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Question Number 101791    Answers: 2   Comments: 0

(1/n^(3 ) )lim_(n→∞) (ne^(−((1/n))^2 ) +2ne^(−((2/n))^2 ) +....∞)

$$\frac{\mathrm{1}}{{n}^{\mathrm{3}\:\:} }\underset{{n}\rightarrow\infty} {\mathrm{lim}}\left({ne}^{−\left(\frac{\mathrm{1}}{{n}}\right)^{\mathrm{2}} } +\mathrm{2}{ne}^{−\left(\frac{\mathrm{2}}{{n}}\right)^{\mathrm{2}} } +....\infty\right) \\ $$

Question Number 101783    Answers: 2   Comments: 0

∫_( 0) ^( 1) ((ln(x^2 + 1))/(x + 1))

$$\int_{\:\mathrm{0}} ^{\:\mathrm{1}} \:\:\frac{\mathrm{ln}\left(\mathrm{x}^{\mathrm{2}} \:+\:\mathrm{1}\right)}{\mathrm{x}\:\:+\:\:\mathrm{1}} \\ $$

Question Number 101779    Answers: 1   Comments: 0

If S_n is the sum of the first n terms of an A.P. Express S_(2k) in terms of S_k and S_(3k)

$$\mathrm{If}\:\:\:\:\mathrm{S}_{\mathrm{n}} \:\:\mathrm{is}\:\mathrm{the}\:\mathrm{sum}\:\mathrm{of}\:\mathrm{the}\:\mathrm{first}\:\:\mathrm{n}\:\:\mathrm{terms}\:\mathrm{of}\:\mathrm{an}\:\mathrm{A}.\mathrm{P}. \\ $$$$\mathrm{Express}\:\:\:\mathrm{S}_{\mathrm{2k}} \:\:\mathrm{in}\:\mathrm{terms}\:\mathrm{of}\:\:\:\mathrm{S}_{\mathrm{k}} \:\:\mathrm{and}\:\:\:\mathrm{S}_{\mathrm{3k}} \\ $$

Question Number 101778    Answers: 2   Comments: 0

Question Number 101777    Answers: 0   Comments: 0

Find a curve passing through pointA(0;1) for which the triangle formed by the axis Oy ,tangent to the curve at its arbitrary point and the radius−vector of the point of contact,issosceles(and base is the segment of the tangent from the point of contact to the axis Oy)

$$\mathrm{Find}\:\mathrm{a}\:\mathrm{curve}\:\mathrm{passing}\:\mathrm{through}\:\mathrm{pointA}\left(\mathrm{0};\mathrm{1}\right) \\ $$$$\mathrm{for}\:\mathrm{which}\:\mathrm{the}\:\mathrm{triangle}\:\mathrm{formed}\:\mathrm{by}\:\mathrm{the}\:\mathrm{axis}\:\mathrm{Oy} \\ $$$$,\mathrm{tangent}\:\mathrm{to}\:\mathrm{the}\:\mathrm{curve}\:\mathrm{at}\:\mathrm{its}\:\mathrm{arbitrary}\:\mathrm{point} \\ $$$$\mathrm{and}\:\mathrm{the}\:\mathrm{radius}−\mathrm{vector}\:\mathrm{of}\:\mathrm{the}\:\mathrm{point}\:\mathrm{of} \\ $$$$\mathrm{contact},\mathrm{issosceles}\left(\mathrm{and}\:\mathrm{base}\:\mathrm{is}\:\mathrm{the}\:\mathrm{segment}\right. \\ $$$$\mathrm{of}\:\mathrm{the}\:\mathrm{tangent}\:\mathrm{from}\:\mathrm{the}\:\mathrm{point}\:\mathrm{of}\:\mathrm{contact} \\ $$$$\left.\mathrm{to}\:\mathrm{the}\:\mathrm{axis}\:\mathrm{Oy}\right) \\ $$

Question Number 101775    Answers: 0   Comments: 1

∫x^x^x ∙x^x ∙xdx=? or it able to solve?

$$\int{x}^{{x}^{{x}} } \centerdot{x}^{{x}} \centerdot{xdx}=? \\ $$$${or}\:{it}\:{able}\:{to}\:{solve}? \\ $$

Question Number 101770    Answers: 1   Comments: 2

lim_(n→∞) ((1^(13) +2^(13) +3^(13) +4^(13) +...+n^(13) )/n^(14) ) ?

$$\underset{{n}\rightarrow\infty} {\mathrm{lim}}\:\frac{\mathrm{1}^{\mathrm{13}} +\mathrm{2}^{\mathrm{13}} +\mathrm{3}^{\mathrm{13}} +\mathrm{4}^{\mathrm{13}} +...+{n}^{\mathrm{13}} }{{n}^{\mathrm{14}} }\:? \\ $$

