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Question Number 9520 Answers: 1 Comments: 0

∫((sinxcosx)/(4+sin^4 x))dx=...?

$$\int\frac{\mathrm{sinxcosx}}{\mathrm{4}+\mathrm{sin}^{\mathrm{4}} \mathrm{x}}\mathrm{dx}=...? \\ $$

Question Number 9517 Answers: 0 Comments: 3

f(1) + f(2) + f(3) + ... + f(n) = n^2 . f(n) n > 1; f(1) = 2016 So, f(2016) = ?

$${f}\left(\mathrm{1}\right)\:+\:{f}\left(\mathrm{2}\right)\:+\:{f}\left(\mathrm{3}\right)\:+\:...\:+\:{f}\left({n}\right)\:=\:{n}^{\mathrm{2}} .\:{f}\left({n}\right) \\ $$$${n}\:>\:\mathrm{1};\:{f}\left(\mathrm{1}\right)\:=\:\mathrm{2016} \\ $$$$\mathrm{So},\:{f}\left(\mathrm{2016}\right)\:=\:? \\ $$

Question Number 9516 Answers: 0 Comments: 2

Σ_(k=1) ^n ⌊log_2 k⌋ = 2018 What is the value of n ?

$$\underset{{k}=\mathrm{1}} {\overset{{n}} {\sum}}\:\lfloor\mathrm{log}_{\mathrm{2}} \:{k}\rfloor\:=\:\mathrm{2018} \\ $$$$\mathrm{What}\:\mathrm{is}\:\mathrm{the}\:\mathrm{value}\:\mathrm{of}\:{n}\:? \\ $$

Question Number 9515 Answers: 1 Comments: 0

cos (x+2y) = a cos (x+y) = b The maximum value of sin^2 (2x+3y) is ...

$$\mathrm{cos}\:\left({x}+\mathrm{2}{y}\right)\:=\:{a} \\ $$$$\mathrm{cos}\:\left({x}+{y}\right)\:=\:{b} \\ $$$$ \\ $$$$\mathrm{The}\:\mathrm{maximum}\:\mathrm{value}\:\mathrm{of}\:\:\:\mathrm{sin}^{\mathrm{2}} \:\left(\mathrm{2}{x}+\mathrm{3}{y}\right)\:\:\mathrm{is}\:... \\ $$

Question Number 9510 Answers: 1 Comments: 0

Prove if a rational times an irrational can result in a rational.

$$\mathrm{Prove}\:\mathrm{if}\:\mathrm{a}\:\mathrm{rational}\:\mathrm{times}\:\mathrm{an}\:\mathrm{irrational} \\ $$$$\mathrm{can}\:\mathrm{result}\:\mathrm{in}\:\mathrm{a}\:\mathrm{rational}. \\ $$

Question Number 9498 Answers: 1 Comments: 1

For y=x^2 , show how to find (dy/dθ)

$$\mathrm{For}\:{y}={x}^{\mathrm{2}} ,\:\mathrm{show}\:\mathrm{how}\:\mathrm{to}\:\mathrm{find}\:\frac{{dy}}{{d}\theta} \\ $$

Question Number 9497 Answers: 0 Comments: 0

Question Number 9495 Answers: 0 Comments: 0

Question Number 9873 Answers: 1 Comments: 0

find the relation between a,b,c and d if the roots of the eqution ax^3 + bx^2 +cx+d=0 are in; (i)A.P (ii)G.P

$$\mathrm{find}\:\mathrm{the}\:\mathrm{relation}\:\mathrm{between}\:\mathrm{a},\mathrm{b},\mathrm{c}\:\mathrm{and}\:\:\mathrm{d} \\ $$$$\mathrm{if}\:\mathrm{the}\:\mathrm{roots}\:\mathrm{of}\:\mathrm{the}\:\mathrm{eqution}\: \\ $$$$\mathrm{ax}^{\mathrm{3}} \:+\:\mathrm{bx}^{\mathrm{2}} +\mathrm{cx}+\mathrm{d}=\mathrm{0}\:\:\mathrm{are}\:\mathrm{in}; \\ $$$$\left(\mathrm{i}\right)\mathrm{A}.\mathrm{P} \\ $$$$\left(\mathrm{ii}\right)\mathrm{G}.\mathrm{P} \\ $$

