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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

$${Solve}:\:{y}^{''} +\mathrm{2}{y}^{'} +\mathrm{2}{y}={secax} \\ $$

Question Number 104028 Answers: 2 Comments: 0

calculate ∫_(20) ^(+∞) (dx/((x−18)^(19) (x−19)^(18) ))

$$\mathrm{calculate}\:\:\int_{\mathrm{20}} ^{+\infty} \:\:\frac{\mathrm{dx}}{\left(\mathrm{x}−\mathrm{18}\right)^{\mathrm{19}} \left(\mathrm{x}−\mathrm{19}\right)^{\mathrm{18}} } \\ $$

Question Number 104016 Answers: 1 Comments: 0

Question Number 104023 Answers: 0 Comments: 1

Question Number 103998 Answers: 3 Comments: 5

if f(x)=x^(3/2) f′(0)=0 or not exist

$${if}\:\:\:\:{f}\left({x}\right)={x}^{\frac{\mathrm{3}}{\mathrm{2}}} \\ $$$${f}'\left(\mathrm{0}\right)=\mathrm{0}\:\:{or}\:{not}\:{exist} \\ $$

Question Number 103986 Answers: 1 Comments: 1

Question Number 103983 Answers: 0 Comments: 0

If α and β are two unequal angle, which satisfy the equation, a cos(α) + b sin(β) = c, show that (i) sin(((α + β)/2)) sec(((α − β)/2)) = (b/c) (ii) tan((α/2)) tan((β/2)) = ((c − a)/(c + a))

$$\mathrm{If}\:\:\alpha\:\:\mathrm{and}\:\:\beta\:\:\mathrm{are}\:\mathrm{two}\:\mathrm{unequal}\:\mathrm{angle},\:\mathrm{which}\:\mathrm{satisfy}\:\mathrm{the}\:\mathrm{equation}, \\ $$$$\:\:\:\:\mathrm{a}\:\mathrm{cos}\left(\alpha\right)\:\:+\:\:\mathrm{b}\:\mathrm{sin}\left(\beta\right)\:\:=\:\:\mathrm{c},\:\:\:\:\mathrm{show}\:\mathrm{that} \\ $$$$\left(\mathrm{i}\right)\:\:\:\:\mathrm{sin}\left(\frac{\alpha\:\:+\:\beta}{\mathrm{2}}\right)\:\mathrm{sec}\left(\frac{\alpha\:\:−\:\:\beta}{\mathrm{2}}\right)\:\:=\:\:\frac{\mathrm{b}}{\mathrm{c}} \\ $$$$\left(\mathrm{ii}\right)\:\:\:\:\:\mathrm{tan}\left(\frac{\alpha}{\mathrm{2}}\right)\:\mathrm{tan}\left(\frac{\beta}{\mathrm{2}}\right)\:\:=\:\:\frac{\mathrm{c}\:\:−\:\:\mathrm{a}}{\mathrm{c}\:\:+\:\:\mathrm{a}} \\ $$

Question Number 103984 Answers: 0 Comments: 1

tan^2 (x)+tan^2 (2x)+tan^2 (4x)=33 x=?

$$\:\:\:\:\boldsymbol{{tan}}^{\mathrm{2}} \left(\boldsymbol{{x}}\right)+\boldsymbol{{tan}}^{\mathrm{2}} \left(\mathrm{2}\boldsymbol{{x}}\right)+\boldsymbol{{tan}}^{\mathrm{2}} \left(\mathrm{4}\boldsymbol{{x}}\right)=\mathrm{33} \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\boldsymbol{{x}}=? \\ $$

Question Number 103979 Answers: 0 Comments: 1

show right and left sid limitation of lim_(x→0) ((ln(b−x))/(ax))

$$\:\:{show}\:{right}\:{and}\:{left}\:{sid}\:\:\:\:{limitation}\:\:\:\: \\ $$$${of}\:\:\:{li}\underset{{x}\rightarrow\mathrm{0}} {{m}}\frac{{ln}\left({b}−{x}\right)}{{ax}} \\ $$$$ \\ $$

Question Number 103974 Answers: 2 Comments: 0

calculate ∫_(−∞) ^(+∞) (dx/((x^2 −x+1)(x^2 +3)^2 ))

$${calculate}\:\:\int_{−\infty} ^{+\infty} \:\:\:\frac{{dx}}{\left({x}^{\mathrm{2}} −{x}+\mathrm{1}\right)\left({x}^{\mathrm{2}} +\mathrm{3}\right)^{\mathrm{2}} } \\ $$

Question Number 103969 Answers: 1 Comments: 0

how do you solve (D^3 +12D^2 +36D)y=0 by constant coefficients

$${how}\:{do}\:{you}\:{solve}\:\left({D}^{\mathrm{3}} +\mathrm{12}{D}^{\mathrm{2}} +\mathrm{36}{D}\right){y}=\mathrm{0} \\ $$$${by}\:{constant}\:{coefficients} \\ $$

Question Number 103961 Answers: 3 Comments: 0

what is 2^(log2x) =3^(log3x)

$${what}\:{is}\: \\ $$$$\mathrm{2}^{{log}\mathrm{2}{x}} =\mathrm{3}^{{log}\mathrm{3}{x}} \\ $$

Question Number 103958 Answers: 1 Comments: 0

what is integrating factor of (xy^2 −y) dx − x dy = 0

$${what}\:{is}\:{integrating}\:{factor} \\ $$$${of}\:\left({xy}^{\mathrm{2}} −{y}\right)\:{dx}\:−\:{x}\:{dy}\:=\:\mathrm{0} \\ $$