Question Number 101768    Answers: 0   Comments: 1

Question Number 101767    Answers: 0   Comments: 2

Question Number 101766    Answers: 0   Comments: 1

Question Number 101762    Answers: 0   Comments: 2

Question Number 101756    Answers: 2   Comments: 0

∫ ((3x^2 −11x+6)/((x^2 +1)(x−5))) dx

$$\int\:\frac{\mathrm{3}{x}^{\mathrm{2}} −\mathrm{11}{x}+\mathrm{6}}{\left({x}^{\mathrm{2}} +\mathrm{1}\right)\left({x}−\mathrm{5}\right)}\:{dx}\: \\ $$

Question Number 101742    Answers: 3   Comments: 0

{ ((ab+a+b = 5)),((bc + b+c = 14)),((ac + a+c = 9)) :} find a+b+c = ___

$$\begin{cases}{{ab}+{a}+{b}\:=\:\mathrm{5}}\\{{bc}\:+\:{b}+{c}\:=\:\mathrm{14}}\\{{ac}\:+\:{a}+{c}\:=\:\mathrm{9}}\end{cases} \\ $$$$\mathrm{find}\:{a}+{b}+{c}\:=\:\_\_\_ \\ $$

Question Number 101747    Answers: 1   Comments: 0

∫(x^((−1)/2) /(1+x^(1/3) ))dx

$$\int\frac{{x}^{\frac{−\mathrm{1}}{\mathrm{2}}} }{\mathrm{1}+{x}^{\frac{\mathrm{1}}{\mathrm{3}}} }{dx} \\ $$

Question Number 101732    Answers: 2   Comments: 1

There are 10 identical mathematics books, 7 identical physics books and 5 identical chemistry books. Find the number of ways to compile the books under the condition that same books are not mutually adjacent.

$$\mathrm{There}\:\mathrm{are}\:\mathrm{10}\:\mathrm{identical}\:\mathrm{mathematics} \\ $$$$\mathrm{books},\:\mathrm{7}\:\mathrm{identical}\:\mathrm{physics}\:\mathrm{books} \\ $$$$\mathrm{and}\:\mathrm{5}\:\mathrm{identical}\:\mathrm{chemistry}\:\mathrm{books}. \\ $$$$\mathrm{Find}\:\mathrm{the}\:\mathrm{number}\:\mathrm{of}\:\mathrm{ways}\:\mathrm{to}\:\mathrm{compile}\: \\ $$$$\mathrm{the}\:\mathrm{books}\:\mathrm{under}\:\mathrm{the}\:\mathrm{condition}\:\mathrm{that} \\ $$$$\mathrm{same}\:\mathrm{books}\:\mathrm{are}\:\mathrm{not}\:\mathrm{mutually}\:\mathrm{adjacent}. \\ $$

Question Number 101715    Answers: 0   Comments: 1

Deleted a few comments.

$$\mathrm{Deleted}\:\mathrm{a}\:\mathrm{few}\:\mathrm{comments}. \\ $$

Question Number 101713    Answers: 3   Comments: 0

Let a and b be positive numbers satisfying a^2 + b^2 = 5, If a cos(θ) − b sin(θ) = 1, find a sin(θ) + b(cosθ)

$$\mathrm{Let}\:\:\:\mathrm{a}\:\:\mathrm{and}\:\:\mathrm{b}\:\:\mathrm{be}\:\mathrm{positive}\:\mathrm{numbers}\:\mathrm{satisfying}\:\:\:\mathrm{a}^{\mathrm{2}} \:\:+\:\:\mathrm{b}^{\mathrm{2}} \:\:=\:\:\mathrm{5}, \\ $$$$\mathrm{If}\:\:\:\:\mathrm{a}\:\mathrm{cos}\left(\theta\right)\:\:−\:\:\mathrm{b}\:\mathrm{sin}\left(\theta\right)\:\:=\:\:\mathrm{1},\:\:\:\:\:\mathrm{find}\:\:\:\:\mathrm{a}\:\mathrm{sin}\left(\theta\right)\:\:+\:\:\mathrm{b}\left(\mathrm{cos}\theta\right) \\ $$

Question Number 105246    Answers: 4   Comments: 0

lim_(x→0) (cosx)^(1/x^2 )

$$\underset{{x}\rightarrow\mathrm{0}} {\mathrm{lim}}\:\left(\mathrm{cos}{x}\right)^{\frac{\mathrm{1}}{{x}^{\mathrm{2}} }} \\ $$

Question Number 101730    Answers: 0   Comments: 3

Version 2.091 is available: - Slightly darker characters are used by default. A preference setting is available to revert to previous font. Change setting and restart app. - A new menu option mark as answered is added. This just mark as answered so that question will not show in unanswered question search.