Question Number 9874 Answers: 0 Comments: 2

if the eqution x^3 +3hx + q=0 has the roots α,β and γ. find the eqution with the roots of; (i)(1/(αβ)) + (1/(βγ)) + (1/(αγ)) (ii)(1/α^2 ) + (1/β^2 ) + (1/γ^2 )

$$\mathrm{if}\:\mathrm{the}\:\mathrm{eqution}\:\mathrm{x}^{\mathrm{3}} +\mathrm{3hx}\:+\:\mathrm{q}=\mathrm{0}\:\mathrm{has}\:\mathrm{the} \\ $$$$\mathrm{roots}\:\alpha,\beta\:\:\mathrm{and}\:\gamma. \\ $$$$\mathrm{find}\:\mathrm{the}\:\mathrm{eqution}\:\mathrm{with}\:\mathrm{the}\:\mathrm{roots}\:\mathrm{of}; \\ $$$$\left(\mathrm{i}\right)\frac{\mathrm{1}}{\alpha\beta}\:+\:\frac{\mathrm{1}}{\beta\gamma}\:+\:\frac{\mathrm{1}}{\alpha\gamma} \\ $$$$\left(\mathrm{ii}\right)\frac{\mathrm{1}}{\alpha^{\mathrm{2}} }\:+\:\frac{\mathrm{1}}{\beta^{\mathrm{2}} }\:+\:\frac{\mathrm{1}}{\gamma^{\mathrm{2}} } \\ $$$$ \\ $$

Question Number 9491 Answers: 1 Comments: 0

find the value of (d/dx)(((acos^3 x)/(3sinx^2 ))) and it is possible or not

$$\mathrm{find}\:\mathrm{the}\:\mathrm{value}\:\mathrm{of}\:\frac{\mathrm{d}}{\mathrm{dx}}\left(\frac{\mathrm{acos}^{\mathrm{3}} \mathrm{x}}{\mathrm{3sinx}^{\mathrm{2}} \:}\right) \\ $$$$\mathrm{and}\:\mathrm{it}\:\mathrm{is}\:\mathrm{possible}\:\mathrm{or}\:\mathrm{not} \\ $$

Question Number 9489 Answers: 0 Comments: 0

Let a, b, c, d are positive real numbers If x = a + b + c + d = 1 The minimum value of (((x−a)(x−b)(x−c)(x−d))/(abcd)) is ...

$$\mathrm{Let}\:{a},\:{b},\:{c},\:{d}\:\mathrm{are}\:\mathrm{positive}\:\mathrm{real}\:\mathrm{numbers} \\ $$$$\mathrm{If}\:{x}\:=\:{a}\:+\:{b}\:+\:{c}\:+\:{d}\:=\:\mathrm{1} \\ $$$$\mathrm{The}\:\mathrm{minimum}\:\mathrm{value}\:\mathrm{of} \\ $$$$\frac{\left({x}−{a}\right)\left({x}−{b}\right)\left({x}−{c}\right)\left({x}−{d}\right)}{{abcd}}\:\:\mathrm{is}\:... \\ $$

Question Number 9488 Answers: 1 Comments: 0

lim_(x→∞) (((√(2016)) + 4(√(2016)) + 9(√(2016)) + ... + x^2 (√(2016)))/x^3 )

$$\underset{{x}\rightarrow\infty} {\mathrm{lim}}\:\frac{\sqrt{\mathrm{2016}}\:+\:\mathrm{4}\sqrt{\mathrm{2016}}\:+\:\mathrm{9}\sqrt{\mathrm{2016}}\:+\:...\:+\:{x}^{\mathrm{2}} \sqrt{\mathrm{2016}}}{{x}^{\mathrm{3}} } \\ $$

Question Number 9487 Answers: 0 Comments: 0

A ticket contains 9 digits serial number, which only composed from number 1, 2, or 3 Ticket can be colored red, blue, or green. If 2 different tickets have different serial numbers, they have different color too John has a blue ticket with serial number 311111111 Ann has a red ticket with serial number 111111111 You have 2 another tickets with serial number 312312312 and 131131131. What color should your tickets have?

$$\mathrm{A}\:\mathrm{ticket}\:\mathrm{contains}\:\mathrm{9}\:\mathrm{digits}\:\mathrm{serial}\:\mathrm{number}, \\ $$$$\mathrm{which}\:\mathrm{only}\:\mathrm{composed}\:\mathrm{from}\:\mathrm{number}\:\mathrm{1},\:\mathrm{2},\:\mathrm{or}\:\mathrm{3} \\ $$$$\mathrm{Ticket}\:\mathrm{can}\:\mathrm{be}\:\mathrm{colored}\:\mathrm{red},\:\mathrm{blue},\:\mathrm{or}\:\mathrm{green}. \\ $$$$\mathrm{If}\:\mathrm{2}\:\mathrm{different}\:\mathrm{tickets}\:\mathrm{have}\:\mathrm{different}\:\mathrm{serial}\:\mathrm{numbers}, \\ $$$$\mathrm{they}\:\mathrm{have}\:\mathrm{different}\:\mathrm{color}\:\mathrm{too} \\ $$$$ \\ $$$$\mathrm{John}\:\mathrm{has}\:\mathrm{a}\:\mathrm{blue}\:\mathrm{ticket}\:\mathrm{with}\:\mathrm{serial}\:\mathrm{number}\:\mathrm{311111111} \\ $$$$\mathrm{Ann}\:\mathrm{has}\:\mathrm{a}\:\mathrm{red}\:\mathrm{ticket}\:\mathrm{with}\:\mathrm{serial}\:\mathrm{number}\:\mathrm{111111111} \\ $$$$\mathrm{You}\:\mathrm{have}\:\mathrm{2}\:\mathrm{another}\:\mathrm{tickets}\:\mathrm{with}\:\mathrm{serial}\:\mathrm{number}\:\mathrm{312312312}\:\mathrm{and}\:\mathrm{131131131}. \\ $$$$\mathrm{What}\:\mathrm{color}\:\mathrm{should}\:\mathrm{your}\:\mathrm{tickets}\:\mathrm{have}? \\ $$