Question Number 103945 Answers: 2 Comments: 0

a cube ABCD.EFGH with length side 4 cm. Given point P is midpoint EF. find the distance of line AP to line HB.

$${a}\:{cube}\:{ABCD}.{EFGH}\:{with}\:{length}\:{side} \\ $$$$\mathrm{4}\:{cm}.\:{Given}\:{point}\:{P}\:{is}\:{midpoint}\:{EF}. \\ $$$${find}\:{the}\:{distance}\:{of}\:{line}\:{AP}\:{to}\:{line} \\ $$$${HB}.\: \\ $$

Question Number 103931 Answers: 2 Comments: 0

(y^2 +2) dx = (xy+2y+y^3 ) dy

$$\left({y}^{\mathrm{2}} +\mathrm{2}\right)\:{dx}\:=\:\left({xy}+\mathrm{2}{y}+{y}^{\mathrm{3}} \right)\:{dy} \\ $$

Question Number 103928 Answers: 1 Comments: 0

Question Number 103921 Answers: 1 Comments: 0

Prove that ∀ x ∈ R^ , ∣ cos x ∣ ≤ 1 − sin^2 x

$$\mathrm{Prove}\:\mathrm{that}\:\forall\:{x}\:\in\:\bar {\mathbb{R}}\:,\:\mid\:\mathrm{cos}\:{x}\:\mid\:\leqslant\:\mathrm{1}\:−\:\mathrm{sin}^{\mathrm{2}} \:{x} \\ $$

Question Number 103914 Answers: 2 Comments: 2

(2+(√3))^x^2 + (2−(√3))^x^2 = 4

$$\left(\mathrm{2}+\sqrt{\mathrm{3}}\right)^{{x}^{\mathrm{2}} } \:+\:\left(\mathrm{2}−\sqrt{\mathrm{3}}\right)^{{x}^{\mathrm{2}} } \:=\:\mathrm{4}\: \\ $$

Question Number 103908 Answers: 2 Comments: 2

Find the least positive integer n for which there exists a set { s_1 ,s_2 ,s_3 ,...,s_n } consisting of n distinct positive integers such that (1−(1/s_1 ))(1−(1/s_2 ))(1−(1/s_3 ))...(1−(1/s_n )) = ((51)/(2010)) .

$${Find}\:{the}\:{least}\:{positive}\:{integer}\:{n}\:{for} \\ $$$${which}\:{there}\:{exists}\:{a}\:{set}\:\left\{\:{s}_{\mathrm{1}} ,{s}_{\mathrm{2}} ,{s}_{\mathrm{3}} ,...,{s}_{{n}} \:\right\} \\ $$$${consisting}\:{of}\:{n}\:{distinct}\:{positive}\:{integers} \\ $$$${such}\:{that}\:\left(\mathrm{1}−\frac{\mathrm{1}}{{s}_{\mathrm{1}} }\right)\left(\mathrm{1}−\frac{\mathrm{1}}{{s}_{\mathrm{2}} }\right)\left(\mathrm{1}−\frac{\mathrm{1}}{{s}_{\mathrm{3}} }\right)...\left(\mathrm{1}−\frac{\mathrm{1}}{{s}_{{n}} }\right) \\ $$$$=\:\frac{\mathrm{51}}{\mathrm{2010}}\:. \\ $$

Question Number 103903 Answers: 1 Comments: 5

a 100 cm long rod should be divided into 3 parts. the length of each part in cm should be integer. in how many different ways can this be done?

$${a}\:\mathrm{100}\:{cm}\:{long}\:{rod}\:{should}\:{be}\:{divided} \\ $$$${into}\:\mathrm{3}\:{parts}.\:{the}\:{length}\:{of}\:{each}\:{part} \\ $$$${in}\:{cm}\:{should}\:{be}\:{integer}.\:{in}\:{how}\: \\ $$$${many}\:{different}\:{ways}\:{can}\:{this}\:{be} \\ $$$${done}? \\ $$

Question Number 104143 Answers: 1 Comments: 1

A Satellite orbits the esrth in a circle of rsdius 8000km. At that distance from the earth g=6.2m/s2. The velocity of the satelliege is?

$$\mathrm{A}\:\mathrm{Satellite}\:\mathrm{orbits}\:\mathrm{the}\:\mathrm{esrth}\:\mathrm{in}\:\mathrm{a}\:\mathrm{circle}\:\mathrm{of}\:\mathrm{rsdius}\:\mathrm{8000km}.\:\mathrm{At}\:\mathrm{that}\:\mathrm{distance}\:\mathrm{from}\:\mathrm{the}\:\mathrm{earth}\:\mathrm{g}=\mathrm{6}.\mathrm{2m}/\mathrm{s2}.\:\mathrm{The}\:\mathrm{velocity}\:\mathrm{of}\:\mathrm{the}\:\mathrm{satelliege}\:\mathrm{is}? \\ $$

Question Number 104176 Answers: 1 Comments: 0

Σ_(n=1) ^∞ (((n/(n+1)))^2 −(2/(n+1))−1)

$$\underset{\mathrm{n}=\mathrm{1}} {\overset{\infty} {\sum}}\left(\left(\frac{\mathrm{n}}{\mathrm{n}+\mathrm{1}}\right)^{\mathrm{2}} −\frac{\mathrm{2}}{\mathrm{n}+\mathrm{1}}−\mathrm{1}\right) \\ $$

Question Number 103894 Answers: 3 Comments: 0

(d/(d((d/dx)sinx)))∙sinx=?

$$\frac{{d}}{{d}\left(\frac{{d}}{{dx}}{sinx}\right)}\centerdot{sinx}=? \\ $$

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