$$\mathrm{Version}\:\mathrm{2}.\mathrm{091}\:\mathrm{is}\:\mathrm{available}: \\ $$$$-\:\mathrm{Slightly}\:\mathrm{darker}\:\mathrm{characters}\:\mathrm{are} \\ $$$$\:\:\:\:\mathrm{used}\:\mathrm{by}\:\mathrm{default}. \\ $$$$\:\:\:\:\mathrm{A}\:\mathrm{preference}\:\mathrm{setting}\:\mathrm{is}\:\mathrm{available} \\ $$$$\:\:\:\:\mathrm{to}\:\mathrm{revert}\:\mathrm{to}\:\mathrm{previous}\:\mathrm{font}. \\ $$$$\:\:\:\:\mathrm{Change}\:\mathrm{setting}\:\mathrm{and}\:\mathrm{restart}\:\mathrm{app}. \\ $$$$-\:\mathrm{A}\:\mathrm{new}\:\mathrm{menu}\:\mathrm{option}\:\mathrm{mark}\:\mathrm{as} \\ $$$$\:\:\:\mathrm{answered}\:\mathrm{is}\:\mathrm{added}.\:\mathrm{This}\:\mathrm{just}\:\mathrm{mark} \\ $$$$\:\:\:\mathrm{as}\:\mathrm{answered}\:\mathrm{so}\:\mathrm{that}\:\mathrm{question}\:\mathrm{will} \\ $$$$\:\:\:\mathrm{not}\:\mathrm{show}\:\mathrm{in}\:\mathrm{unanswered}\:\mathrm{question} \\ $$$$\:\:\:\mathrm{search}. \\ $$

Question Number 101693    Answers: 3   Comments: 2

There are 4 identical mathematics books, 2 identic physics books and 2 identical chemistry books . How many ways to compile the eight books on the condition of the same book are not mutually adjacent?

$$\mathrm{There}\:\mathrm{are}\:\mathrm{4}\:\mathrm{identical}\:\mathrm{mathematics} \\ $$$$\mathrm{books},\:\mathrm{2}\:\mathrm{identic}\:\mathrm{physics}\:\mathrm{books} \\ $$$$\mathrm{and}\:\mathrm{2}\:\mathrm{identical}\:\mathrm{chemistry}\:\mathrm{books} \\ $$$$.\:\mathrm{How}\:\mathrm{many}\:\mathrm{ways}\:\mathrm{to}\:\mathrm{compile}\: \\ $$$$\mathrm{the}\:\mathrm{eight}\:\mathrm{books}\:\mathrm{on}\:\mathrm{the}\:\mathrm{condition} \\ $$$$\mathrm{of}\:\mathrm{the}\:\mathrm{same}\:\mathrm{book}\:\mathrm{are}\:\mathrm{not}\:\mathrm{mutually} \\ $$$$\mathrm{adjacent}? \\ $$

Question Number 101686    Answers: 0   Comments: 5

Question Number 105306    Answers: 1   Comments: 1

(1/(2+(√2))) +(1/(3(√2)+2(√3) ))+(1/(4(√3)+3(√4)))+...+(1/(100(√(99))+99(√(100))))

$$\frac{\mathrm{1}}{\mathrm{2}+\sqrt{\mathrm{2}}}\:+\frac{\mathrm{1}}{\mathrm{3}\sqrt{\mathrm{2}}+\mathrm{2}\sqrt{\mathrm{3}}\:}+\frac{\mathrm{1}}{\mathrm{4}\sqrt{\mathrm{3}}+\mathrm{3}\sqrt{\mathrm{4}}}+...+\frac{\mathrm{1}}{\mathrm{100}\sqrt{\mathrm{99}}+\mathrm{99}\sqrt{\mathrm{100}}} \\ $$

Question Number 101680    Answers: 2   Comments: 0

Question Number 101671    Answers: 0   Comments: 0

What is the set of point M in each case: 1) ∣∣6MG^(→) ∣∣=∣∣−2GC^(→) ∣∣ 2) (6MG^(→) )×(−2GC^(→) )=0

$${What}\:{is}\:{the}\:{set}\:{of}\:{point}\:{M}\:{in}\:{each} \\ $$$${case}: \\ $$$$\left.\mathrm{1}\right)\:\:\:\:\mid\mid\mathrm{6}\overset{\rightarrow} {{MG}}\mid\mid=\mid\mid−\mathrm{2}\overset{\rightarrow} {{GC}}\mid\mid \\ $$$$\left.\mathrm{2}\right)\:\:\:\:\left(\mathrm{6}\overset{\rightarrow} {{MG}}\right)×\left(−\mathrm{2}\overset{\rightarrow} {{GC}}\right)=\mathrm{0} \\ $$

Question Number 105250    Answers: 1   Comments: 0

(1/(1×3))+(2/(1×3×5))+(3/(1×3×5×7))+.........n−terms Find sum

$$\frac{\mathrm{1}}{\mathrm{1}×\mathrm{3}}+\frac{\mathrm{2}}{\mathrm{1}×\mathrm{3}×\mathrm{5}}+\frac{\mathrm{3}}{\mathrm{1}×\mathrm{3}×\mathrm{5}×\mathrm{7}}+.........\mathrm{n}−\mathrm{terms} \\ $$$$\mathrm{Find}\:\mathrm{sum} \\ $$

Question Number 101650    Answers: 2   Comments: 1

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