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Question Number 111850 Answers: 4 Comments: 0

If ∣z∣ = 3 , what is the maximum and minimum value of ∣z−1+i(√3) ∣ ?

$I f ∣ z ∣ = 3, w h a t i s t h e m a x i m u m$ $a n d m i n i m u m v a l u e o f ∣ z - 1 + i \sqrt{3} ∣ ?$

Question Number 112531 Answers: 0 Comments: 4

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.

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

Question Number 111831 Answers: 1 Comments: 0

Let N be the greatest multiple of 36 all of whose digits are even and no two of whose digits are the same. Find the remainder when N is divided by 1000.

$Let N be the greatest multiple of 36 all$ $of whose digits are even and no two of$ $whose digits are the same . Find the$ $remainder when N is divided by 1000 .$

Question Number 112535 Answers: 1 Comments: 5

Let K be the product of all factors (b−a) (not necessarily distinct) where a and b are integers satisfying 1≤a≤b≤10. Find the greatest integer n such that 2^n divides K.

$Let K be the product of all factors$ $(b - a) (not necessarily distinct)$ $where a and b are integers satisfying$ $1 ⩽ a ⩽ b ⩽ 10 . Find the greatest$ $integer n such that 2^{n} divides K .$

Question Number 111818 Answers: 3 Comments: 0

(√(bemath )) (1)∫ ((cos x)/(2−cos x)) dx (2) f(x) = ∣x^3 ∣ ⇒ f ′(x) ?

$\sqrt{b e m a t h}$ $(1) \int \frac{\cos x}{2 - \cos x} d x$ $(2) f (x) = ∣ x^{3} ∣ \Rightarrow f^{'} (x) ?$

