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Question Number 104139 Answers: 2 Comments: 1

what is remainder when 61^(61) divided by 1001

$${what}\:{is}\:{remainder}\:{when}\:\mathrm{61}^{\mathrm{61}} \:{divided} \\ $$$${by}\:\mathrm{1001} \\ $$

Question Number 104122 Answers: 1 Comments: 0

(1/(sin2x))+(1/(sin2^2 x))+.....+(1/(sin2^n x)) Find the value

$$\frac{\mathrm{1}}{\mathrm{sin2x}}+\frac{\mathrm{1}}{\mathrm{sin2}^{\mathrm{2}} \mathrm{x}}+.....+\frac{\mathrm{1}}{\mathrm{sin2}^{\mathrm{n}} \mathrm{x}} \\ $$$$\mathrm{Find}\:\mathrm{the}\:\mathrm{value}\: \\ $$

Question Number 104115 Answers: 1 Comments: 0

Question Number 104110 Answers: 1 Comments: 0

in a box A there are 3 red balls and 2 white balls.while in box B there are 4 red balls and 5 white balls.if from boxA and B each are taken two balls one by one with return . the chance of being taken is one white ball? a.(4/(675)) c.((128)/(675)) e.((218)/(675)) b.((64)/(675)) d.((184)/(675))

$${in}\:{a}\:{box}\:{A}\:{there}\:{are}\:\mathrm{3}\:{red}\:{balls}\:{and}\:\mathrm{2}\:{white} \\ $$$${balls}.{while}\:{in}\:{box}\:{B}\:{there}\:{are}\:\mathrm{4}\:{red}\:{balls}\:{and} \\ $$$$\mathrm{5}\:{white}\:{balls}.{if}\:{from}\:{boxA}\:{and}\:{B}\:{each}\:{are}\: \\ $$$${taken}\:{two}\:{balls}\:{one}\:{by}\:{one}\:{with}\:{return}\:.\:{the}\: \\ $$$${chance}\:{of}\:{being}\:{taken}\:{is}\:{one}\:{white}\:{ball}? \\ $$$${a}.\frac{\mathrm{4}}{\mathrm{675}}\:\:\:\:\:{c}.\frac{\mathrm{128}}{\mathrm{675}}\:\:\:\:\:\:{e}.\frac{\mathrm{218}}{\mathrm{675}} \\ $$$${b}.\frac{\mathrm{64}}{\mathrm{675}}\:\:\:\:\:\:{d}.\frac{\mathrm{184}}{\mathrm{675}} \\ $$$$ \\ $$$$ \\ $$

Question Number 104101 Answers: 1 Comments: 2

Question Number 104100 Answers: 3 Comments: 0

(1)Find all natural pairs of integers (x,y) such that x^3 −y^3 =xy+61. (2)Find gcd(x^4 −x^3 , x^3 −x)

$$\left(\mathrm{1}\right)\mathbb{F}{ind}\:{all}\:{natural}\:{pairs}\:{of} \\ $$$${integers}\:\left({x},{y}\right)\:{such}\:{that}\:{x}^{\mathrm{3}} −{y}^{\mathrm{3}} ={xy}+\mathrm{61}. \\ $$$$\left(\mathrm{2}\right)\mathbb{F}{ind}\:{gcd}\left({x}^{\mathrm{4}} −{x}^{\mathrm{3}} ,\:{x}^{\mathrm{3}} −{x}\right)\: \\ $$

Question Number 104099 Answers: 0 Comments: 0

Question Number 104333 Answers: 2 Comments: 0

if i^( 5n +1 ) = 1 , n∈Z then show that : n = 3+4p ,p∈Z

$${if}\:{i}^{\:\mathrm{5}{n}\:+\mathrm{1}\:} =\:\mathrm{1}\:\:\:\:,\:{n}\in{Z} \\ $$$${then}\:{show}\:{that}\::\:{n}\:=\:\mathrm{3}+\mathrm{4}{p}\:\:\:,{p}\in{Z} \\ $$