Question Number 9486 Answers: 2 Comments: 0

arctan (1/3) + arctan (1/4) + arctan (1/5) + arctan (1/n) = (π/4) So, (1/n) = ?

$$\mathrm{arctan}\:\frac{\mathrm{1}}{\mathrm{3}}\:+\:\mathrm{arctan}\:\frac{\mathrm{1}}{\mathrm{4}}\:+\:\mathrm{arctan}\:\frac{\mathrm{1}}{\mathrm{5}}\:+\:\mathrm{arctan}\:\frac{\mathrm{1}}{{n}}\:=\:\frac{\pi}{\mathrm{4}} \\ $$$$\mathrm{So},\:\frac{\mathrm{1}}{{n}}\:=\:? \\ $$

Question Number 9485 Answers: 0 Comments: 2

a_n = a_(n−1) ^2 + a_(n−2) ^2 If a_1 =1 and a_2 =1, what is the remainder of a_(2016) when divided by 10 ?

$${a}_{{n}} \:=\:{a}_{{n}−\mathrm{1}} ^{\mathrm{2}} \:+\:{a}_{{n}−\mathrm{2}} ^{\mathrm{2}} \\ $$$$\mathrm{If}\:{a}_{\mathrm{1}} =\mathrm{1}\:\mathrm{and}\:{a}_{\mathrm{2}} =\mathrm{1},\:\mathrm{what}\:\mathrm{is}\:\mathrm{the}\:\mathrm{remainder}\:\mathrm{of}\:{a}_{\mathrm{2016}} \:\mathrm{when}\:\mathrm{divided}\:\mathrm{by}\:\mathrm{10}\:? \\ $$

Question Number 9484 Answers: 1 Comments: 0

S(r) = 12 + 12r^1 + 12r^2 + 12r^3 + 12r^4 + ... with −1 < r < 1 If S(a) . S(−a) = 2016, with −1 < a < 1 What is the value of S(a) + S(−a) ?

$$\mathrm{S}\left({r}\right)\:=\:\mathrm{12}\:+\:\mathrm{12}{r}^{\mathrm{1}} \:+\:\mathrm{12}{r}^{\mathrm{2}} \:+\:\mathrm{12}{r}^{\mathrm{3}} \:+\:\mathrm{12}{r}^{\mathrm{4}} \:+\:... \\ $$$$\mathrm{with}\:−\mathrm{1}\:<\:{r}\:<\:\mathrm{1} \\ $$$$ \\ $$$$\mathrm{If}\:\:\mathrm{S}\left({a}\right)\:.\:\mathrm{S}\left(−{a}\right)\:=\:\mathrm{2016},\:\mathrm{with}\:−\mathrm{1}\:<\:{a}\:<\:\mathrm{1} \\ $$$$\mathrm{What}\:\mathrm{is}\:\mathrm{the}\:\mathrm{value}\:\mathrm{of}\:\:\mathrm{S}\left({a}\right)\:+\:\mathrm{S}\left(−{a}\right)\:? \\ $$

Question Number 9482 Answers: 0 Comments: 1

find F_n (−1) or F_n (x). n=1,2,3...... F_n (x)=∫(1+x^3 )^n dx

$${find}\:\:{F}_{{n}} \left(−\mathrm{1}\right)\:{or}\:{F}_{{n}} \left({x}\right). \\ $$$${n}=\mathrm{1},\mathrm{2},\mathrm{3}...... \\ $$$${F}_{{n}} \left({x}\right)=\int\left(\mathrm{1}+{x}^{\mathrm{3}} \right)^{{n}} {dx} \\ $$

