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Question Number 113355 Answers: 1 Comments: 1

A rectangular cardboard is 8cm long and 6cm wide. What is the least number of beads you can arrange on the board such that there are at least two of the beads that are less than (√(10))cm apart.

$$\mathrm{A}\:\mathrm{rectangular}\:\mathrm{cardboard}\:\mathrm{is}\:\mathrm{8cm}\:\mathrm{long} \\ $$$$\mathrm{and}\:\mathrm{6cm}\:\mathrm{wide}.\:\mathrm{What}\:\mathrm{is}\:\mathrm{the}\:\mathrm{least} \\ $$$$\mathrm{number}\:\mathrm{of}\:\mathrm{beads}\:\mathrm{you}\:\mathrm{can}\:\mathrm{arrange}\:\mathrm{on} \\ $$$$\mathrm{the}\:\mathrm{board}\:\mathrm{such}\:\mathrm{that}\:\mathrm{there}\:\mathrm{are}\:\mathrm{at}\:\mathrm{least} \\ $$$$\mathrm{two}\:\mathrm{of}\:\mathrm{the}\:\mathrm{beads}\:\mathrm{that}\:\mathrm{are}\:\mathrm{less}\:\mathrm{than} \\ $$$$\sqrt{\mathrm{10}}\mathrm{cm}\:\mathrm{apart}. \\ $$

Question Number 113354 Answers: 0 Comments: 3

( ((n),(0) )/2)−( ((n),(1) )/3)+( ((n),(2) )/4)−( ((n),(3) )/5)+.....n

$$\frac{\begin{pmatrix}{{n}}\\{\mathrm{0}}\end{pmatrix}}{\mathrm{2}}−\frac{\begin{pmatrix}{{n}}\\{\mathrm{1}}\end{pmatrix}}{\mathrm{3}}+\frac{\begin{pmatrix}{{n}}\\{\mathrm{2}}\end{pmatrix}}{\mathrm{4}}−\frac{\begin{pmatrix}{{n}}\\{\mathrm{3}}\end{pmatrix}}{\mathrm{5}}+.....{n} \\ $$

Question Number 113353 Answers: 1 Comments: 0

What is the maximum number of points to be distributed within a 3×6 to ensure that there are no two points whose distance apart is less than (√2)?

$$\mathrm{What}\:\mathrm{is}\:\mathrm{the}\:\mathrm{maximum}\:\mathrm{number} \\ $$$$\mathrm{of}\:\mathrm{points}\:\mathrm{to}\:\mathrm{be}\:\mathrm{distributed}\:\mathrm{within} \\ $$$$\mathrm{a}\:\mathrm{3}×\mathrm{6}\:\mathrm{to}\:\mathrm{ensure}\:\mathrm{that}\:\mathrm{there}\:\mathrm{are}\:\mathrm{no}\:\mathrm{two} \\ $$$$\mathrm{points}\:\mathrm{whose}\:\mathrm{distance}\:\mathrm{apart}\:\mathrm{is}\:\mathrm{less} \\ $$$$\mathrm{than}\:\sqrt{\mathrm{2}}? \\ $$

Question Number 113346 Answers: 1 Comments: 0

Question Number 113343 Answers: 1 Comments: 1

find the angle between x+3(√(3y))=2,(√(3x))−5y=2 help me sir please

$${find}\:{the}\:{angle}\:{between}\:{x}+\mathrm{3}\sqrt{\mathrm{3}{y}}=\mathrm{2},\sqrt{\mathrm{3}{x}}−\mathrm{5}{y}=\mathrm{2} \\ $$$${help}\:{me}\:{sir}\:{please} \\ $$

Question Number 113341 Answers: 3 Comments: 0

Question Number 113336 Answers: 0 Comments: 0

If 1, a^2 ,a^3 ,...,a^(n−1) are the roots nth of unity , prove that : (1+a)(1+a^2 )(1+a^3 )...(1+a^(n−1) ) = n−2⌊(n/2)⌋

$${If}\:\mathrm{1},\:{a}^{\mathrm{2}} ,{a}^{\mathrm{3}} \:,...,{a}^{{n}−\mathrm{1}} \:{are}\:{the}\:{roots}\: \\ $$$${nth}\:{of}\:{unity}\:,\: \\ $$$${prove}\:{that}\::\:\left(\mathrm{1}+{a}\right)\left(\mathrm{1}+{a}^{\mathrm{2}} \right)\left(\mathrm{1}+{a}^{\mathrm{3}} \right)...\left(\mathrm{1}+{a}^{{n}−\mathrm{1}} \right) \\ $$$$=\:{n}−\mathrm{2}\lfloor\frac{{n}}{\mathrm{2}}\rfloor \\ $$$$ \\ $$