Question Number 111813 Answers: 0 Comments: 3

if ψ(x)=(x/(1−x))+(x^3 /(1−x^2 ))+(x^5 /(1−x^5 ))+(x^7 /(1−x^7 )) show that ∫_0 ^1 [ψ(x)+ψ((x)^(1/3) )+ψ((x)^(1/5) )+ψ((x)^(1/7) )]((ln((1/x)))/x)=((25832π^2 )/(1575)) soltion let the generating series form of ψ(x)=(x/(1−x))+(x^3 /(1−x^2 ))+(x^5 /(1−x^5 ))+(x^7 /(1−x^7 )) be ψ(x)=Σ_(i=0) ^∞ (x^(2i+1) /(1−x^(2i+1) )).........(1) from Ω=∫_0 ^1 ψ(x)+ψ((x)^(1/3) )+ψ((x)^(1/5) )+ψ((x)^(1/7) )((ln((1/x)))/x)dx Ω=−∫_0 ^1 [ψ(x)+ψ(x^(1/3) )+ψ(x^(1/5) )+ψ(x^(1/7) )]((lnx)/x)dx there the series form be Ω=∫_0 ^1 [Σ_(n=0) ^∞ ψ(x^(1/(2n+1)) )((lnx)/x)]dx.........(2) putting equation (1) into (2) Ω=−∫_0 ^1 Σ_(n=0) ^∞ Σ_(i=0) ^∞ [(((x^(2i+1) .x^(1/(2n+1)) )/(1−x^(2i+1) .x^(1/(2n+1)) )))((lnx)/x)]dx Ω=−Σ_(n=0) ^∞ Σ_(i=0) ^∞ ∫_0 ^1 [((x^((2i+1)/(2n+1)) /(1−x^((2i+1)/(2n+1)) )))((lnx)/x)]dx let y=((2i+1)/(2n+1)) Ω=−Σ_(n=0) ^∞ Σ_(i=0) ^∞ ∫_0 ^1 [((x^y /(1−x^y )))((lnx)/x)]dx=−Σ_(n=0) ^∞ Σ_(i=0) ^∞ ∫_0 ^1 ((x^(y−1) /(1−x^y )))lnxdx let the series form of (1/(1−x^y ))=Σ_(m=0) ^∞ x^(my) Ω=−Σ_(n=0) ^∞ Σ_(i=0) ^∞ ∫_0 ^1 (Σ_(m=0) ^∞ x^(my) .x^(y−1) lnx)dx (∂/∂a)∣_(a=0) Ω(a)=−Σ_(n=0) ^∞ Σ_(i=0) ^∞ Σ_(m=0) ^∞ ∫_0 ^1 (x^(my) .x^(y−1) .x^a )dx (∂/∂a)∣_(a=0) Ω(a)=−Σ_(n=0) ^∞ Σ_(i=0) ^∞ Σ_(m=0) ^∞ ∫_0 ^1 x^(my+y+a−1) dx (∂/∂a)∣_(a=0) Ω(a)=−Σ_(n=0) ^∞ Σ_(i=0) ^∞ Σ_(m=0) ^∞ ((1/(my+y+a))) Ω^′ (0)=Σ_(n=0) ^∞ Σ_(i=0) ^∞ Σ_(m=0) ^∞ [(1/((my+y)^2 ))]=(Σ_(n=0) ^∞ Σ_(i=0) ^∞ )(1/y^2 )Σ_(m=0) ^∞ (1/((1+m)^2 )) let z=1+m at m=0 ,z=1 Ω(0)=(Σ_(n=0) ^∞ Σ_(i=0) ^∞ )(1/y^2 )Σ_(z=1) ^∞ (1/z^2 )=(Σ_(n=0) ^∞ Σ_(i=0) ^∞ )(1/y^2 )ζ(2)=(π^2 /6)(Σ_(n=0) ^∞ Σ_(i=0) ^∞ )(1/y^2 ) but y=((2i+1)/(2n+1)) Ω(0)=(π^2 /6)(Σ_(n=0) ^∞ Σ_(i=0) ^∞ (1/((((2i+1)/(2n+1)))^2 )))=(π^2 /6)Σ_(n=0) ^∞ (2n+1)^2 Σ_(i=0) ^∞ (1/((2i+1)^2 )) Ω=(π^2 /6)[1^2 +3^2 +5^2 +7^2 ]×[(1/1^2 )+(1/3^2 )+(1/5^2 )+(1/7^2 )] Ω=(π^2 /6)(1+9+25+49)×[(1/1)+(1/9)+(1/(25))+(1/(49))] Ω=(π^2 /6)•(84)•((12916)/(11025))=((25832π^2 )/(1575)) ∵∫_0 ^1 [ψ(x)+ψ((x)^(1/3) )+ψ((x)^(1/5) )+ψ((x)^(1/7) )]((ln((1/x)))/x)=((25832π^2 )/(1575)) by mathew monday(05/09/2020)

Question Number 111799 Answers: 3 Comments: 4

Question Number 111788 Answers: 1 Comments: 0

Question Number 111781 Answers: 0 Comments: 3

Question Number 111774 Answers: 1 Comments: 0

Question Number 111772 Answers: 1 Comments: 0

calculate lim_(n→+∞) Π_(k=1) ^n (1+((√(k(n−k)))/n^2 ))

$calculate \lim_{n \to + \infty} \prod_{k = 1}^{n} (1 + \frac{\sqrt{k (n - k)}}{n^{2}})$

Question Number 111771 Answers: 0 Comments: 0

calculate lim_(n→+∞) a_n ∫_0 ^1 x^(2n) sin(((πx)/2))dx with a_n =Σ_(k=1) ^n sin(((πk)/(2n)))

$calculate \lim_{n \to + \infty} a_{n} \int_{0}^{1} x^{2 n} \sin (\frac{π x}{2}) dx with a_{n} = \sum_{k = 1}^{n} \sin (\frac{π k}{2 n})$

Question Number 111770 Answers: 0 Comments: 0

f function continue on [0,1] find lim_(n→+∞) (1/n)Σ_(k=0) ^n (n−k)∫_(k/n) ^((k+1)/n) f(x)dx

$f function continue on [0, 1] find \lim_{n \to + \infty} \frac{1}{n} \sum_{k = 0}^{n} (n - k) \int_{\frac{k}{n}}^{\frac{k + 1}{n}} f (x) dx$