Question Number 104093 Answers: 2 Comments: 0

(1)Evaluate the sum ⌊(2^0 /3)⌋+⌊(2^1 /3)⌋+⌊(2^2 /3)⌋+...+⌊(2^(1000) /3)⌋ (2) find 2^(98) (mod 33)

$$\left(\mathrm{1}\right){Evaluate}\:{the}\:{sum}\:\lfloor\frac{\mathrm{2}^{\mathrm{0}} }{\mathrm{3}}\rfloor+\lfloor\frac{\mathrm{2}^{\mathrm{1}} }{\mathrm{3}}\rfloor+\lfloor\frac{\mathrm{2}^{\mathrm{2}} }{\mathrm{3}}\rfloor+...+\lfloor\frac{\mathrm{2}^{\mathrm{1000}} }{\mathrm{3}}\rfloor \\ $$$$\left(\mathrm{2}\right)\:{find}\:\mathrm{2}^{\mathrm{98}} \:\left({mod}\:\mathrm{33}\right)\: \\ $$

Question Number 104092 Answers: 1 Comments: 1

Prove that (1/2) ∙ (3/4) ∙ (5/6) ∙ …∙ ((2005)/(2006)) ∙ ((2007)/(2008)) < (1/(√(2009)))

$${Prove}\:\:{that}\:\: \\ $$$$\frac{\mathrm{1}}{\mathrm{2}}\:\centerdot\:\frac{\mathrm{3}}{\mathrm{4}}\:\centerdot\:\frac{\mathrm{5}}{\mathrm{6}}\:\centerdot\:\ldots\centerdot\:\frac{\mathrm{2005}}{\mathrm{2006}}\:\centerdot\:\frac{\mathrm{2007}}{\mathrm{2008}}\:\:<\:\:\frac{\mathrm{1}}{\sqrt{\mathrm{2009}}} \\ $$

Question Number 104086 Answers: 1 Comments: 3

Solve: log_r 8 + log_3 p = 5 ..... (i) r + p = 11 ..... (ii)

$$\mathrm{Solve}:\:\:\:\:\:\:\mathrm{log}_{\mathrm{r}} \mathrm{8}\:\:\:+\:\:\:\mathrm{log}_{\mathrm{3}} \mathrm{p}\:\:\:=\:\:\mathrm{5}\:\:\:\:\:\:\:\:\:.....\:\left(\mathrm{i}\right) \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\mathrm{r}\:\:\:+\:\:\mathrm{p}\:\:\:=\:\:\mathrm{11}\:\:\:\:\:\:\:\:\:.....\:\left(\mathrm{ii}\right) \\ $$

Question Number 104104 Answers: 4 Comments: 0

∫ ((xtan^(−1) (x))/(√(1+x^2 ))) dx ?

$$\int\:\frac{{x}\mathrm{tan}^{−\mathrm{1}} \left({x}\right)}{\sqrt{\mathrm{1}+{x}^{\mathrm{2}} }}\:{dx}\:? \\ $$

Question Number 104071 Answers: 2 Comments: 0

Question Number 104062 Answers: 3 Comments: 4

{ (((x/y) + (y/x) = ((13)/6))),((x+y = 5)) :} find the solution

$$\begin{cases}{\frac{{x}}{{y}}\:+\:\frac{{y}}{{x}}\:=\:\frac{\mathrm{13}}{\mathrm{6}}}\\{{x}+{y}\:=\:\mathrm{5}}\end{cases} \\ $$$${find}\:{the}\:{solution} \\ $$

Question Number 104060 Answers: 1 Comments: 0

evaluate ∫∫_s (xz+y^2 )dS where S is the surface described by x^2 +y^2 =16 , 0≤z≤3

$${evaluate}\:\int\int_{{s}} \left({xz}+{y}^{\mathrm{2}} \right){dS}\:{where} \\ $$$${S}\:{is}\:{the}\:{surface}\:{described}\:{by}\:{x}^{\mathrm{2}} +{y}^{\mathrm{2}} =\mathrm{16} \\ $$$$,\:\mathrm{0}\leqslant{z}\leqslant\mathrm{3} \\ $$

Question Number 104159 Answers: 0 Comments: 3

How may we plot the graph of f(x)=x+(√((x(x+2))/(x+1))) , with the help of a variation table ?

$$\mathrm{How}\:\mathrm{may}\:\mathrm{we}\:\mathrm{plot}\:\mathrm{the}\:\mathrm{graph}\:\mathrm{of} \\ $$$$\mathrm{f}\left(\mathrm{x}\right)=\mathrm{x}+\sqrt{\frac{\mathrm{x}\left(\mathrm{x}+\mathrm{2}\right)}{\mathrm{x}+\mathrm{1}}}\:,\:\mathrm{with} \\ $$$$\mathrm{the}\:\mathrm{help}\:\mathrm{of}\:\mathrm{a}\:\mathrm{variation}\:\mathrm{table}\:? \\ $$