Question Number 113333 Answers: 2 Comments: 0

Question Number 113330 Answers: 1 Comments: 0

Question Number 113323 Answers: 2 Comments: 0

Question Number 113309 Answers: 2 Comments: 4

Question Number 113375 Answers: 0 Comments: 1

(i) How does one find the equation of the perpendicular to a line? (ii) How does one calculate standard deviation. Explain in details.

$$\left(\mathrm{i}\right)\:\mathrm{How}\:\mathrm{does}\:\mathrm{one}\:\mathrm{find}\:\mathrm{the}\:\mathrm{equation}\:\mathrm{of}\:\mathrm{the} \\ $$$$\mathrm{perpendicular}\:\mathrm{to}\:\mathrm{a}\:\mathrm{line}? \\ $$$$ \\ $$$$\left(\mathrm{ii}\right)\:\mathrm{How}\:\mathrm{does}\:\mathrm{one}\:\mathrm{calculate}\:\mathrm{standard} \\ $$$$\mathrm{deviation}.\:\mathrm{Explain}\:\mathrm{in}\:\mathrm{details}. \\ $$

Question Number 113374 Answers: 1 Comments: 5

Find the next term in this sequence: 26,63,124, __

$$\mathrm{Find}\:\mathrm{the}\:\mathrm{next}\:\mathrm{term}\:\mathrm{in}\:\mathrm{this} \\ $$$$\mathrm{sequence}:\:\mathrm{26},\mathrm{63},\mathrm{124},\:\_\_ \\ $$

Question Number 113372 Answers: 1 Comments: 2

2^a +2^b +2^c +2^d =57, find a+b+c+d. a≠b≠c≠d and a,b,c,d are positive integers.

$$\mathrm{2}^{\mathrm{a}} +\mathrm{2}^{\mathrm{b}} +\mathrm{2}^{\mathrm{c}} +\mathrm{2}^{\mathrm{d}} =\mathrm{57},\:\mathrm{find}\:\mathrm{a}+\mathrm{b}+\mathrm{c}+\mathrm{d}. \\ $$$$\mathrm{a}\neq\mathrm{b}\neq\mathrm{c}\neq\mathrm{d}\:\mathrm{and}\:\mathrm{a},\mathrm{b},\mathrm{c},\mathrm{d}\:\mathrm{are}\:\mathrm{positive} \\ $$$$\mathrm{integers}. \\ $$

Question Number 113278 Answers: 2 Comments: 0

solve ∫((√(x^2 +x+2−(√(4x^2 +4x+4))))/(x(√(x^4 +x^3 +x^2 ))))dx

$${solve} \\ $$$$\int\frac{\sqrt{{x}^{\mathrm{2}} +{x}+\mathrm{2}−\sqrt{\mathrm{4}{x}^{\mathrm{2}} +\mathrm{4}{x}+\mathrm{4}}}}{{x}\sqrt{{x}^{\mathrm{4}} +{x}^{\mathrm{3}} +{x}^{\mathrm{2}} }}{dx} \\ $$

Question Number 113275 Answers: 2 Comments: 0

∫ (dx/(3sin x+sin^3 x)) ?

$$\:\int\:\frac{\mathrm{dx}}{\mathrm{3sin}\:\mathrm{x}+\mathrm{sin}\:^{\mathrm{3}} \mathrm{x}}\:? \\ $$

Question Number 113274 Answers: 4 Comments: 2

(1) lim_(x→0) ((tan x+4tan 2x−3tan 3x)/(x^2 tan x)) (2) lim_(x→0) (((√x)−(√(sin x)))/x^(3/2) ) (3) lim_(x→0) (((√x)+(√(sin x)))/x^(5/2) )

$$\:\left(\mathrm{1}\right)\:\underset{{x}\rightarrow\mathrm{0}} {\mathrm{lim}}\:\frac{\mathrm{tan}\:\mathrm{x}+\mathrm{4tan}\:\mathrm{2x}−\mathrm{3tan}\:\mathrm{3x}}{\mathrm{x}^{\mathrm{2}} \:\mathrm{tan}\:\mathrm{x}} \\ $$$$\:\left(\mathrm{2}\right)\:\underset{{x}\rightarrow\mathrm{0}} {\mathrm{lim}}\:\frac{\sqrt{\mathrm{x}}−\sqrt{\mathrm{sin}\:\mathrm{x}}}{\mathrm{x}^{\frac{\mathrm{3}}{\mathrm{2}}} }\: \\ $$$$\:\left(\mathrm{3}\right)\:\underset{{x}\rightarrow\mathrm{0}} {\mathrm{lim}}\:\frac{\sqrt{\mathrm{x}}+\sqrt{\mathrm{sin}\:\mathrm{x}}}{\mathrm{x}^{\frac{\mathrm{5}}{\mathrm{2}}} } \\ $$