Question Number 111769 Answers: 0 Comments: 0

let u_n =Σ_(k=1) ^n (1/(√((k+n)(k+n+1)))) detremine lim_(n→+∞) u_n

$let u_{n} = \sum_{k = 1}^{n} \frac{1}{\sqrt{(k + n) (k + n + 1)}}$ $detremine \lim_{n \to + \infty} u_{n}$

Question Number 111768 Answers: 0 Comments: 0

explicite f(x) =∫_0 ^(2π) ln(x^2 −2xcosθ +1)dθ (x≠+^− 1)

$explicite f (x) = \int_{0}^{2 π} \ln (x^{2} - 2 xcos θ + 1) d θ (x \neq \bar{+} 1)$

Question Number 111766 Answers: 2 Comments: 0

calculate lim_(n→+∞) Σ_(k=1) ^n ln((n/(n+k)))^(1/n)

$calculate \lim_{n \to + \infty} \sum_{k = 1}^{n} \ln {(\frac{n}{n + k})}^{\frac{1}{n}}$

Question Number 111762 Answers: 1 Comments: 0

caoculate ∫_0 ^(π/4) ln(1+2tanx)dx

$caoculate \int_{0}^{\frac{π}{4}} \ln (1 + 2 tanx) dx$

Question Number 111760 Answers: 0 Comments: 0

find lim_(x→1^+ ) ∫_x ^x^2 ((ln(t))/((t−1)^2 ))dx

$find \lim_{x \to 1^{+}} \int_{x}^{x^{2}} \frac{\ln (t)}{{(t - 1)}^{2}} dx$

Question Number 111756 Answers: 0 Comments: 0

find I_n =∫_0 ^(π/4) (du/(cos^n u))

$find I_{n} = \int_{0}^{\frac{π}{4}} \frac{du}{\cos^{n} u}$

Question Number 111755 Answers: 2 Comments: 0

calculate lim_(n→+∞) Σ_(k=1) ^n (√((n−k)/(n^3 +n^2 k)))

$calculate \lim_{n \to + \infty} \sum_{k = 1}^{n} \sqrt{\frac{n - k}{n^{3} + n^{2} k}}$

Question Number 111754 Answers: 2 Comments: 0

calculate lim_(n→+∞) Σ_(k=1) ^(2n) (k/(k^2 +n^2 ))

$calculate \lim_{n \to + \infty} \sum_{k = 1}^{2 n} \frac{k}{k^{2} + n^{2}}$

Question Number 111887 Answers: 0 Comments: 0

Question Number 111882 Answers: 1 Comments: 0

(√(bemath)) (1)lim_(x→∞) [(x^2 /(x+1)) − (x^2 /(x+3)) ] ? (2) prove that n^2 ≤ 2^n for ∀n∈N by mathematical induction

$\sqrt{b e m a t h}$ $(1) \lim_{x \to \infty} [\frac{x^{2}}{x + 1} - \frac{x^{2}}{x + 3}] ?$ $(2) p r o v e t h a t n^{2} ⩽ 2^{n} f o r \forall n \in N$ $b y m a t h e m a t i c a l i n d u c t i o n$

Question Number 111745 Answers: 0 Comments: 6

Question Number 111826 Answers: 0 Comments: 2

There are two cards; one is yellow on both sides and the other one is yellow in one side and red on the other. The cards have the same probability ((1/2)) of being chosen, and one is chosen and placed on the table. If the upper side of the card on the table is yellow, then the probability that the under side is also yellow is.

$There are two cards; one is yellow on$ $both sides and the other one is yellow$ $in one side and red on the other . The$ $cards have the same probability (\frac{1}{2})$ $of being chosen, and one is chosen and$ $placed on the table . If the upper side$ $of the card on the table is yellow,$ $then the probability that the under$ $side is also yellow is .$

Question Number 111734 Answers: 1 Comments: 0

A chord which is a perpendicular bisector of radius of length 18cm in a circle, has length.

$A chord which is a perpendicular$ $bisector of radius of length 18 cm in a$ $circle, has length .$

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