Question Number 104051 Answers: 3 Comments: 0

(x^2 −1) (dy/dx) + 2y = (x+1)^2

$$\left({x}^{\mathrm{2}} −\mathrm{1}\right)\:\frac{{dy}}{{dx}}\:+\:\mathrm{2}{y}\:=\:\left({x}+\mathrm{1}\right)^{\mathrm{2}} \\ $$

Question Number 104046 Answers: 0 Comments: 6

To do normally his commercial activities in some place situated at 150km from him, a driver use a car. the consumption of gazoil in liters for 10 km is defined by: C(v)=((20)/v)+(v/(45)) where v is the speed of car. How should be the speed of car to reduce minimally the consumption of gazoil?

$$\mathrm{To}\:\mathrm{do}\:\mathrm{normally}\:\mathrm{his}\:\mathrm{commercial} \\ $$$$\mathrm{activities}\:\mathrm{in}\:\mathrm{some}\:\mathrm{place}\:\mathrm{situated}\:\mathrm{at}\:\mathrm{150km} \\ $$$$\mathrm{from}\:\mathrm{him},\:\mathrm{a}\:\mathrm{driver}\:\mathrm{use}\:\mathrm{a}\:\mathrm{car}.\:\mathrm{the}\:\mathrm{consumption} \\ $$$$\mathrm{of}\:\mathrm{gazoil}\:\mathrm{in}\:\mathrm{liters}\:\mathrm{for}\:\mathrm{10}\:\mathrm{km}\:\mathrm{is}\:\mathrm{defined}\:\mathrm{by}: \\ $$$$\mathrm{C}\left(\mathrm{v}\right)=\frac{\mathrm{20}}{\mathrm{v}}+\frac{\mathrm{v}}{\mathrm{45}}\:\mathrm{where}\:\mathrm{v}\:\mathrm{is}\:\mathrm{the}\:\mathrm{speed}\:\mathrm{of}\:\mathrm{car}. \\ $$$$\mathrm{How}\:\mathrm{should}\:\mathrm{be}\:\mathrm{the}\:\mathrm{speed}\:\mathrm{of}\:\mathrm{car}\:\mathrm{to}\:\mathrm{reduce} \\ $$$$\mathrm{minimally}\:\mathrm{the}\:\mathrm{consumption}\:\mathrm{of}\:\mathrm{gazoil}? \\ $$

Question Number 104038 Answers: 1 Comments: 0

what is the largest positive integer n such that n^3 +100 is divisible by n+10 ?

$${what}\:{is}\:{the}\:{largest}\:{positive} \\ $$$${integer}\:{n}\:{such}\:{that}\:{n}^{\mathrm{3}} +\mathrm{100}\:{is} \\ $$$${divisible}\:{by}\:{n}+\mathrm{10}\:?\: \\ $$

Question Number 104037 Answers: 4 Comments: 0

(1) { ((x^3 +y^6 = 91)),((x+y^2 = 7 )) :} find x−y^6 . (2) 2a+(2/a) = 8 ⇒ ((a^6 +1)/a^3 ) ?

$$\left(\mathrm{1}\right)\begin{cases}{{x}^{\mathrm{3}} +{y}^{\mathrm{6}} \:=\:\mathrm{91}}\\{{x}+{y}^{\mathrm{2}} \:=\:\mathrm{7}\:}\end{cases} \\ $$$${find}\:{x}−{y}^{\mathrm{6}} \:. \\ $$$$\left(\mathrm{2}\right)\:\mathrm{2}{a}+\frac{\mathrm{2}}{{a}}\:=\:\mathrm{8}\:\Rightarrow\:\frac{{a}^{\mathrm{6}} +\mathrm{1}}{{a}^{\mathrm{3}} }\:? \\ $$

Question Number 104035 Answers: 1 Comments: 0

Find the image of point OP^(→) = p^ in the line r^ =a^ +λb^ .

$${Find}\:{the}\:{image}\:{of}\:{point}\:\overset{\rightarrow} {{OP}}\:=\:\bar {{p}}\:\:{in} \\ $$$${the}\:{line}\:\:\bar {{r}}=\bar {{a}}+\lambda\bar {{b}}\:. \\ $$

Question Number 104034 Answers: 2 Comments: 0

Solve: y^(′′) +2y^′ +2y=secax