Question Number 113272 Answers: 2 Comments: 0

Solve the following equations: a)(x^2 −a)^2 −6x^2 +4x+2a=0 b)x^4 −4x^3 −10x^3 +37x−14=0,if it known that the left−hand side of the equation can be decomposed into factors with integral coefficients.

$$\mathrm{Solve}\:\mathrm{the}\:\mathrm{following}\:\mathrm{equations}: \\ $$$$\left.\mathrm{a}\right)\left(\mathrm{x}^{\mathrm{2}} −\mathrm{a}\right)^{\mathrm{2}} −\mathrm{6x}^{\mathrm{2}} +\mathrm{4x}+\mathrm{2a}=\mathrm{0} \\ $$$$\left.\mathrm{b}\right)\mathrm{x}^{\mathrm{4}} −\mathrm{4x}^{\mathrm{3}} −\mathrm{10x}^{\mathrm{3}} +\mathrm{37x}−\mathrm{14}=\mathrm{0},\mathrm{if}\:\mathrm{it} \\ $$$$\mathrm{known}\:\mathrm{that}\:\mathrm{the}\:\mathrm{left}−\mathrm{hand}\:\mathrm{side}\:\mathrm{of}\:\mathrm{the} \\ $$$$\mathrm{equation}\:\mathrm{can}\:\mathrm{be}\:\mathrm{decomposed}\:\mathrm{into} \\ $$$$\mathrm{factors}\:\mathrm{with}\:\mathrm{integral}\:\mathrm{coefficients}. \\ $$

Question Number 113271 Answers: 0 Comments: 0

solve the initial boundary value problem of wave equation ((∂^2 u(x,t))/∂t^2 )=9((∂^2 u(x,t))/∂x^2 ),0<x<2,t>0 u(0,t)=1,u(2,t)=3,t>0 u(x,0)=2,0<x<2 (∂u/∂t)(x,0)=sin2x,0<x<2

$${solve}\:{the}\:{initial}\:{boundary}\:{value} \\ $$$${problem}\:{of}\:{wave}\:{equation} \\ $$$$\frac{\partial^{\mathrm{2}} {u}\left({x},{t}\right)}{\partial{t}^{\mathrm{2}} }=\mathrm{9}\frac{\partial^{\mathrm{2}} {u}\left({x},{t}\right)}{\partial{x}^{\mathrm{2}} },\mathrm{0}<{x}<\mathrm{2},{t}>\mathrm{0} \\ $$$${u}\left(\mathrm{0},{t}\right)=\mathrm{1},{u}\left(\mathrm{2},{t}\right)=\mathrm{3},{t}>\mathrm{0} \\ $$$${u}\left({x},\mathrm{0}\right)=\mathrm{2},\mathrm{0}<{x}<\mathrm{2} \\ $$$$\frac{\partial{u}}{\partial{t}}\left({x},\mathrm{0}\right)=\mathrm{sin2}{x},\mathrm{0}<{x}<\mathrm{2} \\ $$

Question Number 113262 Answers: 2 Comments: 0

find the complex form of equation 4x^2 −2y^2 =5

$$\mathrm{find}\:\mathrm{the}\:\mathrm{complex}\:\mathrm{form}\:\mathrm{of}\: \\ $$$$\mathrm{equation}\:\mathrm{4x}^{\mathrm{2}} −\mathrm{2y}^{\mathrm{2}} =\mathrm{5} \\ $$

Question Number 113261 Answers: 1 Comments: 0

(((√((√5)+2))+(√((√5)−2)))/( (√((√5)+1))))−(√(3−2(√2)))

$$\frac{\sqrt{\sqrt{\mathrm{5}}+\mathrm{2}}+\sqrt{\sqrt{\mathrm{5}}−\mathrm{2}}}{\:\sqrt{\sqrt{\mathrm{5}}+\mathrm{1}}}−\sqrt{\mathrm{3}−\mathrm{2}\sqrt{\mathrm{2}}} \\ $$

Question Number 113246 Answers: 1 Comments: 1

Question Number 113243 Answers: 1 Comments: 3

Question Number 113241 Answers: 1 Comments: 0

Question Number 113237 Answers: 1 Comments: 0

Question Number 121155 Answers: 0 Comments: 0

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