrm{Find}\:\mathrm{the}\:\mathrm{maximum}\:\mathrm{area}\:\mathrm{of}\:\mathrm{a}\:\mathrm{triangle} \\ $$$$\mathrm{whose}\:\mathrm{vertices}\:\mathrm{lie}\:\mathrm{on}\:\mathrm{a}\:\mathrm{regular} \\ $$$$\mathrm{hexagon}\:\mathrm{of}\:\mathrm{unit}\:\mathrm{area}. \\ $$

Question Number 110772 Answers: 0 Comments: 0

lim_(n→∞) (1+Σ_(r=1) ^n (1/(3^r r!))Π_(k=1) ^r (2k−1))

$$\underset{{n}\rightarrow\infty} {\mathrm{lim}}\left(\mathrm{1}+\underset{{r}=\mathrm{1}} {\overset{{n}} {\sum}}\frac{\mathrm{1}}{\mathrm{3}^{{r}} {r}!}\underset{{k}=\mathrm{1}} {\overset{{r}} {\prod}}\left(\mathrm{2}{k}−\mathrm{1}\right)\right) \\ $$

Question Number 110760 Answers: 2 Comments: 0

Question Number 110845 Answers: 1 Comments: 0

(x^2 +3x−10)^(x^3 −9x) = (x^2 +3x−10)^(3x^2 −8x)

$$\left(\mathrm{x}^{\mathrm{2}} +\mathrm{3x}−\mathrm{10}\right)^{\mathrm{x}^{\mathrm{3}} −\mathrm{9x}} \:=\:\left(\mathrm{x}^{\mathrm{2}} +\mathrm{3x}−\mathrm{10}\right)^{\mathrm{3x}^{\mathrm{2}} −\mathrm{8x}} \\ $$

Question Number 110749 Answers: 1 Comments: 0

please evaluate : Ω=∫_0 ^( (1/2)) ((ln^2 (1−x))/x) dx=??? M.N.July 1970# .... Good luck....

$$\:\:\:\:\:\:\:\:\:\:\: \\ $$$$\:\:{please}\:{evaluate}\:: \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\Omega=\int_{\mathrm{0}} ^{\:\frac{\mathrm{1}}{\mathrm{2}}} \frac{{ln}^{\mathrm{2}} \left(\mathrm{1}−{x}\right)}{{x}}\:{dx}=???\:\:\:\: \\ $$$$ \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\mathscr{M}.\mathscr{N}.\mathscr{J}{uly}\:\mathrm{1970}# \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:....\:\mathscr{G}{ood}\:\:{luck}.... \\ $$$$ \\ $$$$ \\ $$

Question Number 110775 Answers: 0 Comments: 0

Let a,b and c be positive integers such that ab+1∣bc+1 and bc+1∣ca+1. Show that ab+1 is the sum of two squares.

$$\mathrm{Let}\:\mathrm{a},\mathrm{b}\:\mathrm{and}\:\mathrm{c}\:\mathrm{be}\:\mathrm{positive}\:\mathrm{integers}\:\mathrm{such} \\ $$$$\mathrm{that}\:\mathrm{ab}+\mathrm{1}\mid\mathrm{bc}+\mathrm{1}\:\mathrm{and}\:\mathrm{bc}+\mathrm{1}\mid\mathrm{ca}+\mathrm{1}. \\ $$$$\mathrm{Show}\:\mathrm{that}\:\mathrm{ab}+\mathrm{1}\:\mathrm{is}\:\mathrm{the}\:\mathrm{sum}\:\mathrm{of}\:\mathrm{two} \\ $$$$\mathrm{squares}. \\ $$

Question Number 110742 Answers: 0 Comments: 0

please give me a result of E(tanx)

$${please}\:{give}\:{me}\:{a}\:{result}\:{of}\:{E}\left({tanx}\right) \\ $$

Question Number 110741 Answers: 1 Comments: 0

solve: y^(′′) +2y^′ +y=48e^(−x) cos(4x) help me sir

$${solve}:\:{y}^{''} +\mathrm{2}{y}^{'} +{y}=\mathrm{48}{e}^{−{x}} {cos}\left(\mathrm{4}{x}\right) \\ $$$${help}\:{me}\:{sir} \\ $$

Question Number 110739 Answers: 1 Comments: 1

Question Number 110736 Answers: 0 Comments: 1

given f(x)=ax^2 +bx+c and f(x) is negative on a<x<b. The value of lim_(x→a) (((x−a)^2 )/(1−cos (f(x))))

$$\mathrm{given}\:{f}\left({x}\right)={ax}^{\mathrm{2}} +{bx}+{c}\:\mathrm{and}\:{f}\left({x}\right) \\ $$$$\mathrm{is}\:\mathrm{negative}\:\mathrm{on}\:{a}<{x}<{b}.\:\mathrm{The}\:\mathrm{value}\:\mathrm{of} \\ $$$$\underset{{x}\rightarrow{a}} {\mathrm{lim}}\:\frac{\left({x}−{a}\right)^{\mathrm{2}} }{\mathrm{1}−\mathrm{cos}\:\left({f}\left({x}\right)\right)} \\ $$

Question Number 110729 Answers: 1 Comments: 0

How many ways can the letters in the word MATHEMATICS be rearranged such that the word formed neither starts nor ends with a vowel, and any four consecutive letters must contain at least a vowel?

$$\mathrm{How}\:\mathrm{many}\:\mathrm{ways}\:\mathrm{can}\:\mathrm{the}\:\mathrm{letters}\:\mathrm{in} \\ $$$$\mathrm{the}\:\mathrm{word}\:\boldsymbol{\mathrm{MATHEMATICS}} \\ $$$$\mathrm{be}\:\mathrm{rearranged}\:\mathrm{such}\:\mathrm{that}\:\mathrm{the}\:\mathrm{word} \\ $$$$\mathrm{formed}\:\mathrm{neither}\:\mathrm{starts}\:\mathrm{nor}\:\mathrm{ends}\:\mathrm{with}\:\mathrm{a}\: \\ $$$$\mathrm{vowel},\:\mathrm{and}\:\mathrm{any}\:\mathrm{four}\:\mathrm{consecutive}\: \\ $$$$\mathrm{letters}\:\mathrm{must}\:\mathrm{contain}\:\mathrm{at}\:\mathrm{least}\:\mathrm{a}\:\mathrm{vowel}? \\ $$

Question Number 110728 Answers: 1 Comments: 0

a+b=9 , ab=20 a−b=?

$${a}+{b}=\mathrm{9}\:\:,\:\:{ab}=\mathrm{20}\:\:\:\:\:{a}−{b}=? \\ $$

Question Number 110727 Answers: 1 Comments: 0

a^2 +b^2 =10 , ab=13 , a^3 +b^3 =?

$${a}^{\mathrm{2}} +{b}^{\mathrm{2}} =\mathrm{10}\:\:\:,\:\:{ab}=\mathrm{13}\:\:\:,\:{a}^{\mathrm{3}} +{b}^{\mathrm{3}} =? \\ $$

Question Number 110706 Answers: 0 Comments: 0