Question Number 9478 Answers: 0 Comments: 0

Question Number 9474 Answers: 0 Comments: 0

Question Number 9471 Answers: 1 Comments: 0

prove by mathematical induction a + (a + d) + (a + 2d) + ... + [a + (n − 1)d] = (1/2)n[2a + (n − 1)d]

$$\mathrm{prove}\:\mathrm{by}\:\mathrm{mathematical}\:\mathrm{induction} \\ $$$$\mathrm{a}\:+\:\left(\mathrm{a}\:+\:\mathrm{d}\right)\:+\:\left(\mathrm{a}\:+\:\mathrm{2d}\right)\:+\:...\:+\:\left[\mathrm{a}\:+\:\left(\mathrm{n}\:−\:\mathrm{1}\right)\mathrm{d}\right]\:=\:\frac{\mathrm{1}}{\mathrm{2}}\mathrm{n}\left[\mathrm{2a}\:+\:\left(\mathrm{n}\:−\:\mathrm{1}\right)\mathrm{d}\right]\: \\ $$

Question Number 9470 Answers: 1 Comments: 2

prove by mathematical induction. (1/(1.3)) + (1/(2.5)) + (1/(3.7)) + ... + (1/((2n − 1)(2n + 1))) = (n/(2n + 1))

$$\mathrm{prove}\:\mathrm{by}\:\mathrm{mathematical}\:\mathrm{induction}. \\ $$$$\frac{\mathrm{1}}{\mathrm{1}.\mathrm{3}}\:+\:\frac{\mathrm{1}}{\mathrm{2}.\mathrm{5}}\:+\:\frac{\mathrm{1}}{\mathrm{3}.\mathrm{7}}\:+\:...\:+\:\frac{\mathrm{1}}{\left(\mathrm{2n}\:−\:\mathrm{1}\right)\left(\mathrm{2n}\:+\:\mathrm{1}\right)}\:=\:\frac{\mathrm{n}}{\mathrm{2n}\:+\:\mathrm{1}} \\ $$

Question Number 9467 Answers: 1 Comments: 0

∫x^2 ((√(1 − x^2 ))) dx

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

Question Number 9458 Answers: 0 Comments: 1

If the zeta function of 2 is 𝛇(2) = Σ_(n=1) ^∞ (1/n^2 ) 𝛇(2) = (𝛑^2 /6) the sum of infinite rational numbers, why converges for (𝛑^2 /6), an irrational number?

$$\mathrm{If}\:\mathrm{the}\:\mathrm{zeta}\:\mathrm{function}\:\mathrm{of}\:\mathrm{2}\:\mathrm{is} \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\boldsymbol{\zeta}\left(\mathrm{2}\right)\:=\:\underset{\boldsymbol{{n}}=\mathrm{1}} {\overset{\infty} {\sum}}\:\frac{\mathrm{1}}{\boldsymbol{{n}}^{\mathrm{2}} } \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\boldsymbol{\zeta}\left(\mathrm{2}\right)\:=\:\frac{\boldsymbol{\pi}^{\mathrm{2}} }{\mathrm{6}} \\ $$$$\mathrm{the}\:\mathrm{sum}\:\mathrm{of}\:\mathrm{infinite}\:\boldsymbol{\mathrm{rational}}\:\mathrm{numbers}, \\ $$$$\mathrm{why}\:\mathrm{converges}\:\mathrm{for}\:\frac{\boldsymbol{\pi}^{\mathrm{2}} }{\mathrm{6}},\:\mathrm{an}\:\boldsymbol{\mathrm{irrational}} \\ $$$$\mathrm{number}? \\ $$

Question Number 9455 Answers: 1 Comments: 0

Solve the following system of equations 3x−8z=−11 2y−3x=−3 y−4z=−7 See the comment of Q#9439.

$$\mathrm{Solve}\:\mathrm{the}\:\mathrm{following}\:\:\mathrm{system}\:\mathrm{of}\:\mathrm{equations} \\ $$$$\mathrm{3x}−\mathrm{8z}=−\mathrm{11} \\ $$$$\mathrm{2y}−\mathrm{3x}=−\mathrm{3} \\ $$$$\mathrm{y}−\mathrm{4z}=−\mathrm{7} \\ $$$${See}\:{the}\:{comment}\:{of}\:{Q}#\mathrm{9439}. \\ $$

Question Number 9453 Answers: 1 Comments: 0

find dc′s dr′s of a mormal to the plane 2x+(5/2)y+(7/8)z=23

$$\mathrm{find}\:\mathrm{dc}'\mathrm{s}\:\mathrm{dr}'\mathrm{s}\:\mathrm{of}\:\mathrm{a}\:\mathrm{mormal}\:\mathrm{to}\:\mathrm{the}\: \\ $$$$\mathrm{plane}\:\mathrm{2x}+\frac{\mathrm{5}}{\mathrm{2}}\mathrm{y}+\frac{\mathrm{7}}{\mathrm{8}}\mathrm{z}=\mathrm{23} \\ $$

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